interest in relativistic hydrodynamics has heightened in recent years due to the explosion in the field of gamma - ray bursts ( grbs costa et al .1997 , van paradijs et al .1997 , frail et al .1997 , macfadyen & woosley 1999 , aloy et al .2000 , frail et al .2001 , fox et al .2005 , gehrels et al . 2005 , bloom et al .2006 , obrien et al .the current paradigm for grbs involves the extraction of energy from a newly formed black hole and collimation into a relativistic jet , which then propagates along the line of sight to the observer .the emission is thus strongly beamed and doppler boosted .the interaction of the jet with the circumstellar medium produces afterglow . for newtonian hydrodynamicsthe density contrast across a strong shock is given by , where is the polytropic index ; in relativistic hydrodynamics , where is the lorentz factor ( blandford & mckee 1976 ) .for putative values thought to be required for grb jets , a relativistic shock can have extremely high density and be very narrow due to lorentz contraction .this poses a severe test for standard finite difference ( fd ) methods , and necessitates adaptive mesh refinement ( zhang & macfadyen 2006 , morsony , lazzati , & begelman 2007 ) .adaptive refinement techniques also present challenges , as it has yet to be demonstrated that increased levels of refinement on a complex , multidimensional problem , lead to convergent solutions . the desired test of showing that a standard performance metric integrated over the computational volume asymptotes to a constant value with increasing level of refinement has yet to be carried out ( e.g. , zhang & macfadyen 2006 ) .traditional methods for calculating hydrodynamical evolution of a relativistic fluid have relied on finite differencing , i.e. , discretizing the differential equations ( norman & winkler 1986 , mart & mller 2003 , del zanna & bucciantini 2002 , lucas - serrano et al .figure [ bm_soln ] presents an example of smearing inherent in standard fd methods .it shows the evolution of lorentz factor in a spherical relativistic blast wave calculation initialized with a blandford & mckee ( 1976 ) solution , taking initially .each panel shows the same initial conditions , with increasing grid resolution along the .we use a three dimensional cartesian grid and utilize the method described in del zanna & bucciantini ( 2002 ) .our implementation of their method is detailed in cannizzo , gehrels , & vishniac ( 2004 ) . within each panelthe number of grid points along the direction of propagation is increased by a factor of 4 . in the fourth panel ,for which there are 64 grid points per small tick mark , one can see the clear development of a forward / reverse shock feature .the inherent smearing behavior of the technique is evident by comparing successive panels .glimm ( 1965 ) presented the theoretical basis for the random choice , or glimm s method .it relies on first idealizing the solution in over grid points as consisting of piecewise constant states , and then solving the local riemann problem times between adjacent grid points .second , a random location is selected within a cell , the exact solution evaluated at that point , and then that is used as the starting solution for the next time step .chorin ( 1976 , 1977 ) developed glimm s method into a numerical algorithm for problems that could be formulated in terms of nonlinear hyperbolic conservation laws .sod ( 1978 ) reviewed several techniques for newtonian hydrodynamics and found glimm s method to be superior in terms of preserving the sharpness of shock edges . in the early studies using glimm s methodone sees clear deficiencies in the solutions , however , both in terms of shock front localization and overall stability .a breakthrough came from colella ( 1982 ) who proposed using the van der corput sequence instead of a standard random number generator for determining the solution evaluation location with cells in each time step .this sequence is generated by a simple manipulation of the digits in the binary representation of consecutive integers .the application of glimm s method to relativistic hydrodynamics became possible when balsara ( 1994 ) and mart & mller ( 1996 ) generalized the solution of the riemann solution for relativistic hydrodynamics .wen , panaitescu , & laguna ( 1997 ) used the results of mart & mller to implement a relativistic hydrodynamics glimm s method . their study and panaitescu et al .( 1997 ) are the only works to date that employ glimm s method for relativistic hydrodynamics .the basic method is demonstrated in wen et al .( 1997 , see their fig .2 ) . in one half time step the exact solution to the local riemann problem is calculated between two grid points , at a position determined by the van der corput sequence . as explained in colella ( 1982 ) , the sequence is determined by taking the binary representation of the positive integers , , , , , , , , , etc . , and then flipping the binary digits with respect to the ( binary ) decimal point , yielding the sequence , , , , , , , , etc .note that the sequence alternates between the two half - unit intervals ( 0 , 0.5 ) and ( 0.5 , 1 ) , which helps minimize spurious shock propagation .furthermore the series can be shown to be optimal in terms of uniform coverage of the unit interval ( 0 , 1 ) .the value adopted in a given time step is the same for all inter - grid points .the exact solution at a given grid point is alternately taken to be either the left or right solution between two adjacent grid points . the solution is evaluated in each alternating half - time step at a time , where with cfl ( courant , friedrichs , & lewy 1967 ) number . thus the pure glimm s method effectively adopts a cfl number of 0.5 for the full time step .although most of our results use a simple one dimensional cartesian grid , wen et al .( 1997 ) also present geometrical correction terms for carrying out one dimensional calculations in cylindrical or spherical symmetry .an important advance since wen et al .( 1997 ) are the studies generalizing the relativistic riemann solution to include tangential flow ( pons , mart , & mller 2000 , rezzolla , zanotti , & pons 2003 ) .this allows one to extend glimm s method to problems involving shear , and to begin to envision a two dimensional glimm s method .pons et al .obtain a solution by solving ( 1 ) the jump conditions across shocks , and ( 2 ) a differential equation that comes from a self - similarity condition along rarefaction waves .rezzolla et al .present an integral solution to the equation derived by pons et al . and they propose an efficient gaussian quadrature technique for solving it . to solve the local riemann problem between adjacent grid points we use the publicly available code riemann_vt.f written by j .-mart and e. mller ( cf .mart & mller 2003 ) which uses the formalism described in pons et al .( 2000 ) and rezzolla et al .shock tube problems used in testing hydrodynamical codes are a subset of the riemann problems , for which for all .one dimensional riemann problems are typically run on a grid such that , and the thermodynamic variables , , and are discontinuous across initially .starting the simulation is equivalent to removing a diaphragm between left ( l ) and right ( r ) states .the strong gradients across result in four constant states separated by three elementary waves : rarefaction , contact discontinuity , and shock wave .analytical solutions for the time evolution of these problems for ( special ) relativistic hydrodynamics are given by mart & mller ( 1994 ) for nonshearing problems , and by pons et al .( 2000 ) for riemann problems with added shear ( i.e. , non - zero ) .  the level of agreement between the exact , analytical solutions and the numerical ones is quantified by the norm error , defined for 1d problems as , where is the coordinate of grid point , is the analytical value , and the numerical value .the grid spacing is . for consistency with previous groups ,we take the solution in proper density .the analytical and numerical solutions are calculated on the same grids , and the number of grid points in the solutions are varied between trials .the values in the initial left and right states are and . the adiabatic index .the result at is compared to the analytical one .the gradient in pressure produces in the subsequent evolution a rarefaction wave moving left and a shock wave moving right , with a contact discontinuity between .the flow is mildly relativistic , with post - shock velocity .figure [ f.ri1 ] shows a comparison of the glimm solution with the exact one , computed on a grid with .the small inset panels show a detail of the leading and trailing edges of the density spike associated with the shock .for the time step shown , the leading edge of the glimm solution is off the analytical solution by one grid point , and the trailing edge is exact .table [ t.r1 ] presents the errors in density between 3 methods , flash ( from morsony et al .2007 ) , weno ( weighted essentially nonoscillatory , from zhang & macfadyen 2006 ) , and glimm .( this test problem has been studied by many workers previously e.g. , hawley , smarr , & wilson 1984 , schneider et al .1993 , mart & mller 1996 , wen , panaitescu , & laguna 1997 , mart et al .1997 , aloy et al . 1999 . )the asterisked values indicate those trials for which the leading and trailing shock edge positions of the glimm solutions are in agreement with the analytical ones .  for a small sample of individual glimm trials , the error is not always a consistent indicator of success .although shock propagation speeds are expected to be accurate in an averaged sense , within a given time step specific features in the glimm solution can be one or two grid points off from their correct location . for problems with sharp edges , such as shocks, the error will be large ( locally ) at such a position .most the rest of the error is introduced by idealizing the curved state ( riemann fan ) to be composed of a series of piecewise constant states .even if a shock edge location is incorrect at a given time step , at a slightly later time step , or at the same time step for a run with a different number of grid points , the glimm solution may have the correct location of the shock front edges .therefore a better way to measure the success of the method is to plot the errors for a large number of different trials , all compared at the same time step with the analytical solution for the same . for problems which are typically dominated by one large density enhancement, one observes bands of solutions representing those for which the calculated edges are ( 1 ) exact , ( 2 ) off by one grid point ( leading or trailing edge ) , ( 3 ) off by two grid points total , ( 4 ) off by three grid points total , etc .we denote the cumulative grid point error in shock front localization by .this effect is shown in figure [ f.r1 ] where we plot the error versus .the black and blue points indicate values for the 6 grid points shown in table [ t.r1 ] , and the red points show a much larger sample drawn from equi - logarithmically spaced values for the glimm solutions .there is a large scatter vertically in the glimm errors according to the degree of matching of the shock edges .riemann problem 2 has a more extreme pressure contrast between the l and r states initially than problem 1 , and therefore drives a faster and higher density shock .the values in the initial left and right states are and . the adiabatic index .the result at is compared to the analytical one .the flow is relativistic , with post - shock velocity .the shock speed is 0.986 .the width of the shock is at and for is covered by 4.2 grid points ( in the analytical solution ) .the asterisked values indicate those trials for which .table 2 compares the errors for the three methods .figure [ f.r2 ] shows the error plot for riemann problem 2 , with the values given in table [ t.r2 ] plus glimm values for additional values . due to the thinness of the shock compared to problem 1, there is now a clear banded structure to the glimm solutions .the lowest striation , which also contains the first and sixth values from table [ t.r2 ] , corresponds to solutions for which both leading and trailing shock edge positions are exact , .the next highest striation , containing glimm entries from the table , corresponds to , and the third striation , containing the second glimm entry from the table , corresponds to .the first striation lies about two orders of magnitude below the f errors , while the second and third are within a factor of f.   riemann problem 3 starts with a strong negative pressure gradient that launches a reverse shock , and a positive flow speed in the left state that initiates a forward shock .thus there is no riemann fan .the values in the initial left and right states are and . the adiabatic index .the result at is compared to the analytical one .table [ t.r3 ] compares the errors for the three methods .figure [ f.r3 ] shows the error plot for riemann problem 3 , with the values given in table [ t.r3 ] plus glimm values for more values .none of the values in the table lie in the band for .the three upper limit triangles indicate solutions for which , i.e. , all three shock edge locations in the problem are exact at .the only limiting precision is the machine epsilon ( ) .for one dimensional problems consisting only of constant states , glimm s method finds the exact values , therefore the only error is introduced by shock edge location inaccuracies . in traditional methods this test problem produces postshock pressure oscillations in the reverse shock ( e.g. , lucas - serrano et al . 2004 ,see their fig . 1 ;zhang & macfadyen 2006 , see their fig .lucas - serrano et al .( 2004 ) note , however , that the oscillations completely disappear when the cfl number is reduced below 0.3 .we now proceed to one dimensional problems involving shear .the `` easy '' shear problem takes riemann problem 2 and adds constant background shear in the r state , .the adiabatic index , and the result at is compared to the analytical one .the highest lorentz factor in the resulting flow .unlike purely newtonian flows in which orthogonal components of the velocity field are decoupled from each other ( aside from dissipation ) , with special relativity we now add the condition that .this effectively limits the component of velocity along the direction of the flow , and also the degree of density enhancement relative to background within the shock .in addition , now includes a contribution from the shear .there is also a back reaction in terms of the evolution of on the initially constant values .table [ t.easy ] compares the errors for the three methods .figure [ f.easy ] shows the errors for the values given in table [ t.easy ] , plus additional values for g. as with figs .[ f.r2 ] and [ f.r3 ] , the banded structure associated with the precision in the shock edge localization is evident .the locus of solutions for lies below the f errors , while the second striation , corresponding to , lies within a factor of 10 of the f errors .the `` hard '' shear problem starts with riemann problem 2 and adds background shear in both the r and l states , .the adiabatic index , and the result at is compared to the analytical one .the highest lorentz factor in the resulting flow is .table [ t.hard ] compares the errors for the three methods .the asterisked value indicates the trial for which .this problem poses a severe challenge for the traditional methods , but is well - handled by the glimm method .in fact , the error for f for the highest values shown are equal to those for the lowest values for g. zhang & macfadyen ( 2006 ) present results of the hard shear test for up to 51,200 grid points , either uniform or the adaptive mesh equivalent ( see their table 7 and fig .their errors for of are comparable to those in our test for .the challenge of relativistic 1d shearing problems for standard fd techniques is also evident in morsony et al .( 2007 , see their fig .the profiles of and for the fd shearing experiments shown in mignone , plewa , & bodo ( 2005 ) , zhang & macfadyen ( 2006 ) , and morsony et al .( 2007 ) all exhibit a strong displacement and skewing of the shock density spike with respect to the analytical solutions .figure [ f.hard ] shows the errors for the values given in table [ t.hard ] , plus additional values for g. the striation lies below the f errors , and the higher striations are still a factor below f.    the previous problems contained sharp gradients produced by shocks .we now look at a problem with smooth flow , the isentropic flow problem .this consists of an initial state with smooth profiles in , , and .a pulse of moving fluid is superposed on top of a constant density , zero velocity state .the velocity of each individual element is constant in time .therefore the `` exact '' solution at a later time is found by advancing each element in time at its known velocity , which yields a grid with irregular spacing , and then interpolating the result back onto a uniform grid .the initial structure is given by ,\ ] ] where is the density of the constant background state , and the function for , and for .the width of the pulse is and the amplitude is .the initial velocity profile within the pulse is set by taking one of the two riemann invariants to be constant , where ) ] _ open hexagons _ ) for the first aloy & rezzolla ultrarelativistic problem at with the exact solution ( _ solid line _ ) , where both are computed with .the small insert panel shows a detail of the density spike .a dashed line connects the glimm density points.[f.referee1 ]   a comparison of the pressure ( _ open triangles _ ) , density ( _ open squares _ ) , velocity ( _ open pentagons _ ) , and lorentz factor ( ^{-1/2}$ ] _ open hexagons _ ) for the second aloy & rezzolla ultrarelativistic problem at with the exact solution ( _ solid line _ ) , where both are computed with .a snapshot of the first three variables is shown in the top panel , and the bottom panel shows .the small insert within the top panel presents a detail of the density spike .a dashed line connects the glimm density points.[f.referee3 ]   the evolution of ( _ from top to bottom _ ) lorentz factor , density , and pressure for a spherically symmetric 1d test run of a thin - shell , relativistic blast wave to compare with wen et al ( 1997 , see their fig .5 ) . for this run over the entire computational domain ( ) , or 3800 grid points overthe domain plotted .a blandford - mckee profile with lorentz factor is taken initially for a thin spherical shell extending from to , where .the frame of reference is continually adjusted so that the origin corresponds to the position of the contact discontinuity .there is a rightward moving forward shock and a leftward moving reverse shock.[kz_soln ]

Abstract: we present the results of standard one - dimensional test problems in relativistic hydrodynamics using glimm s ( random choice ) method , and compare them to results obtained using finite differencing methods . for problems containing profiles with sharp edges , such as shocks , we find glimm s method yields global errors orders of magnitude smaller than the traditional techniques . the strongest differences are seen for problems in which a shear field is superposed . for smooth flows , glimm s method is inferior to standard methods . the location of specific features can be off by up to two grid points with respect to an exact solution in glimm s method , and furthermore curved states are not modeled optimally since the method idealizes solutions as being composed of piecewise constant states . thus although glimm s method is superior at correctly resolving sharp features , especially in the presence of shear , for realistic applications in which one typically finds smooth flows plus strong gradients or discontinuities , standard fd methods yield smaller global errors . glimm s method may prove useful in certain applications such as grb afterglow shock propagation into a uniform medium .


vehicular traffic on highways as well as in cities tends to suffer from a jamming transition when the global traffic density exceeds a critical threshold value .the phenomena related to traffic jams have attracted the attention of engineers and physicists since many years and there exists a large variety of experimental observations , see  and references therein .the formation of traffic jams has been investigated by many authors using various methods .as early as 1955 lighthill and whitham  described the spontaneous formation of regions of increased car concentrations and shock waves using their approach of kinematic waves , i.e. wave motion where the spatial value of the flow is a function of the spatial concentration distribution .prigogine and herman  investigated traffic jams on highways by performing a stability analysis of hydrodynamic equations .one dimensional cellular automata models  for the simulation of traffic flow have been proposed e.g. in  .this type of models has been studied intensively using massively parallel computers  .schadschneider and schreckenberg  solved a one dimensional cellular automaton model analytically in the mean field approximation , csahk and vicsek  investigated their automaton in the presence of quenched noise .nonlinear wave descriptions have been proposed in .al apply a ballistic aggregation process to model the kinetics of clustering in one dimensional traffic flow .hydrodynamic approaches have been investigated by various authors , e.g.  .these models are based on certain assumptions concerning the `` hydrodynamic '' properties of traffic flows such as the velocity versus density relation  , and the viscous terms  .one dimensional traffic has been studied extensively and the developed models are sufficiently sophisticated to reproduce experimental observations , i.e. the fundamental diagram ( throughput versus flow density ) of one lane roads .some of the authors ( e.g.  ) claim that the size distribution of the jams obeys a power law , but very large scale computations have shown that the lifetimes of the jams and hence their sizes reveal a characteristic cut off which seems to be no finite size effect of the simulation  .there are several models describing traffic flow in more than one dimension .the analysis of traffic on a network with multiple sources and sinks can be found e.g. in .nagatani  describes a cellular automaton representing a two lane roadway .an interesting new approach was currently proposed by bando et al . who express the rules which govern the behavior of the cars ( i.e. acceleration and deceleration ) by a dynamic equation for each car .the interaction of neighboring cars is expressed in terms of the function of the distance between the cars . using this approach the authors connect in some respect the model of the cellular automata with molecular dynamics .two dimensional cellular automata designed to simulate traffic in a city have been investigated by some authors .biham et al . proposed a three state cellular automaton model where at a given time step each site can be occupied by a car moving from south to north , by a car moving from west to east , or the site may be empty .the time behavior is ruled by synchronous traffic lights at each site , allowing alternatively for vertical or horizontal traffic .obviously there are situations when the traffic jams , i.e. when the traffic light allows for driving but the next site is occupied . the model given in  andparticularly the spatial correlations in the jamming phase have been numerically investigated by tadaki and kikuchi  . a similar model but with `` faulty '' traffic lights was studied in  .nagatani  investigated the spreading of a jam which is induced by an accident using an extremely simple automaton rule .numerically he finds scaling laws for the size of the spreading jam as a function of time elapsed since the occurrence of the accident which can not be derived analytically so far .fukui et al . numerically investigated the evolution of ensemble averages of the jamming process for the simple automaton model described in  .for the same model  chau et al . recently gave an analytic upper bound ( depending on the dimension ) for the critical car density , i.e. when the system transits from the `` moving phase '' into the `` jamming phase '' . in two dimensions they found .obviously this upper bound is not close to the value which was observed in numerical simulations . in the case of nagatani and seno  is a set of parallel one way streets , all oriented in the same direction ( ) , which is intersected by a single perpendicular one way street in .  for fixed car density on the perpendicular street find that the flux of cars rises linearly with the car density until a characteristic critical threshold is reached .when further increasing the density the flux approximately remains constant while the average velocity of the cars drops .this rather sharp change in the overall behavior is due to the formation of traffic jams where the flow is irregular and discontinuous . when further increasing the density the discontinuous character of the flow disappears at a second threshold and the flux declines linearly .this behavior corresponds to the symmetry of the car density in and the spaces between the cars .indeed , one finds that the critical densities and are almost exactly symmetric , e.g. for one finds and .cuesta et al . have been the first who reported on simulations of a two dimensional automaton where the cars are allowed to change their direction . in their modeleach crossing can be occupied by one car or it can be empty . each car is assigned a preferred direction ; the parameter is the probability that the car at the position moves in the next time step horizontally and is the probability to move in vertical direction .horizontal motion is allowed at even time steps , vertical motion at odd times .there have been defined two variants , a : there are only one way streets directed from south to north and from east to west .half of the cars is given the trend and the other half is given . in the second variant bthere are one way streets pointing alternatively south north and north south , and west east and east west , respectively .the trends of the cars which are subdivided into four equal groups , point into one of the four possible directions .  we should mention that several authors claim that at least the one dimensional traffic flow problem is closely related to sand flowing in pipes , e.g.  .  in the present paper we are concerned with a two dimensional cellular automaton designed to simulate traffic in a city .as a special feature of our model we emphasize the fact that we use rather realistic traffic rules which will cause a slowing down of the average traffic velocity and finally will lead to the collapse .in our model the city is represented by a set of streets in horizontal direction crossing streets in vertical direction .cars are allowed to move in both directions , i.e. we have no restrictions to one way streets .the crossings define a two dimensional cellular automaton , each of the crossings is represented by a site of the automaton .we assume periodic boundary conditions in both directions .[ fig : kreuzung ] shows a schematic plot of an automaton site ( a crossing ) . at each crossing there are four queues of maximum length filled by cars coming from one of the four directions , representing the finite space where cars can move freely between crossings in realistic urban traffic systems .hence , at each crossing there are allowed at most cars .each of the cars has a desired direction , i.e. to the right , to the left or straight on , which has to be chosen according to a certain rule .we will discuss this point in detail below . within each time step the first of the cars in each queue can go to the next crossing provided that it does not have to give way to one of the other cars at the same site . in caseit has to give way to another car it will not move during this time step . in our model we assume the simple and realistic rule that a car has to give way in the case that another car occupies the same crossing at its right hand side .moreover a car has to stop if it intends to turn left and there is a car at the opposite side of the crossing which goes straight on or turns to the right . if all of the four top positions of a crossing are occupied by four cars one of the cars will be selected randomly and will then be allowed to move in the current time step .insofar we have chosen realistic rules which are current law in many countries .each of the cars in the system starts at a randomly selected crossing , its desired direction ( left , straight on , right ) will be determined according to some rule ( discussed below ) . within each timestep a car maximally moves a distance of one lattice cell , i.e. it tries to transit to one of the next neighbors of the current site provided the following three conditions hold :  1 .  it does not have to give way .it is placed at the top position of its queue .3 .  there are less than cars standing in the destination queue .otherwise it rests . obviously , if there is only one car in town it never has to give way nor will it find another car in front of itself .therefore the average velocity by the symbol , whereas the time average by . ] in units of lattice cells per time steps .as mentioned above we have to discuss the rule which determines the desired direction within each time step .there are at least three reasonable rules :  1 .  the desired direction for each car is selected at random within each time step not regarding whether the car moved in the previous time step or stopped .this rule is rather artificial since provided a jam occurs it can dissolve due to the fact that the cars might give up their previously desired direction when they can not move .each car is assigned a destination site on the lattice .once a car reached its destination it will be assigned a new randomly chosen destination .this rule is problematical since one has to define the detailed path a car follows to reach its destination .a natural choice would be to chose a path so that the car has to change its direction only once on its way between its starting position and its destination . in this case , however , the overall behavior of the system would depend on the size of the lattice since one can check that the probability to be allowed to move straight on is higher than to be allowed to turn . cars which intend to turn either left or right ( on the average ) have to give way more frequently than those which go straight on .the larger the system the smaller becomes the probability per time step to turn . for the case the probability for a car in a given time step to turn approaches zero .hence this rule does not allow for size independent results , and the results of a simulation using this rule apply only for one fixed lattice size .each car is assigned a desired direction during the initialization .but now , only after moving one step to the aimed destination a new desired direction will be chosen at random , i.e. the primarily chosen desired direction is maintained by each car until all of the three conditions hold which allow the car to move .this rule has several advantages compared with the other rules .first we avoid the size dependent behavior of the previous rule , and second we do not have long range correlations between the lattice sites for the case of low car density , which would be caused by the second rule .hence the behavior of the system can be described by a markov process , which will be the basic assumption of our analytical calculation in section [ statisticaldescription ] . an analytical description which takes into account the long range correlation between the sites coming from the rules which determine the path of the cars seems to be very difficult , except perhaps for the ( trivial ) case , when the probability of the cars to turn , and hence the traffic rules connected with the turn , can be neglected . in the following we always refer to the third rule .in this section we present the results found by numerical simulations of a cellular automaton which represents the traffic rules described above .the two dimensional automaton consists of sites on a rectangular lattice . to eliminate boundary induced effectswe have chosen periodic boundary conditions in both directions , i.e. we assume the topology of a torus .we simulated systems of size , , , and no striking difference in the behavior of the system was observed .hence , we exclude finite size effects . the fact that already comparatively small lattices exhibit this independence from their size seems to indicate the existence of only weak spatial correlations between sites separated by a large distance . for high densities , i.e. close to or above the critical density , this statement does not remain valid .in fact , we expect the existence of long range correlations .however , in the present paper we are mostly concerned with the low density region and we will not devote ourselves extensively to the behavior of the automaton for high car density .each of the automaton sites had its own four queues of length connecting the site with one of its four neighboring sites as described in the previous section .for each value of the density , where is the number of cars , we started the system at randomly chosen initial conditions .each car was assigned the initial site , the queue in this site ( i.e. whether it comes from north , east , south or west ) and the desired direction ( left , straight , right ) for the next step .when a car transits to an empty site from a certain direction it occupies the first empty place in the queue . if it is at the top position of the queue it can transit to the next site or it has to wait , depending on the cars coming from the other directions of the same site , and depending on the traffic rules which apply to the given situation . if a car transits to another site and if there are other cars located in the previously mentioned queue , all of them advance by one position in this queue hence , moving up `` closer to the crossing '' .after starting the simulation for a given number of cars , i.e. for a given density , we evolved the system 10,000 time steps without recording the statistics in order to let the system forget about the initial preparation .typically a `` stationary '' behavior was found after only a few hundred time steps . in density regions very close to the transition valuehowever , the relaxation of the system lasts longer which hints at some phase transition like phenomenon .the mean velocity of the cars was then computed using 2,000 time steps .we varied the car density beginning from to in steps .the results of the simulations are shown in fig .[ figsim ] . like in simulations of cellular automata performed by other authors with different traffic rules we found a smoothly decaying function in the low density region . at a certain threshold value the character of the traffic flow changesabruptly and the system transits into the clogged regime .but different from other systems we find a remnant velocity in the clogged regime .this is due to the rule that in case all four of the queues are occupied there will be one car selected randomly which then will be allowed to move .therefore , in our model there exist situations where a jamming at a crossing with four queues occupied may still dissolve .the curve in the small density region is very smooth , i.e. there are almost no fluctuations .hence , we can assume that long range correlations do not play a major role there .in the following section [ statisticaldescription ] we will present a statistical description appropriate to model the flow in the low density regime , i.e. for , which was inspired by this observation .let the number of cars be denoted by and the number of cells respectively crossings building the torus by where is the common perimeter of the torus .for the global density we choose the symbol .the quantity we focus our attention on is a time averaged is chosen sufficiently large in order to scan the state space respectively the limit set with sufficient accuracy . ] and normalized velocity where means the total number of cars prevented from moving by traffic rules at time which depends on the exact situation experienced during the simulation .  for a description of the dynamics in the low density region we apply the following arguments :  1 .an effective description is possible which separates the process of clustering at the possible crossings from the process of obeying the traffic rules .due to this independence assumption the time average ( [ ta ] ) extracted from a simulation can be decomposed into two averages : first a time and cluster average related to the traffic rules and second a time average with respect to the clustering process .given cars meeting at one crossing a variety of situations is possible .every car is approaching the crossing from either north , east , south or west and intends to go straight , to turn left or to turn right . in the long runall possible situations will have equal probability .hence , the average number of cars which must wait , denoted by with , can be computed by combinatorial reasoning .the result of this computation is collected in table [ waitingcars ] for .a sketch of the derivation is given in appendix [ appendixa ] .+ for clusters of size the are not shown since their contributions are suppressed by very small cluster probabilities .central to our description is the identification of cluster distributions , i.e. we write meaning that denotes the number of cells in the system where cars are meeting .of course , there are two boundary conditions , namely and we now assume that for the low density regime the process of clustering is ergodic which means for almost all initial configurations the time average can equivalently be represented by an average according to an invariant probability distribution .note that this probability distribution is defined over the space of all configurations compatible with the boundary conditions ( [ bc1 ] ) and ( [ bc2 ] ) . in order to derive this distributionwe additionally assume that the dynamics effectively is equivalent to a process of independently distributing cars ( balls ) among cells with crossings ( urns ) according to the equidistribution .this means we assume the discrete dynamics to be a bernoulli process .then the probability distribution reads } \ ; . \label{understand}\ ] ] to understand formula ( [ understand ] ) in detail one has to realize that there exist +  ] ways to index the cars with numbers taking care of the fact that cars within an identical cell are not distinguishable .finally the equidistribution with respect to all configurations yields the factor .this result was obtained by von mises ( 1939 )  .  now the time average is replaced by the ensemble average the sum with respect to reflects the average delay due to the traffic rules at the crossings and the sum with respect to all configurations obeying ( [ bc1 ] ) and ( [ bc2 ] ) which is indicated by the symbol  accounts for the cluster statistics .the factor accounts for the fact that the dynamics of the real process differs from a bernoulli process . in generalthere may exist spatial as well as temporal correlations .the spatial correlations become important for densely filled lattices .the dynamics which control the traffic within one cell is strongly influenced by the cars moving in neighboring cells .finally the emergence of a traffic jam clearly expresses these correlations . on the contrary temporal correlationsplay the dominant role for very sparse system , i.e. in the low density region .consecutive situations are not completely statistically independent .let us briefly explain this effect considering only cluster of size : when the cars are randomly distributed amongst the lattice sites there is a certain probability that two cars and occupy the same site and hence form a cluster of size .assuming statistical independence this probability is time invariant but for the cellular automaton the situation is quite different .if two cars meet at a site there exist two possibilities : either both cars continue , or one of them continues and the other one has to give way . there is no chance that both cars meet again one time step later . in the extreme case of only two cars ,i.e. , the probability for them to meet by random distribution is thus effectively reduced by a factor of two in comparison with the automaton dynamics . given only a small number of cars , i.e , and provided only two of them and meet occasionally one has to consider in the following time step a system where only cars feel the interaction with ( in our case : can meet occasionally ) other cars and cars which can meet only cars , i.e. is not allowed to meet and can not meet .hence , the system has some kind of memory and therefore it is not ideally bernoullian .obviously this effect plays a crucial role only for small since otherwise the relative difference between or is negligible . speaking in terms of densitythis means that the effect is substantial only for very small densities .then however , clusters larger than are suppressed by small likelihood .the net effect of these correlations can be taken into account through counting all possible configurations which can be realized after two cars have met .this procedure is similar to the method employed for the determination of the ( as explained in appendix [ appendixa ] ) . neglecting contributionsrelated to clusters of size larger than three , which is safe for very small densities ( see fig [ figcluster ] , explanation below ) means for . from simple geometric considerations we find , and .inserting expression eq.([timeaverage ] ) into ( [ ta ] ) yields and performing the sum over all ( ) we find the distribution  declared for  can be calculated using the inclusion exclusion principle  .this derivation is performed in appendix [ appendixb ] and the result reads we see from ( [ moment ] ) that the ingredients we actually need are the first moments of the cluster distribution .they can be derived calculating a generating function for the ( descending factorial ) moments , denoted  .this calculation is performed in appendix [ appendixc ] .the result reads   in fig .[ figcluster ] we have depicted the expectation value of the cluster distribution ( eq . ( [ momentsofclusterdistr ] ) ) divided by the system size .the solid lines correspond to the function given in eq .( [ momentsofclusterdistr ] ) and the points represent related data taken from our simulation on a torus .the values for give the probability for a site to be empty .for and the analytical and numerical results agree perfectly , for larger the discrepancies between the solid lines and points are a direct consequence of the deviations from the assumed independent behavior as explained above .note that the probabilities in fig .[ figcluster ] are plotted using a logarithmic scale .the observed discrepancies for clusters of size larger than 3 will hardly have any influence on the overall behavior of the automaton .for values the simulation data illustrate a breakdown for the small occupation numbers ( except ) due to the emergence of jams which are clusters of high order . at the transitionpoint the occupation rates abruptly drop by three orders of magnitude .the precise position of this transition point is masked by the finite size of used in the simulation .increasing the car density beyond the blurred critical zone results in a slow rise of the occupation number for the small clusters and simultaneously , in a slow decline of the number of empty sites .this is due to the increasing number of cars still moving around the jammed crossings .inserting equation ( [ momentsofclusterdistr ] ) into ( [ moment ] ) yields the following formula the last step will be to substitute in ( [ eqf ] ) the density for the number of cars according to which results in the main result of our statistical description , namely the mean velocity as a function of the car density ( eq . ( [ eqff ] ) ) , is plotted in fig .[ figeqff ] .the solid line shows the function given by eq .( [ eqff ] ) and the points are the data taken from the simulation on the torus ( compare fig.[figsim ] . in the low density regionwe find a nice agreement .the closer one approaches the transition point the more eq .( [ eqff ] ) overestimates the simulation results .this can be understood by recalling the fact that for densities close to the critical value the effective decomposition of the automaton dynamics into the cluster dynamics and the single site dynamics becomes inappropriate due to formation and dissolution of short lived jams ( like critical fluctuations ) .they tend to increase the occupation number of larger clusters and hence , lead to an effective decrease of the average velocity .  in the thermodynamic limit which means and keeping constant we drop terms of order and end up with where the exponential corresponds to the dominant contribution of the last factor in ( [ eqff ] ) .this formula can be expressed by a series expansion in powers of  note that there is no constant term in the double sum since .equation ( [ expansion ] ) can be truncated at a given order of hence , giving rise to an approximation scheme . in fig .[ figexpansion ] we plotted the linear ( dashed ) , the quadratic ( dotted ) and cubic ( long dashed ) approximations together with the full formula ( fat solid ) .the range of reliability visibly increases with increasing the order of approximation . on the other handthis scheme clearly illustrates that the functional relation between mean velocity and global density definitely is _ nonlinear _ , except for very small car densities .we investigated the behavior of a two dimensional cellular automaton with periodic boundary conditions to simulate traffic flow in cities .the automaton mimics realistic traffic rules which apply to our everyday experience of vehicular traffic . in agreement with similar models we found a slow decay for the mean velocity as a function of the global traffic density . at a critical threshold value the mean velocity collapses abruptly and the system transits into another regime of global behavior which we call the jamming regime . beyond the critical density the average velocity is very small and it declines further when increasing the density . by simulating different sizes of automatawe could exclude the influence of finite size effect provided the density value is not rather close to the critical density .  applying combinatorics and statistical methods for the description of the system in the low density regime the analytical calculation performed in section [ statisticaldescription ] yielded the average car velocity as a function of the car density  which nicely agrees with the values of the numerical simulations in section [ numerics ] . since the derived functional relationship between meanvelocity and global traffic density was based on very general assumptions ( conservation of the number of cars , weak spatial correlations in the low density regime , ergodicity ) we expect this description to be valid for a wider class of traffic systems .note that the specific structure of traffic rules enter the description only through the average delays .the only nonlocal ingredient arises from the restriction that cars have to stop in case the next desired queue at a neighboring site is totally filled up .but this nonlocal character only becomes substantial for densities close to the critical value where it causes long range spatial correlations .several authors ( e.g.  ) assert that the average velocity in the low density regime is a linear function of the density and indeed the simulation results seem to support this observation .a more detailed analysis however reveals that this function definitely is nonlinear .an analytic expansion shows ( eq . ( [ expansion ] ) , fig . [ figexpansion ] )that the results become dramatically wrong when truncating the formula after the linear term .surprisingly the analytic description gives good results even for densities not too small where the nonlocal effects of the dynamics and hence , long range correlations spoil the basic assumptions of our treatment .  for still higher values of the density the results of our simulation agree with results reported in literature ( e.g.  and many others ) , but no longer with our approximate theory . above a critical densitywe observe an abrupt transition of the system into the jammed regime where the averaged velocity is close to zero .this regime however is beyond the scope of our statistical description and has to be investigated starting from other approaches ( e.g. nucleation processes ) .we thank h. herzel and l. schimansky geier for helpful discussion .we greatly acknowledge u. kchler for drawing our attention to ref . and for discussing the combinatorics in detail .in table [ waitingcars ] we present the average number of cars which are prevented from moving given cars meeting at a crossing . obviously the number is confined to the interval  ] .insertion of this probability into the inclusion exclusion formula ( [ inclexcl ] ) yields & = & { n!\over   m^n}\:\sum\limits_{j = k_i}^m\ : ( -1)^{(j - k_i)}\left({j\atop   k_i}\right)\ : { ( m - j)^{(n - ji)}\over ( i!)^j~ ( n - ji)!}\end{aligned}\ ] ] because of the generalized definition of factorials using the gamma function the sum effectively only ranges from up to which is sensible .in the following we will derive the mean value of the cluster distribution which is given in eq .( [ momentsofclusterdistr ] ) .the generating function for the ( descending ) factorial moments of the cluster distribution we use is where is the probability to find a value for the stochastic variable when trying with balls .the benefit of such a complicated looking generating function is that the sums can be performed yielding an analytical expression namely ^m\ ] ] for the explicit derivation the reader is referred to the book by johnson and kotz\equiv p(k_i)$ ] ] p. 116ff  .moreover , this function is related to the ( descending ) factorial moments of the stochastic variable according to hence , the first moment is given as the coefficient of in consequently we calculate ^m \right ) \right|_{x=1 } & = & { n!\over m^{(n-1 ) } } \ :  \exp^{\left[z(m-1)\right]}\ : { z^i\over i!}\nonumber\\[.2 cm ] & & \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!=  \sum\limits_{k=0}^{\infty}\ ; { n!\over i!\ ; k!}\ ; { ( m-1)^k\over   m^{(n-1)}}\ ; z^{(k+i)}\end{aligned}\ ] ] and finally arrive at                   k. nagel and s. rasmussen . in r.a. r. a. brooks and p. maes , editors , _ artificial life iv : proceedings of the fourth international workshop on the synthesis and simulation of living systems , cambridge , 1994 _ , page 222 .mit press , cambridge , ma , 1994 .

Abstract: a two dimensional cellular automaton is introduced to model the flow and jamming of vehicular traffic in cities . each site of the automaton represents a crossing where a finite number of cars can wait approaching the crossing from each of the four directions . the flow of cars obeys realistic traffic rules . we investigate the dependence of the average velocity of cars on the global traffic density . at a critical threshold for the density the average velocity reduces drastically caused by jamming . for the low density regime we provide analytical results which agree with the numerical results .   vehicular traffic , kombinatorics


a cauliflower is a variety of cabbage with an edible head of condensed flowers and flower stems .it is a form of cabbage in the mustard family , consisting of a compact terminal mass of greatly thickened , modified , and partially developed flower structures , together with their embracing fleshy stalks .this terminal cluster has been known as a typical fractal among living organisms .it is clear that the dimensionality of a fractal is the most fundamental concept of fractal analysis .nevertheless , the fractal dimension of a white cauliflower has not known yet .grey and kjems discussed the fractal dimension of a cauliflower , and they just suggested the possibility that the fractal dimension of a cauliflower could be larger than that of a broccoli .romera et .al . suggested a mathematical model of a cauliflower as a sequence of a baby mandelbrot set , and its fractal dimension was not obtained , either .  in this article , we measure the fractal dimension of the cross - section of a white cauliflower .then , the fractal dimension of the bulk cauliflower is introduced .next , we create a mathematical model for the cauliflower , and compare it with our experimental measurement .the condition of creating the fractal in our mathematical model is discussed , too .there are many definitions of fractal dimensions , and the basic concept in our model has almost the property as follows : let us introduce that is a measurement scale , and that the measurement is , then the fractal dimension of a set is determined by the power law if is a constant as , has a dimension of .  the term capacity dimension " or box - counting dimension " is most widely used , because it can be easily applied to fractal objects .it is defined as where is the smallest number of sets of diameter at most which can cover .fractal objects embedded in one- or two - dimension are easy to measure relatively compared to other higher dimensions , but most of fractals in nature that are embedded in three dimensions is difficult to measure even in the capacity dimension . however , the dimensions of a cross - section is generally known to be related with that of the bulk .let us think of a bulk of three dimensions , and assume the dimension of its cross - section is two , that is , it is 2/3 of the bulk .we can propose an _ansaz _ from the above property : let be a fractal dimension of a cross - section embedded in two dimensions .then , the fractal dimension of the bulk imbedded in three dimensions can be written as where represents every possible cross - section . as the fractal dimension of the cross - section is independent of directions , eq .( [ 30 ] ) is simply written as .we prepared several white cauliflowers of about in mass and in diameter .we cut half of them in horizontal ( h ) direction , and the other half in vertical ( v ) direction .then , we scanned the cross - section by a scanner and read the image into black and white in fig . 1 . here , the two figures by the two perpendicular directions look totally different .next , we read the image into matrix of numbers .the numbers of the matrix are the reflectivity of each pixels .the black backgrounds produces 0 s , and the white images produce large numbers .let us call the non - zero number 1 for convenience .the size of the matrix is about .note that the size of a pixel is an order of 100 .  in order to measure the fractal dimension of the cross - sections in eq .( [ 20 ] ) , we count the non - zero numbers by the box - counting method .it becomes the smallest number of sets to cover the white images by . in a half reduction procedure of the matrix ,a component becomes 1 if it contains any non - zero number , otherwise it becomes 0 .then , the number of 1 s becomes the smallest number of sets to cover the white images by . and so on .in the reducing steps in half , that is , the conversion from the reduced matrices to images is plotted in fig . 2 for the first four steps .we see that the basic structure of the cross - sections remains unchanged .3(a ) is the log - log plot for the h - direction , and fig .3(b ) is for the v - direction .we plotted it for a couple of different cauliflower samples .the slopes are the capacity dimensions of the cross - sections .surprisingly , we found the two slopes for different directions are very similar .it is . repeating the procedure for different white cauliflowers , we observed similar values of as for the two different directionstherefore , from eq .( [ 30 ] ) we predict that the capacity dimension of the white cauliflower is about 2.8 .converting a fractal found in nature into a corresponding mathematical model is a hope of a theorist , but most of the work is far away from the real world or extremely complicated in analysis .the modeling of nature should be as simple as it can , and at the same time , it should contain the basic structure of nature .furthermore , if some physical quantities of the model can match the real systems , it will guarantee more credits to the mathematical model .we modeled the cross - section of a cauliflower in v - direction as a rectangular tree that has three equilateral sides in fig .this is the simplest model of a self - similar set that has a single scale factor .note that the cross - section of a broccoli in v - direction has modeled into a self - similar set of triangular tree or pythagoras tree .the the scale factor of the rectangular tree in fig .4 have the following relation where is the angle of the rectangular tree .downsizing the tree by a ratio , it creates three branches .therefore , the fractal dimension of fig .4 is   the areas of the rectangular tree have the following series where . because the fractal is the limit of the series, it should converge as increases .the condition of convergence of the series is , or .it is clear from .this condition corresponds to by eq .( [ 40 ] ) .since the in the cauliflower is 1.88 , the scale factor and the angle in the model is obtained as 0.56 and from eqs .( [ 40 ] ) and ( [ 45 ] ) .we recognize that the cross - section in v - direction of the cauliflower is pretty close to two dimensions .as the approaches , the dimension of the tree goes to two because the limit bents to cover the two dimensional surface . on the other hand ,as the approaches , the dimension of the tree goes to one because the limit goes to a long rod .the head of the real cauliflower is rounded not like the model in fig .4 . it gives a possibility that the real cauliflower is not a self - similar set of single scale factor , but a set of multi - scale factors .we measured the fractal dimension of the cross - section of a white cauliflower by the direct scanning method .it was , and almost independent of the directions of the cross - sections . from these results, we predict that the fractal dimensions of the bulk cauliflower is about 2.8 .we created a mathematical model for the v - direction of a cauliflower with only one scale factor .it is a rectangular tree of three equilateral sides .we suggested the condition of creating fractals in our model , and we showed that the angle of the model is 67 degrees and the scale factor is 0.56 comparing with experiment .this method of a scanning cross - sections and mathematical model from a polygonal tree can be widely applied to many complex bulk fractals in nature .b. mandelbrot , _ the fractal geometry of nature _ ( freeman , new york , 1977 ) . f. grey and j. k. kjems , physica d. * 38 * , 154 ( 1989 ) .m. romera , g. paster , g. lvarez , and f. montoya , computers graphics , * 28 * , 437 ( 2004 ) .k. falconer , _ fractal geometry _( john wiley & sons , new york , 1990 ) .k. falconer , _ techniques in fractal geometry _ ( john wiley sons , new york , 1996 ) .peitgen , h. jurgens , and d. saupe , _ chaos and fractals _ ( springer , new york , 1992 ) p.126 .

Abstract: the fractal structure of a white cauliflower is investigated by box - counting method of its cross - section . the capacity dimension of the cross - section is independent of directions . from the result , we predict that the capacity dimension of the cauliflower is about 2.8 . the vertical cross - section of the cauliflower is modeled into a self - similar set of a rectangular tree . we discuss the condition of the fractal object in the tree , and show that the vertical cross - section has an angle of 67 degrees in our model .


receding horizon control is a popular paradigm for designing control policies . in the context of deterministic systemsit has received a considerable amount of attention over the last two decades , and significant advancements have been made in terms of its theoretical foundations as well as industrial applications .the motivation comes primarily from the fact that receding horizon control yields tractabile control laws for deterministic systems in the presence of constraints , and has consequently become popular in the industry .the counterpart in the context of stochastic systems , however , is a relatively recent development . in this articlewe solve the problem of stochastic receding horizon control for linear systems subject to additive ( possibly ) unbounded disturbances and hard norm bounds on the control inputs , over a class of feedback policies .methods for guaranteeing hard bounds on the control inputs , within our context , while ensuring tractability of the underlying optimization problem are , to our knowledge , not available in the current literature .preliminary results in this direction were reported in .  in the deterministic setting , the receding horizon control scheme is dominated by worst - case analysis relying on robust control and robust optimization methods , see , for example , and the references therein .the central idea is to synthesize a controller based on the bounds of the noise such that a certain target set becomes invariant with respect to the closed - loop dynamics .however , such an approach tends to yield rather conservative controllers and large infeasibility regions .moreover , assigning an a priori bound to the noise seems to demand considerable insight .a stochastic model of the noise is a natural alternative approach to this problem : the conservativeness of worst - case controllers may be reduced , and one may not need to impose any a priori bounds on the maximum magnitude of the noise . in  ,the authors reformulate the stochastic programming problem as a deterministic one with bounded noise and solve a robust optimization problem over a finite horizon , followed by estimating the performance when the noise is unbounded but takes high values with low probability ( as in the gaussian case ) . in  a slightly different problem is addressed in which the noise enters in a multiplicative manner , and hard constraints on the states and control inputs are relaxed to constraints resembling the integrated chance constraints of or risk measures in mathematical finance .similar relaxations of hard constraints to soft probabilistic ones have also appeared in  for both multiplicative and additive noise inputs , as well as in  for additive noise inputs .there are also other approaches , e.g. , those employing randomized algorithms as in  .related lines of research can be found in  dealing with constrained model predictive control ( mpc ) for stochastic systems motivated by industrial applications , in  dealing with stochastic stability , in dealing with q - design , in , e.g. , dealing with alternative approaches to control under actuator constraints and neural - network approximation .the articles deal with a formulation that allows probabilistic state constraints but not hard input constraints , and is hence complementary to the approach in the present article , and treats the case of output feedback . .finally , note that probabilistic constraints on the controllers naturally raise difficult questions on what actions to take when such constraints are violated , see  and  for partial solutions to these issues .the main contributions of the article are as follows : we give a tractable , convex , and globally feasible solution to the finite - horizon stochastic linear quadratic ( lq ) problem for linear systems with possibly unbounded additive noise and hard constraints on the elements of the control policy . within this framework onehas two directions to pursue in terms of controller design , namely , a posteriori bounding the standard lqg controller , or employing certainty - equivalent receding horizon controller . while the former direction explicitly incorporates some aspects of feedback ,the synthesis of the latter involves control constraints and implicitly incorporates the notion of feedback .our choice of feedback policies explores the middle ground between these two choices : we explicitly incorporate both the control bounds and feedback at the design phase .more specifically , we adopt a policy that is affine in certain bounded functions of the past noise inputs .the optimal control problem is lifted onto general vector spaces of candidate control functions from which the controller can be selected algorithmically by solving a convex optimization problem .our novel approach does not require artificially relaxing the hard constraints on the control input to soft probabilistic ones ( to ensure large feasible sets ) , and still provides a globally feasible solution to the problem .minimal assumptions of the noise sequence being i.i.d and having finite second moment are imposed .the effect of the noise appears in the convex optimization problem as certain fixed cross - covariance matrices , which may be computed off - line and stored .once tractability of the optimization problem is ensured , we employ the resulting control policy in a receding horizon scheme . under our policiesthe closed - loop system is in general not necessarily markovian , and as a result stability of the closed - loop system is not immediate .in fact , we can no longer appeal directly to standard foster - lyapunov methods .we establish that our receding horizon control scheme provides stability under the assumption that the zero - input and zero - noise system is asymptotically stable .we provide examples that demonstrate the effectiveness of our policies with respect to standard methods such as certainty - equivalent mpc , standard unconstrained lqg and saturated lqg control .these examples show that our policies perform no worse than the standard unconstrained lqg controller in the absence of control constraints , and outperform the certainty - equivalent mpc as well as the saturated lqg control by a significant margin .our mechanism for selection of a policy consists of two steps : the first concerns the structure of our policies , and is motivated by preceding work in robust optimization and mpc .the second concerns the procedure for selection of an optimal policy from a general vector space of candidate control functions , and is inspired by approximate dynamic programming techniques . with respect to the first step ,our policies are more general compared to those in . with respect to the second , the selection procedure of our policies consists of a one - step tractable static optimization program .  the rest of this article is organized as follows . in section [ s : ps ]we state the main problem to be solved in the most general form . in section[ s : mainres ] we provide a tractable solution to the finite horizon optimization problem on general vector spaces .this result is specialized to various classes of noise and input constraint sets in section [ s : if ] .stability of receding horizon implementations of the obtained closed - loop policy is shown in section [ sec : stability ] , and input - to - state stability properties are discussed in section [ sec : iss ] .we provide a host of numerical examples that illustrate the effectiveness of our approach in section [ sec : examples ] .finally , we conclude in section [ sec : conclusions ] with a discussion on future research directions .hereafter , is the set of natural numbers , , is the set of all the integers , is the set of nonnegative real numbers , and denotes the set of complex numbers .we let denote the indicator function of a set , and and denote the -dimensional identity matrix and -dimensional zeros matrix , respectively .let denote the standard euclidean norm , and denote the usual norms .also , let  ] for the matrix obtained by stacking the columns of followed by the columns of . for a given symmetric -dimensional matrix with real entries , let be the set of eigenvalues of , and let and . finally , for a random vector let denote the matrix  ] .consider the following discrete - time stochastic dynamical system : where is the state , is the control input taking values in some given control set to be defined later , , , and is a sequence of stochastic noise vectors with .we assume that the initial condition is given and that , at any time , is observed perfectly .we do not assume that the components of the noise are uncorrelated , nor that they have zero mean ; this effectively means that may be of the form for some noise whose components are uncorrelated or mutually independent , , and .without loss of generality we shall stick to the simpler notation of  throughout this article .the results readily extend to the general case of , as can be seen in  .generally , a _control policy _ is a sequence of borel measurable maps .policies of finite length such as will be denoted in the sequel by .fix an optimization horizon and let us consider the following objective function at time given the state : ,\ ] ] where are some given symmetric matrices of appropriate dimension . at each timeinstant , we are interested in minimizing over the class of causal state feedback strategies defined as : } = \left [ \begin{array}{l } \pi_t(x_t ) \\\pi_{t+1}(x_t , x_{t+1 } ) \\ \vdots \\\pi_{t+n-1}(x_t , x_{t+1},\cdots , x_{t+n-1})\end{array}\right],\ ] ] for some measurable functions , while satisfying for each .the _ receding horizon control _ procedure for a given control horizon and time can be described as follows :  * measure the state ; * determine an admissible optimal feedback control policy , say , that minimizes the -stage cost function starting from time , given the measured initial condition ; * apply the first elements of the policy ; * increase to , and go back to step ( a ) .  in this context , if then this is usual mpc , and if , then it is usually known as rolling horizon control .since both the system and cost are time - invariant , it is enough to consider the problem of minimizing the cost for . in view of the above we consider the problem : if feasible , the problem generates an optimal sequence of feedback control laws .the evolution of the system ( [ eq : system ] ) over a single optimization horizon can be described in a compact form as follows : where  using the compact notation above , the optimization problem  can be rewritten as follows : \,\big|\ , \text{dynamics \eqref{eq : compactdyn } } , u\in\u\bigr\ } , \end{aligned}\ ] ] where , , and .we require that our controller is selected from a vector space of candidate controllers spanned by a given set of `` simple '' basis functions .the precise algorithmic selection procedure is based on the solution to an optimization problem .the basis functions may represent particular types of control functions that are easy or inexpensive to implement , e.g. , minimum attention control , or may be the only ones available for a specific application .for instance , piecewise constant policy elements with finitely many elements in their range may be viewed as controllers that can provide only finitely many values ; this may be viewed as an extended version of a bang - bang controller , or as a hybrid controller with a finite control alphabet .more formally , let be a nonempty separable vector space of functions with the control set as their range , i.e. , is the linear span of measurable functions , where - an ordered countable index set ( see for more details ) . as mentioned above , the elements of may be linear combinations of typical `` simple '' controller functions for .we are interested in policies of the form , where is an -dimensional vector and each component of the -dimensional vector - valued function is a member of . although this feedback function is directly from the noise , since the state is assumed to be perfectly measured , from the system dynamics  it follows at once that this controller is actually a feedback from all the states .indeed , in the spirit of  we have in other words , by construction , is generally a nonlinear feedback controller depending on the past states .is non - markovian as it is a function of the state vectors at all the previous times and not just on . ] also by construction , it is causal .  our general control policy can now be expressed as the vector where ,  * , * for is the -th random noise vector , * is an -dimensional vector for , * for is an -dimensional vector , and * each function belongs to the linear span of the basis elements , and thus has a representation as a linear combination , , , where are matrices of coefficients of appropriate dimension .analogous to fourier coefficients in harmonic analysis , we call the the -th fourier coefficient of the function . therefore , whenever for every , we have the finite representation where , , },\;\;\theta_{t , i}^\nu\in\r^{m\times n } , \quad { \rm and}\quad { { \mathfrak e}}(w_{i}){\coloneqq}{\left[\begin{matrix } { { \mathfrak e}}^1(w_i)\\\vdots \\ { { \mathfrak e}}^{|\mathcal i|}(w_i ) \end{matrix}\right]},\quad { \forall\,}i=0,1,\cdots , n-2.\ ] ] in this notation the policy  can be written as where is now the matrix of fourier coefficients having dimension . this fourier coefficient matrix and the vector play the role of the optimization parameters in our search for an optimal policy .note that does not include the noise vector , and that is strictly lower block triangular to enforce causality . in what follows , as a matter of notation , by shall denote the formal -th block - row of the matrix in , i.e. , , for , with being the identically row .we make the following assumption :  [ a : sys ] the sequence of noise vectors is i.i.d with  ] , the control input vector is bounded in -norm at each instant of time , i.e. , for  ] is convex with respect to the decision variables defined in . for , and it admits convex tractable versions with tighter domains of , given by where , &  \sigma_{{{\mathfrak e } } } ' & { \coloneqq}\ee\bigl [ w { { \mathfrak e } } ( w ) { ^\mathsf{t}}\bigr],\\       \mu_w & { \coloneqq}\ee [ w ] , \quad \quad \mu_{{\mathfrak e}}{\coloneqq}\ee[{{\mathfrak e } } ( w ) ] , & c & { \coloneqq}x_0{^\mathsf{t } } { a } { ^\mathsf{t}}q  a x_0 + 2\xz{^\mathsf{t } } { a } { ^\mathsf{t}}q  d  \mu + \operatorname{\mathbf{tr}}\bigl ( { d } { ^\mathsf{t}}q  d \sigma _ { w } \bigr ) .\end{aligned}\ ] ]  it is easy to see that is convex nondecreasing , and both and are affine functions of the design parameters for any realization of the noise .hence , is convex in since taking expectations of a convex function retains convexity ( * ? ? ?* section 3.2 ) .moreover , the control constraint sets in are convex in .this settles the first claim .the objective function  is given by + \ee_{\xz}\bigl [ { u } { ^\mathsf{t}}r  u \bigr]\\       & = \ee_{\xz}\bigl[\bigl ( a  x_0 +  b ( \eta + \theta { { \mathfrak e } } ( w ) ) +  d  w \bigr){^\mathsf{t}}q \bigl ( a  x_0 +  b ( \eta + \theta { { \mathfrak e } } ( w ) ) +  d  w \bigr)\bigr]\\       & \quad + \ee_{\xz}\bigl[(\eta + \theta { { \mathfrak e } } ( w ) ) { ^\mathsf{t}}r ( \eta + \theta { { \mathfrak e } } ( w ) ) \bigr]\\       & = x_0{^\mathsf{t } } { a } { ^\mathsf{t}}q  a x_0 + 2\xz{^\mathsf{t } } { a } { ^\mathsf{t}}q  b \eta + \eta{^\mathsf{t}}\bigl ( { b } { ^\mathsf{t}}q  b  +  r \bigr)\eta \\       & \quad + 2\bigl ( a \xz +  b \eta\bigr){^\mathsf{t}}q \bigl ( b \theta\ee_{\xz } [ { { \mathfrak e } } ( w ) ] +  d \ee_{\xz } [ w ] \bigr)\\       & \quad + \ee_{\xz}\bigl[\bigl ( b \theta { { \mathfrak e } } ( w ) +  d  w \bigr){^\mathsf{t}}q \bigl ( b \theta { { \mathfrak e } } ( w ) +  d  w \bigr)\bigr ] + \ee_{\xz}\bigl[(\theta { { \mathfrak e } } ( w ) ) { ^\mathsf{t}}r \theta { { \mathfrak e } } ( w ) \bigr]\\       & = x_0{^\mathsf{t } } { a } { ^\mathsf{t}}q  a x_0 + 2\xz{^\mathsf{t } } { a } { ^\mathsf{t}}q  b \eta + \eta{^\mathsf{t}}\bigl ( { b } { ^\mathsf{t}}q  b  +  r \bigr)\eta + 2\eta{^\mathsf{t}}r \theta\ee [ { { \mathfrak e } } ( w ) ] \\       & \quad +  2\bigl ( a \xz +  b \eta\bigr){^\mathsf{t}}q \bigl (  d \ee_{\xz } [ w ] +  b \theta\ee [ { { \mathfrak e } } ( w ) ] \bigr ) + \operatorname{\mathbf{tr}}\bigl ( { d } { ^\mathsf{t}}q  d \ee_{\xz}\bigl [ w { ^\mathsf{t}}\bigr]\bigr)\\       & \quad + \operatorname{\mathbf{tr}}\bigl(\theta{^\mathsf{t}}\bigl ( { b } { ^\mathsf{t}}q  b + r \bigr)\theta\ee_{\xz}\bigl [ { { \mathfrak e } } ( w ) { { \mathfrak e } } ( w ) { ^\mathsf{t}}\bigr]\bigr ) + 2\operatorname{\mathbf{tr}}\bigl(\theta{^\mathsf{t } } { b } { ^\mathsf{t}}q  d \ee_{\xz}\bigl [ w  { { \mathfrak e } } ( w ) { ^\mathsf{t}}\bigr]\bigr ) .\end{aligned}\ ] ] incorporating the definitions , , , , and , the right - hand side above equals since the matrix is positive semidefinite , it can be expressed as a finite nonnegative linear combination of matrices of the type , for vectors of appropriate dimension ( * ? ? ?* theorem 1.10 ) .accordingly , if , then defining and adjoining these equalities to the constraints of the optimization program , we arrive at the optimization program we see immediately that  is a convex program in the parameters , and , and is equivalent to the cost in .it only remains to consider the last constraint in .first we consider the cases of .using the notation above , an application of the triangle inequality immediately shows that the constraints can be written as it follows that the objective function in is quadratic and the constraints in - are affine in the optimization parameters , , and . as such, for our problem is a quadratic program .  for the case of , note that , and by definition of it is clear that .this immediately translates to , which is the third constraint in problem [ e : probspan ] and it is a quadratic constraint in the optimization parameters .therefore , for our problem is a quadratically constrained quadratic program .the optimization problem simplifies if we assume that = 0 ] if and only if = 0 ] , * , for , and * {^\mathsf{t}} ] with , where is defined in .then for the problem  under the control policy  is a convex optimization program with respect to the decision variables , given by where and are the -th rows of and , respectively , ,\cdots,\ee\bigl[\varphi_{n-1}(w_{n-1})\varphi_{n-1}(w_{n-1}){^\mathsf{t}}\bigr]\bigr\},\\         \gamma_2 & = \mathrm{diag}\bigl\{\ee\bigl[\varphi_0(w_0 ) w_0{^\mathsf{t}}\bigr],\cdots,\ee\bigl[\varphi_{n-1}(w_{n-1})w_{n-1}{^\mathsf{t}}\bigr]\bigr\}.       \end{aligned}\ ] ] the resulting policy is guaranteed to satisfy the control constraint set for .  a complete proof may be found in  ; it proceeds along the lines of the proof of theorem [ t : gen ] .note that the program  exactly solves  under the policy  and is neither a restriction nor a relaxation .problem  is a quadratic program in the optimization parameters ( see the discussion following theorem [ t : gen ] ) .the matrices and capture the statistics of the noise in the presence of the functions and can be computed numerically _ off - line _ using monte carlo techniques ( * ? ? ?* section 3.2 ) .this method will be utilized in the examples in section [ sec : examples ] . however , in some instances it is actually possible to compute these matrices in closed form ; this is shown in the next three examples .[ ex : sigmoids ] let us consider  when the noise process is an i.i.d sequence of gaussian random vectors of mean and covariance and standard sigmoidal policy functions , i.e. , . assume further that the components of are mutually independent , which implies that is a diagonal matrix . then from the identities in fact [ facts ] in [ s : facts ] , we have for and , & = \frac{1}{\sqrt{2\pi}\sigma_i}\int_{-\infty}^\infty { \widetilde}\varphi(t)^2 { \mathrm e^{-\frac{t^2}{2\sigma_i^2 } } } { \mathrm}dt = 2\cdot\frac{1}{\sqrt{2\pi}\sigma_i}\int_{0}^\infty \frac{t^2}{1+t^2 } { \mathrm e^{-\frac{t^2}{2\sigma_i^2}}}\\         & = \sqrt{2\pi}\sigma_i - \pi{\mathrm e^{-\frac{1}{2\sigma_i^2}}}\operatorname{erfc}\bigl(\frac{1}{\sqrt 2\sigma_i}\bigr ) .\end{aligned}\ ] ] this shows that the matrix in corollary [ c : sat ] is , where similarly , since & = \frac{1}{\sqrt{2\pi}\sigma_i}\int_{-\infty}^\infty t{\widetilde}\varphi(t ) { \mathrm e^{-\frac{t^2}{2\sigma_i^2 } } } { \mathrm}dt = 2\cdot\frac{1}{\sqrt{2\pi}\sigma_i}\int_{-\infty}^\infty \frac{t^2}{\sqrt{1+t^2 } } { \mathrm e^{-\frac{t^2}{2\sigma_i } } } { \mathrm}dt\\         & = \frac{\sigma_i}{\sqrt 2}u\bigl(\frac{1}{2 } , 0 , \frac{1}{2\sigma_i^2}\bigr ) ,       \end{aligned}\ ] ] where is the confluent hypergeometric function ( defined in the appendix ) , the matrix in corollary [ c : sat ] is , where therefore , given the system , the control policy , and the description of the noise input as above , the matrices and derived above complete the set of hypotheses of corollary [ c : sat ] .the problem  can now be solved as the quadratic program .[ ex : sigmoidsgen ] consider the setting of example [ ex : sigmoids ] ( with a standard sigmoid ) under the assumption that is a not necessarily diagonal matrix .to wit , the components of may be correlated at each time ; however , the random vector sequence is assumed to be i.i.d .this is equivalent to the knowledge of the correlations between the random variables , which are constant over . then  ] and the -th entry of being  ] ,  ] . in both the above cases the resulting policiesare guaranteed to satisfy the control constraint set for .( sketch . ) evaluating the objective function in  gives the objective function in .recall that is the -th block row of the formal matrix , and is the sub - row of the block row , where and .applying the triangle inequality for any , we get by orthogonality of the basis elements .the right - hand side of the last equality appears as the constraint in .for the finite - dimensional case , we note that the objective function is identical to the one in , and the constraint in  follows from the fact that , and .this leads to a quadratically constrained quadratic program in the finite- dimensional decision variables .let us illustrate the usage of corollary [ c : hspolicy ] through the following example .[ ex : unif ] consider the system , and suppose that the components of the noise vector are independent uniform random variables taking values in  ] .it is a standard fact in fourier analysis that the system is an orthonormal basis for the hilbert space of square - integrable functions on  ] ,  ] is given by = \frac{1}{2a}\int_{-a}^a { \mathrm e^{2\pi\ii v t}}\,{\mathrm}dt = \operatorname{sinc}(2\pi va) ] , } ] .suppose that the above -horizon optimal policy is computed as in corollary [ c : sat ] .we define the receding horizon policy corresponding to the consecutive concatenation of this -horizon optimal policy as   [ prop : rhmain ] consider the system , and suppose that assumptions [ a : sys ] and [ a : schur ] hold . for andany control horizon the receding horizon control policy renders the closed loop system  mean - square bounded , i.e. , < \infty ] for some than an estimate of  ] .the property expressed by  is one possible iss - type property for stochastic systems .one can come up with alternative stochastic analogs of the iss property , such as the following : , 1[ ] and such that is comparatively better suited .we shall however stick with the iss in property in this article .the following proposition can be established with the aid of proposition [ prop : rhmain ] for ; the proofs for and are also similar in spirit .consider the system , and suppose that assumptions [ a : sys ] and [ a : schur ] hold .then the closed - loop system  is iss in under the policy in for any .in this section we present several numerical examples to illustrate the theoretical results in the preceding sections .we start in example [ ex : unconstrained ] by comparing the performance of our policy to that of the standard finite horizon lqg controller whenever the control inputs set , i.e. , there are no bounds on the norm of the inputs .then we compare the performance of our policy against a saturated lqg controller in example [ ex : satlqgvsrhc ] .finally , in example [ ex : constrained ] we illustrate the effectiveness of our policy compared to the certainty - equivalent receding horizon control .  [ ex : unconstrained ] a natural question that may arise whenever the control inputs in our setup are not constrained , i.e. , , is the following : how does the policy compare to the globally optimal controller , which in this case is the standard finite - horizon lqg controller ?one would expect our policy to perform worse on the average since we restrict to a class of feedback policies that may not contain the globally optimal controller .we compared our policy against that of the lqg problem in simulation for two controllable -dimensional single - input linear systems . in each casewe solved an unconstrained finite - horizon lq optimal control problem corresponding to state and control weights and for every .we selected an optimization horizon , and simulated the system responses starting from different initial conditions selected at random uniformly from the cube ^ 3 ] , the nonlinear bounded term in our policy was a vector of scalar standard saturation functions applied to each coordinate of the vector , and the lqg control input was saturated at .the covariance matrices and required to solve the optimization problem were computed empirically via classical monte carlo integration methods ( * ? ? ?* section 3.2 ) using i.i.d samples .we simulated the system starting from the same initial condition }{^\mathsf{t}} ] .we simulated the system in both cases starting from the same initial condition }{^\mathsf{t}} ] , we see that for every  & = x_{kn_c}{^\mathsf{t}}(a^\ell){^\mathsf{t}}p a^\ell x_{kn_c}+2x_{kn_c}{^\mathsf{t}}(a^\ell){^\mathsf{t}}pb_\ell \ee_{x_{kn_c}}\bigl[\pi^*_{kn_c : kn_c + \ell - 1}(x_{kn_c})\bigr]\\     & \quad + \ee_{x_{kn_c}}\bigl[{\left\lvert{b_\ell\pi^*_{kn_c : kn_c + \ell - 1}(x_{kn_c } ) + d_\ell\tilde w_{kn_c : kn_c + \ell - 1}}\right\rvert}_p^2\bigr ] ,   \end{aligned}\ ] ] where .using the fact that by construction , we obtain the following bound : { \leqslant}x_{kn_c}{^\mathsf{t}}(a^\ell){^\mathsf{t}}p a^\ell x_{kn_c } + 2 c_{1\ell } { \left\lvert{x_{kn_c}}\right\rvert}_\infty + c_{2\ell } ,   \end{aligned}\ ] ] where ,\\     & \text{and } \quad \upsilon(x_{kn_c } ) { \coloneqq}{\left[\begin{matrix } \theta^*_{1}(x_{kn_c})\\ \cdots\\\theta^*_{n_c-1}(x_{kn_c } ) \end{matrix}\right]}.   \end{aligned}\ ] ] since is a schur stable matrix ( and hence so is ) there exists ( * ? ? ?* proposition 11.10.5 ) a matrix with real - valued entries that satisfies ; in particular , we have .therefore , with in we arrive at { \leqslant}x_{kn_c}{^\mathsf{t}}p_\ell x_{kn_c } - { \left\lvert{x_{kn_c}}\right\rvert}^2 + 2 c_{1\ell } { \left\lvert{x_{kn_c}}\right\rvert}_\infty + c_{2\ell}.   \end{aligned}\ ] ] for \max\{0,1-\lambda_{\max}(p_\ell)\ } , 1[ ] , and $ ] .from we get & = \ee_{\xz}\bigl[\ee\bigl[v(x_{kn_c+\ell})\,\big|\ , x_{kn_c}\bigr]\bigr ] { \leqslant}\ee_{\xz}\bigl[\rho'v(x_{kn_c})+b'{\mathbf 1_{k'}}(x_{kn_c})\bigr]{\nonumber}\\     & { \leqslant}\ee_{\xz}\bigl[\rho'\ee\bigl[v(x_{kn_c})\,\big|\ , x_{(k-1)n_c}\bigr]+b'{\mathbf 1_{k'}}(x_{kn_c})\bigr]{\nonumber}\\     & { \leqslant}\ee_{\xz}\bigl[\rho'\rho_{n_c } v(x_{(k-1)n_c})+b{\mathbf 1_{k_{n_c}}}(x_{(k-1)n_c } ) + b'{\mathbf 1_{k'}}(x_{kn_c})\bigr]{\nonumber}\\     & \cdots{\nonumber}\\     & { \leqslant}\rho'\rho_{n_c}^k v(x)+\sum_{i=0}^{k-1}b\rho_{n_c}^{k-1-i } \ee_{\xz}\bigl[{\mathbf 1_{k_{n_c}}}(x_{in_c})\bigr ] + b'\ee_{\xz}\bigl[{\mathbf 1_{k'}}(x_{kn_c})\bigr]{\nonumber}\\     & { \leqslant}\rho'\rho_{n_c}^k v(x)+\frac{b\bigl(1-\rho_{n_c}^k\bigr)}{1-\rho_{n_c}}+b ' .\label{eqn : finalbound }   \end{aligned}\ ] ] note that the conditioning in the first few steps of  is well - defined because it is performed every steps starting from , and the structure of our policy makes the process markovian .therefore , it follows from  that for all , & { \leqslant}\frac{1}{\lambda_{\min}(p_{n_c})}\sup\limits_{t\in\nz}\ee_{\xz}\bigl[v(x_{kn_c+\ell})\bigr]\\     & { \leqslant}\frac{1}{\lambda_{\min}(p_{n_c})}\left(\rho'\rho_{n_c}^k v(x)+\frac{b}{1-\rho_{n_c}}+b'\right)\\     & < \infty ,   \end{aligned}\ ] ] where the last step follows from the fact that .this completes the proof .

Abstract: we design receding horizon control strategies for stochastic discrete - time linear systems with additive ( possibly ) unbounded disturbances , while obeying hard bounds on the control inputs . we pose the problem of selecting an appropriate optimal controller on vector spaces of functions and show that the resulting optimization problem has a tractable convex solution . under the assumption that the zero - input and zero - noise system is asymptotically stable , we show that the variance of the state is bounded when enforcing hard bounds on the control inputs , for any receding horizon implementation . throughout the article we provide several examples that illustrate how quantities needed in the formulation of the resulting optimization problems can be calculated off - line , as well as comparative examples that illustrate the effectiveness of our control strategies .   = 1