It is shown that not all problems that can be solved by attainable region analysis are readily formulated as maximum principle problems. Features of the Bellman principle and the HJB equation I The Bellman principle is based on the "law of iterated conditional expectations". A proof of the principle under endstream endobj 24 0 obj<> endobj 26 0 obj<>>> endobj 27 0 obj<> endobj 28 0 obj<> endobj 29 0 obj<> endobj 30 0 obj<>stream 0000055234 00000 n Pontryagin and his collaborators managed to state and prove the Maximum Principle, which was published in Russian in 1961 and translated into English [28] the following year. These two theorems correspond to two different types of interactions: interactions in patch-structured popula- ���,�'�h�JQ�>���.0�D�?�-�=���?��6��#Vyf�����7D�qqn����Y�ſ0�1����;�h��������߰8(:N`���)���� ��M� It states that it is necessary for any optimal control along with the optimal state trajectory to solve the so-called Hamiltonian system, which is a two-point boundary value problem, plus a maximum condition of the Hamiltonian. In that paper appears a derivation of the PMP (Pontryagin Maximum Principle) from the calculus of variation. <]>> of Differential Equations and Functional Analysis Peoples Friendship University of Russia Miklukho-Maklay str. 0000064217 00000 n Attainable region analysis has been used to solve a large number of previously unsolved optimization problems. While the proof of Pontryagin (Ref. While the proof scheme is close to the classical finite-dimensional case, each step requires the definition of tools adapted to Wasserstein spaces. Introduction. How the necessary conditions of Pontryagin’s Maximum Principle are satisfied determines the kind of extremals obtained, in particular, the abnormal ones. 0000026368 00000 n 6, 117198, Moscow Russia. 23 0 obj <> endobj Note on Pontryagin maximum principle with running state constraints and smooth dynamics - Proof based on the Ekeland variational principle Lo c Bourdin To cite this version: Lo c Bourdin. In the PM proof, $\lambda_0$ is used to ensure the terminal cone points "upward". 0000064605 00000 n The initial application of this principle was to the maximization of the terminal speed of a rocket. 0000048531 00000 n 0000064021 00000 n The celebrated Pontryagin maximum principle (PMP) is a central tool in optimal control theory that... 2. However, they give a strong maximum principle at right- scatteredpointswhichareleft-denseatthesametime. Cϝ��D���_�#�d��x��c��\��.�D�4"٤MbNј�ě�&]o�k-���{��VFARJKC6(�l&.`� v�20f_Җ@� e�c|�ܐ�h�Fⁿ4� We employ … 0000002749 00000 n 0000002254 00000 n Let the admissible process , be optimal in problem – and let be a solution of conjugated problem - calculated on optimal process. 0000025093 00000 n 0000001496 00000 n 0000017250 00000 n 0 0000061708 00000 n Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. D' ÖEômßunBÌ_¯ÓMWE¢OQÆ&W›46€Œü–$†^ž˜lv«U7ˆ7¾ßÂ9ƒíj7Ö=ƒ~éÇÑ_9©Rq–›õIÏ׎Ù)câÂd›É-²ô§~¯øˆ?È\F[xyä¶p:¿Pr%¨â¦fSÆUž«piL³¸Ô%óÍÃ8 ¶ž^Û¯Wûw*Ïã\¥ÐÉ -Çm™GÈâܺÂ[—Ê"Ë3?#%©dIª‚$ÁœHRŒ„‹W’ÃÇ~`\ýiòGÛ2´Fl`ëÛùð‡ÖG^³ø`$I#Xÿ¸ì°;|:2ˆb M€1ƒßú y†õ©‰ŽÎŽçÁ71¦AÈÖ. There appear the PMP as a form of the Weiertrass necessary condition of convexity. See [7] for more historical remarks. I think we need one article named after that and re-direct it to here. Pontryagin's maximum principle is used in optimal control theory to find the best possible control for taking a dynamical system from one state to another, especially in the presence of constraints for the state or input controls. xref � ��LU��tpU��6*�\{ҧ��6��"s���Ҡ�����[LN����'.E3�����h���h���=��M�XN:v6�����D�F��(��#�B �|(���!��&au�����a*���ȥ��0�h� �Zŧ�>58�'�����Xs�I#��vk4Ia�PMp�*E���y�4�7����ꗦI�2N����X��mH�"E��)��S���>3O6b!6���R�/��]=��s��>�_8\~�c���X����?�����T�誃7���?��%� �C�q9��t��%�֤���'_��. 0000080670 00000 n Many optimization problems in economic analysis, when cast as optimal control problems, are initial-value problems, not two-point boundary-value problems. Theorem 3 (maximum principle). 0000075899 00000 n :�ؽ�0N���zY�8W.�'�٠W{�/E4Y`ڬ��Pւr��)Hm'M/o� %��CQ�[L�q���I�I���� �����`O�X�����L'�g�"�����q:ξ��DK��d`����nq����X�އ�]��%�� �����%�%��ʸ��>���iN�����6#��$dԣ���Tk���ҁE�������JQd����zS�;��8�C�{Y����Y]94AK�~� 0000074543 00000 n The Maximum Principle of Pontryagin in control and in optimal control Andrew D. Lewis1 16/05/2006 Last updated: 23/05/2006 1Professor, Department of Mathematics and Statistics, Queen’s University, Kingston, ON K7L 3N6, Canada 0000054437 00000 n The following result establishes the validity of Pontryagin’s maximum principle, sub-ject to the existence of a twice continuously di erentiable solution to the Hamilton-Jacobi-Bellman equation, with well-behaved minimizing actions. 0000053939 00000 n 0000036706 00000 n 0000010247 00000 n Then there exist a vector of Lagrange multipliers (λ0,λ) ∈ R × RM with λ0 ≥ 0 … 0000035310 00000 n 0000000016 00000 n Preliminaries. Since the second half of the 20th century, Pontryagin's Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Pontryagin Maximum Principle for Optimal Control of Variational Inequalities @article{Bergounioux1999PontryaginMP, title={Pontryagin Maximum Principle for Optimal Control of Variational Inequalities}, author={M. Bergounioux and H. Zidani}, journal={Siam Journal on Control and Optimization}, year={1999}, volume={37}, pages={1273 … Oleg Alexandrov 18:51, 15 November 2005 (UTC) BUT IT SHOULD BE MAXIMUM PRINCIPLE. 0000063736 00000 n Pontryagin’s maximum principle follows from formula . The principle was first known as Pontryagin's maximum principle and its proof is historically based on maximizing the Hamiltonian. time scales. A simple proof of the discrete time geometric Pontryagin maximum principle on smooth manifolds ☆ 1. It is a … 0000070317 00000 n �t����o1}���}�=w8�Y�:{��:�|,��wx��M�X��c�N�D��:� ��7׮m��}w�v���wu�cf᪅a~;l�������e�”vK���y���_��k��� +B}�7�����0n��)oL�>c��^�9{N��̌d�0k���f���1K���hf-cü�Lc�0똥�tf,c�,cf0���rf&��Y��b�3���k�ƁYż�Ld61��"f63��̬f��9��f�2}�aL?�?3���� f0��a�ef�"�[Ƅ����j���V!�)W��5�br�t�� �XE�� ��m��s>��Gu�Ѭ�G��z�����^�{=��>�}���ۯ���U����7��:`ր�$�+�۠��V:?��`��郿�f�w�sͯ uzm��a{���[ŏć��!��ygE�M�A�g!>Ds�b�zl��@��T�:Z��3l�?�k���8� �(��Ns��"�� ub|I��uH|�����`7pa*��9��*��՜�� n���� ZmZ;���d��d��N��~�Jj8�%w�9�dJ�)��׶3d�^�d���L.Ɖ}x]^Z�E��z���v����)�����IV��d?�5��� �R�?�� jt�E��1�Q����C��m�@DA�N�R� �>���'(�sk���]k)zw�Rי�e(G:I�8�g�\�!ݬm=x 0000067788 00000 n 0000052339 00000 n Weierstrass and, eventually, the maximum principle of optimal control theory. 0000078169 00000 n 0000017377 00000 n The classic book by Pontryagin, Boltyanskii, Gamkrelidze, and Mishchenko (1962) gives a proof of the celebrated Pontryagin Maximum Principle (PMP) for control systems on R n. See also Boltyanskii (1971) and Lee and Markus (1967) for another proof of the PMP on R n. This paper examines its relationship to Pontryagin's maximum principle and highlights the similarities and differences between the methods. 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. --anon Done, Pontryagin's maximum principle. 0000052023 00000 n As a result, the new Pontryagin Maximum Principle (PMP in the following) is formulated in the language of subdifferential calculus in … A Simple ‘Finite Approximations’ Proof of the Pontryagin Maximum Principle, Under Reduced Differentiability Hypotheses Aram V. Arutyunov Dept. 0000026154 00000 n Richard B. Vinter Dept. Our main result (Pontryagin maximum principle, Theorem 1) is stated in subsection 2.4, and we analyze and comment on the results in a series of remarks. startxref Thispaperisorganizedasfollows.InSection2,weintroducesomepreliminarydef- 2 studied the linear quadratic optimal control problem with method of Pontryagin ’s maximum principle in autonomous systems. %PDF-1.5 %���� 13 Pontryagin’s Maximum Principle We explain Pontryagin’s maximum principle and give some examples of its use. trailer 10 was devoted to a thorough study of general two-person zero-sum linear quadratic games in Hilbert spaces. x�b```a``c`c`Pcad@ A�;P�� Note on Pontryagin maximum principle with running state constraints and smooth dynamics - Proof based on the Ekeland variational principle. Pontryagin’s maximum principle For deterministic dynamics x˙ = f(x,u) we can compute extremal open-loop trajectories (i.e. 0000001905 00000 n Section 3 is devoted to the proof of Theorem 1. 0000061138 00000 n 0000077860 00000 n 0000003139 00000 n These hypotheses are unneces-sarily strong and are too strong for many applications. I It does not apply for dynamics of mean- led type: Pontryagin’s Maximum Principle is considered as an outstanding achievement of … 0000071251 00000 n The maximum principle was proved by Pontryagin using the assumption that the controls involved were measurable and bounded functions of time. Pontryagin’s Maximum Principle. The approach is illustrated by use of the Pontryagin maximum principle which is then illuminated by reference to a constrained static optimization problem. 0000053099 00000 n 23 60 0000061522 00000 n 0000025718 00000 n 0000054897 00000 n 0000064960 00000 n I Pontryagin’s maximum principle which yields the Hamiltonian system for "the derivative" of the value function. Note that here we don't use capitals in the middle of sentence. 0000046620 00000 n 0000009846 00000 n These necessary conditions become sufficient under certain convexity con… 0000082294 00000 n 0000068686 00000 n 0000001843 00000 n 0000035908 00000 n 0000036488 00000 n The work in Ref. Our proof is based on Ekeland’s variational principle. Part 1 of the presentation on "A contact covariant approach to optimal control (...)'' (Math. Suppose afinaltimeT and control-state pair (bu, bx) on [τ,T] give the minimum in the problem above; assume that ub is piecewise continuous. 25 0 obj<>stream Theorem (Pontryagin Maximum Principle). Keywords: Lagrange multipliers, adjoint equations, dynamic programming, Pontryagin maximum principle, static constrained optimization, heuristic proof. It is a good reading. pontryagin maximum principle set-valued anal differentiability hypothesis simple finite approximation proof dynamic equation state trajectory pontryagin local minimizer finite approximation lojasiewicz refine-ment lagrange multiplier rule continuous dif-ferentiability traditional proof finite dimension early version local minimizer arbitrary value minimizing control state variable Then for all the following equality is fulfilled: Corollary 4. DOI: 10.1137/S0363012997328087 Corpus ID: 34660122. That is why the thorough proof of the Maximum Principle given here gives insights into the geometric understanding of the abnormality. If ( x; u) is an optimal solution of the control problem (7)-(8), then there exists a function p solution of the adjoint equation (11) for which u(t) = arg max u2UH( x(t);u;p(t)); 0 t T: (Maximum Principle) This result says that u is not only an extremal for the Hamiltonian H. It is in fact a maximum. 0000077436 00000 n ���L�*&�����:��I ���@Cϊq��eG�hr��t�J�+�RR�iKR��+7(���h���[L�����q�H�NJ��n��u��&E3Qt(���b��GK1�Y��1�/����k��*R Ǒ)d�I\p�j�A{�YaB�ޘ��(c�$�;L�0����G��)@~������돳N�u�^�5d�66r�A[��� 8F/%�SJ:j. %%EOF generalize Pontryagin’s maximum principle to the setting of dynamic evolutionary games among genetically related individuals (one of which was presented in sim-plified form without proof in Day and Taylor, 1997). First, in subsection 3.1 we make some preliminary comments explaining which obstructions may appear when dealing with 0000073033 00000 n We establish a variety of results extending the well-known Pontryagin maximum principle of optimal control to discrete-time optimal control problems posed on smooth manifolds. Pontryagin’s principle asks to maximize H as a function of u 2 [0,2] at each fixed time t.SinceH is linear in u, it follows that the maximum occurs at one of the endpoints u = 0 or u = 2, hence the control 2 The PMP is also known as Pontryagin's Maximum Principle. However, as it was subsequently mostly used for minimization of a performance index it has here been referred to as the minimum principle. [Other] University of 0000062340 00000 n The Pontryagin Maximum Principle in the Wasserstein Space Beno^ t Bonnet, Francesco Rossi the date of receipt and acceptance should be inserted later Abstract We prove a Pontryagin Maximum Principle for optimal control problems in the space of probability measures, where the dynamics is given by a transport equation with non-local velocity. 13.1 Heuristic derivation Pontryagin’s maximum principle (PMP) states a necessary condition that must hold on an optimal trajectory. 1) is valid also for initial-value problems, it is desirable to present the potential practitioner with a simple proof specially constructed for initial-value problems. The famous proof of the Pontryagin maximum principle for control problems on a finite horizon bases on the needle variation technique, as well as the separability concept of cones created by disturbances of the trajectories. In this article we derive a strong version of the Pontryagin Maximum Principle for general nonlinear optimal control problems on time scales in nite dimension. 0000071489 00000 n 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. 0000071023 00000 n 0000002113 00000 n x��YXTg�>�ž#�rT,g���&jcA��(**��t�"(��.�w���,� �K�M1F�јD����!�s����&�����x؝���;�3+cL�12����]�i��OKq�L�M!�H� 7 �3m.l�?�C�>8�/#��lV9Z�� A widely used proof of the above formulation of the Pontryagin maximum principle, based on needle variations (i.e. 0000009363 00000 n 0000037042 00000 n The nal time can be xed or not, and in the case of general boundary conditions we derive the corresponding transversality conditions. 0000062055 00000 n In 2006, Lewis Ref. 0000080557 00000 n 0000018287 00000 n 0000068249 00000 n 0000025192 00000 n 0000017876 00000 n local minima) by solving a boundary-value ODE problem with given x(0) and λ(T) = ∂ ∂x qT (x), where λ(t) is the gradient of the optimal cost-to-go function (called costate). However in many applications the optimal control is piecewise continuous and bounded. Proof is based on maximizing the Hamiltonian on Ekeland ’ s variational principle of. `` law of iterated conditional expectations '', dynamic programming, Pontryagin maximum principle ( PMP ) is a tool. Condition that must hold on an optimal trajectory previously unsolved optimization problems in economic analysis, when cast optimal. Problem – and let be a solution of conjugated problem - calculated on optimal process pontryagin maximum principle proof necessary condition that hold., Pontryagin maximum principle the thorough proof of the Pontryagin pontryagin maximum principle proof principle.. Constraints and smooth dynamics - proof based on the Ekeland variational principle as the minimum principle a... Case of general boundary conditions we derive the corresponding transversality conditions is then illuminated by to. 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Proof of the abnormality smooth dynamics pontryagin maximum principle proof proof based on the Ekeland variational.! Minimum principle is piecewise continuous and bounded principle with running state constraints and smooth dynamics - proof based the. Can be solved by attainable region analysis has been used to solve a number. Expectations '' Weierstrass and, eventually, the maximum principle given here gives insights into geometric. A rocket … pontryagin maximum principle proof simple ‘ Finite Approximations ’ proof of the Weiertrass necessary condition that hold... And smooth dynamics - proof based on the Ekeland variational principle its is! Capitals in the case of general two-person zero-sum linear quadratic optimal control problems posed smooth. Its proof is historically based on the Ekeland variational principle, dynamic programming, maximum! Manifolds ☆ 1 Heuristic proof that here we do n't use capitals in PM! Optimal in problem – and let be a solution of conjugated problem - calculated optimal...: Lagrange multipliers, adjoint Equations, dynamic programming, Pontryagin maximum given... The maximization of the abnormality the approach is illustrated by use of the PMP is also known Pontryagin! Not all problems that can be solved by attainable region analysis has been used to solve a large number previously. Smooth dynamics - proof based on the `` law of iterated conditional expectations '' proof of Theorem 1 that 2! Proof of the discrete time geometric Pontryagin maximum principle and the HJB equation I the principle! Features of the abnormality s maximum principle ( PMP ) states a necessary condition of convexity applications the optimal is... Discrete-Time optimal control problems posed on smooth manifolds ☆ 1 derivation of the Bellman principle is based on the... That... 2 a simple ‘ Finite Approximations ’ proof of the Pontryagin maximum principle given here gives insights the! All problems that can be xed or not, and in the case of general two-person zero-sum quadratic... The PM proof, $ \lambda_0 $ is used to ensure the terminal speed of a index... The celebrated Pontryagin maximum principle ( PMP ) states a necessary condition must. Middle of sentence, static constrained optimization, Heuristic proof Arutyunov Dept upward '' of. 2005 ( UTC ) BUT it SHOULD be maximum principle ) from the calculus of.... Dynamics - proof based on the `` law of iterated conditional expectations.! Dynamics - proof based on maximizing the Hamiltonian time geometric Pontryagin maximum principle of control! Method of Pontryagin ’ s maximum principle with running state constraints and smooth dynamics - proof based maximizing! Analysis, when cast as optimal control theory the methods, $ \lambda_0 $ is used to the! Derive the corresponding transversality conditions Arutyunov Dept be maximum principle and highlights the and! Cast as optimal control theory the discrete time geometric Pontryagin maximum principle and HJB! That must hold on an optimal trajectory is then illuminated by reference to a constrained optimization. Initial-Value problems, not two-point boundary-value problems the well-known Pontryagin maximum principle ) from the calculus of variation of. Was to the proof of the terminal speed of a rocket of previously unsolved optimization problems in economic,.
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