the hormone auxin plays a crucial role in plant development  , yet the mechanisms through which it accumulates in certain cells and interacts with cell growth mechanisms remain largely unclear .the patterns formed during the growth of a plant are controlled by the local auxin concentration .for example , it is known that the distribution of auxin maxima in the shoot apex gives rise to the formation of primordia  .similarly , the distribution of auxin in the root tip coordinates cell division and cell expansion  . in models of root hair initiation , intra - cellular levels and gradients of auxin concentration influence the localisation of g - proteins , which in turn promote hair formation  .in addition , it is known that the distribution of auxin in the leaf primordia mediates vascular patterning  . in recent years, many aspects of the molecular basis of these mechanisms have been unraveled and mathematical models of auxin transport have been proposed to explain growth and development  .computer simulations are often used to compare the model output with observed data such as auxin distribution , venation patterns , growth or development . at cellular level, carriers such as pin - formed ( pin ) proteins , which are localised in the cell membrane , determine the rate and direction of auxin transport .the coordinated activity of many cells can create peaks of auxin that drive differentiation and growth .various models that implement and refine these ideas have been proposed  .such models differ primarily in the specifics of the transport and the coupling to the cell growth and division , but a common feature is that they generate spatially - extended patterns of auxin concentration , which have also been observed experimentally .existing transport models can be classified into two main categories , _ flux - based _ and _ concentration - based _ , depending on how auxin influences the localization of transport mediators ( pins ) to form patterns . in flux - based models ,first proposed in  , the polarization depends on the net auxin flux between neighbouring cells : the higher the net flux towards the neighbours , the more pin will accumulate at the membrane , and changes in the pin distribution determine changes in auxin fluxes .by contrast , in concentration - based models it is assumed that the pin accumulation on the membrane is caused by differences in auxin concentration between neighbouring cells .this type of models was introduced in  and  . for other reviews on flux- and concentration - based models we refer the reader to  .  in the models mentioned above, patterns are found by direct numerical simulation , upon choosing control parameters within a plausible biological range .however , there is still a large uncertainty on many of the parameter values which are often approximated  , adopted from different systems  or estimated with large error margins  .furthermore , it is unclear what is the effect of systematic parameter variations on the generated patterns and how this relates to the behaviour of the biological system : understanding the formation of auxin peaks from a dynamical system standpoint is still an open problem , therefore a mathematical exploration of the parameter variations may generate new , experimentally testable hypotheses , thereby gaining insight into pattern formation mechanisms  .  in spite of the uncertainty on experimental parameters, it is believed that active transport is a key player in auxin patterning  .transport models posses an inherent time - scale separation : the growth hormone dynamics involve short time scales ( of the order of seconds ) , while changes in cellular shapes and proliferation of new cells occur on much slower time scales ( hours or days ) . in order to determine the distribution of auxin in the plant , it is then possible to concentrate on the fast time scale of the hormone transport , assume a static cell structure and study the plant tissue as a dynamical system , subject to variations in the active transport parameter .  in this paperwe perform such exploration on concentration - based auxin models , which are studied using standard bifurcation analysis techniques  . in particular , we find steady states of the system and explore their dependence upon control parameters , investigating how patterns lose or gain stability in response to parameter changes .the aim is to predict qualitatively the distribution patterns that can occur for a certain parameter range and to understand transitions between different types of patterns .  at present , only a few papers regard transport models as dynamical systems : among them , reference  stands out for being a _ systematic _ analysis of flux- and concentration - based models , whose auxin patterns are studied by considering local interactions between cells . in the present paper , we take the analysis one step further , by finding steady states simultaneously in the whole tissue , and by studying the important effects of its boundedness on the auxin patterns .the main result of our analysis is that , in a large class of concentration - based models posed on finite tissues , peaks do not arise from an instability of the homogeneous steady state , as it was previously reported for unbounded tissues  : on regular bounded domains , the geometry of the tissue drives the formation of small auxin peaks , which nucleate without instabilities near the boundary .further , we investigate the effects of changes in the active transport coefficient , in the diffusivity coefficient and in the auxin production coefficient for two specific examples : the concentration - based model proposed by smith et al . and a more recent modification studied by chitwood et al . . in these systemsthe localised peaks , determined analytically for small values of the active transport coefficient , persist also for moderate and large values of this control parameter .we found that , owing to their boundedness , realistic tissues can select from a multitude of patterned configurations , characterised by a variable number of localised peaks and organised in a characteristic snaking bifurcation diagram .  snaking bifurcation diagramsare commonly found for localised states  arising in ( systems of ) nonlinear partial differential equations posed in one  , two  and three  spatial dimensions , as well as in discrete  and nonlocal systems  .however , this mechanism has never been reported for auxin models : solutions with localised peaks undergo a series of saddle - node bifurcations , giving rise to a hierarchy of steady states with an increasing number of bumps .a direct consequence of this mechanism is that the resulting patterns are robust against changes in the transport parameter and other control parameters as found , for instance , by sahlin et al .we argue that this mechanism could be a robust feature in several other types of concentration - based auxin models .the paper is structured as follows : in section [ sec : resultssummary ] we summarise our working hypotheses and describe the main results of the paper ; in section [ subsec : theexistenceof ] we present our calculations for a simple 1d tissue , giving a primer on bifurcation analysis for auxin models ; in section [ subsec:2ddom ] we generalise our results to 2d tissues , which are further discussed in section [ sec : discussion ] ; finally , we provide a more formal presentation of our general asymptotic results in section [ sec : materialsandmethods ] .( a blue arrow indicates time evolution ) ; when is increased above a critical value , the flat state becomes unstable and the system evolves towards a fully patterned state ( 2 ) ; states with low auxin peaks ( 1 ) may be attracting or repelling , depending on the bifurcation type . snaking mechanism ( right ) : on bounded domains with free boundary conditions ,the flat state ( 0 ) exists only for ; upon increasing , we find an attracting state with low auxin peaks at the boundary ( 1 ) , followed by several other attracting states with variable number of peaks ( 2 , 3 ) ; the tissue is able to select from different patterns , depending on the initial condition and on the value of . ]we begin by giving a generic definition of concentration - based models , and a summary of the main results of the paper . in concentration - based models ,cells are identified with an index and to each cell it is associated a set of neighbours , containing neighbours , and a vector of time - dependent state variables .for instance , may contain the auxin concentration ( ) or both auxin and pin - formed1 concentrations ( ) .generically , the rate of change of the concentrations is expressed as a balance between production and decay within the cell , diffusion towards neighbouring cells and active transport , hence we study generic concentration - based auxin models of the form where , are the production and decay functions , respectively , is a diffusion matrix , is the active transport parameter and are the active transport functions . in this paperwe concentrate on the fast time scale of hormone transport and hence consider plant organs as static cell structures , so the number of cells remains constant in time .we make two key assumptions :  [ hyp : domain ] cells are arranged in a regular domain , that is , they have all the same shape and size and they tessellate the tissue .we do not make any assumption on the dimensionality of the domain .[ hyp : nu ] the active transport functions can be expressed as where the functions , depend on the model choices .hypothesis [ hyp : nu ] is a factorisation of the active transport functions that is met by several models in literature  : active transport between cell and is influenced by the respective concentrations and , but also by concentrations in the neighbours of cell . in section [ subsec : twomodels ]we introduce examples of concrete models satisfying hypothesis [ hyp : nu ] and in section [ sec : materialsandmethods ] we derive explicit expressions for the corresponding functions , .it is established in literature that concentration - based models are capable of reproducing auxin patterns that are found in sam experiments  , for biologically realistic values of the transport parameters . in this paperwe will address the following questions : what type of patterns are generated by the class of models described above ?do they all predict the occurrence of auxin peaks ?what are the instabilities that lead to the formation of auxin peaks ?are auxin patterns robust to changes in the control parameters and initial conditions ?these question have been partially addressed in previous papers  , where analytical results have been obtained only for particular models and only for regular domains without boundaries , where all cells have the same number of neighbours .in such domains a homogeneous steady state , satisfying the balancing condition , is known to exist for all values of  . in domains without boundaries ,the formation of peaks has been explained in terms of a turing bifurcation in the active transport parameter ( see figure [ fig : sketch1d ] ) : peaks are formed all at once as is varied , and they derive from an instability of the homogeneous steady state . from a dynamical system viewpoint, however , we expect that boundary conditions and finite domain sizes affect the formation and selection of auxin patterns .the main result of our investigation is that the mechanism for the formation of peaks is radically different in tissues of finite size , and in particular :  [ res : homstate ] in finite domains , concentration - based models  satisfying hypotheses [ hyp : domain][hyp : nu ] support a homogeneous equilibrium for but , generically , this homogeneous state is not present when .this result is a direct consequence of the geometry of the domain : an inspection of equation  shows that the active transport terms contain a nested sum over the neighbours of the cell and , in a finite domain , the number of neighbours varies from cell to cell , namely cells at the boundary of the regular tissue have fewer neighbours than cells in the interior ; consequently , is generally different from at the boundary , even when .the conclusion is that , on finite domains , peaks can not form with a turing bifurcation , since the homogeneous steady state exists for , but not for ( however small ) . for regular domains where the number of neighbours is the same for all cells , the turing mechanism is still possible . in addition , a turing bifurcation is also possible in any domain , provided that and diffusion is used as a bifurcation parameter . in this regime, however , the active transport is absent and the tissue is a standard medium with reaction - diffusion mechanisms , which is not a biologically valid hypothesis for auxin models .  [ res : smallt ] to understand how peaks are formed from the homogeneous state we study the case of small active transport coefficients . for and in the absence of passive transport , ,the models above generate steady states with small peaks .such states take the form ^{-1 }      { \boldsymbol{\psi}}({\bm{y}}^*,{\bm{y}}^ * ) + \mathcal{o}(t^2 )  \label{eq : pert}\ ] ] where the coefficients depend on the number of neighbours at distance 2 from the cell are neighbours of the neighbours of cell .] , namely equation  predicts that peaks are formed as small perturbations of the homogeneous steady state ( see figure [ fig : sketch1d ] ) .the amplitude of small peaks is proportional to and .importantly , in the interior of regular domains , therefore peaks localise at the boundary and without bifurcations , as opposed to the turing scenario where they form everywhere owing to an instability of : the mechanism on finite domains is purely geometric , as the location of the peaks is determined by the factors .  [res : diffusion ] for small active transport and at the presence of passive transport , , we still obtain solutions with localised peaks , similar to the case discussed above .the location of the peaks depends again on .[ rem : generality ] results [ res : homstate][res : diffusion ] are valid for generic models of the form , provided they satisfy hypotheses [ hyp : domain][hyp : nu ] . in particular , these results are valid for regular cellular arrays in any spatial dimension .the main implication of this result is that a wide class of concentration - based models are able to generate spontaneously auxin peaks in various geometries , irrespective of the model specifics .a similar derivation can be done for irregular domains , albeit the coefficients depend in this case on the local cellular volumes as well as the number of neighbours .  the analytical theory described above , which is presented in more detail in section [ sec : materialsandmethods] , is valid only for small values of the active transport coefficient and does not explain the formation of peaks in the interior of the domain  . while it is difficult to make general analytical predictions for larger values of , it is possible to explore the solution landscape of specific models via numerical methods .we investigated regular domains in two concrete models by smith et al . and chitwood et al . ( henceforth called the smith model and the chitwood model , respectively ) which satisfy hypotheses [ hyp : domain][hyp : nu ] and will be described in detail in section [ subsec : twomodels ] .our computations confirm the theoretical results [ res : homstate][res : diffusion ] and provide numerical evidence for the following conclusions :  [ res : snaking ] as the active transport rises , the two models by smith and chitwood predict the formation of peaks in the interior .peaks are formed progressively , from the boundary towards the interior via saddle - node bifurcations , with a characteristic snaking bifurcation diagram . from a biological perspective, this means that the tissue can form peaks that are robust with respect to changes in the control parameters .in addition , the tissue is capable of selecting from a variety of auxin patterns , with a variable number of spots , depending on the initial conditions and on the value of the auxin transport parameter ( see figure [ fig : sketch1d ] ) .[ res : robustness ] the scenario above is robust to perturbations to secondary parameters , that is , solutions with localised peaks at the boundary should be observable in experiments for which the two models are applicable , if the active transport is inhibited with respect to passive transport .furthermore , in numerical calculations we are able to violate hypotheses [ hyp : domain][hyp : nu ] and see how this affects our results . an important conclusion is the following :  [ res : irregular ] when the smith and the chitwood models are posed on irregular domains , the bifurcation structure presented above persists .auxin peaks are formed progressively via saddle - node bifurcations , albeit they may in principle nucleate in the interior of the domain before the boundary is filled . furthermore , this scenario is robust to changes in secondary parameters , so result [ res : robustness ] is still valid on irregular domains for both models .we now present two models that will be analysed in detail using the asymptotic theory of section [ sec : materialsandmethods ] and numerical simulations . as a first example we consider the smith model  , which features 2 state variables per cell , namely the indole-3-acetic acid ( iaa ) concentration , , and the pin - formed1 ( pin1 ) amount , .the model features iaa production , decay , active and passive transport terms , whereas for pin1 only production and decay are included .this results in the following set of coupled nonlinear odes ,  \end{split }  \label{eq : smithmodel1 }  \\\frac{dp_i}{dt } = & \frac{\rho_{{\textnormal{{\tiny pin}}}_0 } + \rho_{\textnormal{{\tiny pin}}}a_i}{1+\kappa_{\textnormal{{\tiny pin}}}p_i } - \mu_{\textnormal{{\tiny pin}}}p_i ,  \label{eq : smithmodel2}\end{aligned}\ ] ] for . in this model is a diffusion coefficient , is the cellular volume , is the ratio between the contact area of the adjacent cells and , and the sum of the corresponding cellular wall thicknesses and .in addition , is the active transport coefficient and is the number of pin1 proteins on the cellular membrane of cell facing cell , the smith model posed on a regular domain satisfies hypotheses [ hyp : domain][hyp : nu ] in section [ sec : resultssummary ] and the reader can find explicit expressions for the functions , in section [ sec : one - dimensional-2-comp ] .more details on the model and simulations of realistic phyllotactic patterns can be found in  .  the second concentration - based transport model that will be studied below is the more recent chitwood model  .this modification of the smith model is able to produce stable spiral phyllotactic patterns once cell division is included .the system also features 2 variables per cell , the iaa concentration and the pin1 amount , and it is given by the following set of coupled nonlinear odes ,  \end{split }  \label{eq : chitwoodmodel1 }  \\\frac{dp_i}{dt } = & \frac{\rho_{{\textnormal{{\tiny pin}}}_0 } + \rho_{\textnormal{{\tiny pin}}}a_i}{1+\kappa_{\textnormal{{\tiny pin}}}p_i } -  \mu_{\textnormal{{\tiny pin}}}p_i , \label{eq : chitwoodmodel2}\end{aligned}\ ] ] for , where are given by  and the only new parameter , , controls the exponential transport .the chitwood model posed on a regular domain also satisfies hypotheses [ hyp : domain][hyp : nu ] , as shown in section [ sec : one - dimensional-2-comp ] .  in the reminder of the paper we shall fix most parameters in the smith and the chitwood models , and we will examine variations in the active transport parameter , auxin diffusion coefficient and auxin production coefficient . in table [ table : parameters ]we report a brief description of parameters for both models , together with characteristic values and units , which are taken from  ..control parameters for the smith and chitwood models ( parameter values are taken from  and ) .we examine variations in , and , for which we report a range of values in the second part of the table . [ cols="^,<,<,^,^,^ " , ]   [ table : parameters ]  [ rem : pars ] the model equations [ eq : smithmodel1][eq : chitwoodmodel2 ] show that the transport parameters and are scaled by the cellular volumes : comparisons to experimental parameters and other computer simulations available in literature should be based on the ratios and . through these ratioswe implicitly specify , so as to account for competition between active and passive transports .[ rem : tissuetype ] we model 3 plant tissues : identical cubic cells arranged on a line of finite length ( here and henceforth , 1d regular ) , identical hexagonal prismic cells tessellating a finite square ( 2d regular ) and irregular prismic cells tessellating an almost - circular domain ( 2d irregular , taken from merks et al .we stress that the nomenclature 1d and 2d refers to the domain , not the cells , which are assumed to have consistently assigned volumes and contact areas .we note that cellular volumes and contact areas may change between different domains ( see also remark [ rem : pars ] ) ., but the next flux with cell is 0 ; at we have a physical boundary , therefore no cells exist to the right of cell and .bottom : predicted approximate solution pattern in the proximity of the boundary for the smith model with and various values of the diffusion coefficient . parameters : , , other parameters as in table [ table : parameters ] . ]  in this section , we present in detail results [ res : homstate][res : robustness ] for the smith model posed on a 1d regular tissue , which is represented in figure [ fig : finitedomainwithdiffusion ] .the tissue consists of a file of identical cubic cells with volume and .we assume that there exist cells to the left of cell and that the net proximal flux is zero , hence we prescribe neumann boundary conditions at .we impose that there are no cells to the right of cell , so we use a free boundary condition should not be confused with a zero dirichlet boundary condition , for which we would prescribe and .] at by setting . in this geometry, the tissue has a physical boundary only at , as illustrated in figure [ fig : finitedomainwithdiffusion ] , so each cell has neighbours , except cell , which has only neighbour .this geometry ( with periodic boundary conditions ) has been studied in other auxin - patterning simulations : it was used in reference  as an approximation of a root tissue and in reference  to model part of a leaf , between the midvein and the margin . in this paper, we use this geometry only as a primer to illustrate how the asymptotic and numerical calculations are used to predict the formation of auxin spots in concentration - based models . while the focus is on the smith model , several of the results presented in this section , are valid in more general models and spatial configurations ( we refer the reader to remark [ rem : generality ] and the whole section [ sec : resultssummary ] for further comments on the generality of these results ) .the smith model posed on a regular domain satisfies hypotheses [ hyp : domain][hyp : nu ] ( as shown in section [ sec : one - dimensional-2-comp ] ) , so we can apply the asymptotic theory in section [ sec : materialsandmethods ] ( in particular , lemma [ lemma1 ] ) . on an unbounded array ( or on a bounded array with periodic boundary conditions )every cell has the same number of neighbours , therefore the model admits the following homogeneous steady state for . however the 1d regular domain considered here is finite , so the homogeneous solution exists only for , as for positive the sums in the transport term in  do not vanish in general ( result [ res : homstate ] ) . for and ,lemma [ lemma1 ] shows the existence of patterns in the form of small deviations from the homogeneous steady state ( see section [ sec : one - dimensional-2-comp ] for a full derivation )   a^* +   t   \dfrac{p^*}{2v }  \bigg [  \dfrac{\rho_{\textnormal{{\tiny iaa}}}\kappa_{\textnormal{{\tiny iaa}}}}{1 + \kappa_{\textnormal{{\tiny iaa}}}(a^*)^2 }  + \mu_{\textnormal{{\tiny iaa}}}\bigg]^{-1 }  \dfrac{(a^*)^2}{1 + \kappa_t ( a^*)^2 }  & \text{for , } \\[1em ]  a^ * -  t  \dfrac{p^*}{2v }  \bigg [  \dfrac{\rho_{\textnormal{{\tiny iaa}}}\kappa_{\textnormal{{\tiny iaa}}}}{1 + \kappa_{\textnormal{{\tiny iaa}}}(a^*)^2 }  + \mu_{\textnormal{{\tiny iaa}}}\bigg]^{-1 }  \dfrac{(a^*)^2}{1 + \kappa_t ( a^*)^2 }  & \text{for . } \end{cases}\\ p_i & =   \begin{cases }   p^ *  & \text{for },\\   p^ * + \left[\dfrac{(\rho_{{\textnormal{{\tiny pin}}}_0 } + \rho_{\textnormal{{\tiny pin}}}a^*)\kappa_{\textnormal{{\tiny pin } } } }    { ( 1 + \kappa_{\textnormal{{\tiny pin}}}p^*)^2}+ \mu_{\textnormal{{\tiny pin}}}\right]^{-1}\!\!\!\dfrac{\rho_{\textnormal{{\tiny pin}}}}{1+\kappa_{\textnormal{{\tiny pin}}}p^*}(a_i - a^ * ) & \text{for },\\     p^ * - \left[\dfrac{(\rho_{{\textnormal{{\tiny pin}}}_0 } + \rho_{\textnormal{{\tiny pin}}}a^*)\kappa_{\textnormal{{\tiny pin } } } }    { ( 1 + \kappa_{\textnormal{{\tiny pin}}}p^*)^2}+ \mu_{\textnormal{{\tiny pin}}}\right]^{-1}\!\!\!\dfrac{\rho_{\textnormal{{\tiny pin}}}}{1+\kappa_{\textnormal{{\tiny pin}}}p^*}(a_i - a^ * ) & \text{for }.  \end{cases } \end{aligned } \label{eq : asymptoticsolutions}\ ] ]  the equations above predict that , for infinitesimal values of the transport coefficient , the perturbed solution coincides with the homogeneous solution , except for a small peak at cell and a small dip at cell ( see also bold curve in figure [ fig : finitedomainwithdiffusion ] ) . in other words , peaks are present where the number of neighbours differs from the number of neighbours in the unbounded domain , that is , where the sum in the active transport in equation  is nonzero ( result [ res : smallt ] ) . the analysis above applies also in the limit of slow dynamics of the pin1 proteins : upon assuming constant and homogeneous in the tissue , we find that is as in equation [ eq : asymptoticsolutions ] , with replaced by ( see section [ sec : one - dimensional ] for details ) .  we have thus established that , in regular domains , a small auxin transport coefficient elicit low auxin peaks .such correlation was previously reported in numerical experiments on various models  and we now provide a mathematical explanation of this phenomenon .asymptotic calculations can also be carried out in the presence of diffusion ( ) , leading to a linear system for the perturbations . in section [ subsec : modelswithdiffusion ] we present a derivation for generic models in generic tissues , which is then specialised for the smith model as an example .quantitative results of this calculation are shown in figure [ fig : finitedomainwithdiffusion ] , where we plot approximate steady states for the smith model towards the boundary for and various values of the diffusion coefficient .we notice that , in the regime of small active transport and comparatively much bigger diffusion coefficient , a peak is still present at the boundary . inspecting the solid line ( ) andthe dashed lines ( and ) we see that the peaks decrease in amplitude and are more spread out , as expected ( result [ res : diffusion ] ) . as a concluding remark , we point out that these results are not influenced by the no - flux boundary conditions specified at : the only physical boundary is at , where the number of neighbours differs from the interior .we now turn to the more interesting question of how the tissue develops high auxin peaks ( result [ res : snaking ] ) which are observed in experiments . once again, we illustrate our findings in the 1d regular case and generalise in the following sections .  in realistic simulationsthe transport coefficient is not necessarily small , therefore it is interesting to explore the solution landscape when is increased at the presence of diffusion .this is done using numerical bifurcation analysis , that is , equilibria of systems  are followed in parameter space using newton raphson method and pseudo - arclength continuation  .linear stability is then inferred computing the spectrum of the jacobian at the steady state .identical cells with neumann boundary conditions at and free boundary conditions at ( see figure [ fig : finitedomainwithdiffusion ] ) . left : 2-norm of auxin concentration versus active transport parameter .right : as the snaking bifurcation diagram is ascended , new peaks are formed from the boundary towards the interior .bottom - right : stable segments of the branch are found between hopf bifurcations and , and , etc .other secondary instabilities ( not shown ) are present along the unstable branches .parameters : , ; other parameters as in table [ table : parameters ] . ]  in figure [ fig : bifdiagramr0850 ] we show a branch of solutions of the smith model for the 1d regular domain obtained with the parameter set in table [ table : parameters ] .we start the computation from the homogeneous solution at and follow the pattern for increasing values of . as changes , we plot the -norm of the auxin vector , , which is a measure of the spatial extent of the solution ( the lower , the more localised the pattern ) and denote stable ( unstable ) branches with solid ( dashed ) or thick ( thin ) lines .the low peak found close to the boundary persists for increasing values of and grows steadily until we meet a first turning point ( tp1 ) . before exploring the diagram in full, we compare our numerical findings with the analytical predictions of the asymptotic theory , valid for small .the analytic asymptotic profile  gives a relative error less than for , after which higher - order terms become predominant .this is shown in figure [ fig : analyticvsnumericalbd ] where we compare a branch of approximate solutions ( magenta ) to a branch of solutions to the full nonlinear problem ( blue ) for and small values of .solutions ( magenta ) and the corresponding branch of solutions to the full 2-component nonlinear system by smith ( blue ) for a row of 150 identical cells with homogeneous neumann boundary conditions at and free boundary conditions at .the plot shows the 2-norm of the iaa concentration vector versus the continuation parameter .parameters as in figure [ fig : bifdiagramr0850 ] . ]as we ascend the bifurcation diagram in figure [ fig : bifdiagramr0850 ] , new peaks are formed on the left side of the existing peaks , that is , towards the interior of the domain , until the whole domain is filled with peaks .we note that peaks are regularly spaced , as it was also observed in .this bifurcation diagram resembles the one found for reaction diffusion pdes posed on the real line  except that here peaks are formed at the boundary rather than at the core of the domain .when peaks fill the entire domain , the branch enters an unstable irregular regime without snaking ( not shown ) .branches of solutions with peaks covering the entire domain are also present ( not shown ) and are partially discussed in section [ subsec : influenceofsecondaryparameters ] .the diagram in figure [ fig : bifdiagramr0850 ] makes a plausible biological prediction for the formation of large peaks .a tissue composed of a string of cells , in the presence of passive diffusion , selects auxin patterns depending on the value of the active transport .our analysis of the smith model predicts that there exist two main regimes : for small there is a single low auxin peak at the boundary .as becomes larger , we enter a regime where the tissue can select from a large variety of auxin patterns . if , for instance , the tissue is able to select pattern 1 , 2 or 3 , which have a variable number of peaks .therefore , the pattern selected in experiments depends highly upon the initial conditions of the system , similarly to what was reported by jnsson and krupinski  .as we shall see , a slanted version of the snaking bifurcation diagram is also present in 2d domains ( both regular and irregular and for both the smith and the chitwood models ) .  .middle panels : spectra at selected bifurcation points along the snaking bifurcation branch of figure [ fig : bifdiagramr0850 ] .bottom panels : unstable eigenfunctions at and .parameters as in figure [ fig : bifdiagramr0850 ] . ]to understand if a solution pattern in stable to small perturbations we study in detail the eigenvalues of the jacobian matrix ( result [ res : snaking ] ) .the jacobian matrix for the spatially - extended system  is sparse with a characteristic block structure determined by transport and diffusion terms ( we refer the reader to  for a detailed description ) and , for relatively small systems such as this one , eigenvalues are computed with dense linear algebra routines .  in this example, the solution with one small peak at the boundary becomes unstable at a hopf bifurcation ( ) at , closely followed by other oscillatory instabilities and a saddle - node bifurcation ( ) at , after which the solution remains unstable . on the snaking branch, we find that saddle - node and hopf bifurcations alternate regularly , as documented in figure [ fig : bifdiagramr0850 ] : saddle - node bifurcations align at and , while hopf bifurcations depart from each other as patterns become less localised . in this parameter setting , stable portions of the branchare delimited by two hopf bifurcations , which , to the best of our knowledge , has not been reported before for snaking in reaction diffusion systems .it should be noted , however , that burke and dawes  found hopf bifurcations at the bottom of the snaking branch for an extended swift hohenberg equation , which may lead to a bifurcation structure similar to the one in figure [ fig : bifdiagramr0850 ] if secondary parameters are varied .  in figure[ fig : spectrum ] we show spectra of solutions at selected points on the branch . overall these spectra resemble those found in discretised advection - diffusion pdes , with largely negative real eigenvalues associated with diffusion terms of the governing equations . in this context , however , increasing the number of cells does not alter the cell spacing , hence the spectrum does not grow in the negative real direction for larger system sizes .  .we set ( and other parameters as in figure [ fig : bifdiagramr0850 ] ) and use as initial condition a steady state with one peak at the boundary , obtained for .a long - time periodic auxin wave travels ( and new auxin peaks are recruited ) from the boundary towards the interior of the domain . ]we monitored spectra of localised solutions as the snaking branch was ascended ( see figure [ fig : spectrum ] ) : immediately after the hopf bifurcation hp1 , multiple eigenvalues cross the imaginary axis , therefore several oscillatory instabilities exist between and ( and , etc . ) . in the bottom panel of figure [ fig : spectrum ]we show that the hopf eigenfunction at has a maximum near the boundary and the one for is also spatially localised .we expect that branches of time - periodic ( possibly spatially - localised ) solutions emerge from the hopf bifurcations .we have not observed stable small - amplitude oscillations in direct numerical simulations , but we report the existence of stable periodic states in which a temporal oscillation of the peak at initiates a wave of auxin moving towards the boundary at , with long oscillation periods .  in fig .[ fig : timeevolution ] we show such a periodic solution obtained via time simulation in the neighbourhood of ( which is also visible in figure [ fig : analyticvsnumericalbd ] ) .we set and use as initial condition a steady state ( with one peak at the boundary ) , obtained for .in the resulting periodic state , auxin peaks are dynamically formed from the tip towards the interior of the domain : we point out that the period of oscillations ( about 377 hours ) is much greater than the period associated to the unstable hopf eigenvalues . in addition , on such long time scales it is reasonable to assume that new cells are formed , so the geometry of the problem should change as well .it was recently shown by farcot and yuan that , in one - dimensional flux - based models with no - flux boundary conditions , active transport is sufficient to elicit auxin oscillations  . in the concentration - based model considered here , oscillatory states in regular one - dimensional arraysare also found in a regime where active transport dominates over diffusion .( other parameters as in figure [ fig : bifdiagramr0850 ] ) . the lower the , the wider the snaking width and the stable branches .for sufficiently large values of the snaking disappears and peaks form at once on a fully stable branch . ]we conclude this primer on the 1d tissue by investigating the robustness of the snaking scenario described above ( results [ res : robustness ] ) . in it was shown that the auxin production parameter has a significant influence on the solution profiles , therefore it is interesting to study how changes in this parameter affect the bifurcation structure .we repeated the numerical continuation of the smith model for values of in the interval  i \in \mathcal{b} i = n-1 i = n i = n-1 i = n i = n-1 i = n i = n-1 i = n i = n-1 i = n$. } \end{cases}\ ] ]   lemma [ lemma1 ] also applies when the smith or the chitwood model are posed on a two - dimensional array of identical hexagonal cells . in this case , the computations for are identical to the previous example and the asymptotic derivation is also straightforward . instead of writing the full expressions for , and , we refer the reader to figure [ fig : xihexagons ] , where the values of are shown for two corners of the domain .as claimed in section [ subsec : twodimensionaldomain ] , the highest peaks occur at the top - left and right - bottom corners.   we can extend the definition of concentration - based models to the case where diffusion is present .a concentration - based model with diffusion is a set of odes of the form for , where is a diagonal diffusion matrix and all other quantities are as in definition [ def : conbasedmodelnodiff ]  reasoning like in lemma [ lemma1 ] , we obtain where satisfy \begin{bmatrix } { \boldsymbol{\eta}}_1 \\ \vdots \\ { \boldsymbol{\eta}}_n \end{bmatrix } =  \begin{bmatrix } \xi_1 { \boldsymbol{\psi}}({\bm{y}}^*,{\bm{y}}^ * ) \\ \vdots \\ \xi_n { \boldsymbol{\psi}}({\bm{y}}^*,{\bm{y}}^ * ) \end{bmatrix},\ ] ] is block - diagonal with blocks , is the laplacian matrix associated to the graph with neumann boundary conditions and denotes the kronecker product between matrices .the operator is negative semi - definite and it has a zero eigenvalue , corresponding to a constant eigenvector .however , summing this matrix to , makes the resulting linear operator non - singular .  in the presence of diffusionwe can not directly apply the formula , even for regular cellular arrays : owing to diffusion , cells in will also deviate from the homogeneous state , hence peaks and dips are not necessarily formed within , but may occur in interior cells that are close to the boundary .first - order corrections for these cases can be computed analytically using chebyshev polynomials  or numerically using linear algebra routines . even though we report below an example of this calculation , we point out that in practice this is not necessary , since the numerical bifurcation software gives access to the full nonlinear solution and to its linear stability .we return to the smith model with , posed on a row of identical cells , and we now add diffusion only in the auxin component .the expressions for and are unchanged from section [ sec : one - dimensional-2-comp ] , as is the first order approximation of ( since there is no diffusion for ) .expressions for the first - order approximations in are more involved : proceeding as explained above for generic models with diffusion , we obtain , to leading order \nonumber \\ \cdot\left(\begin{array}{c } \eta_1 \\ \eta_2 \\ \vdots \\ \vdots \\\eta_n \end{array}\right ) = \left(\begin{array}{c } \xi_1  \psi\left({\bm{a}^*},{\bm{p}^*}\right ) \\\xi_2\psi_2\left({\bm{a}^*},{\bm{p}^*}\right ) \\ \vdots \\ \vdots\\ \xi_n \psi_n\left({\bm{a}^*},{\bm{p}^*}\right ) \end{array}\right ) = \left(\begin{array}{c } 0 \\ \vdots \\ 0 \\  \frac{1}{2}\psi_{n-1}\left({\bm{a}^*},{\bm{p}^*}\right ) \\-\frac{1}{2}\psi_n\left({\bm{a}^*},{\bm{p}^*}\right ) \end{array}\right ) .\label{eq : linsystemdiffusion}\end{aligned}\ ] ]  solving this linear equation above led to the approximate solution profiles in figure [ fig : finitedomainwithdiffusion ] and the red solution branch in figure [ fig : analyticvsnumericalbd ] .the same derivation and figures can be obtained for the chitwood model ( not shown ) .we acknowledge fruitful discussions with gerrit t.s .beemster , jan broeckhove , dirk de vos , abdiravuf dzhurakhalov , etienne farcot and przemyslaw klosiewicz .dd acknowledges financial support from the department of mathematics and computer science of the university of antwerp .da acknowledges the university of nottingham research development fund , supported by the engineering and physical sciences research council ( epsrc ) .band l , wells d , larrieu a , sun j , middleton a , french a , et al .root gravitropism is regulated by a transient lateral auxin gradient controlled by a tipping - point mechanism .2012;109(12):46684673 .de reuille p , bohn - courseau i , ljung k , morin h , carraro n , godin c , et al .computer simulations reveal properties of the cell cell signaling network at the shoot apex in arabidopsis .2006;103(5):16271632 .heisler m , ohno c , das p , sieber p , reddy g , long j , et al .patterns of auxin transport and gene expression during primordium development revealed by live imaging of the arabidopsis inflorescence meristem . current biology. 2005;15:18991911 .stoma s , lucas m , chopard j , schaedel m , traas j , godin c. flux - based transport enhancement as a plausible unifying mechanism for auxin transport in meristem development .plos comput biol .2008;4(10):e1000207 .swarup r , kramer e , perry p , knox k , leyser h , haseloff j , et al .root gravitropism requires lateral root cap and epidermal cells for transport and response to a mobile auxin signal .nature cell biology .2005;7:10571065 .band l , fozard j , godin c , jensen o , pridmore t , mj b , et al .multiscale systems analysis of root growth and development : modeling beyond the network and cellular scales .the plant cell online . 2012;24(10):38923906 .feugier f , mochizuki a , iwasa y. self - organization of the vascular system in plant leaves : inter - dependent dynamics of auxin flux and carrier proteins .journal of theoretical biology .2005;236:366375 .bilsborough g , runions a , markoulas m , jenkins h , hasson a , galinha c , et al . model for the regulation of arabidopsis thaliana leaf margin development .proceedings of the national academy of sciences . 2011;108:34243429 .chitwood dh , headland lr , ranjan a , martinez cc , braybrook sa , koenig dp , kuhlemeier c , smith rs , sinhaa nr .leaf asymmetry as a developmental constraint imposed by auxin - dependent phyllotactic patterning . the plant cell .

Abstract: we study the formation of auxin peaks in a generic class of concentration - based auxin transport models , posed on static plant tissues . using standard asymptotic analysis we prove that , on bounded domains , auxin peaks are not formed via a turing instability in the active transport parameter , but via simple corrections to the homogeneous steady state . when the active transport is small , the geometry of the tissue encodes the peaks amplitude and location : peaks arise where cells have fewer neighbours , that is , at the boundary of the domain . we test our theory and perform numerical bifurcation analysis on two models which are known to generate auxin patterns for biologically plausible parameter values . in the same parameter regimes , we find that realistic tissues are capable of generating a multitude of stationary patterns , with a variable number of auxin peaks , that can be selected by different initial conditions or by quasi - static changes in the active transport parameter . the competition between active transport and production rate determines whether peaks remain localised or cover the entire domain . we relate the occurrence of localised patterns to a _ snaking _ bifurcation structure , which is known to arise in a wide variety of nonlinear media but has not yet been reported in plant models .  _ _


optimal feedback controls for evolutionary control systems are of significant practical importance . differently from open - loop optimal controls ,they do not rely on knowledge of the initial condition and they can achieve design objectives , as for instance stabilisation , also in the presence of perturbations .furthermore , the online synthesis of feedback control can be implemented in a real - time setting .it is well - known that their construction relies on special hamilton - jacobi - bellman ( hjb ) equations , see for instance .the solution of the hjb equation is the value function associated to the optimal control problem , and its gradient is used to construct the optimal feedback control . in the very special , but important case of a linear control system with quadratic cost without constraints on the control or the state variables , the hjb equation reduces to a riccati equation which has received a tremendous amount of attention , both for the cases when the control system is related to ordinary or to partial differential equations .otherwise one has to deal with the hjb equation which is a partial differential equation whose spatial dimension is that of the control system .thus optimal feedback control for partial differential equations leads to hjb equations in infinite dimensions .after semi - discretization in space of the controlled partial differential equation ( pde ) , the hjb equation is posed in a space of dimension corresponding to the spatial discretization of the pde . for standard finite element or finite difference discretizations this leads to high - dimensional hjb equations .this is one of the instances which is referred to as _ the curse of dimensionality _ .many attempts to tackle the difficulties posed for numerically solving the hjb equations arising in optimal control have been made in the past or are currently being investigated .we refer , for instance , to , which mainly focuses on semi - lagrangian schemes , and further references given there .a related approach to numerical optimal feedback control of pdes is to semi - discretize the dynamics and to add a model order reduction step , either with balanced truncation or proper orthogonal decomposition , in order to reduce the dimension of the dynamics to a number that is tractable for grid - based , semi - lagrangian schemes .this approach has been successfully explored , for instance , in and references therein .it strongly relies on a trustworthy representation of the dynamics via low - dimensional manifolds .such a low - dimensional representation may deteriorate when nonlinear and/or advection effects are relevant .thus , it is important to strive for techniques , or combinations of techniques , which allow to solve higher dimensional problems .another direction of research evolves around generalizing the riccati - based approach to allow for nonlinearities in the state equation .one such technique is termed state - dependent riccati equation . herethe coefficients in the ordinary riccati equation are functions of the state rather than constants as in the case of linear state equations .another approach realizes the fact that the riccati equation can be interpreted as the equation satisfied by the first term arising in the power series expansion of the value function , and attempts to improve by realizing also higher order terms in the expansion .these methods are succinctly explained in .yet another technique which has received a considerable amount of attention is termed successive galerkin approximation .roughly speaking , the nonlinear hjb equation associated to the continuous - time optimal control problem is solved by means of a newton method . at each iteration ,the control law is fixed .this leads to a generalized hamilton - jacobi equation ( ghjb ) which is linear .the iteration is closed by an update of the control law based on the gradient of the value function .this method was intensively investigated in , see also , and the references given in these citations .it is worth to mention that the discrete - time counterpart of this method corresponds to the well - known policy iteration or howards algorithm .the numerical examples in do not go beyond dimension five , and most , if not all , of the published numerical results for nonlinear hjb equations do not exceed dimension eight .an alternative sparse grid approach for high - dimensional approximation of hjb equations based on open - loop optimal control has been presented in , with tests up to dimension six .numerical methods relying on tensor calculus have been shown to perform well in high - dimensional settings where the associated hjb equation is a linear pde .in the present paper , to solve optimal control problems for certain classes of semilinear parabolic equations we shall proceed as follows . to accommodate the curse of dimensionality , the discretization of the pde is based on a pseudospectral collocation method , allowing a higher degree of accuracy with relatively few collocation points . to solve the resulting hjb we utilize a newton method based on the ghjb equation as described above .next , the discretization of the ghjb equation is addressed through a galerkin approximation with polynomial , globally supported , ansatz functions .while this mitigates the curse of dimensionality in terms of removing the mesh structure , it leads to high - dimensional integrals .we therefore resort to separable representations for the system dynamics and for the basis set of the polynomial approximation .the separability assumption reduces the computation of the galerkin residual equation to products of one - dimensional integrals .the combination of these procedures allowed us to solve hjb equations related to nonlinear control systems up to dimension fourteen by means of basic parallelization tools .the successful use of the newton procedure requires to provide a feasibly initialization , i.e. a sub - optimal , stabilizing control .since we do not consider constraints , this is not restrictive for finite horizon problem , but can be challenging for infinite horizon problems , and specifically for the stabilization problems which are considered in the present paper . in this respectwe developed a continuation procedure based on the use of a discount factor .specifically , we consider a nested iterative procedure : within the outer loop the value of a positive discount factor is driven to zero , within the inner loop the hjb equation is solved approximately for a fixed discount factor . with this approach , which , is summarized in algorithms [ alg : sga1 ] and [ alg : sga2 ] below, we managed to solve optimal feedback stabilization problems for semilinear parabolic equations with different stability behavior of the desired steady state .let us give a brief outline of the paper .section 2 sets the stage and provides the discussion of a special case to facilitate the understanding of the following material . in section 3the solution process of the hjb equation is detailed . in section 4we provide the formulas which are needed to numerically realize the discretized hjb equation after a separable basis has been chosen .numerical experiments are documented in section 5 . therewe can also find comparisons to suboptimal feedback strategies based on riccati and asymptotic expansion techniques .we consider the following undiscounted infinite horizon optimal control problem :   subject to the nonlinear dynamical constraint where we denote the state , the control , with , the state running cost , and the control penalization .furthermore , we assume the running cost and the system dynamics and to be . throughout itis assumed that and .our focus is therefore asymptotic stabilization to the origin .it is well - known that the optimal value function characterizing the solution of this infinite horizon control problem is the unique viscosity solution of the hamilton - jacobi - bellman equation with .here we follow the convention of dropping the subscript of .we study this equation in the unconstrained case , i.e. , , where the explicit minimizer of is given by note that by inserting this expression for the optimal control in , we obtain the equivalent hjb equation which under further assumptions can be simplified to the riccati equation associated to linear - quadratic infinite horizon optimal feedback control .the methodology we present in this work is applicable to systems fitting the aforedescribed setting , although for the sake of simplicity we restrict the presentation by the following choices :  * the control is a scalar variable , i.e. . *the running cost is quadratic , i.e. , with positive - definite , * the control term is a constant vector in .  at this point ,our setting differs from the linear - quadratic case as it allows nonlinear dynamics , and nonquadratic state costs . for the numerical scheme that we develop ,the following assumption is crucial :  [ as:1 ] the free dynamics are separable in every coordinate  where is a tensor - valued function . in the case , then we shall also assume a similar separable structure for .   in the following ,we illustrate how the presented framework sets the grounds for a computational approach for approximate optimal feedback controllers for nonlinear pdes .we consider the following optimal stabilization problem :    subject to the semilinear parabolic equation \ , , t\in{\mathbb{r}}^+,\\ { \partial_{\xi } x}(-1,t)&={\partial_{\xi } x}(1,t)=0\,,\quad x(\xi,0)=x_0\,.\notag\end{aligned}\ ] ] in this case , the scalar control acts through the indicator function , with . at the abstract level , this corresponds to an infinite - dimensional optimal control problem .a first step towards the application of the proposed framework is the space discretization of the system dynamics , leading to finite - dimensional state space representation .the use of the pseudospectral collocation methods for parabolic equations has been studied in , and leads to a state space representation of the form where the discrete state corresponds to the approximation of at collocation points , and is the coordinatewise power .the matrices and are finite - dimensional approximations of the laplacian and control operators , respectively .such a discretization of the dynamics directly fulfills the separability required in assumption [ as:1 ] , as the i - th equation of the dynamics reads with a separability degree .it is very important to note that semidiscretization in space of a wide class of time - dependent pdes will lead to finite - dimensional state space representations of this type , thus the applicability of the presented framework is only limited by the dimensionality of the associated hjb equation .this motivates the choice of a pseudospectral collocation method for the discretization , as it is possible to obtain a meaningful representation of the dynamics with considerably fewer degrees of freedom than classical low - order schemes .however , if pseudospectral collocation is not a suitable discretization method for the dynamics , model reduction procedures such as balanced truncation , proper orthogonal decomposition , or reduced basis techniques shall also lead to separable state - space representations .once the finite - dimensional state state space representation is obtained , we proceed to approximate the solution of the associated hjb equation , leading to the optimal feedback controller .we now present a preview of the numerical results of the proposed approach .further details of the numerical scheme will be developed in the forthcoming sections .the system dynamics in , are approximated in 12 collocation points ( 14 with b.c.s ) , and therefore our approximation scheme seeks for a solution of a 12-dimensional hjb equation , which allows the computation of online optimal feedback controllers .we compare our hjb - based controller ( hjb ) to the linear - quadratic controller ( lqr ) obtained by linearization of the system dynamics , and to an approximation method for the hjb equation based on power series expansion ( pse ) . in figure[ fig : ex1 ] we observe the basic features of the dynamics and the control schemes .the uncontrolled system dynamics ( diffusion+dissipative source term ) are stable , but stabilization is extremely slow .the control algorithms considerably reduce the transient phase. however , the control signals are different , and the hjb - based controller generates a feedback control with reduced overall cost . observe that at the beginning of the time horizon even the signs of the lqr- , pse- , and hjb - based controls differ .  .dynamics are stable but slow .total closed - loop costs : * i ) * uncontrolled : 13.45 , * ii ) * lqr : 7.39 , * iii ) * pse : 9.43 , * iv ) * hjb : 6.56 .,title="fig:",scaledwidth=45.0% ] .dynamics are stable but slow .total closed - loop costs : * i ) * uncontrolled : 13.45 , * ii ) * lqr : 7.39 , * iii ) * pse : 9.43 , * iv ) * hjb : 6.56 .,title="fig:",scaledwidth=45.0% ] .dynamics are stable but slow .total closed - loop costs : * i ) * uncontrolled : 13.45 , * ii ) * lqr : 7.39 , * iii ) * pse : 9.43 , * iv ) * hjb : 6.56 .,title="fig:",scaledwidth=45.0% ] .dynamics are stable but slow .total closed - loop costs : * i ) * uncontrolled : 13.45 , * ii ) * lqr : 7.39 , * iii ) * pse : 9.43 , * iv ) * hjb : 6.56 .,title="fig:",scaledwidth=45.0% ]in this section , we construct a numerical scheme for the approximation of the hjb equation where .we recall two additional features in this equation which render the application of classical approximation techniques difficult : the absence of a variational formulation , and the minimization with respect to the control variable , which makes the hjb equation fully nonlinear .the simplest numerical approach to these problems is the use of monotone , grid - based discretizations ( finite differences , semi - lagrangian ) , in conjunction with a fixed point iteration for the value function which typically depends on the use of a discount factor .the so - called `` value iteration '' procedure was first presented by bellman in , and although it has become a standard solution method for low - dimensional hjb equations , it suffers from three major drawbacks .first , the grid - based character of the scheme makes it inapplicable for high - dimensional dynamics , as the total number of degrees of freedom scales exponentially with respect to the dimension of the dynamical system .this corresponds to the most classical statement of the so - called _ curse of dimensionality_. second , the contractive mapping includes a minimization procedure which needs to be solved for every grid point at every iteration .third , the lipschitz constant of the contractive mapping goes to 1 when the discretization parameter goes to 0 , becoming extremely slow for fine - mesh solutions .in order to circumvent these limitations , we develop a numerical scheme combining an iteration on the control variable rather than the value function , together with a polynomial expansion for the value function to mitigate the computational burden associated to mesh - based schemes .   in the following ,we revisit the method presented in , which is referred as successive approximation algorithm .we begin by defining the set of admissible controls .we say that a feedback mapping is admissible on , denoted as , if , , and for all .starting from an admissible initial guess , the successive approximation algorithm ( algorithm [ alg : sga1 ] below ) generates the pair which solves equation .algorithm [ alg : sga1 ] corresponds to a newton method for solving equation , and in the linear - quadratic setting it is equivalent to the newton - kleinmann iteration for solving the riccati equation .it can be also directly identified with the policy iteration algorithm for hjb equations ( see and references therein ) , although in this context the usual setting includes a discount factor which relaxes the admissibility assumption , as well as discrete - time dynamics .consequently , it is applied to a bellman equation with no continuous gradient . in both cases ,the core ingredient of the algorithm is to generate a decreasing sequence of values by solving an associated sequence of linear problems . in our casethis translates into solving , for a given at each iteration , the generalized hamilton - jacobi - bellman ( ghjb ) equation the following result from summarizes relevant properties of the ghjb equation .[ tghjb ] if is a compact subset of , is lipschitz continuous on and , is strictly increasing in , , and , then :  1 .there exists a unique satisfying . 2 . is a lyapunov function of the controlled system .3 .  , for all .the update satisfies .if satisfies , then for all .a critical aspect of the successive approximation algorithm [ alg : sga1 ] is its initialization , which requires the existence of an admissible control which in view of means that it asymptotically stabilizes all the initial conditions in . forasymptotically stable dynamics , this is trivially satisfied by . for more general cases ,the computation of stabilizing feedback controllers is a challenging task .a partial answer is to consider the stabilizing feedback associated to the linearized system dynamics .however , this feedback is only locally stabilizing , and therefore the identification of a suitable domain where this control law is admissible becomes relevant . for low dimensional dynamics , this has been studied in the context of zubov s method in . an alternative solution that we propose is to consider a discounted infinite horizon control problem   where the inclusion of the discount factor relaxes the admissibility condition .recently , in , the link between discounted optimal control and asymptotic stabilization has been discussed , and under certain conditions , the discounted control problem can generate optimal controls that are also admissible for the undiscounted problem .we recall that the associated hjb equation for the infinite horizon optimal control problem is given by and the associated ghjb reads we consequently modify the successive approximation algorithm in order to embed it within a path - following iteration with respect to the discount factor :  for a sufficiently large , this algorithm can be initialized with .continued reduction of the discount factor using hotstart every time when is called with a reduced -value , leads to an approximate solution of equation .so far we have discussed the iterative aspects of a computational method for solving hjb equations .we now address the numerical approximation of the ghjb equation . for this purpose , we consider an expansion of the form where , with belonging to a complete set of basis functions in , and .in particular , we shall often generate from a multidimensional monomial basis as illustrated in figure [ fig : pol2d ] , which directly satisfies the boundary condition .the coefficients are obtained by imposing the galerkin residual equation   the convergence of has been studied thoroughly in .it follows a power series argument , and requires conditions for uniform convergence of pointwise convergent series , in order to guarantee that for sufficiently large . in our particular case, we further assume that the dynamics are polynomial ( as illustrated in section [ pdesub ] ) .therefore , under the assumptions of theorem 26 in , by choosing a multidimensional monomial basis ( of degree ) and an admissible control , it can be established that , , such that for , , and .we now focus on the different terms involved in the approximation of the ghjb equation .since this equation is meant to be solved within the iterative loop described in the previous section , we assume that can be expressed in the form where corresponds to the value function of the previous iteration , approximated with the expansion below we shall write for .we proceed by expanding case by case the different terms of the galerkin residual equation   1 . : it is directly verifiable that 2 . : by inserting the expansion we obtain and therefore 3 . : the relation leads to such that 4 . : we further assume that 5 . : note that leading to  is given by   after discretization , the ghjb reduces to a parameter - dependent linear system for  where is given by the expansion of ( terms * 4 ) * and * 5 ) * in the list above ) .under assumption [ as:1 ] concerning the separability of the free dynamics , and with the construction of a separable set of basis functions by taking the tensor product of one - dimensional basis functions as shown in figure [ fig : pol2d ] , the calculation of the -dimensional inner products of the galerkin residual equation of the previous section is reduced to the product of one - dimensional integrals . in the following , we provide further details of this procedure .  . adding terms of higher orderallows a more accurate solution for nonlinear control problems .we construct the high - order terms by limiting the degree of the monomials . ]the multi - dimensional basis functions + for the expansion of are generated as follows .we start by choosing a polynomial degree , and a one - dimensional polynomial basis . for the sake of simplicity, we consider the monomial basis , but the same ideas apply for other basis , such as orthogonal polynomials .the multidimensional basis is generated as a subset of the -dimensional tensor product of one - dimensional basis , such that i.e. , we construct a full multidimensional tensorial basis and then we remove elements according to the approximation degree . the elimination step is fundamental and is twofold .if no elimination is performed , the cardinality of would be , and again one would face the _ curse of dimensionality _ that also affects grid - based schemes . by reducing the set to multdimensional monomials of degree at most , the cardinality of the set given by which replaces the exponential dependence on by a combinatorial one .this formula is evaluated in table [ tab : count ] for different values of interest for and . by considering globally defined polynomial basis functions , the dependence on the dimensionis replaced by the combinatorial expression .the dimensional reduction of the basis is particularly significant for low order polynomial approximation ( up to degree 6 ) . a second justification for the way in which we generate the basis set has a control - theoretical inspiration .a well - known result in optimal feedback control is that if the dynamics are linear , and the running cost is quadratic , the value function associated to the infinite horizon control problem ( in the unconstrained case and other technical assumptions ) is a quadratic form , i.e. is of the form , which fits precisely the elements generated for with a monomial basis when and linear elements are eliminated .therefore , our basis can be interpreted as a controlled increment , accounting for the nonlinear dynamics , of the basis required to recover the solution of the control problem associated to the linearized dynamics around the equilibrium point .\ & 2 & 4 & 6 & 8 & 2 & 4 & 6 & 8 +  ' '' ''  6&27 & 209 & 923&3002&21 & 147 & 609&1896 + 8 & 44 & 494&3002&12869&36 & 366 & 2082&8517 + 10 & 65 & 1000&8007&43757&55 & 770 & 5775&30085 + 12 & 90 & 1819&18563&125969&78 & 1443 & 13819&89401 + 14 & 119 & 3059 & 38759 & 319769 & 105 & 2485 &29617 & 233107 +  [ oddeven ] theorem 7.1 in states parity conditions to reduce the polynomial basis . under the assumptions , and ,if  * is a symmetric rectangle around the origin , i.e. , \times\ldots\times[-l_d , l_d]\,,$ ] * the free dynamics are odd - symmetric on , i.e. , for all   then is an even - symmetric function , i.e. , , and therefore odd - degree monomials are excluded from the basis .a direct corollary is that in the linear quadratic case , where the linear dynamics are trivially odd - symmetric , is a quadratic form .finally , for the calculation presented in the following , it is important to note that due to the construction procedure , the basis elements directly admit a separable representation where each component .we begin by recalling that where is a tensor - valued function , and that .  as in the previous section ,we proceed term by term , to obtain the summands in .the integration is carried over the hyperrectangle .1 .  : this term is directly assembled from the calculation of 2 . : this term involves the calculation of which is expanded by using the separable structure of the free dynamics where 3 . : in this case , we need to work on the expression which is obtained directly from the computations for in * 5 ) * below . : where with a similar argument as in the previous term we expand 5 . : this term requires the computation of the inner product with given by by using the separable representation of the basis functions we expand the inner product   [ [ initialization ] ] * initialization * + + + + + + + + + + + + + + + +  the first iteration , with a stabilizing initial guess , requires special attention . if it is obtained via a riccati - type argument , then initialization follows directly from .otherwise we shall relax this requirement , and only assume that the initial stabilizing controller is given in separable form   in this case , we must recompute the term :  *  + as for the term , which needs to be computed differently in the first iteration , we can proceed in the same way as for , since both and have the same separable structure , it just takes to assign .   among the expressions developed in the previous subsection, the overall computational burden is governed by the approximation of which requires the assembly of the 5-dimensional tensor .each entry of this tensor is a -dimensional inner product , which under the aforementioned separability assumptions is computed as the product of , one - dimensional integrals .thus , the total amount of one - dimensional integrals required for the proposed implementation is .a positive aspect of our approach is that the assembly of tensors like falls within the category of _ embarrassingly parallelizable _ computations , so the cpu time scales down almost directly with respect to the number of available cores .furthermore , can be entirely computed in an offline phase , before entering the iterative loops in algorithms [ alg : sga1 ] and [ alg : sga2 ] . however , for values of interest of and , such as and , table [ tab : count ] indicates that is indeed a very large number .a rough estimate of the cpu time required for the assembly of is given by where corresponds to the time required for the computation of a one - dimensional integral .therefore , it is fundamental for an efficient implementation to reduce to a bare minimum . from closer inspection of the expression, we observe that all the terms can be identified as elements of the tensors  both and can be computed exactly with a computer algebra system , or approximated under suitable quadrature rules .we follow this latter approach , implementing an 8-point gauss - legendre quadrature rule . after having computed and ,the assembly of reduces to calls to properly indexed elements of these tensors .this approach requires a careful bookkeeping of the separable components of each multidimensional basis function . in this way, an entry of takes of the order of 1e-7 seconds and the overall cpu time is kept within hours for problems of dimension up to 12 .we assess the convergence of the polynomial approximation in a 1d test , with such that the exact solution of equation is given by we implement the path - following version ( algorithm 2 ) , starting with , and a threshold value , a parameter , and an internal tolerance . the relative error for table [ tabtestconv ]is defined as and number of iterations for different polynomial degree approximations are shown in table [ tabtestconv ] and figure [ figconv1d ] .ccccc + & & + ( lr)2 - 3(lr)4 - 5 + (degree ) & error & iterations & error & iterations + ( lr)1 - 1(lr)2 - 2(lr)3 - 3(lr)4 - 4(lr)5 - 5 + 2 & 1.1539 & 53 & 1.4127 & 52 + 4 & 0.2541 & 49 & 0.3643 & 58 + 6 & 0.015 & 52 & 0.0206 & 52 + 8 & 5.01e-4 & 55 & 6.41e-4 & 53 + 10 & 8.33e-6 & 55 & 1.072e-5 & 55 +  +   denotes the total number of basis functions.,title="fig : " ]  denotes the total number of basis functions.,title="fig : " ]   similarly as in section [ pdesub ] , we consider the following optimal control problem subject to the semilinear dynamics where the linear operator is of the form with , and is a nonlinear operator such that .the scalar control acts through the indicator function , with the the system is closed under suitable boundary conditions .we choose , , , and .the nonlinearity covers both advective , burgers-type , and polynomial source terms . in order to generate a low - dimensional state space representation of the dynamics ,we resort to a pseudospectral collocation method with chebyshev polynomials as in ( for further details we also refer to . by considering collocation points , the continuous state is discretized into , where .the semilinar pde dynamics are thus approximated by the nonlinear ode system where the operators correspond to the finite - dimensional realization of through the chebyshev pseudospectral method .therefore , the number of collocation points governs the dimension of the resulting nonlinear ode system , and consequently determines the dimension of the domain where the associated hjb equation is solved . in the following ,tests 1 - 3 are computed in 14 collocation points , which after including boundary conditions lead to a 12 dimensional domain for the hjb equation .test 4 is solved in 14 dimensions .the high - dimensional solver was implemented in matlab , parallelizing the tensors assembly , and tests were run on a muti - core architecture 8x intel xeon e7 - 4870 with 2,4ghz , 1 tb of ram .the matlab pseudoparallelization distributes the tasks among 20 workers .representative performance details are shown in table [ tab : cpu ] .the assembly of high - dimensional tensor that enter the iterative algorithm accounts for over 80% of the total cpu time .this percentage increases when algorithm [ alg : sga1 ] is implemented for asymptotically stable dynamics , as it requires a much lower number of iterations .note that much of the work done during the assembly phase is independent of the dynamics ( see for instance ) , and therefore can be re - used in latter problems , mitigating the overall computational burden .cccc + test & dimension&cpu - assembly&cpu - iterative ( # ) + ( lr)1 - 1(lr)2 - 2(lr)3 - 3(lr)4 - 4 + 1 & 10 & 2.061e3[s ] & 4.221e2[s](32 ) + 1 & 12 & 1.945e4[s ] & 3.377e3[s](32 ) + 4 & 14 & 1.557e5[s ] & 3.102e4[s](37 ) +  +  we now turn to the specification of parameters for the solution of the hjb equation .we set , and consider a monomial basis up to order 4 as described in section [ sec : comp ] .depending on the dynamics of every example , we will neglect odd - degree basis functions as in remark [ oddeven ] .all the integrals are approximated with an 8 point gauss - legendre quadrature rule .whenever system dynamics are stable at the origin , the value function is obtained from the undiscounted algorithm [ alg : sga1 ] , initialized with .when the dynamics are unstable over , we implement algorithm [ alg : sga2 ] , with , , and .the initializing controller is given by the solution of the associated linear - quadratic optimal feedback , as described below .for both implementations , the tolerance of the algorithm is set to . in the following tests , we compare the hjb - based feedback control with respect to the uncontrolled dynamics ( ) , the linear - quadratic optimal feedback ( lqr ) , and the power series expansion type of controller ( pse ) .we briefly describe these controllers .the well - known lqr feedback controller corresponds to the hjb synthesis applied over the linearized system around the origin and results in the optimal feedback control law given by where is the unique self - adjoint , positive - definite solution of the algebraic riccati equation and corresponds to the finite - dimensional approximation of .once this controller has been computed , the high - order pse feedback is obtained as   where corresponds to the lowest order term of the nonlinearity .variations of such feedback laws have been discussed in previous publications , see eg . and references therein . for the burgers equationit was observed numerically in that this suboptimal nonlinear controller leads to an increased closed - loop stability region with respect to the lqr feedback applied for the linearized dynamics .   in this first testwe address nonlinear optimal stabilization of advective - reactive phenomena , by considering a 1d burgers-like model with given by   .total costs : * i ) * uncontrolled : , * ii ) * lqr : 7.55 , * iii ) * pse : 6.87 , * iv ) * hjb : 6.25,title="fig:",scaledwidth=45.0% ] .total costs : * i ) * uncontrolled : , * ii ) * lqr : 7.55 , * iii ) * pse : 6.87 , * iv ) * hjb : 6.25,title="fig:",scaledwidth=45.0% ] .total costs : * i ) * uncontrolled : , * ii ) * lqr : 7.55 , * iii ) * pse : 6.87 , * iv ) * hjb : 6.25,title="fig:",scaledwidth=45.0% ] .total costs : * i ) * uncontrolled : , * ii ) * lqr : 7.55 , * iii ) * pse : 6.87 , * iv ) * hjb : 6.25,title="fig:",scaledwidth=45.0% ]  .total costs : * i ) * uncontrolled : , * ii ) * lqr : 7.55 , * iii ) * pse : 6.87 , * iv ) * hjb : 6.25,scaledwidth=45.0% ]  the feedback stabilization of burgers equation ( without the exponential source term ) has been thoroughly studied in different contexts , including the work of , and the recent work . since our interest is the study of optimal stabilization , we consider an additional source term such that the origin is not asymptotically stable .this can be appreciated in the numerical results shown in figure [ fig : test1 ] .for this model , we consider a reduced - order state space representation of 12 states , solving a hjb equation over .the value function is approximated with a monomial basis including both even and odd - degree polynomials up to degree 4 . in figure [ fig : test1 ] we can compare the uncontrolled solution to the lqr- and hjb - controlled solutions , where the lqr decay is significantly slower that the one of the hjb synthesis .the hjb controller stabilizes at a higher speed , which is reflected both in the plots and in the total costs .the hjb controller obtains a reduction of approximately 18% with respect to the lqr cost .more importantly , the control signals differ in sign , magnitude , and speed .such a behavior illustrates the nonlinear character of both the control problem and the feedback law .we now turn our attention to a diffusion equation with nonlinearity ( the case with the reversed inequality sign in front of the cubic term was already treated in subsection [ pdesub ] ) , we close the system with neumann boundary conditions .the origin is an unstable equilibrium of the uncontrolled dynamics .any other initial condition is unstable with finite time blow - up . in this case, feedback controls can only provide local stabilization , and the purpose of this numerical test is to show that hjb - based synthesis leads to an increased closed - loop asymptotic stability region when compared to lqr , and pse controllers .for this purpose , we compute feedback controls with the lqr , pse and hjb approaches , for initial conditions of the form , with .the hjb feedback is computed with , algorithm [ alg : sga2 ] , initialized with nonlinear feedback control law provided by the pse approach .the test is carried out over , and the value function is approximated with monomial basis elements of degree 2 and 4 .numerical results are presented in figure [ fig : test2 ] , for and for a series of increased values of in table [ tabtestcubic ] .as the magnitude of the initial condition grows , the locally stabilizing lqr and pse controllers are not able to prevent the finite blow - up of the dynamics .this eventually also happens for the hjb feedback , but at a much larger value of ( we report the last value until which the hjb control stabilizes the dynamics ) . , +  cccc + controller & && + ( lr)1 - 1(lr)2 - 2(lr)3 - 3(lr)4 - 4 + uncontrolled & & & + lqr & 4.14 & & + pse & 4.09 & 14.09 & + hjb & 4.06 & 13.98 & 50.36 +  +   .the diffusion - reaction equation corresponds to a particular case of the so - called schlgl model , whose feedback stabilization has been studied in .this is a special case of a bistable system with as stable and as unstable equilibria .here we use in an essential manner that we consider neumann boundary conditions . for dirichlet conditions the only equilibrium is the origin .such systems arise for instance in rayleigh - benard convection and describe excitable systems such as neurons or axons . as in the previous example, the reduced state - space is chosen as , and the basis elements for the hjb approach are even degree monomials of degree 2 and 4 .numerical results for the different controllers are shown in figure [ fig : test3 ] .while all the feedback laws effectively stabilize the initial condition to the origin , the hjb feedback has the smallest overall cost . as in test 1, it can be observed that the three feedback strategies have a considerably different transient behavior .note that the lqr controller , which neglects the effect of the nonlinearity , has an increased control magnitude with respect to the nonlinear controllers which are able to account the dissipative effect of the nonlinearity .  .uncontrolled dynamics are attracted by the stable equilibrium .total costs * i ) * uncontrolled : , * ii ) * lqr : 10.17 , * iii ) * pse:9.69 , * iv ) * hjb : 8.85,title="fig:",scaledwidth=45.0% ] .uncontrolled dynamics are attracted by the stable equilibrium .total costs * i ) * uncontrolled : , * ii ) * lqr : 10.17 , * iii ) * pse:9.69 , * iv ) * hjb : 8.85,title="fig:",scaledwidth=45.0% ] .uncontrolled dynamics are attracted by the stable equilibrium .total costs * i ) * uncontrolled : , * ii ) * lqr : 10.17 , * iii ) * pse:9.69 , * iv ) * hjb : 8.85,title="fig:",scaledwidth=45.0% ] .uncontrolled dynamics are attracted by the stable equilibrium .total costs * i ) * uncontrolled : , * ii ) * lqr : 10.17 , * iii ) * pse:9.69 , * iv ) * hjb : 8.85,title="fig:",scaledwidth=45.0% ]  for the sake of completeness , we also consider this test case with a switch of the sign of nonlinearity , i.e. , .this case is more demanding than test 2 , as now the linear part is .however , the performance of the controllers is similar as in test 2 , and the results are summarized in table [ tabtest22 ] .again , the hjb feedback law has an increased closed - loop stability region compared to the lqr and pse controllers . , +  cccc + controller & && + ( lr)1 - 1(lr)2 - 2(lr)3 - 3(lr)4 - 4 + uncontrolled & & & + lqr & 5.09 & & + pse & 4.92 & 20.02 & + hjb & 4.89 & 17.35 & 31.02 +  +   in this last test case , we consider the model given by this equation , which arises for instance in combustion theory , has as stable and as unstable equilibria .for this case , we increase the dimension of the hjb domain to 14 , i.e. , , and the basis functions are monomials of odd and even degree up to 4 .numerical results are shown in figure [ fig : test4 ] , where it can be seen that the hjb controller yields the smaller overall cost .note that the pse controller for this case has a diminished performance as compared even to the lqr controller .this can be explained by the fact that the pse controller only takes into account the lowest order nonlinearity , in this case , neglecting the cubic term .this is a well - known drawback of this controller , and therefore justifies the need of more complex synthesis methods for nonlinear feedback design , such as the proposed hjb approach .  .total costs * i ) * uncontrolled : , * ii ) * lqr : 9.45 , * iii ) * pse : 11.25 , * iv ) * hjb : 8.91 , title="fig:",scaledwidth=45.0% ] .total costs * i ) * uncontrolled : , * ii ) * lqr : 9.45 , * iii ) * pse : 11.25 , * iv ) * hjb : 8.91 , title="fig:",scaledwidth=45.0% ]. total costs * i ) * uncontrolled : , * ii ) * lqr : 9.45 , * iii ) * pse : 11.25 , * iv ) * hjb : 8.91 , title="fig:",scaledwidth=45.0% ]. total costs * i ) * uncontrolled : , * ii ) * lqr : 9.45 , * iii ) * pse : 11.25 , * iv ) * hjb : 8.91 , title="fig:",scaledwidth=45.0% ]a systematic technique for the computational approximation of hjb equations in optimal control problems related to semilinear parabolic equations was presented . to partially circumvent the curse of dimensionality ,the dynamics of the parabolic equation are approximated by a pseudospectral collocation method , and the generalized hjb equation is approximated by separable multi - dimensional basis functions of a given order .the numerical results show that the feedback controls obtained by the proposed methodology differ and improve upon applying riccati approaches to the linearized equations .the generalized hjb approach has been addressed in earlier publications , reporting on numerical results with lower dimensions than here and in part restrained enthusiasm about the numerical performance , possibly due to the lack of a systematic initialization procedure . for the class of problems considered in this paper the results were consistently better than riccati approaches .the use of the discount factor path - following technique as proposed in algorithm 2 is essential for stabilizing to unstable equilibria .the authors gratefully acknowledge support by the erc advanced grant 668998 ( ocloc ) under the eu s h2020 research program .a. alla and m. falcone ._ an adaptive pod approximation method for the control of advection - diffusion equations _ , control and optimization with pde constraints , internatnumer . math . * 164*(2013 ) , 117 .d. kalise and a. krner ._ reduced - order minimum time control of advection - reaction - diffusion systems via dynamic programming , _ proc .21st international symposium on mathematical theory of networks and systems , 1196 - 1202 ( 2014 ) .i. lasiecka and r. triggiani ._ control theory for partial differential equations : continuous and approximations theories , _ encyclopedia of mathematics and its applications 74 , cambridge university press , 2000 .r. postoyan , l. busoniu , d. nesic , and j. daafouz ._ stability of infinite - horizon optimal control with discounted cost ._ proc . of the 53rd ieee conference on decision and control , 39033908 , los angeles ,california , usa , december 2014 .e. stefansson and y.p ._ sequential alternating least squares for solving high dimensional linear hamilton - jacobi - bellman equations , _ proc .of ieee / rsj international conference on intelligent robots and systems ( iros ) , doi : 10.1109/iros.2016.7759553 , 2016 .

Abstract: a procedure for the numerical approximation of high - dimensional hamilton - jacobi - bellman ( hjb ) equations associated to optimal feedback control problems for semilinear parabolic equations is proposed . its main ingredients are a pseudospectral collocation approximation of the pde dynamics , and an iterative method for the nonlinear hjb equation associated to the feedback synthesis . the latter is known as the successive galerkin approximation . it can also be interpreted as newton iteration for the hjb equation . at every step , the associated linear generalized hjb equation is approximated via a separable polynomial approximation ansatz . stabilizing feedback controls are obtained from solutions to the hjb equations for systems of dimension up to fourteen .