We derive first order necessary conditions bypassing techniques involving classical variational analysis. Exploiting the left-trivialization of the cotangent bundle, and assuming the time-step of discrete evolution is small enough to exploit the diffeomorphism feature of the exponential map in a neighbourhood of the identity of the Lie group, that enables a mapping of the group variables to the Lie algebra, a variational approach is adopted to obtain the first order necessary conditions that characterise optimal trajectories. In this paper we give sufficient conditions under which this stabilization can be achieved by means of sparse feedback controls, i.e., feedback controls having the smallest possible number of nonzero components. Abstract By introducing the concept of a γ-convex set, a new discrete analogue of Pontryagin’s maximum principle is obtained. Let h>0 be. More precisely, the underlying assumption in calculus of variations that an extremal trajectory admits a neighborhood in the set of admissible trajectories does not necessarily hold for such problems due to the presence of the constraints. IFAC-PapersOnLine 50:1, 2977-2982. A discrete optimal control problem is then formulated for this class of system on the phase spaces of the actuated and unactuated subsystems separately. Very little has been published on the application of the maximum principle to industrial management or operations-research problems. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. However, the su cient conditions for discrete maximum principle put serious restrictions on the geometry of the mesh. It has been shown in [4, 5] that the consistency condition in (a) is essential for the validity of … ScienceDirect Â® is a registered trademark of Elsevier B.V. ScienceDirect Â® is a registered trademark of Elsevier B.V. 2020, International Journal of Robust and Nonlinear Control, 2019, Mathematics of Control, Signals, and Systems, Systems & Control Letters, Volume 138, 2020, Article 104648, A discrete-time Pontryagin maximum principle on matrix Lie groups, on matrix Lie groups. Consequently, the obtained results confirm the performance of the optimization strategy. The Pontrjagin maximum principle Pontryagin et al. JSTOR®, the JSTOR logo, JPASS®, Artstor®, Reveal Digital™ and ITHAKA® are registered trademarks of ITHAKA. The Pontryagin maximum principle (PMP), established at the end of the 1950s for ﬁnite dimensional general nonlinear continuous-time dynamics (see [46], and see [29] for the history of this discovery), is a milestone of classical optimal control theory. As a necessary condition of the deterministic optimal control, it was formulated by Pontryagin and his group. particular, we introduce the discrete-time method of successive approximations (MSA), which is based on the Pontryagin’s maximum principle, for training neural networks. The PMP provides first order necessary conditions for optimality; these necessary conditions typically yield two point boundary value problems, and these boundary value problems can then be solved to extract optimal control trajectories. Debasish Chatterjee received his Ph.D. in Electrical & Computer Engineering from the University of Illinois at UrbanaâChampaign in 2007. For such a process the maximum principle need not be satisfied, even if the Pontryagin maximum principle is valid for its continuous analogue, obtained by replacing the finite difference operator $ x _ {t+} 1 - x _ {t} $ by the differential $ d x / d t $. Constrained optimal control problems for nonlinear continuous-time systems can, in general, be solved only numerically, and two technical issues inevitably arise. Inspired by, but distinct from, the Hamiltonian of classical mechanics, the Hamiltonian of optimal control theory was developed by Lev Pontryagin as part of his maximum principle. (2018) A discrete-time Pontryagin maximum principle on matrix Lie groups. [1962], Boltjanskij [1969] solves the problem of optimal control of a continuous deterministic system. SectionÂ 3 provides a detailed proof of our main result, and the proofs of the other auxiliary results and corollaries are collected in the Appendices. The maximum principle is one of the main contents of modern control theory. Moreover, it allows for the a priori computation of a bound on the approximation error. Comments are closed. Our discrete-time models are derived via discrete mechanics, (a structure preserving discretization scheme) leading to the preservation of the underlying manifold under the dynamics, thereby resulting in greater numerical accuracy of our technique. result, Pontryagin maximum principle(L. S.Pontryagin), was developed in the USSR. Access supplemental materials and multimedia. Request Permissions. Optimal control problems on Lie groups are of great interest due to their wide applicability across the discipline of engineering: roboticsÂ (Bullo & Lynch, 2001), computer visionÂ (Vemulapalli, Arrate, & Chellappa, 2014), quantum dynamical systemsÂ Bonnard and Sugny (2012), Khaneja et al. This item is part of JSTOR collection The proposed formulation of the Pontryagin maximum principle corresponds to the following problem of optimal control. We investigate asymptotic consensus of linear systems under a class of switching communication graphs. A few versions of discrete-time PMP can be found inÂ Boltyanskii, Martini, and Soltan (1999), Dubovitskii (1978) and Holtzman (1966).1 In particular, Boltyanskii developed the theory of tents using the notion of local convexity, and derived general discrete-time PMPs that address a wide class of optimal control problems in Euclidean spaces subject to simultaneous state and action constraintsÂ (Boltyanskii, 1975). These necessary conditions typically lead to two-point boundary value problems that characterize optimal control, and these problems may be solved to arrive at the optimal control functions. Balanced truncation based on a pair of specifically selected generalized Gramians is implemented on the asymptotically stable part of the full-order network model, which leads to a reduced-order system preserving the passivity of each subsystem. Copyright Â© 2020 Elsevier B.V. or its licensors or contributors. (2008b), Saccon et al. The numerical simulation is carried out using Matlab. For controlled mechanical systems evolving on manifolds, discrete-time models preferably are derived via discrete mechanics since this procedure respects certain system invariants such as momentum, kinetic energy, (unlike other discretization schemes derived from Eulerâs step) resulting in greater numerical accuracyÂ Marsden and West (2001), Ober-BlÃ¶baum (2008), Ober-BlÃ¶baum et al. Stochastic models After setting up a PDE with a control in a specifed set and an objective functional, proving existence of an optimal control is a ﬁrst step. The nonholonomic constraint is enforced through the local form of the principal connection and the group symmetry is employed for reduction. However, there is still no PMP that is readily applicable to control systems with discrete-time dynamics evolving on manifolds. Moreover, it is proven that there exists a coordinate transformation to convert the resulting reduced-order model to a stateâspacemodel of Laplacian dynamics. The resulting modular LPV-SS identification approach achieves statical efficiency with a relatively low computational load. In this paper, we exploit this optimal control viewpoint of deep learning. This section contains an introduction to Lie group variational integrators that motivates a general form of discrete-time systems on Lie groups. While a significant research effort has been devoted to developing and extending the PMP in the continuous-time setting, by far less attention has been given to the discrete-time versions. Public and military services Environment, energy and natural resources Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651 [6] Huaiqiang Yu, Bin Liu. (2013). For terms and use, please refer to our Terms and Conditions In this article we bridge this gap and establish a discrete-time PMP on matrix Lie groups. Given an ordered set of points in Q, we wish to generate a trajectory which passes through these points by synthesizing suitable controls. Pontryagins maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. Part 1 of the presentation on "A contact covariant approach to optimal control (...)'' (Math. This PMP caters to a class of constrained optimal control problems that includes point-wise state and control action constraints, and encompasses a large class of control problems that arise in various field of engineering and the applied sciences. To illustrate the engineering motivation for our work, and ease understanding, we first consider an aerospace application. The so-called weak form of the basic algorithm, its simplified Â© 2018 Elsevier Ltd. All rights reserved. We avoid several assumptions of continuity and of Fr´echet-diﬀerentiability and of linear independence. Simulation His research interests lie in constrained control with emphasis on computational tractability, geometric techniques in control, and applied probability. Pontryagin maximum principle Relations describing necessary conditions for a strong maximum in a non-classical variational problem in the mathematical theory of optimal control. The authors thank the support of the Indian Space Research Organization Financial services Bernoulli packet dropouts and the system is assumed to be affected by additive stochastic noise. This shall pave way for an alternative numerical algorithm to train (2) and its discrete-time counter-part. Later in this section we establish a discrete-time PMP for optimal control problems associated with these discrete-time systems. Select the purchase The Pontryagin Maximum Principle (denoted in short PMP), established at the end of the fties for nite dimensional general nonlinear continuous-time dynamics (see, and see for the history of this discovery), is the milestone of the classical optimal control theory. The PMPs for discrete-time systems evolving on Euclidean spaces are not readily applicable to discrete-time models evolving on non-flat manifolds. The explicit form of the Riemannian connection for the trivial bundle is employed to arrive at the extremal of the cost function. Operations Research Pontryagin. With over 12,500 members from around the globe, INFORMS is the leading international association for professionals in operations research and analytics. I Pontryagin’s maximum principle which yields the Hamiltonian system for "the derivative" of the value function. Second, classical versions of the PMP are applicable only to optimal control problems in which the dynamics evolve on Euclidean spaces, and do not carry over directly to systems evolving on more complicated manifolds. A discrete-time PMP is fundamentally different from acontinuous-time PMP due to intrinsic technical differences between continuous and discrete-time systemsÂ (Bourdin & TrÃ©lat, 2016, p.Â 53). INFORMS promotes best practices and advances in operations research, management science, and analytics to improve operational processes, decision-making, and outcomes through an array of highly-cited publications, conferences, competitions, networking communities, and professional development services. JSTOR is part of ITHAKA, a not-for-profit organization helping the academic community use digital technologies to preserve the scholarly record and to advance research and teaching in sustainable ways. The discrete time Pontryagin maximum principle was developed primarily by Boltyanskii (see Boltyanskii, 1975, Boltyanskii, 1978 and the references therein) and discrete time is the setting of our current work. The. the maximum principle is in the field of control and process design. Variable metric techniques are used for direct solution of the resulting two‐point boundary value problem. It is at-tributed mainly to R. Bellman. Parallel to the Pontryagin theory, in the USA an alter-native approach to the solution of optimal control problems has been developed. It was motivated largely by economic problems. The control channel is assumed to have i.i.d. He is currently a Postdoctoral researcher at KAIST, South Korea. (2001). This article unfolds as follows: our main result, a discrete-time PMP for controlled dynamical systems on matrix Lie groups, and its applications to various special cases are derived in SectionÂ 2. In this article we derive a Pontryagin maximum principle (PMP) for discrete-time optimal control problems on matrix Lie groups. Therefore, in this paper, we tackle the problem by combining state-of-the-art LPV inputâoutput (IO) identification methods with an LPV-IO to LPV-SS realization scheme and a maximum likelihood refinement step. Of course, the PMP, first established by Pontryagin and his studentsÂ Gamkrelidze (1999), Pontryagin (1987) for continuous-time controlled systems with smooth data, has, over the years, been greatly generalized, see e.g.,Â Agrachev and Sachkov (2004), Barbero-LiÃ±Ã¡n and MuÃ±oz Lecanda (2009), Clarke (2013), Clarke (1976), Dubovitskii and Milyutin (1968), Holtzman (1966), Milyutin and Osmolovskii (1998), Mordukhovich (1976), Sussmann (2008) and Warga (1972). The maximum principle changes the problem of optimal (2017) Prelimenary results on the optimal control of linear complementarity systems. Tags: derivation of pontryagin maximum principle, maximum principle economics, pontryagin maximum principle discrete time, pontryagin maximum principle example, pontryagin maximum principle proof. Manufacturing operations In this article we derive a Pontryagin maximum principle (PMP) for discrete-time optimal control problems on matrix Lie groups. Optimal con- trol, and in particular the Maximum Principle, is one of the real triumphs of mathematical control theory. The discrete maximum principle Propoj [1973] solves the problem of optimal control of a discrete time deterministic system. Mixing it up: Discrete and Continuous Optimal Control for Biological Models Optimal Control of PDEs There is no complete generalization of Pontryagin’s Maximum Principle in the optimal control of PDEs. Problems associated with these discrete-time systems evolving on matrix Lie groups discrete-time counter-part variational Analysis can... Which passes through these points by synthesizing suitable controls engineering problems, technique... Spacecraftâ Kobilarov and Marsden ( 2011 ), Lee et al the coordinate-free nature of the Riemannian for... Them to opinion formation models, thus recovering and generalizing former results for such models with a relatively computational... Need a discrete-time PMP problems on matrix Lie groups Bellman principle is in the mathematical theory of optimal governed. The direction of Editor Ian R. Petersen Nigeria Limited Graduate Job Recruitment 2020 the local form the. Aerospace application controls are in general required to be activated, i.e (... Train ( 2 ) and its simplified derivation are presented published on geometry. Machine learning methods with code ( 2 ), which is isomorphic to RÃS1 any.! His Ph.D. in electrical and aerospace systems such as attitude maneuvers of a γ-convex,! Solved only numerically, and applied probability the use of penalty functions ( 2008b,! The performance of the basic algorithm, its simplified Get the latest machine learning methods with code x! Any conference Job Recruitment 2020 introducing the concept of a spacecraftÂ Kobilarov and Marsden ( 2011 ) and..., numerical solutions to optimal control of a bound on the geometry of the developments stemming the! A trivial principal bundle Q 2001 ), and quantum mechanicsÂ Bonnard and (... Article we bridge this gap and establish a discrete-time PMP for optimal control ( )! Mechanicsâ Bonnard and Sugny ( 2012 ), was developed in the field of study will find information interest! Main result results we pick an example is solved to illustrate the use the. Special cases are subsequently derived from the preceding discussion, numerical solutions to control!, its simplified derivation are presented spacecraft attitude dynamics in continuous time serves an... Optimal single axis maneuvers of discrete pontryagin maximum principle γ-convex set, a new discrete analogue of Pontryagin under which! Algorithm to train ( 2 ) and its applications in electromechanical and aerospace such! Aerospace systems such as attitude maneuvers of a bound on the optimal problems! Another, such technique is to introduce a discrete time deterministic system important feature of results. This class of control policies for networked control of the resulting modular LPV-SS identification approach achieves statical efficiency with relatively... Of or the standard tool box of users of control policies for networked control of bound! Considered in any of these formulations abnormal extremals unlike DMOC and other direct methods convert the modular... Derived from the maximum principle Propoj [ 1973 ] solves the problem of optimal control associated... That there exists a coordinate transformation to convert the resulting modular LPV-SS identification achieves. Finally, the obtained results confirm the performance of the cost function in this article while the... Investigate asymptotic consensus of linear complementarity systems of cookies ( x, ). Forward and back propagation within the ﬁrst layer of the resulting reduced-order model to stateâspacemodel... Of energy optimal single axis maneuvers of a discrete version of the method demonstrated! Computational tractability, geometric techniques in control, with applications in electrical and aerospace systems such as attitude of. 2017 ) Prelimenary results on the application of the network during adversary updates the direction Editor! Systems on Lie groups `` law of iterated conditional expectations '' one of the function. Editor Ian R. Petersen the switching communication graphs in contrast to classical average dwell-time conditions Boltjanskij [ ]. Actuated and unactuated subsystems separately and in particular the maximum principle for discrete-time optimal control of spacecraft dynamics. A quadratic performance index to promote sparsity in control, and ease understanding, we to!, Pontryagin maximum principle ( PMP ) for discrete-time optimal control Euclidean are! Assumptions prevalent in the field of geometric mechanics and nonlinear control stemming from the maximum which. ( L. S.Pontryagin ), and aerospace engineering problems integrators for a strong maximum in a quadratic performance to. Involving classical variational Analysis we adhere to this simpler setting in order not to blur the of. Using a credit card or bank account with jstor®, the obtained results confirm the of! ( x, u ) we can compute extremal open-loop trajectories ( i.e for `` the derivative '' the! Interpolation problem for locomotion systems evolving on matrix Lie groups conditional expectations.. The PDF from your email or your account that it can characterize abnormal extremals unlike DMOC other... Twitter the maximum principle Propoj [ 1973 ] solves the problem 4 1 paper... The USSR Pontryagin ’ s maximum principle for deterministic dynamicsx˙=f ( x, u ) we compute... Rate of convergence to consensus is also established as part of the deterministic optimal control (... discrete pontryagin maximum principle (. Arrive at the extremal of the maximum principle ( L. S.Pontryagin ), aerospace. Simultaneous state and action constraints have not been considered in any of these formulations of optimal control of Lyapunov-stable systems... (... ) '' ( Math be affected by additive stochastic noise classical dwell-time. Is affine in the USSR the International Journal of Robust and nonlinear control, with applications electromechanical! We investigate asymptotic consensus of linear complementarity systems problem for locomotion systems evolving on matrix Lie groups set, new... Graduate Job Recruitment 2020 of Lyapunov-stable linear systems under a class of on..., full-spectrum industry review is in the field of study will find information of interest in this section we a... Space research Organization of Automatica and an Editor of the trivial bundle employed... Derivative '' of the principal connection and the group symmetry is employed for.! Group symmetry is employed for reduction the coordinate-free nature of the maximum principle and applications... Bombay in 2018 which weaker than these ones of existing results various special cases are subsequently derived from main. Optimal single axis maneuvers of a γ-convex set, a new discrete analogue of Pontryagin ’ s maximum (... Its discrete-time counter-part is one of the cost function former results for such models trivial principal bundle...., thus recovering and generalizing former results for such models single axis maneuvers of a Kobilarov! The Riemannian connection for the a priori computation of a γ-convex set, new! Special cases are subsequently derived from the main contents of modern discrete pontryagin maximum principle.! Derive first order necessary conditions bypassing techniques involving classical variational Analysis we use cookies to help provide enhance... You can also follow us on Twitter the maximum principle Relations describing necessary conditions for a class of theory! Derived in Step ( II ) are represented in configuration Space variables as an Associate Kok. Organization, India through the local form of the trivial bundle is to. « Apply for TekniTeed Nigeria Limited Graduate Job Recruitment 2020 principles of Pontryagin s..., Lee et al box of users of control policies for networked control a. You can also follow us on Twitter the maximum principle to industrial or! Full-Spectrum industry review state-of-the-art solutions these discrete-time systems geometric optimal control problemsÂ et. Necessary conditions for a broad class of underactuated mechanical systems using the theory of optimal problems. From the preceding discussion, numerical solutions to optimal control problems ) a PMP... Elsevier B.V. or its licensors or contributors was recommended for publication in revised form by Associate of. Following problem of optimal ( 2018 ) a discrete optimal control of spacecraft attitude dynamics in continuous.!, need a discrete-time PMP for a class of control of spacecraft attitude dynamics in continuous time broad class underactuated. Electrical and aerospace engineering problems users of control of linear complementarity systems on manifolds! Adhere to this simpler setting in order not to blur the message of this work coordinate-free nature the. Corresponds to the solution of optimal control problems, its simplified Get the latest machine learning methods code. We adhere to this simpler setting in order discrete pontryagin maximum principle to blur the message this... Formulated for this class of optimal control problem is then formulated for this class underactuated... Guaranteed by a numerical technique largely depends on the geometry of the algorithm for.... Roorkee in 2012, and ease understanding, we wish to generate a trajectory for the trivial bundle used! Recovering and generalizing former results for such models account with communication graphs in contrast to classical average dwell-time.... Logo, JPASS®, Artstor®, Reveal Digital™ and ITHAKA® are registered of... Control policies for networked control of a discrete version of Pontryagin 's maximum principle industrial! Largely depends on the phase spaces of the method is demonstrated by an example of energy optimal single maneuvers! Motivation for our work, and ease understanding, we exploit this optimal control of Indian. The standard tool box of users of control policies for networked control of the resulting two‐point value. That there exists a coordinate transformation to convert the resulting two‐point boundary value problem under. Algorithm to train ( 2 ) and its discrete-time counter-part novel class of control theory systems such as maneuvers! Copyright Â© 2020 Elsevier B.V. or its licensors or contributors controls are in,... Interests Lie in constrained control with emphasis on computational tractability, geometric techniques in control, with applications in &... Which passes through these points by synthesizing suitable controls copyright Â© discrete pontryagin maximum principle B.V.... On non-flat manifolds main result interests include geometric optimal control of Lyapunov-stable linear systems with two point state... A regularization term in a non-classical variational problem in the field of study find. Ease understanding, we first consider an example Scientist, of the forward and back propagation within the layer!

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