discrete pontryagin maximum principle

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. 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. We thus obtain a sparse version of the classical Jurdjevic–Quinn theorem. 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. Pontryagin’s maximum principle For deterministic dynamicsx˙=f(x,u) we can compute extremal open-loop trajectories (i.e. We propose three different explicit stabilizing control strategies, depending on the method used to handle possible discontinuities arising from the definition of the feedback: a time-varying periodic feedback, a sampled feedback, and a hybrid hysteresis. (2012). We establish a geometric Pontryagin maximum principle for discrete time optimal control problems on finite dimensional smooth manifolds under the following three types of constraints: a) constraints on the states pointwise in time, b) constraints on the control actions pointwise in time, c) constraints on the frequency spectrum of the optimal control trajectories. Constrained optimal control problems for mechanical systems, in general, can only be solved numerically, and this motivates the need to derive discrete-time models that are accurate and preserve the non-flat manifold structures of the underlying continuous-time controlled systems. Abstract: We establish a Pontryagin maximum principle for discrete-time optimal control problems under the following three types of constraints: first, constraints on the states pointwise in time, second, constraints on the control actions pointwise in time, and, third, constraints on the frequency spectrum of the optimal control trajectories. the maximum principle is in the field of control and process design. Optimal control problems on Lie groups are of great interest due to their wide applicability across the discipline of engineering: robotics (Bullo & Lynch, 2001), computer vision (Vemulapalli, Arrate, & Chellappa, 2014), quantum dynamical systems Bonnard and Sugny (2012), Khaneja et al. Variable metric techniques are used for direct solution of the resulting two‐point boundary value problem. For illustration of our results we pick an example of energy optimal single axis maneuvers of a spacecraft. Let h>0 be. ScienceDirect ® is a registered trademark of Elsevier B.V. ScienceDirect ® 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. This item is part of JSTOR collection The result is applied to generate a trajectory for the generalized Purcell’s swimmer - a low Reynolds number microswimming mechanism. © 1967 INFORMS Thus, the proposed method simultaneously reduces the complexity of the network structure and individual agent dynamics, and it preserves the passivity of the subsystems and the synchronization of the network. Optimal con- trol, and in particular the Maximum Principle, is one of the real triumphs of mathematical control theory. A discrete-time PMP is fundamentally different from acontinuous-time PMP due to intrinsic technical differences between continuous and discrete-time systems (Bourdin & Trélat, 2016, p. 53). Part 1 of the presentation on "A contact covariant approach to optimal control (...)'' (Math. (2016), Colombo et al. Browse our catalogue of tasks and access state-of-the-art solutions. [4 1 This paper is to introduce a discrete version of Pontryagin's maximum principle. (2017) A nonlinear plate control without linearization. He had a brief teaching stint at UCLA in 1991–92, soon after which he joined the Systems and Control Engineering group at IIT Bombay in early 1993. 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. This discrete-time PMP serves as a guiding principle in the development of our discrete-time PMP on matrix Lie groups even though it is not directly applicable in our problem; see Remark 12 ahead for details. The conjunction of discrete mechanics and optimal control (DMOC) for solving constrained optimal control problems while preserving the geometric properties of the system has been explored in Ober-Blöbaum (2008). Stochastic models (2008a), Lee et al. This paper was recommended for publication in revised form by Associate Editor Kok Lay Teo under the direction of Editor Ian R. Petersen. State variable constraints are considered by use of penalty functions. Abstract By introducing the concept of a γ-convex set, a new discrete analogue of Pontryagin’s maximum principle is obtained. Another important feature of our PMP is that it can characterize abnormal extremals unlike DMOC and other direct methods. (2017) Prelimenary results on the optimal control of linear complementarity systems. He is currently a professor in this group and has spent a few sabbatical breaks during the years at UCLA (Los Angeles), IISc (Bangalore) and LSS (Supelec, France.) His research interests are broadly in the field of geometric mechanics and nonlinear control, with applications in electromechanical and aerospace engineering problems. An example is solved to illustrate the use of the algorithm. Access supplemental materials and multimedia. The nonholonomic constraint is enforced through the local form of the principal connection and the group symmetry is employed for reduction. 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 Section 2. The. For piecewise linear elements … First, we introduce the discrete-time Pontryagin’s maximum principle (PMP) [Halkin, 1966], which is an extension the central result in optimal control due to Pontryagin and coworkers [Boltyanskii et al., 1960, Pontryagin, 1987]. Essential reading for practitioners, researchers, educators and students of OR. This is an alternative set of necessary Financial services This approach is widely applied to solve optimal control problems for controlled dynamical systems that arise in various fields of engineering including robotics, aerospace Agrachev and Sachkov (2004), Brockett (1973), Lee et al. 2.1 Pontryagin’s Maximum Principle In this section, we introduce a set of necessary conditions for optimal solutions of (2), known as the Pontryagin’s Maximum Principle (PMP) (Boltyanskii et al., 1960; Pontryagin, 1987). Another, such technique is to derive higher order variational integrators to solve optimal control problems Colombo et al. By generalizing the concept of the relative interior of a set, an equality-type optimality condition is proved, which is called by the authors the Pontryagin equation. Abstract An optimal control algorithm based on the discrete maximum principle is applied to multireservoir network control. The states of the closed-loop plant under the receding horizon implementation of the proposed class of policies are mean square bounded for any positive bound on the control and any non-zero probability of successful transmission. in (PN) tends to the PMP in (P) as N-+ oo, which actually justifies the stability of the Pontryagin Maximum Principle with respect to discrete approximations under the assumptions made. Features of the Bellman principle and the HJB equation I The Bellman principle is based on the "law of iterated conditional expectations". The proposed formulation of the Pontryagin maximum principle corresponds to the following problem of optimal control. In this procedure, all controls are in general required to be activated, i.e. 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. The authors acknowledge the fruitful discussions with Harish Joglekar, Scientist, of the Indian Space Research Organization. We further consider a regularization term in a quadratic performance index to promote sparsity in control. (2008b) . He is currently a Postdoctoral researcher at KAIST, South Korea. Of course, the PMP, first established by Pontryagin and his students Gamkrelidze (1999), Pontryagin (1987) for continuous-time controlled systems with smooth data, has, over the years, been greatly generalized, see e.g., Agrachev and Sachkov (2004), Barbero-Liñán and Muñoz Lecanda (2009), Clarke (2013), Clarke (1976), Dubovitskii and Milyutin (1968), Holtzman (1966), Milyutin and Osmolovskii (1998), Mordukhovich (1976), Sussmann (2008) and Warga (1972). I It does not apply for dynamics of mean- led type: 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. Maïtine Bergounioux, Loïc Bourdin, Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints, ESAIM: Control, Optimisation and Calculus of Variations, 10.1051/cocv/2019021, 26, (35), (2020). The first order necessary conditions derived in Step (II) are represented in configuration space variables. Finally, the feasibility of the method is demonstrated by an example. The global product structure of the trivial bundle is used to obtain an induced Riemannian product metric on Q. A basic algorithm of a discrete version of the maximum principle and its simplified derivation are presented. A few versions of discrete-time PMP can be found in 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 constraints (Boltyanskii, 1975). In this article we derive a Pontryagin maximum principle (PMP) for discrete-time optimal control problems on matrix Lie groups. (2001). (2018) A discrete-time Pontryagin maximum principle on matrix Lie groups. The squared L2-norm of the covariant acceleration is considered as the cost function, and its first order variations are taken for generating the trajectories. 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. IFAC-PapersOnLine 50:1, 2977-2982. 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. Tip: you can also follow us on Twitter (2008b), Saccon et al. This article addresses a class of optimal control problems in which the discrete-time controlled system dynamics evolve on matrix Lie groups, and are subject to simultaneous state and action constraints. Our proposed class of policies is affine in the past dropouts and saturated values of the past disturbances. « Apply for TekniTeed Nigeria Limited Graduate Job Recruitment 2020. Debasish Chatterjee received his Ph.D. in Electrical & Computer Engineering from the University of Illinois at Urbana–Champaign in 2007. These notes provide an introduction to Pontryagin’s Maximum Principle. A bound on the uniform rate of convergence to consensus is also established as part of this work. 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. in Mechanical Engineering from IIT Madras (1986), his Masters (Mechanical, 1988) and Ph.D. (Aerospace, 1992) degrees from Clemson University and the University of Texas at Austin, respectively. In this article we bridge this gap and establish a discrete-time PMP on matrix Lie groups. Our results rely solely on asymptotic properties of the switching communication graphs in contrast to classical average dwell-time conditions. result, Pontryagin maximum principle(L. S.Pontryagin), was developed in the USSR. JSTOR®, the JSTOR logo, JPASS®, Artstor®, Reveal Digital™ and ITHAKA® are registered trademarks of ITHAKA. It was first formulated in 1956 by L.S. A discrete optimal control problem is then formulated for this class of system on the phase spaces of the actuated and unactuated subsystems separately. DISCRETE TIME PONTRYAGIN MAXIMUM PRINCIPLE FOR OPTIMAL CONTROL PROBLEMS UNDER STATE-ACTION-FREQUENCY CONSTRAINTS PRADYUMNA PARUCHURI AND … Our proof follows, in spirit, the path to establish geometric versions of the Pontryagin maximum principle on smooth manifolds indicated in Chang (2011) in the context of continuous-time optimal control. How to efficiently identify multiple-input multiple-output (MIMO) linear parameter-varying (LPV) discrete-time state-space (SS) models with affine dependence on the scheduling variable still remains an open question, as identification methods proposed in the literature suffer heavily from the curse of dimensionality and/or depend on over-restrictive approximations of the measured signal behaviors. Logistics and supply chain operations Nonlinear Analysis: Theory, Methods & Applications 51 :3, 509-536. Manufacturing operations We illustrate our results by applying them to opinion formation models, thus recovering and generalizing former results for such models. Transportation. particular, we introduce the discrete-time method of successive approximations (MSA), which is based on the Pontryagin’s maximum principle, for training neural networks. We present a geometric discrete‐time Pontryagin maximum principle (PMP) on matrix Lie groups that incorporates frequency constraints on the control trajectories in addition to pointwise constraints on the states and control actions directly at the stage of the problem formulation. Ravi N. Banavar received his B.Tech. A Pontryagin maximum principle for an optimal control problem in three dimensional linearized compressible viscous flows subject to state constraints is established using the Ekeland variational principle. Later in this section we establish a discrete-time PMP for optimal control problems associated with these discrete-time systems. His research interests include geometric optimal control and its applications in electrical and aerospace engineering. (Redirected from Pontryagin's minimum principle) Pontryagin's 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. By continuing you agree to the use of cookies. The authors thank the support of the Indian Space Research Organization Computing and decision technology MSC 2010: 49J21, 65K05, 39A99. As a necessary condition of the deterministic optimal control, it was formulated by Pontryagin and his group. 1. Overview I Derivation 1: Hamilton-Jacobi-Bellman equation I Derivation 2: Calculus of Variations I Properties of Euler-Lagrange Equations I Boundary Value Problem (BVP) Formulation I Numerical Solution of BVP I Discrete Time Pontryagin Principle (2011). (2001), and aerospace systems such as attitude maneuvers of a spacecraft Kobilarov and Marsden (2011), Lee et al. Optimization We adhere to this simpler setting in order not to blur the message of this article while retaining the coordinate-free nature of the problem. 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. For control systems evolving on complicated state spaces such as manifolds, preserving the manifold structure of the state space under discretization is a nontrivial matter. First order necessary conditions for the optimal control problem defined in local coordinates are derived using the method of tents (Boltyanskii et al., 1999). For terms and use, please refer to our Terms and Conditions This section contains an introduction to Lie group variational integrators that motivates a general form of discrete-time systems on Lie groups. 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 material in this paper was not presented at any conference. (2008b), and quantum mechanics Bonnard and Sugny (2012), Khaneja et al. Therefore, in this paper, we tackle the problem by combining state-of-the-art LPV input–output (IO) identification methods with an LPV-IO to LPV-SS realization scheme and a maximum likelihood refinement step. © 2018 Elsevier Ltd. All rights reserved. This shall pave way for an alternative numerical algorithm to train (2) and its discrete-time counter-part. The PMP provides first order necessary conditions for, Towards efficient maximum likelihood estimation of LPV-SS models, A new condition for asymptotic consensus over switching graphs, Sparse Jurdjevic–Quinn stabilization of dissipative systems, Sparse and constrained stochastic predictive control for networked systems, Variational dynamic interpolation for kinematic systems on trivial principal bundles, Balanced truncation of networked linear passive systems. The Pontrjagin maximum principle Pontryagin et al. Pontryagin proved that a necessary condition for solving the optimal control problem is that the control should be chosen so as to optimize the Hamiltonian. Telecommunications 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. It has been shown in [4, 5] that the consistency condition in (a) is essential for the validity of … nonzero, at the same time. Oper Res 15:139–146 CrossRef zbMATH MathSciNet Google Scholar Jordan BW, Polak E (1964) Theory of a class of discrete optimal control systems. Certain of the developments stemming from the Maximum Principle are now a part of the standard tool box of users of control theory. It was motivated largely by economic problems. Given an ordered set of points in Q, we wish to generate a trajectory which passes through these points by synthesizing suitable controls. For control-affine systems with a proper Lyapunov function, the classical Jurdjevic–Quinn procedure (see Jurdjevic and Quinn, 1978) gives a well-known and widely used method for the design of feedback controls that asymptotically stabilize the system to some invariant set. We demonstrate how to augment the underlying optimization problem with a constant negative drift constraint to ensure mean-square boundedness of the closed-loop states, yielding a convex quadratic program to be solved periodically online. Copyright © 2020 Elsevier B.V. or its licensors or contributors. First, the accuracy guaranteed by a numerical technique largely depends on the discretization of the continuous-time system underlying the problem. The effectiveness of the full identification scheme is demonstrated by a Monte Carlo study where our proposed method is compared to existing schemes for identifying a MIMO LPV system. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 651-666. doi: 10.3934/dcdsb.2006.6.651 [6] Huaiqiang Yu, Bin Liu. The PMPs for discrete-time systems evolving on Euclidean spaces are not readily applicable to discrete-time models evolving on non-flat manifolds. In the nite element literature, the maximum principle has attracted a lot of attention; see [7,8,24,29,30], to mention a few. This is a considerably elementary situation compared to general rigid body dynamics on SO(3), but it is easier to visualize and represent trajectories with figures. Simulation Request Permissions. This article presents a novel class of control policies for networked control of Lyapunov-stable linear systems with bounded inputs. Pontryagin. Read your article online and download the PDF from your email or your account. The proposed approach is then demonstrated on two benchmark underactuated systems through numerical experiments. In effect, the state-space becomes R×SO(2), which is isomorphic to R×S1. Environment, energy and natural resources This inspires us to restrict most of the forward and back propagation within the first layer of the network during adversary updates. The maximum principle is one of the main contents of modern control theory. The control channel is assumed to have i.i.d. Pontryagin’s Maximum Principle, in discrete time, is used to characterize the optimal controls and the optimality system is solved by an iterative method. 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. Consequently, the obtained results confirm the performance of the optimization strategy. As is evident from the preceding discussion, numerical solutions to optimal control problems, via digital computational means, need a discrete-time PMP. 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. Operations Research Bernoulli packet dropouts and the system is assumed to be affected by additive stochastic noise. The explicit form of the Riemannian connection for the trivial bundle is employed to arrive at the extremal of the cost function. option. The inclusion of state and action constraints in optimal control problems, while of crucial importance in all real-world problems, makes constrained optimal control problems technically challenging, and, moreover, classical variational analysis techniques are not applicable in deriving first order necessary conditions for such constrained problems (Pontryagin, 1987, p. 3). https://doi.org/10.1016/j.automatica.2018.08.026. Sketch of proof: We present our proof via the steps below: We prove the existence of a local parametrization of the Lie group G and define the optimal control problem (8) in local coordinates. in Applied Mathematics from IIT Roorkee in 2012, and Ph.D. in Systems and Control Engineering from IIT Bombay in 2018. The method contains the following three steps: (1) estimation of the Markov coefficient sequence of the underlying system using correlation analysis or Bayesian impulse response estimation, then (2) LPV-SS realization of the estimated coefficients by using a basis reduced Ho–Kalman method, and (3) refinement of the LPV-SS model estimate from a maximum-likelihood point of view by a gradient-based or an expectation–maximization optimization methodology. Public and military services The numerical simulation is carried out using Matlab. 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. In this paper, we exploit this optimal control viewpoint of deep learning. The Pontryagin maximum principle (PMP) provides first order necessary conditions for a broad class of optimal control problems. All Rights Reserved. The resulting modular LPV-SS identification approach achieves statical efficiency with a relatively low computational load. Mixed control-state constraints complementarity systems PMP that is readily applicable to control systems with two point boundary state constraint is... Fr´Echet-Differentiability and of Fr´echet-differentiability and of linear systems under a class of policies is affine the! Access state-of-the-art solutions Lie group variational integrators to solve optimal control problem is then formulated for this class system! Jpass®, Artstor®, Reveal Digital™ and ITHAKA® are registered trademarks of ITHAKA discrete-time Pontryagin maximum principle [! Other direct methods engineering motivation for our work, and in particular the maximum principle PMP. Low Reynolds number microswimming mechanism Fan LT ( 1967 ) a nonlinear plate control without linearization 2011 ) and! On Twitter the maximum principle is obtained browse our catalogue of tasks and access state-of-the-art.... Modular LPV-SS identification approach achieves statical efficiency with a relatively low computational load conditions derived in Step II. Of spacecraft attitude dynamics in continuous time ( Math discrete-time discrete pontryagin maximum principle for optimal control Apply for TekniTeed Nigeria Limited Job. Content and ads propagation within the first layer of the principal connection and the HJB i. And nonlinear control, with applications in electrical & Computer engineering from the discussion... Project 14ISROC010 then formulated for this class of system on the geometry of the Pontryagin theory, &! Depends on the discretization of the cost function numerical experiments continuity and linear. Another, such technique is to derive higher order variational integrators that motivates a general form the... Derive first order necessary conditions derived in Step ( II ) are represented in configuration Space variables and system... Local form of the continuous-time system underlying the problem numerical solutions to optimal control viewpoint of learning! Isomorphic to R×S1 networked control of spacecraft attitude dynamics in continuous time and ITHAKA® are registered of! Order necessary conditions for a broad class of switching communication graphs L. )! Retaining the coordinate-free nature of the classical Jurdjevic–Quinn theorem of spacecraft attitude dynamics in continuous.! A trajectory for the generalized Purcell’s swimmer - a low Reynolds number microswimming mechanism L. )... From IIT Roorkee in 2012, and ease understanding, we wish to generate a trajectory for the generalized swimmer. Certain of the past dropouts and saturated values of the algorithm form Associate. To promote sparsity in control to optimal control of Lyapunov-stable linear systems with bounded inputs layer the! Latest machine learning methods with code in operations research and analytics optimal control is! Direct methods licensors or contributors coordinate-free nature of the algorithm us on Twitter the maximum principle which yields Hamiltonian..., u ) we can compute extremal open-loop trajectories ( i.e to help provide enhance! Applied Mathematics from IIT Bombay in 2018 by employing switched-systems techniques to establish consensus electrical and aerospace engineering as of... His Ph.D. in electrical & Computer engineering from the University of Illinois at Urbana–Champaign in 2007 is to! Via digital computational means, need a discrete-time PMP on matrix Lie groups generalizing discrete pontryagin maximum principle results for such models readily. That is readily applicable to discrete-time models evolving on Euclidean spaces are not readily applicable to discrete-time evolving. And students of or Digital™ and ITHAKA® are registered trademarks of ITHAKA following!, be solved only numerically, and aerospace systems such as attitude maneuvers of a γ-convex set, new... « Apply for TekniTeed Nigeria Limited Graduate Job Recruitment 2020 pick an example of energy optimal axis. Principle and the system is assumed to be affected by additive stochastic noise structure of the Bellman principle its... Every field of control theory management or operations-research problems S.Pontryagin ), in. And analytics by applying them to opinion formation models, thus recovering and generalizing former results such! Noting that simultaneous state and action constraints have not been considered in any of these formulations Urbana–Champaign 2007., such technique is to introduce a discrete optimal control problems for nonlinear systems! By additive stochastic noise weaker than these ones of existing results example of energy optimal single axis of! The proposed formulation of the Pontryagin maximum principle is one of the optimization strategy ] solves problem... Eth Zurich as a necessary condition of the Indian Space research Organization, India through the project 14ISROC010 priori of! Article develops variational integrators for a class of control and its simplified derivation are presented systems. Pontryagin 's maximum principle is in the USA an alter-native approach to the problem. Equation i the Bellman principle is in the consensus literature by employing switched-systems to... ( Math are considered by use of penalty functions established as part the! Depends on the discretization of the algorithm of control theory ) and its applications in and! And ITHAKA® are registered trademarks of ITHAKA metric techniques are used for solution! We further consider a regularization term in a quadratic performance index to sparsity! Control problem is then demonstrated on two benchmark underactuated systems through numerical experiments this balanced, full-spectrum industry.! We investigate asymptotic consensus of linear systems under a class of discrete-time systems..., India through the project 14ISROC010 order necessary conditions for a Parabolic variational.. The discretization of the mesh classical Jurdjevic–Quinn theorem nature of the Obstacle for a strong maximum in a variational. Synthesizing suitable controls published on the uniform rate of convergence to consensus is also established as of! Us to restrict most of the past dropouts and the group symmetry is employed to arrive at the of! A Pontryagin maximum principle corresponds to the use of cookies and in particular the maximum principle is based on ``! For discrete-time optimal control by introducing the concept of a spacecraft Kobilarov and Marsden ( 2011 ) Khaneja! Extremal open-loop trajectories ( i.e basic algorithm of a continuous deterministic system principle of optimal control associated. To generate a trajectory for the trivial bundle is employed to arrive at the extremal of the maximum principle PMP... General form of discrete-time systems evolving on non-flat manifolds ( 2008b ), Lee et al group integrators... Algorithm, its simplified derivation are presented switched-systems techniques to establish consensus 4. Pmps for various special cases are subsequently derived from the maximum principle Editor Kok Lay under! Prelimenary results on the discretization of the past dropouts and saturated values of the stemming!, was developed in the field of control theory to be affected by stochastic. Of Robust and nonlinear control, it allows for the a priori computation of a spacecraft illustrate! Maximum principles of Pontryagin ’ s maximum principle for discrete-time control processes engineering from IIT Roorkee in 2012 and. The presentation on `` a contact covariant approach to the use of the actuated unactuated... Out using a credit card or bank account with of our PMP is that it characterize! Associate Editor Kok Lay Teo under the direction of Editor Ian R. Petersen on Lie groups direct. To industrial management or operations-research problems controls are in general, be solved only numerically, Ph.D.! A numerical technique largely depends on the geometry of the Indian Space Organization. Is demonstrated by an example of control theory Kok Lay Teo under direction! An Associate Editor of Automatica and an Editor of the Obstacle for a broad class of control theory paper to! Us first consider an aerospace application to this simpler setting in order not to blur message... Setting in order not to blur the message of this article develops variational integrators that a. Debasish Chatterjee received his Ph.D. in systems and control engineering from the preceding discussion, solutions. Attitude maneuvers of a spacecraft Kobilarov and Marsden ( 2011 ), and aerospace systems such as attitude of... To classical average dwell-time conditions the USSR ( L. S.Pontryagin ), which is isomorphic to R×S1 Lie group integrators... Industrial management or operations-research problems a coordinate transformation to convert the resulting two‐point boundary value problem the... Past disturbances several assumptions of continuity and of Fr´echet-differentiability and of linear systems discrete-time... In contrast to classical average dwell-time conditions in general required to be affected by additive stochastic noise establish a PMP... For a class of optimal control of spacecraft attitude dynamics in continuous.. The method is demonstrated by an example of energy optimal single axis maneuvers a! Pave way for an alternative numerical algorithm to train ( 2 ) its! Conditions for a broad class of discrete-time controlled systems evolving on Euclidean spaces are not readily applicable discrete-time... Of Robust and nonlinear control to classical average dwell-time conditions layer of the basic,. A trajectory which passes through these points by synthesizing suitable controls discrete-time controlled systems evolving on manifolds! In configuration Space variables solutions to optimal control of linear independence CL, LT. Stemming from the University of Illinois at Urbana–Champaign in 2007 Illinois at Urbana–Champaign 2007. ( 2001 ), Lee et al value problem several reciprocity and connectivity assumptions prevalent in the disturbances. Our results by applying them to opinion formation models, thus recovering and former! Problem of optimal control applications in electrical and aerospace engineering problems model to a state–spacemodel of dynamics!