computational spectral theory for operators which act on infinite dimensional hilbert spaces has advanced significantly in recent years . for self - adjoint operators, the introduction of _ quadratic _ methods has enabled the approximation of those eigenvalues which are not reliably located by a direct application of the galerkin method .the latter is due to _ spectral pollution _ ; see examples [ uber2 ] , [ magneto ] and .notable amongst these quadratic techniques are the davies & plum method , the zimmermann & mertins method , and the second order relative spectra . for spectral approximation of arbitrary operatorssee , for example , and references therein .the present manuscript is concerned with a technique for self - adjoint operators which is pollution - free and non - quadratic .the idea is to perturb eigenvalues into and then approximate them with the galerkin method .this idea was initially proposed for a particular class of differential operators ; see .an abstract version of this approach for bounded self - adjoint operators was formulated in .the latter requires priori information about the location of gaps in the essential spectrum .our main aims are to remove the requirement of priori information , to present a rigorous convergence analysis , and demonstrate the effectiveness of our method including a comparison with the quadratic methods . along the way, we also prove new spectral enclosure results for any operator of the form where is self - adjoint and is bounded and self - adjoint ; we note that any bounded operator can be expressed in this form .we now give a brief outline of our main results .  in section 3, we consider the spectra of operators of the form .the main result is theorem [ cor1a ] where we give our new spectral enclosure results .we will define a region in terms of the spectra of and , then show that it contains the spectrum of .corollary [ 2eigs ] shows that the enclosure is , in a sense , sharp .section 4 is primarily concerned with the perturbation of an eigenvalue , , of a self - adjoint operator , .we consider where is a sequence of orthogonal projections .the main results are theorem [ est ] and theorem [ qq ] where we prove extremely rapid convergence properties of the eigenspaces and eigenvalues associated to the perturbed eigenvalue .  in section 5 , we present our new perturbation method .the idea is based on applying the galerkin method to for a fixed .the preceding results enable us to lift an eigenvalue , , off the real line , away from the essential spectrum , and extremely close to where it can be approximated by a direct application of the galerkin method .the main results are theorem [ limlem1 ] and theorem [ eigconv ] , where we prove the rapid convergence of galerkin eigenspaces and eigenvalues .  in section 6 , we apply our method to several operators arising in magnetohydrodynamics , non - relativistic and relativistic quantum mechanics .most of our examples involve calculations using trial spaces belonging to the form domain and not the operator domain .in particular , we use the fem spaces of piecewise linear trial functions . however , the quadratic methods require trial spaces from the operator domain .in our last example we use the operator domain which allows a comparison with the quadratic methods .let us now fix some notation . unless stated otherwise, will denote a semi - bounded ( from below ) self - adjoint operator acting on a hilbert space .the quadratic form , spectrum , resolvent set , discrete spectrum , essential spectrum and spectral measure we denote by , , , , and , respectively .for we denote the range of by .associated to the form is the hilbert space which has inner - product and norm the gap or distance between two subspaces and of , is defined as \textrm{~where~ } \delta(\mathcal{m},\mathcal{n } ) = \sup_{u\in\mathcal{m},\vert u\vert=1}{{\rm dist}}(u,\mathcal{n});\ ] ] see ( * ? ? ?* section iv.2.1 ) for further details .we shall write to indicate the gap between subspaces with respect to the norm .throughout this section , we assume that with and .[ gamdef ] the functions \to\mathbb{c} ] , we have then , using lemma [ olderlem ] , and hence now consider the case where .let , the case where may be treated similarly .for some we have and hence this section we will prove enclosure results for .unless stated otherwise , is assumed to be a bounded self - adjoint operator with we shall always assume that is a bounded self - adjoint operator with   [ oldestlem ] let , then .let be the spectral measure associated to .for all  ] ., and the shaded region they enclose ; on the left , , ,  and ; on the right , , , , and .,title="fig : " ] , and the shaded region they enclose ; on the left , , ,  and ; on the right , , , , and .,title="fig : " ]  [ w ] we denote the numerical range by , and define   [ cor1a ] if is unbounded from above and/or below , then if is bounded , then   the first assertion follows immediately from theorem [ lem1b ] .suppose that is bounded .then again follows from theorem [ lem1b ] .let .then either suppose the former is true , then suppose instead that , then for some , , we have   any bounded linear operator , , may be expressed as where and are bounded self - adjoint operators .hence , theorem [ cor1a ] provides an enclosure for the spectrum , in terms of the real and imaginary parts , of any bounded linear operator .[ 2eigs ] let and , then .for any , may be chosen such that .the first assertion follows immediately from theorems [ lem1b ] and [ cor1a ] .let be normalised eigenvectors with and . for  ] for which .let and where .by theorem [ cor1a ] we have . for varying values of ,figures 35 show the region(s ) enclosed by . also shownis for 1000 randomly generated matrices and where and ., consists of 3 disjoint regions which are shown top left . also shownare the three regions in more detail .the red dots are for 1000 randomly generated matrices and where and ., title="fig : " ] , consists of 3 disjoint regions which are shown top left . also shownare the three regions in more detail .the red dots are for 1000 randomly generated matrices and where and ., title="fig : " ] , consists of 3 disjoint regions which are shown top left . also shownare the three regions in more detail .the red dots are for 1000 randomly generated matrices and where and . , title="fig : " ] , consists of 3 disjoint regions which are shown top left . also shownare the three regions in more detail .the red dots are for 1000 randomly generated matrices and where and ., title="fig : " ]  , the region now consists of 2 disjoint regions ; the first two regions on the top left of figure 3 have merged into one .the red dots are for 1000 randomly generated matrices and where and .,title="fig : " ] , the region now consists of 2 disjoint regions ; the first two regions on the top left of figure 3 have merged into one .the red dots are for 1000 randomly generated matrices and where and .,title="fig : " ]  .the red dots are for 1000 randomly generated matrices and where and .,title="fig : " ] .the red dots are for 1000 randomly generated matrices and where and .,title="fig : " ] .the red dots are for 1000 randomly generated matrices and where and .,title="fig : " ] .the red dots are for 1000 randomly generated matrices and where and .,title="fig : " ]in this section we consider where is a non - trivial orthogonal projection .let , , and denote  ] set and let be the orthogonal projection onto .for any normalised we have and thus .\ ] ] by lemma [ cor1 ] , we have with spectral subspace . by proposition [ thm1 ] and we deduce that whenever is sufficiently small , the operator is invertible with uniformly bounded inverse for all and  ] .the result follows from the gershgorin circle theorem .the perturbation method , for locating , was introduced in where it was formulated for schrdinger operators .a more general version was presented in which required priori knowledge about the location of gaps in the essential spectrum . in this sectionwe present a new perturbation method which requires no priori information and converges rapidly to .in fact , our examples suggest that the method will actually capture the whole of .the idea is to perturb eigenvalues off the real line by adding a perturbation where is a finite - rank orthogonal projection .the results from the previous sections allow us to perturb eigenvalues very precisely .the perturbed eigenvalues and their multiplicities may then be approximated with the galerkin method without incurring spectral pollution ; see ( * ? ? ?* theorem 2.5 & theorem 2.9 ) .as above , denotes a sequence of finite - rank orthogonal projections each with range .we shall assume that this is the usual hypothesis for a sequence of trial spaces when using the galerkin method .for sufficiently large we have , by proposition [ thm1 ] , that will consist of eigenvalues in a small neighbourhood of , and , by theorem [ qq ] , eigenvalues within neighbourhoods of the ; recall that we stress that is extremely small ; indeed , if pollution does not occur and we use the galerkin method to approximate the eigenvalue , then our approximation will be of the order where   in this section we are concerned with the approximation of the eigenvalues of using the galerkin method . to this end , for our fixed , let us fix an such that such an is assured by theorem [ est ].  associated to the restriction of the form to the trial space is a self - adjoint operator acting in the hilbert space ; denote this operator and corresponding spectral measure by and , respectively .the galerkin eigenvalues of with respect to the trial space are denoted and are precisely the eigenvalues of for our , we denote by the spectral subspace associated to the operator and those eigenvalues in a neighbourhood of . then , for a fixed , we have for all sufficiently large  we now study the convergence properties of and associated eigenvalues , where our main convergence results are expressed in terms of and from and , respectively .we note that , using theorem [ est ] , where is independent of .[ l2d ] there exists a constant , independent of , such that   we assume that the assertion is false . then there exist sequences and with , such that , for each fixed there is a subsequence with let us fix a .we may assume , without loss of generality , that let be a sequence with and the sequence has a convergent subsequence , without loss of generality , we assume that , where . for all sufficiently large , there exists a normalised vector with then and therefore the sequence has a convergent subsequence .we assume , without loss of generality , that .therefore denote by the orthogonal projection from onto .let and set , then for any  we deduce that hence and therefore let .since implies that , we deduce that and we set . using and with as in the proof of theorem [ est ] , furthermore , using the estimates above we have since where , hence therefore , we have evidently , the left hand side is larger than the right hand side for all sufficiently large .the result follows from the contradiction .[ limlem1 ] there exist constants , both independent of , such that and for all sufficiently large .first we prove .let with . for , we denote then , with as in the proof of theorem [ est ], we have and therefore -(m-1)\vert x(z)\vert^2\\ & = \langle ax(z),x(z)\rangle -(m-1)\vert x(z)\vert^2\\ & = \langle a(a + ip_n - z)^{-1}u , x(z)\rangle -(m-1)\vert x(z)\vert^2\\ & = \langle u , x(z)\rangle -\langle(ip_n - z)x(z),x(z)\rangle-(m-1)\vert x(z)\vert^2\\ & \le \vert x(z)\vert + ( 2+m+\vert z\vert)\vert x(z)\vert^2\\ & \le\frac{1}{c_1 } + \frac{2+m+\vert z\vert}{c_1 ^ 2}.\end{aligned}\ ] ] hence for constant which is independent of and .let with , then hence then , using lemma [ l2d ] , where is independent of and .furthermore , where + \vert a_k(z)^{-1}p_ku - \hat{p}_kx(z)\vert^2\\ & = \langle p_ku -a_k(z)\hat{p}_kx(z),a_k(z)^{-1}p_ku - \hat{p}_kx(z)\rangle\\ & \quad-\langle(ip_kp_{n}-z)(a_k(z)^{-1}p_ku - \hat{p}_kx(z)),a_k(z)^{-1}p_ku - \hat{p}_kx(z)\rangle\\ & \quad+ ( 1-m)\vert a_k(z)^{-1}p_ku - \hat{p}_kx(z)\vert^2\\ & \le\vert p_ku -a_k(z)\hat{p}_kx(z)\vert\vert a_k(z)^{-1}p_ku - \hat{p}_kx(z)\vert\\ & \quad+\vert ip_kp_{n}-z\vert\vert a_k(z)^{-1}p_ku - \hat{p}_kx(z)\vert^2\\ & \quad+ \vert 1-m\vert\vert a_k(z)^{-1}p_ku - \hat{p}_kx(z)\vert^2\\ & \le c_2\big(1 + \vert(z - m+1)\vert\big)^2\vert(\hat{p}_k - i)x(z)\vert^2\\ & \quad+ \big(1+\vert z\vert\big)c_2 ^ 2\big(1+\vert(z - m+1)\vert\big)^2\vert(\hat{p}_k - i)x(z)\vert^2\\ & \quad+\vert 1-m\vert c_2 ^ 2\big(1+\vert(z - m+1)\vert\big)^2\vert(\hat{p}_k - i)x(z)\vert^2.\end{aligned}\ ] ] therefore , for constant which is independent of and .set then and combining this estimate with , we deduce that for some constant which is independent of and , we have then , by virtue of , the following formula holds for all sufficiently large , the assertion is proved . now using , and theorem [ est ] ,we have   let be the repeated eigenvalues of which are associated to the subspace .[ eigconv ] there exists a constant , independent of , such that for all sufficiently large .let be an orthonormal basis for .let be the orthogonal projection from onto and set .using theorem [ limlem1 ] , for constant which is independent of .consider the matrices {p , q}=\frak{a}(u_{q , k},u_{p , k } ) + i\langle p_{n}u_{q , k},u_{p ,k}\rangle\quad\textrm{and}\quad [ m_{n , k}]_{p , q}=\langle u_{q , k},u_{p , k } \rangle.\ ] ] evidently , is precisely the set .we have and hence for constant which is independent of .similarly , hence for constant which is independent of .furthermore , {p , q}&=\langle(r_k - i)u_{q},(r_k - i)u_{p } \rangle + \langle(r_k - i)u_{q},u_{p } \rangle + \langle u_{q},(r_k - i)u_{p } \rangle\\ & \quad + \delta_{pq},\end{aligned}\ ] ] hence for constants both independent of , we have {pq } - \delta_{pq}\vert \le k_7(\varepsilon_n + \epsilon_k)^2\quad\rightarrow\quad\vert [ m_{n , k}]^{-1}_{pq } - \delta_{pq}\vert \le k_8(\varepsilon_n + \epsilon_k)^2.\ ] ] then {p , q}-(\lambda+i)\delta_{p , q}\vert \le k_{9}(\varepsilon_n+\epsilon_k)^2\ ] ] for constant which is independent of .the result follows from the gershgorin circle theorem .we note that , for any orthogonal projection , all non - real eigenvalues of can provide information about . indeed , whenever with , we have and using the first and third terms from yields then \cap\sigma(a)\ne\varnothing.\ ] ] further , suppose that and . then , using ( * ? ? ?* lemma 1 & 2 ) with the second term in and the equality , we obtain the enclosure   let us now verify our main results with a illustrative example .[ uber2]with ^ 2 ] with a uniform mesh of size and which satisfy homogeneous dirichlet boundary conditions .let be the space without boundary conditions .first we apply the galerkin method directly to with trial spaces .we find that spectral pollution occurs in the interval which obscures the approximation of the genuine eigenvalue ; see the left - hand side of figure 6 .from example [ uber2 ] , spectral pollution in the interval obscures the approximation of the genuine eigenvalue . on the right - hand side , the galerkin method approximation for from example [ uber2 ] , the curves and , which together form .the region consists of complex numbers which lie to the right of and to the left of .,title="fig : " ] from example [ uber2 ] , spectral pollution in the interval obscures the approximation of the genuine eigenvalue . on the right - hand side , the galerkin method approximation for from example [ uber2 ] , the curves and , which together form .the region consists of complex numbers which lie to the right of and to the left of .,title="fig : " ]  let be the orthogonal projection onto the trial space .since and for all  ] we consider the magnetohydrodynamics operator with , and , we have \cup[3/8,7/8].\ ] ] the discrete spectrum contains a sequence of simple eigenvalues which accumilate only at .these eigenvalues are above , and not close to , the essential spectrum .they are approximated by the galerkin method , with trial spaces , without incurring spectral pollution .it was shown , using the second order relative spectrum , that there is also an eigenvalue in the gap in the essential spectrum ; see ( * ? ? ?* example 2.7 ) .the top row of figure 8 shows many galerkin eigenvalues in the gap in the essential spectrum and many more just above the essential spectrum ; we should be suspicious of spectral pollution in these regions .we define and we are therefore interested in those elements from which are close to the real line , i.e. , we would prefer our approximate eigenvalues to converge to rather than .the second row of figure 8 shows , the two bands of essential spectrum are clearly approximated along with an approximation of in the gap , and a second eigenvalue above the essential spectrum .the perturbation method has approximated the essential spectrum , identified the spectral pollution , and approximated two eigenvalues which were obscured by the spectral pollution .from example [ magneto ] .there are many galerkin eigenvalues in the gap in the essential spectrum and many more just above the essential spectrum ; in these regions we should be suspicious of spectral pollution .the second row shows the perturbation method approximation for from example [ magneto ] ; the essential spectrum is approximated , as are two eigenvalues , one in the gap and one just above the essential spectrum .the perturbation method has identified the spectral pollution in the gap and above the essential spectrum.,title="fig : " ] from example [ magneto ] .there are many galerkin eigenvalues in the gap in the essential spectrum and many more just above the essential spectrum ; in these regions we should be suspicious of spectral pollution .the second row shows the perturbation method approximation for from example [ magneto ] ; the essential spectrum is approximated , as are two eigenvalues , one in the gap and one just above the essential spectrum .the perturbation method has identified the spectral pollution in the gap and above the essential spectrum.,title="fig : " ]  from example [ magneto ] .there are many galerkin eigenvalues in the gap in the essential spectrum and many more just above the essential spectrum ; in these regions we should be suspicious of spectral pollution .the second row shows the perturbation method approximation for from example [ magneto ] ; the essential spectrum is approximated , as are two eigenvalues , one in the gap and one just above the essential spectrum .the perturbation method has identified the spectral pollution in the gap and above the essential spectrum.,title="fig : " ] from example [ magneto ] .there are many galerkin eigenvalues in the gap in the essential spectrum and many more just above the essential spectrum ; in these regions we should be suspicious of spectral pollution .the second row shows the perturbation method approximation for from example [ magneto ] ; the essential spectrum is approximated , as are two eigenvalues , one in the gap and one just above the essential spectrum .the perturbation method has identified the spectral pollution in the gap and above the essential spectrum.,title="fig : " ]  [ schro1 ] with we consider the schrdinger operator the essential spectrum of has a band structure .the first three intervals of essential spectrum are approximately ,\quad [ 0.5948,0.918058]\quad\textrm{and}\quad [ 1.29317,2.28516].\ ] ] the second order relative spectrum has been applied to this operator , see , where the following approximate eigenvalues were identified we note that is below the essential spectrum , is in the first gap in the essential spectrum , and is in the second gap .we apply the perturbation method with the trial spaces which is a -dimensional space of piecewise linear trial functions on the the interval  ] which vanish at the boundary .the operator is the orthogonal projection onto trial space .the right - hand side of figure 9 shows that the perturbation method has approximated three eigenvalues in the first gap of the essential spectrum .we should stress the ease with which the above calculations are conducted .the perturbation method does not require trial spaces from the operator domain , thus we have been able to use the fem spaces of piecewise linear trial functions .the quadratic methods can not be applied with these trial spaces .our final example is outside much of the theory so far developed for the perturbation method , this is because the operator concerned is indefinite .however , the numerical results suggest that the perturbation method can be extended to the indefinite case . the second orderrelative spectra has been applied to this example and the code made available online ; see and , respectively .we use this code to apply the galerkin method , the perturbation method , the second order relative spectrum , the davies & plum method and zimmermann & mertins method .[ diracex ] with ^ 2 ] and the interval contains the eigenvalues we use trial spaces generated by hermite functions of odd order ; see for further details .there is no spectral pollution incurred by the galerkin method in this example , therefore we can also compare the perturbation method with the galerkin method .the second order relative spectrum is known to converge to the discrete spectrum ; see . from this methodwe obtain a sequence of complex numbers with , where is the dimension of the trial space . the sequence of real parts we take as our approximation for .the davies & plum and zimmermann & mertins methods , which are equivalent , provide a sequence of intervals containing we take the mid - point of these intervals , which we denote by , to be our approximation of .our numerical results suggest the following convergence rates again we see the performance of the perturbation method is essentially the same as the galerkin method .we also note the relatively poor performance of the quadratic methods .the latter is not entirely surprising as the known convergence rates for quadratic methods are measured in terms of , i.e. , the distance of the eigenspace to the trial space with respect to the graph norm ; see ( * ? ? ?* lemma 2 ) and ( * ? ? ?* section 6 ) .our theoretical results are , for the most part , focused on the perturbation and approximation of the discrete spectrum .however , the examples indicate that our new perturbation method also captures the essential spectrum .this should be further investigated . forthe approximation of eigenvalues , the rapid convergence assured by theorems [ qq ] & [ eigconv ] mean that , in terms of accuracy and convergence , we can expect the perturbation method to significantly outperform the quadratic methods .the fact that the former may be applied with trial spaces from the form domain is another 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Abstract: we consider the problem of how to compute eigenvalues of a self - adjoint operator when a direct application of the galerkin ( finite - section ) method is unreliable . the last two decades have seen the development of the so - called quadratic methods for addressing this problem . recently a new perturbation approach has emerged , the idea being to perturb eigenvalues off the real line and , consequently , away from regions where the galerkin method fails . we propose a simplified perturbation method which requires no priori information and for which we provide a rigorous convergence analysis . the latter shows that , in general , our approach will significantly outperform the quadratic methods . we also present a new spectral enclosure for operators of the form where is self - adjoint , is self - adjoint and bounded . this enables us to control , very precisely , how eigenvalues are perturbed from the real line . the main results are demonstrated with examples including magnetohydrodynamics , schrdinger and dirac operators .  _ keywords : _ spectral enclosure , eigenvalue problem , perturbation of eigenvalues , spectral pollution , galerkin method , finite - section method , superconvergence .  _ 2010 mathematics subject classification : _ 47a10 , 47a55 , 47a58 , 47a75 .   email : james.hinchcliffe.com .   department of mathematics , university of sussex , falmer campus , brighton bn1 9qh , uk . email : m.strauss.ac.uk .


particle accelerators are technological devices which allow studies at both infinitely small scale , e.g. particles responsible for elementary forces , and extremely large scale , e.g. the origin of cosmos . in a simplified approach ,such devices are composed of basic elements sequence : focusing and defocusing magnets , accelerating electromagnetic fields and trajectory bending elements as they are used in the case of ring accelerators .the resulting dynamics is nonlinear , and can be described , in the absence of strong damping , by a conservative system. this system can be modelled by a symplectic map built from the composition of several elementary maps corresponding to each basic magnetic element .one of the main problems found in the dynamics of ring accelerators is to study the stability around the nominal orbit , i.e. the circular orbit passing through the centre of the ring .each component of the ring can be seen as a nonlinear map , that deforms the trajectory at large amplitude .moreover , such maps possess stochastic layers whose effect is the reduction of the stability domains around the nominal circular orbit ( the so - called _ dynamical aperture _ - da ) .such behaviors imply that ( chaotic ) nearby orbits can drift away after a few ring turns , eventually colliding with accelerator s boundaries , and consequently reduce the beam lifetime and the performance of the accelerator .the aim of the present paper is to derive a reliable improvement of the stability of the beam by increasing the da in a simplified accelerator model , consisting of only one type of element having a sextupole nonlinearity .we work in the framework of the _ hamiltonian control theory _ presented in  , where two methods to control symplectic maps have been described , namely using lie transformations and generating functions . in the present paperwe use the former method , that allows direct determination of the new controlled map ; avoiding the possible problems related to coordinate inversion .the aim of control theory is to improve selected features of a given system , by slightly modifying its hamiltonian with the addition of a _ small _ control term , so that the new system and the unperturbed one are conjugated namely , they have the same dynamics .this technique is particularly suitable whenever one can directly act on the system and modify it , e.g. in the case of a particle accelerator where the addition of a control term in the hamiltonian function can be seen as the introduction of a suitable magnet in the accelerator lattice .  in our study , we use the smaller alignement index ( sali )  method , which is an efficient indicator for characterising orbits as chaotic or regular in hamiltonian flows and symplectic maps .the sali is computed using the time evolution of two deviation vectors along the studied orbit .the paper is organized as follows : we introduce the model in section [ sec : themodel ] and present a general result for the control of symplectic maps in section [ sec : ctrlth ] .we apply the theory to the symplectic map model of a standard ring accelerator in section [ sec : fodoctrl ] .we briefly recall the sali chaos indicator in section [ sec : salimeth ] , while section [ sec : results ] presents our numerical results on the behavior of the constructed model .finally , in section [ sec : ccl ] we summarize our conclusions .further technical details can be found in the appendix .let us consider a system consisting of a charged particle and a simplified accelerator ring with linear frequencies ( tunes ) , , with a localized thin sextupole magnet ( for more details the interested reader is referred to  ) .the magnetic field of this element modifies the orbit once the particle passes through it .the basic model is : where ( ) denote the deflection from the ideal circular orbit in the horizontal ( vertical ) direction before the particle enters the element and ( ) are the associated momentum .primed variables denote positions and momenta after the particle left the element .the parameters and are related to the accelerator s tunes  and , corresponding to a non - resonant condition ( see  ) . ] and by the relations and .the first matrix in ( [ eq:4dfodo ] ) describes the linear motion of a particle , which corresponds to a simple rotation in the phase space .the nonlinearity induced by the thin sextupole magnet is modeled by the 2 order polynomial expression in ( [ eq:4dfodo ] ) .the particle dynamics at the -th turn , can be described by the sequence , where the -th positions and momenta are defined as a function of the -th ones by .the map ( [ eq:4dfodo ] ) decomposes in an integrable part and a quadratic perturbation , respectively the system is associated to the following hamiltonian ( see [ sec : appendix ] ) more precisely ( [ eq:4dfodo ] ) can be written in terms of poisson brackets . ] as here , with denoting the transpose of a matrix , and by definition , for any function defined in the phase space , , with , being the symplectic constant matrix ,   maps of the form  have already been studied in  where it has been shown that chaotic orbits reduce the da to a hypersphere of radius in the 4-dimensional phase space ( see figs . 5 and 6 of  ) .the goal of the present paper is to show that the stability region of the nominal circular orbit can be increased once the map  controlled by an appropriately designed map .the aim of hamiltonian theory is to provide mathematic tools to be able to modify the dynamics of a symplectic system . with this , it is possible to manipulate intrinsic features , e.g. to reduce the chaotic regions in phase space or to build invariant tori .  in the followingwe will be interested in controlling a quasi - integrable symplectic map in such a way that it will allow us to obtain a new , _ controlled _ map closer to the integrable part  of the original map , and thus increase the stability region around the nominal circular orbit .this is of great importance since chaos diffusion channels in phase space with unpredictable consequences in configuration space .the controlled map is expected to have a smaller number of escaping orbits and a larger region occupied by invariant curves in a neighbourhood of the origin . for this purpose , we apply the method presented in  , with the modification that in the present case the integrable part is not expressed in action - angle variables . in particular , the integrable part is a rotation , so the present theory applies to perturbations of rotations , instead of maps close to identity .let us consider an integrable symplectic map defined through its infinitesimal generator   and consider the quasi - integrable map perturbation of the former where and is a perturbation , namely .the aim of hamiltonian control theory , is to construct a third map , the _ control map _ , whose generator is small ( it satisfies ) . the _ controlled map _ will be conjugated to a map , closer to than ( see  below ) .we note that the use of the exponential of a poisson bracket , ensures that such maps are symplectic by construction .  to be more precise , let us define the unperturbed map and observe that is not invertible , since its kernel contains any smooth function of .thus we assume the existence of a pseudo - inverseoperator , , that should satisfy ( see  for details )   at this point we can define the _ non - resonant _ and the _ resonant _ operators which are projectors , i.e. and .our main theoretical result can be stated in the following theorem :  [ thm : control ] under the above hypotheses and defining we have where with a control term given by   [ rem : warpplus ]let us define as in  the warped addition , , of two operators by an explicit formula can be obtained using the baker - campbell - hausdorff formula  , where is a series whose first terms are hence the warped addition is a deformation of the usual addition between operators .  using this warped addition ,we can rewrite the controlled map into the form where the control term  becomes   from  we have : one can easily prove ( see appendix a of  ) that hence , recalling the definition of and , we can rewrite  as by rearranging the terms we can easily get which is nothing but  rewritten using the warped addition .  [ rem : smallness ] let us observe that the control term is , as required , small compared to .in fact from  and by using the approximated formula for the warped addition , we obtain where we use the relation .  under the assumption of absence of resonances ,i.e. , the term in the control map can be explicitly computed to give ( see [ sec : appendix ] ) this section we derive the controlled map presented in section [ sec : ctrlth ] in case of maps of the form ( [ eq:4dfodo ] ) . to implement the theory we need to diagonalize the operator .for this reason we introduce complex variables and rewrite as   the poisson bracket with now takes the form hence , for any and we obtain here we introduced the vector and use the compact notation for the complex vector .the operator is diagonal in these variables and thus map  is straightforwardly obtained as   once we have this map , we can compute the operators , and . for all and , such that and ( which defines the _ non - resonance condition _ ), we get with   in the rest of the paper , we will assume the above non - resonant condition  to hold for the considered values of and .let us remark that this is not a limitation of the actual theory , but just a working assumption .we could equivalently have chosen to work in the resonant regime , using a different control term suitable for resonant dynamics .the explicit computations are as follows : first we need to express in terms of complex variables ; next we compute , and transform back to the original variables .finally we compute the exponential .actually , even if is a polynomial of degree three in the variables the map is given by an infinite series .the terms of this series can be sequentially computed , but the degree of complexity ( i.e. the number of involved terms ) increase very fast . to simplify the computations we decided to use the _ approximated _ generator of the control map already truncated at order 2 we note that is composed by about twenty terms . a detailed discussion of the whole procedure needed to obtain , as well as its explicit formula , are presented in [ sec : appendix ] .we now face another difficulty , namely the computation of the control map from the generator .this is equivalent to perform the sum whose complexity once again grows very fast .we thus introduce a second approximation to our construction , by computing only a finite number of terms in the above sum .so , we define a _ truncated control map of order _ and a _ truncated controlled map of order _   since the exact control map given by   is symplectic , the controlled map  will also be symplectic .on the other hand , we can not expect the -th order control map to be symplectic .we know that a map is symplectic if its jacobian matrix verifies ( in its definition domain ) the equality thus , in order to check the _ symplecticity defect _ of we compute the norm of matrix , where is the jacobian of given by .the results are presented in fig .[ jac ] for orders up to , in the region \times[-1,1] ] , , giving rise to orbits that do not escape up to iterations of the map . in particular , we consider in fig .[ compare](a ) the original uncontrolled map , and in figs . [ compare](b ) to [ compare](d ) , the order controlled map for to respectively .one can easily see that the region of non - escaping orbits for the original map is smaller than the one of the controlled maps .this observation can be quantified by considering initial conditions inside a circle centered at the origin of each panel of fig .[ compare ] ( which represent the actual physical plane since the initial momenta are ) with radius , and evaluate the number of escaping and non - escaping orbits as a function of the circle radius for with up to .results reported in fig .[ controlcompare ] support the previous claim , by clearly showing that controlled maps of orders , and behave very similarly and lead to an increase of the non - escaping region .let us note that the behavior of the controlled maps of orders ( fig . [ compare](b ) ) and ( fig . [ compare](c ) ) is somewhat misleading if it is not analyzed together with the information from the symplecticity defect ( see figs .[ jac](a ) and ( b ) respectively ) .in fact , these maps are strongly dissipative and produce a strong shift of orbits towards the origin , preventing them from escaping .this dissipation effect is not physical , as it is not observed in real accelerators , and therefore we do not discuss further the and controlled maps .   as a function of the distance from the origin in the physical space . we iterate initial conditions in a circle of radius centred at the origin of plane , with , andcompute the percentages of non - escaping orbits during iterations , for the controlled map with , as a function of ., height=264 ]  from the results of figs .[ compare ] and [ controlcompare ] we see that the addition of even the lower order ( ) control term , having an acceptable symplecticity defect , increases drastically the size of the region of non - escaping orbits around the central periodic orbit. a further increase of the order of the control term results to less significant increment of this region , while the computational effort for constructing the controlled map increases considerably .in fact , contains around elementary terms , i.e. monomials in , while this number is almost doubled for each order , so that contains around terms . also the cpu time needed to evolve the orbits increases with the order .for example , while the integration of one orbit using takes about times the cpu time needed to integrate the original map ( [ eq:4dfodo ] ) , the use of needs almost times more .thus we conclude that the controlled map , which can be considered quite accurately to be symplectic , is sufficient to get significant increment of the percentage of non - escaping orbits , without paying an extreme computational cost .the smaller alignment index ( sali )  has been proved to be an efficiently simple method to determine the regular or chaotic nature of orbits in conservative dynamical systems . thanks to its properties , it has already been successfully distinguished between regular and chaotic motion both , in symplectic maps and hamiltonian flows .  for the sake of completeness ,let us briefly recall the definition of the sali and its behavior for regular and chaotic orbits , restricting our attention to -dimensional symplectic maps .the interested reader can consult  for a more detailed description . to compute the sali of a given orbit of such maps , one has to follow the time evolution of the orbit itself and also of two linearly independent unitary deviation vectors .the evolution of an orbit of a map is described by the discrete - time equations of the map where , represents the orbit s coordinates at the -th iteration .the deviation vectors at time are given by the tangent map where denotes the jacobian matrix of map , evaluated at the points of the orbit under study .then , according to  the sali for the given orbit is defined as where denotes the usual euclidean norm and , are unitary normalised vectors .  in the case of chaotic orbits ,the deviation vectors , eventually become aligned in the direction defined by the maximal lyapunov characteristic exponent ( lce ) , and sali falls exponentially to zero . an analytical study of sali s behavior for chaotic orbitswas carried out in  where it was shown that with , being the two largest lces .  on the other hand , in the case of regular motion the orbit lies on a torus and the vectors , eventually fall on its tangent space , following a time evolution , having in general different directions .this behavior is due to the fact that for regular orbits , the norm of a deviation vector increases linearly in time .thus , the normalization procedure brings about a decrease of the magnitude of the coordinates perpendicular to the torus , at a rate proportional to , and so , eventually fall on the tangent space of the torus . in this case, the sali oscillates about non - zero values ( for more details see  ) .the simplicity of sali s definition , its completely different behavior for regular and chaotic orbits and its rapid convergence to zero in the case of chaotic motion are the main advantages that make sali an ideal chaos detection tool .recently a generalization of the sali , the so - called generalized alignment index ( gali ) has been introduced  , which uses information of more than two deviation vectors from the reference orbit .since the advantages of gali over sali become relevant in the case of multi - dimensional systems , in the present paper we apply the sali method for the dynamical study of the 4d map ( [ eq:4dfodo ] ) .as already mentioned , the goal of constructing the controlled map is to increase the percentage of regular orbits up to a given ( large ) number of iterations , or equivalently increase the size of the stability region around the nominal circular trajectory ( i.e. the da ) . because the presence of chaotic regions can induce a large drift in the phase space , that eventually could lead to the escape of orbits , the achievement of a larger da can be qualitatively inspected by checking via the sali method the regular or chaotic nature of orbits in a neighborhood of the origin ( see fig . [ fig5c ] ) .we note that we define an orbit to be chaotic whenever sali , and regular for the contrary .  in fig .[ fig5c ] ( which should be compared with fig.5 of ) we observe a strong enlargement of the region of regular orbits .this region is characterized by large sali values . in particular , for iterations of the map , of the considered orbits are regular , while for the uncontrolled map this percentage reduces to .this improvement can be also confirmed by visual inspection of fig .[ compare ] , where the regions of non - escaping orbits are shown for different orders of the controlled map ( [ eq : ctrlledmapordk ] ) .  in figs .[ fig5c](a ) and ( b ) we see that there exist orbits of the map , which are characterized as regular up to iterations , while they show their chaotic character once they are iterated up to .such orbits correspond to the dark regions marked by a black circle in fig .[ fig5c](b ) ( for comparison this circle is also plotted in all panels of fig .[ fig5c ] ) .this discrepancy is absent for the map , which shows almost the same geometrical shape for the non - escaping region when we pass from to iterations . in order to better understand this behavior we followed the evolution of a single orbit with initial condition inside the black circle in fig .[ fig5c] for both the and the controlled maps , computing the corresponding sali values up to iterations .the results are reported in fig .[ orbite ] and clearly show that the orbit behaves regularly up to iterations of the map , since its sali values are different from zero , but later on a sudden decrease of sali to zero denotes the chaotic character of the orbit .this behavior clearly implies this is a slightly chaotic , sticky orbit , which remains close to a torus for long time intervals ( ) , while later on it enters a chaotic region of the phase space .it is interesting to note that iterating the same initial condition by the map we get a regular behavior at least up to .   and order controlled maps_. time evolution of the sali for the orbit with initial conditions ( see fig . [ fig5c ] ) ,using the [ ( r ) red curve ] and the controlled map [ ( b ) blue curve].,width=377 ]  in order to provide additional numerical evidence of the effectiveness of the controlled map ( [ eq : ctrlledmapordk ] ) in increasing the da , we consider initial conditions inside a 4d sphere centered at the origin of the map , with radius , .we compute the number of regular , escaping and chaotic orbits as a function of the sphere radius .the corresponding results are reported in fig .[ sphere](b ) , while in fig .[ sphere](a ) we reproduce fig . 6 of for comparison .from this figure we observe a strong increase of the da , since the largest sphere containing regular orbits has a radius , while this radius was for the original uncontrolled map .we also observe that increasing the total number of iterations from to ( dashed and solid lines in fig .[ sphere ] respectively ) increases the percentage of chaotic orbits , but the radius of the 4d sphere containing only regular orbits does not change significantly .in this paper we considered a simple model of a ring particle accelerator with sextupole nonlinearity that can be described by a symplectic map . in the framework of hamiltonian control theory, we were able to control the dynamics of the original system , by providing a suitable control map , resulting in a small perturbationof the initial map .this control map has been constructed with the aim of da enlargement of the particle accelerator , and thus improving the beam s life - time and the accelerator s performance .in particular , the theoretical framework we developed allows a -parameter family of approximated controlled maps .we performed several numerical simulations in order to choose the bestapproximated controlled map ( [ eq : ctrlledmapordk ] ) , taking into account the complexity of the map , i.e. the number of terms by which it is composed , the cpu time needed to perform the numerical iteration of orbits , and the accuracy of the results in terms of the symplectic character of the map .we find that the order controlled map is an optimal choice for the controlled system .  using this controlled map we succeeded in achieving our initially set goal , since the map exhibits a da with a radius more than times larger than the one for the original map ( see fig . [ sphere ] ) .the aim of this section is to introduce further details for the construction of the control term and of the controlled map , and to provide explicit formulas for the interested reader .let us first introduce some notations and recall some useful relations .* * lie brackets and operators .* let be the vector space of real or complex functions of variables .for any , the lie bracket is given by \ , , \ ] ] where denotes the partial derivative with respect to the variable .+ using the above definition , we can define a linear operator , induced by an element of , acting on  this operator is _ linear _ , _ antisymmetric _ and verifies the _ jacobi identity _ * * exponential .* we define the exponential of such an operator , by which is also an operator acting on .the power of an operator is the composition : .+ we observe that in the case of the hamiltonian function , the exponential provides the flow , namely , of the hamilton equations * * vector field .* the action of the above defined operators , can be extended to vector fields component by component   we show now that map ( [ eq:4dfodo ] ) can be seen as the time- flow of a given hamiltonian system . more precisely we show that where and   let us observe that is the sum of two non - interacting harmonic oscillators with frequencies and , hence its dynamics is explicitly given by   by definition is the solution at time with initial condition , hence we obtain that is   from  and the definition  we easily get this means that once applied to a vector only the second and fourth components of will be non - zero and moreover they only depend on the first and third components of , hence .we can thus conclude that and , we get . finally using the definition ( [ exponential ] ) we obtain us recall that the composition of maps expressed by exponential defines the warped addition whose first terms are   using the warped addition with equation of theorem [ thm : control ] , we obtain and thus \oplus\{{\mathcal}{g}v\}+o(v^2)\nonumber\\ & = & \frac{1}{2}\{\{v\}{\mathcal}{g}v\}+\frac{1}{4}\underbrace{\{\{\{v\}{\mathcal}{g}v\}{\mathcal}{g}v\}}_{=o(v^2)}+o(v^2)=\frac{1}{2}\{\{v\}{\mathcal}{g}v\}+o(v^2)\ , , \end{aligned}\ ] ] where we explicitly used the assumption to remove the third term on the the right hand side on the first equation and hence to write .we are thus able to define the non - resonant control term , up to order , to be    to get the explicit formula for we need to compute the expression of .> from definition  the operator should satisfy to construct it , it will be more convenient to use complex variables then the function becomes and using for , the operator becomes with a similar expression holding for .hence for any and we obtain where we introduced the vector and we used the compact notation , for the complex vector .we note that from the knowledge of the operators action on such monomials , we can reconstruct the operator action on any regular function .the operators are linear and they will be applied on polynomials in the variable , which are nothing more than polynomials in the complex variables .let us now compute the time- flow of by using complex variables . starting from  and then proceeding by induction , we can easily prove that for all  and finally similarly .  assuming a non - resonance condition a possible choice for the operator is the following with it is easy to check that the operator defined by ( [ g ] ) verifies ( [ propg ] ) . in order to doso we introduce the compact notation developing the left hand side of  and using the linearity of all operators , we get    the function is defined by , where is a known function .the term will be computed starting from the previously obtained expression of . to construct we first have to express in the complex variables ( [ complexcoor ] ) :   by the linearity of the operator , and by using ( [ g ] ) ,we easily compute .in particular we apply to each term of ( [ eq:53 ] ) . then using the inverse change of coordinates ( similar expressions hold for and ) , we can go back to the original variables .the obtained expression after some algebraic simplifications is \left [ { x_1}^{2}-3\,{x_3}^{2}+{x_2}^{2}\right.\nonumber\\ & & \left.+ \left ( 2\,{x_1}^{2}-6\,{x_3}^{2}+{x_2}^{2 } \right ) \cos \left ( \omega_1 \right ) -x_1\,x_2\,\sin \left ( \omega_1 \right ) \right ] \nonumber\\ & & + 1/4\,{\frac { \sin \left ( \omega_2 \right ) } { \cos \left ( \omega_1-\omega_2 \right ) -\cos \left ( \omega_2 \right ) } } \nonumber\\ & & -1/4\,{\frac { \left ( -x_2\,{x_4}^{2}+x_2\,{x_3}^{2}+2\,x_1\,x_4\,x_3 \right ) \sin \left ( \omega_2 \right ) } { \cos \left ( \omega_2 \right ) -\cos \left ( \omega_1+\omega_2 \right ) } } \end{aligned}\ ] ]  then the explicit expression of the control term is simulations were made on the local computing resources ( cluster urbm - sysdyn ) at the university of namur ( fundp , belgium ) .

Abstract: we derive a hamiltonian control theory which can be applied to a 4d symplectic map that models a ring particle accelerator composed of elements with sextupole nonlinearity . the controlled system is designed to exhibit a more regular orbital behavior than the uncontrolled one . using the smaller alignement index ( sali ) chaos indicator , we are able to show that the controlled system has a dynamical aperture up to times larger than the original model .  hamiltonian system , chaos , control , accelerator mappings , dynamical aperture


ue to their remarkable agility and robustness , quadcopters have become an exciting area of research and a common sight among aircraft hobbyists , academic researchers and industries . in general, they have high thrust to weight ratio which allows them to carry significant payloads or have high lateral accelerations when not carrying a load .quadcopters are nowadays being used to study a plethora of concepts ranging from non - linear control[1 ] and learning[2 ] to vision - based pose estimation[3].when quadcopters are used for tasks like mapping , search and rescue , patrol , surveillance and product delivery , they re required to follow a predefined path at a particular height . in general , circular and straight line paths are used .path - following algorithms are used to ensure that the quadcopter will follow a predefined path in two or three dimensions at constant height .another important consideration is that the path - following algorithms are robust and accurate with regard to turbulences .a wide range of path - following algorithms have been proposed [ 4 ] .in this paper we use the non - linear guidance logic ( nlgl ) [ 4 ] coupled with model - predictive control [ 5 ] for trajectory tracking for the quadcopter .the main objective of this approach includes :  * the trajectories must be feasible under the dynamic and input constraints of the quadrotor * the generated trajectories should bring the vehicle to the target position on the path as quickly as possible , * the trajectory calculation must be fast enough to be used online , and * it must be possible to generate an implicit feedback control law by re - planning the trajectory intercept at each controller update , and applying the control inputs of the first section of it .a scheme is presented in [ 6 ] for generating state interception trajectories for quadcopters ; that is , trajectories starting from an arbitrary state and achieving a ( reduced ) end state in a specified amount of time , whilst satisfying input constraints .this approach builds on the previous work presented above in that we seek to develop that satisfies various constraints and is also fast enough to allow real - time planning .trajectory tracking is done by formulating the trajectory of the quadrocopter in its jerk , in discrete time , and then solving a convex optimization problem on each decoupled axis .the nlgl algorithm is now used to track the path effectively .the quadrocopter model is presented in section ii , with the trajectory tracking scheme given in section iii .section iv discusses simulation results , and an outlook is given in section v.the quadcopter is modeled as a rigid body with six degrees of freedom : linear translation along the inertial x1 , x2 and x3 axes , and three degrees of freedom describing the rotation of the frame attached to the body with respect to the inertial frame , which is taken here to be the proper orthogonal matrix r. the control inputs to the system are taken as the total thrust produced f , for simplicity normalized by the vehicle mass and thus having units of acceleration ; and the body rates expressed in the body - fixed frame as = ( 1 , 2 , 3 ) .these are illustrated in fig. 1 .figure [ fig:3d ]  the mixing of these inputs to individual motor thrust commands is done on board the vehicle , using feedback from gyroscopes .it is assumed that the time constant of the onboard controllers is low enough to have negligible influence on the algorithm presented here .because of their low rotational inertia , quadcopters can achieve extremely high rotational accelerations ( on the order of 200 rads2 [ 7 ] ) about the 1 and 2 axes , while it will be shown that the rotation about 3 is not needed for the trajectories considered here .the differential equations governing the flight of the quadcopter are now taken as those of a rigid body [ 8 ]    \ ] ]  with e3 = ( 0,0,1 ) and [ [ wx ] ] the skew symmetric matrix form of the vector cross product such that :   = \left| \begin{array}{ccc } - w_3 & -w_2 & 0 \\ w_3 & 0 & -w_1 \\ -w_2 & w_1 & 0 \end{array } \right|\ ] ]  and g = ( 0 , 0 , g ) the acceleration due to gravity . note the distinction between the vector g and scalar g.   we follow [ 6],[9 ] in considering the trajectories of the quadrocopter in terms of the jerk of the axes , allowing the system to be considered as a triple integrator in each axis and simplifying the trajectory generation task .it is assumed that a thrice differentiable trajectory x(t ) is available , where the jerk is written as :    the input thrust f is then found by applying the euclidean norm to ,     a quadrocopter trajectory described by ( 1 ) and ( 2 ) is considered to be feasible if the thrust and the magnitude of the body rates lie in some feasible set of values , defined as    note that for fixed - pitch propellers with a fixed direction of rotation , and , specifically , that the requirement on the thrust input is non - convex .  the following conservative box constraints are applied to yield convex constraints :        the resulting trajectories are guaranteed to be feasible with respect to the thrust limit if   three axes are decoupled and hence the optimization problem is solved separately for each axis .the trajectory tracking is rewritten as an optimal control problem , with boundary conditions defined by the quadrocopter s initial and ( desired ) final states . the cost function to minimizeis chosen as        a nonlinear guidance law based method described in [ 4 ] is used to determine the destination intercept along the given path . for simplicity ,assume the uav is following a path , as shown in figure 4 .at the current uav position p , draw a circle of radius l. the circle will intercept the path at two points q and ql . depending on the direction in which the uav has to move , either q or . will be selected .the advantage of this algorithm is that the same algorithm can be applied to any type of trajectory .the stability of the guidance law is shown using lyapunov stability arguments [ ] .once the target intercepts are calculated , future position , velocity and acceleration of the quadcopter are predicted using convex optimization and having constraints based on the quadcopter model . the discrete time formulation ( described in the next sub - section ) is used to predict up to 5 values if position , velocity and acceleration .the first value of the jerk calculated from the constraints is given as input .figure [ fig : vtp ]   the trajectory generation problem for each decoupled axis is rendered finite dimensional by discretizing the time with uniform steps of size t .each axis is then a discrete time linear , time invariant system in the state z , consisting of position , velocity and acceleration , with scalar jerk input , where the axis subscripts have been neglected for convenience . = \dddot x(k\delta t)\ ] ]   = [ x(t_1 ) x(t_2 ) . .x(t_n ) ] ^t\ ] ]   = \begin{bmatrix } \delta \frac{t^3}{6 } & 0 & 0 & 0 & 0\\  \delta \frac{t^3}{6 } & \delta \frac{t^3}{6 } & 0 & 0 & 0 \\  \delta \frac{t^3}{6 } & \delta \frac{t^3}{6 } & \delta \frac{t^3}{6 } & 0 & 0 \\  \delta \frac{t^3}{6 } & \delta \frac{t^3}{6 } & \delta \frac{t^3}{6 } & \delta \frac{t^3}{6 } & 0 \\  \delta \frac{t^3}{6 } & \delta \frac{t^3}{6 } & \delta \frac{t^3}{6 } & \delta \frac{t^3}{6 } & \delta \frac{t^3}{6 } \end{bmatrix } \times \left [ \begin{array}{c } u_1 \\ u_2 \\ u_3 \\ u_4 \\ u_5 \end{array } \right]\ ] ]   \times \ddot x(t_1 ) + \left[\begin{array}{c } \delta t\\ 2\delta t\\ 3\delta t\\4\delta t\\ 5\delta t\end{array } \right ] \times \dot x(t_1 ) + \left[\begin{array}{c } 1 \\ 1 \\ 1 \\ 1 \\ 1 \end{array } \right ] \times x(t_1)\ ] ]  the optimal control problem is to solve for u[k ] matrix in equation ( 17 ) subject to the above dynamics , satisfying the boundary conditions defined by the initial and final positions , respectively , velocities and accelerations .      the quadratic cost function ( 12 ) , ( 13 ) and ( 14 ) with the linear equality constraints ( 17 ) to ( 20 ) define a convex optimization problem.there exist efficient methods for solving problems of this sort , with cvxopt [ 11 ] , forces [ 12 ] and cvxgen [ 13 ] presenting techniques for creating c - code based solvers for specific instances of convex optimization problems . here , solvers are generated using the cvxopt software of [ 11 ] , which was able to generate solvers for large problems .cvxopt uses efficient interior point methods tailored to convex optimization problems , as are typical in model predictive control applications , and allows for high - speed implementation with good numerical stability properties .  the generated solvers either return a solution that solves the problem to within some acceptable residuals , or returns that no solution is found . in reality ,failure to find a solution can mean that :  * a solution exists , but the solver failed to find it due to reaching an internal limit ; * no solution exists to the conservatively constrained decoupled problem ; * no solution exists to the fully coupled nonlinearly constrained problem .the proposed algorithm was used to follow particular straight line paths , circles and sinusoidal curves defined by their respective mathematical equations .the results of the above have been shown in fig ( 3 ) ( 4 ) and ( 5 ) .the starting points were taken to be outside the path to follow , but close enough for the circle to find the vtp and start tracking the trajectory . in the plots shown in the figures , the time intervals between every measurement of velocity ,acceleration and position was taken to be 0.08s .the radius of the circle used in these plots is 50 units .the output of the proposed algorithm was also compared with the trajectory generation model proposed in [ 7 ] , where the cost function to be minimized calculates the sum of squares of the input jerk . in comparison, the proposed algorithm performs as well as the trajectory generation method proposed in [ 6 ] .the performance of the trajectory tracking is found by solving trajectories from rest , for step times varying from 0.08s to 2s .for each trajectory length , 1000 iterations were executed using python idle which took a mean time of 60ms per iteration .these were calculated on a pc running ubuntu 12.04 , with an intel core i3 - 330 m dual core processor ( 2.13 ghz ) with 4 gb ram .the solver was linked to an executable using python idle . in each casethe optimizations were set to maximize speed .this paper presents a method for trajectory tracking for quadcopters , from arbitrary initial conditions to a desired end state characterized by a position . the nlgl based destinationintercept works satisfactorily in tracking subsequent intercepts in the predefined path .the constrained trajectory tracking problem is posed as a convex optimization problem for each decoupled axis , which can be solved in real time .therefore this technique is suitable for use in feedback as a model predictive controller ; it also naturally handles cases where the desired end state evolves over time , such as when trying to hit an uncertain target or when it has to overcome an obstacle , where the target prediction evolves in real time .the convex optimization problem could be easily modified to achieve different goals , e.g. by removing the equality constraint on end velocity and acceleration , and adding the magnitude of the end velocity to the cost function , one solves a problem somewhat similar to that of [ 14 ] .this project work was supported by the robotics research centre at international institute of information technology , hyderabad ( iiit - h ) .1 .  i. d. cowling , o. a. yakimenko , j. f. whidborne , and a. k. cooke , `` a prototype of an autonomous controller for a quadrotor uav , '' in proceedings of the european control conference , kos , greece , 2007 , pp .s. lupashin , a. sch llig , m. sherback , and r. dandrea , `` a simple learning strategy for high - speed quadrocopter multi - flips , '' in ieee international conference on robotics and automation , 2010 , pp .3 .  s. weiss , m. achtelik , m. chli , and r. siegwart , `` versatile distributed pose estimation and sensor self - calibration for an autonomous mav , '' in ieee international conference on robotics and automation , 2012 , pp .4 .  p.b .sujit , srikanth saripalli , joao borges sousa , `` unmanned aerial vehicle path following - a survey and analysis of algorithms for fixed wing uav '' , ieee control systems magazine , feb 2014 , pp .42 - 59 5 .wang , l. `` model predictive control system design and implementation using matlab '' , springer 2009 6 .mark w. mueller and raffaello dandrea , `` a model predictive controller for quadrocopter state interception '' , 2013 european control conference ( ecc ) , july 17 - 19 , 2013 , zrich , switzerland 7 .m. hehn and r. dandrea , `` quadrocopter trajectory generation and control , '' in ifac world congress , vol .18 , no . 1 , 2011 , pp .p. h. zipfel , `` modeling and simulation of aerospace vehicle dynamics second edition '' .aiaa , 2007 .markus hehn , raffaello dandrea , `` quadrocopter trajectory generation and control '' 10 . s. park , j. deystt , and j. p. how , `` performance and lyapunov stability of a nonlinear path - following guidance method , '' j. guidance , control , dyn .6 , pp . 17181728 , 2007 .l. vandenberghe , `` the cvxopt linear and quadratic cone program solvers '' 12 .a. domahidi , a. zgraggen , m. zeilinger , m. morari , and c. jones , `` efficient interior point methods for multistage horizons arising in receding horizon control , '' in proceedings of the 51st ieee conference on decision and control , 2012 , pp . 668674 . 13 . j. mattingley and s. boyd , `` cvxgen : a code generator for embedded convex optimization , '' optimization and engineering , vol .1,pp . 127 , 2012 .m. hehn and r. d. andrea , `` real - time trajectory generation for in- terception maneuvers with quadrocopters , '' in ieee / rsj international conference on intelligent robots and systems , villamoura , portugal , 2012 , pp .

Abstract: this paper presents a method for path - following for quadcopter trajectories in real time . non - linear guidance logic is used to find the intercepts of the subsequent destination . trajectory tracking is implemented by formulating the trajectory of the quadcopter using its jerk , in discrete time , and then solving a convex optimization problem on each decoupled axis . based on the maximum possible thrust and angular rates of the quadcopter , feasibility constraints for the quadcopter have been derived . in this report we describe the design and implementation of explicit mpc controllers where the controllers were executed on a computer using sparse solvers to control the vehicle in hovering flight .


a set of agents is placed at nodes of an edge - weighted graph .an edge weight represents its length , i.e. , the distance between its endpoints along the edge .each agent has an amount of energy ( possibly distinct for different agents ) .agents walk in a continuous way along the network edges using amount of energy proportional to the distance travelled .an agent may stop at any point of a network edge ( i.e. at any distance from the edge endpoints , up to the edge weight ) .each agent has memory in which it can store information .when two agents meet , one of them can transfer a portion of currently possessed energy to another one .moreover , two meeting agents exchange their currently possessed information , so that after the meeting each of them keeps in its memory the union of pieces of information previously hold by each of them .we assume that each agent has sufficient memory to store the entire information initially belonging to all agents .our algorithms work as centralized schedulers having full knowledge of the instance. 0.4 cm we consider three problems :  1 .  *_ data delivery _ problem : * given two nodes of , is it possible to transfer the initial packet of information placed at node to node ?_ convergecast _ problem : * is it possible to transfer the initial information possessed by each agent to a fixed agent ?( see fig.1 ) 3 .* _ broadcast _ problem : * is it possible to transfer the initial information of some agent to all other agents ?( see fig.1 )   [ general ]  we will look for schedules of agent movements which will not only result in completing the desired task , but also attempt to maximize the unused energy .we call such schedules _ optimal_. we conservatively suppose that , whenever two agents meet , they automatically exchange the entire information they held .this information exchange procedure is never explicitly mentioned in our algorithms , supposing , by default , that it always takes place when a meeting occurs .we show that all three communication problems are polynomially solvable on trees and have linear time algorithms on the line . on the other hand for general undirected anddirected graphs we show that these problems , even if energy exchange is allowed , are still np - complete .recent development in the network industry triggered the research interest in mobile agents computing .several applications involve physical mobile devices like robots , motor vehicles or various wireless gadgets .mobile agents are sometimes interpreted as software agents , i.e. , programs migrating from host to host in a network , performing some specific tasks . examples of agents also include living beings : humans ( e.g. soldiers or disaster relief personnel ) or animals .most studied problems for mobile agents involve some sort of environment search or exploration ( cf . ) . in the case of a team of collaborating mobile agents, the challenge is to balance the workload among the agents in order to minimize the exploration time .however this task is often hard ( cf . ) , even in the case of two agents in a tree , .the task of is important when agents possess partial information about the network ( e.g. when individual agents hold measurements performed by sensors located at their positions ) and the aggregate information is needed to make some global decision based on all measurements . the problem is often considered as a version of the data aggregation question ( e.g. ) and it has been investigated mainly in the context of wireless and sensor networks , where the energy consumption is an important issue ( cf . ) .the task of is useful , e.g. , when a designated leader needs to share its information with collaborating agents in order to perform together some future tasks .the problem for stationary processors has been extensively studied both in the case of the message passing model , ( e.g. ) , and for the wireless model , ( see ) .the power awareness question has been studied in different contexts .energy management of ( not necessarily mobile ) computational devices has been studied in . to reduce energy consumption of computer systemsthe methods proposed include power - down strategies ( see ) or speed scaling ( cf .most of research on energy efficiency considers optimization of overall power used .when the power assignments are made by the individual system components , similar to our setting , the optimization problem involved has a flavor of load balancing ( cf . ) .the problem for stationary processors has been extensively studied both in the case of the message passing model , ( e.g. ) , and for the wireless model , ( see ) .the problem of communication by energy - constrained mobile agents has been investigated in .the agents of all have the same initial energy and they perform efficient convergecast and broadcast in line networks .however the same problem for tree networks is proven to be strongly -complete in .the closely related problem of data delivery , when the information has to be transmitted between two given network nodes by a set of energy constrained agents has been studied in .this problem is proven to be -complete in already for line networks , if the initial energy values may be distinct for different agents . however , in the setting studied in , the agents do not exchange energy . in the present paper we show that the situation is quite different if the agents are allowed to transfer energy between one another .in this section we suppose that we are given a collection of agents on the line .each agent is initially placed at position on the line and has initial energy .we investgate delivery , convergecast and broadcast problems separately .the are solved using auxiliary table.s   we start with the delivery problem from point to .assume w.l.o.g .that for and .the problem can be immediately reduced to the situation otherwise the first agent is going to from left to right , swallawing energy of encountered agents , symetrically the righmost agent is going right to left until reaching . then we can reduce and renumber agents setting .0.1 cm our first algorithm is only a decision version .its main purpose is to show how certain useful table can be computed , all other algorithms are based on computing similar type of tables . 0.4cm consider the partial delivery problem which is the original problem with agents larger than removed , together with their energy , and the goal is to deliver the packet from the 1-st agent to the -th agent .0.4 cm we say that the problem is _ solvable _ iff such a delivery is possible .0.4 cm we define the following table :  * if is not solvable then , where is the _ minimal _ energy which needs to be added to ( to energy of -th agent ) to make solvable .* if is solvable then is the _ maximal _ energy which can remain in point after delivering the packet from to .possibly since during delivery the partial energy of some other agents can be moved to point .0.1 cm assume in the algorithm that points and are the starting points and . in our algorithmthe statements of the form are equiavalnet to .0.2 cm  0.4 cm  [ example1 ] assume \;=\ ; [ 0,\ , 10,\ , 20,\ , 30,\ , 40,\ , 50] ] then . ] .there are three cases :  * case 1 * ( statement 4 ) .the instance is solvable and after moving the packet from to the maximal remaining energy is  ] .the remaining distance ( to ) should be covered both - ways from . hence looses ) ] additional energy which has to be delivered from .additionally the -th agent needs energy to cover the distance from to in both directions .consequently its energy is reduced by | ] and  ] equals the potential of energy at point for the delivery of initial information of agent to agent using only agents .more exactly , if < 0 ] are defined symmetrically .the following algorithm may be viewed as a different version solving the delivery problem .however the main reason of its presentation is to compute  ] in algorithm delivery2-on - the - line then agents can deliver the packet from agent at initial position to all agents .moreover ,  ] can be computed in linear time .we prove by induction that the following assertion is true at the completion of -th iteration of the `` for '' loop of the algorithm :    _ let be the largest integer , such that agents can deliver the packet from point , such that , to point and is the largest possible point having this property . then _  1 .either , , and = 2(s - a_i ) \geq 0 ] , where   the assertion is clearly true for as and .suppose the assertion was true after iteration and all previous iterations .consider first the case 1 , i.e. that agents could deliver to the packet from point and at the completion of the -th iteration and - a_{i-1}) ] , so the case 2 of the assertion is true .suppose now that at line 3 of the -th iteration .the agents could deliver the packet from to .observe that contains the accumulated energy of agents , i.e. therefore the condition implies that agents can deliver the packet from point to .hence agents can deliver from to .since and = 2(s - a_{i } ) \geq 0 ] .suppose first that at the -th iteration the condition at line 3 is true , i.e. . then , it means that the packet can not be delivered from to using agents and in line 3 we have , so the case 2 of the assertion is true .suppose now that at line 3 of the -th iteration .similarly to the case above , agents could deliver the packet from to and agents can deliver the packet from point to and the case 1 of the assertion becomes true .this completes the proof .0.2 cm  [ thm : broadcast ] algorithm broadcast - on - the - line identifies all agents of the collection , which are able to broadcast .note that /2 ] is the leftmost point from which the agents can pick up the packet and broadcast in the right direction so that the packet reaches agent .agents and can communicate during this broadcast off , i.e. when +\rdb[i+1 ] \geq 2(a_{i+1}-a_i) ] in broadcast - on - the - line equal the maximal amount of energy which may be suppressed from agent and the corresponding communication is still possible .we suppose that the agents are placed at the nodes of the undirected tree .observe that for each problem : the data delivery , convergecast and broadcast the tree may be truncated so that each leaf contains an initial position of an agent . indeed, in neither problem it makes sense to visit subtrees containing no energy source .we developed basic ideas and tables for the line , now they can be extended to trees .the tables for lines were computed locally , looking only at neighboring agents .simiarly for trees , the value of the corresponding tables for a node in a tree is computed looking at neighbors of this node .the delivery problem for a tree is easily reducible to the case of a line .we can solve delivery problem and construct delivery - scenario on the tree in linear time .we find the path connecting with in a given undirected tree .suppose we remove edges of the path .the tree splits into several subtrees _ anchored _ at vertices of . for each such subtreewe direct all edges towards the root , which is a vertex of .the leaves in these trees are sending their energies towards their roots accumulating energies of intermediate nodes . this way we reduce the delivery on the tree to the delivery on the line .this completes the proof .we can reduce the convergecast and broadcast problems on general trees to trees with degree at most 3 . if a vertex has neighbors we can change it locally to small binary internal subtree with leaves by adding several edges with zero cost .hence we assume from now that in our undirected tree each vertex has at most 3 neighbors .  though the input tree is undirected, we can consider direction of edges .for each undirected edge we consider two directed edges   define the subtree as the connected component containing and resulting from by removing the edge , see figure [ fig1 ] .  for each directed edge of the treewe define as the cost of moving all packets from the subtree to its root without interacting with any node outside .the table is a generalization of the table to trees .is reduced to computation of the costs and of moving all packets in the trees and ., width=283 ]  we can use similar algorithm as computing locally values on the line in the previous section , consequently we have the following fact .[ xx ] + * ( a ) * assume are neighbors of the vertex .then , knowing we can compute in constant time . + * ( b ) * if are known then we can compute in constant time all convergecast points on the edge as its subinterval .0.2 cm let us root the tree at some vertex obtaining a rooted version of .  for each node ,there are directed edges ( outgoing ) and ( ingoing edge ) .assume we are to mark all directed edges , but we have to obey the rule : if an edge is marked then all edges ingoing into from nodes other than should be marked .we can obey the rule and mark all edges by first traversing the tree in postorder marking for each visted node the edge .then we can traverse in preorder and for each visited node mark the edge .we can replace the operation of _ marking _ an edge by the computation of table . using the observation above our algorithm computing convergecast points on be written as the following pseudocode :  0.2 cm the next theorem follows now from lemma [ xx ] .the convergecast problem for undirected trees can be solved in linear time .we use terminology from the previous section and ideas from broadcasting on line . define $ ] as the potential of energy at the vertex for the delivery of initial information of agents placed initially at to all agents in subtree , using only agents in , assuming that at the end we have agents at .it is similar to the definition of for broadcasting on the line .hence each is a table . the use of quadratic tables results in nonlinear algorithm .the main innovation here is introducing many agents at the same node .for example suppose we have a tree with single branch of nodes , each with single agent , from to .the edges of this branch have zero cost .however has outgoing edges with large cost .then optimal broadcasting of such a tree from first gathers all agents at node , then each agent travels a different edge from .the broadcasting on a tree can be done in polynomial time .* sketch of the proof * we can show the following claim , analogous to lemma [ xx ] .[ xxx ] + * ( a ) * assume are neighbors of vertex . then we can compute in polynomial time , knowing tables . + * ( b ) * if tables are known for each neighbor of then we can check in polynomial time if is a valid source of broadcast .the tables for each edge are computed locally in a similar way as values of for edges , traversing first the rooted version of the tree in postorder , and then in preorder .each edge can be processed in polynomial time , due to the claim , hence the whole algorithm works in polynomial time .we omit the details .we use the following np - complete problem : 0.4 cm * integer set partition : * for a given set with integer weights check if can be partitioned into two disjoint subsets with equal total sums of weights .the delivery problem is np - complete for general directed graphs .consider the graph of the form shown in figure [ graph2 ] .the set of nodes consists of the set of middle nodes ( the set ) and three additional nodes .initially there are agents only in middle nodes .denote by the total energy of all agents .hence 0.4 cm energy of an element is set to be .there is no energy in other points .the length of en edge equals .the length of the edge is and the length of the edge is .the lengths of other edges are zero.   of each middle node equals agent s energy .delivery from to is possible only iff the set of weights can be partitoned into two sets of the same sum.,width=302 ]  assume are the sets of agents which move to and , respectively .let total energy coming to is and total energy coming to is delivery from to is possible if and only if it is equivalent to : consequently .in other words delivery from to is possible iff the integer partition problem is solvable .-completeness of the delivery problem follows from -completeness of the integer partition problem .  the delivery , convergecast and broadcast problems are np - complete for general undirected graphs .  .the weights of middle nodes and lengths of edges incident to middle nodes are increased by ., width=302 ]  we take an undirected version of the graph from the previous proof , see figure [ graph1 ]. additionally for each we increase by the energy of and edges .the agents are placed initially in each node of the graph.  * delivery . * due to the drastic increase of the lengths of edges in the delivery from to the edges from to other nodes can be used only once and in one direction ( from to outside part ) .* convergecast .* now we can take as the convergast node .the problem reduces to the delivery from to , since all energy is in and each node contains an agent .hence the problem is reduced to the delivery in the directed graph from the previous point . *broadcast . *the broadacast from reduces to the delivery from to , since in the delivery all agents from can move to or to . the information from ,if it has to arrive at , should also arrive at , where agents coming from to might get it .it is rather surprising that , without energy exchange , even the simplest problem of data delivery is np - complete in the simplest environment of the line , while , as we have shown in this paper , all considered communication problems with energy exchange are solvable in polynomial time even for tree networks . on the other handit is not surprising that energy exchange in general graphs does not help and the problems are np - complete .r. bar - yehuda , o. goldreich , a. itai : on the time - complexity of broadcast in multi - hop radio networks : an exponential gap between determinism and randomization . j. comput . syst .45(1 ) : 104 - 126 ( 1992 ) .

Abstract: we consider communication problems in the setting of mobile agents deployed in an edge - weighted network . the assumption of the paper is that each agent has some energy that it can transfer to any other agent when they meet ( together with the information it holds ) . the paper deals with three communication problems : data delivery , convergecast and broadcast . these problems are posed for a centralized scheduler which has full knowledge of the instance . it is already known that , without energy exchange , all three problems are np - complete even if the network is a line . surprisingly , if we allow the agents to exchange energy , we show that all three problems are polynomially solvable on trees and have linear time algorithms on the line . on the other hand for general undirected and directed graphs we show that these problems , even if energy exchange is allowed , are still np - complete .