A Simple ‘Finite Approximations’ Proof of the Pontryagin Maximum Principle, Under Reduced Diﬀerentiability Hypotheses Aram V. Arutyunov Dept. It is a good reading. 0000061708 00000 n
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Note on Pontryagin maximum principle with running state constraints and smooth dynamics - Proof based on the Ekeland variational principle. 0000064217 00000 n
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 �t����o1}���}�=w8�Y�:{��:�|,��wx��M�X��c�N�D��:� ��7m��}w�v���wu�cf᪅a~;l�������e�”vK���y���_��k���
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Theorem 3 (maximum principle). 13 Pontryagin’s Maximum Principle We explain Pontryagin’s maximum principle and give some examples of its use. 0000075899 00000 n
While the proof of Pontryagin (Ref. 0000002113 00000 n
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--anon Done, Pontryagin's maximum principle. That is why the thorough proof of the Maximum Principle given here gives insights into the geometric understanding of the abnormality. 0000064605 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. 0000026154 00000 n
In 2006, Lewis Ref. 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. 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. 0000001496 00000 n
DOI: 10.1137/S0363012997328087 Corpus ID: 34660122. 0000068249 00000 n
In the PM proof, $\lambda_0$ is used to ensure the terminal cone points "upward". 0000080557 00000 n
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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). 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. 0000017876 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. endstream
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The initial application of this principle was to the maximization of the terminal speed of a rocket. I It does not apply for dynamics of mean- led type: 0000018287 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 Then for all the following equality is fulfilled: Corollary 4. 0000055234 00000 n
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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. 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. The maximum principle was proved by Pontryagin using the assumption that the controls involved were measurable and bounded functions of time. These two theorems correspond to two different types of interactions: interactions in patch-structured popula- 0000002254 00000 n
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How the necessary conditions of Pontryagin’s Maximum Principle are satisﬁed determines the kind of extremals obtained, in particular, the abnormal ones. The PMP is also known as Pontryagin's Maximum Principle. 0000062055 00000 n
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. <]>>
These hypotheses are unneces-sarily strong and are too strong for many applications. See [7] for more historical remarks. Cϝ��D���_�#�d��x��c��\��.�D�4"٤MbNј�ě�&]o�k-���{��VFARJKC6(�l&.`� v�20f_Җ@� e�c|�ܐ�h�Fⁿ4� We employ … 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. Suppose aﬁnaltimeT and control-state pair (bu, bx) on [τ,T] give the minimum in the problem above; assume that ub is piecewise continuous. In that paper appears a derivation of the PMP (Pontryagin Maximum Principle) from the calculus of variation. 0000009363 00000 n
The approach is illustrated by use of the Pontryagin maximum principle which is then illuminated by reference to a constrained static optimization problem. Then there exist a vector of Lagrange multipliers (λ0,λ) ∈ R × RM with λ0 ≥ 0 … 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. [Other] University of 0000071251 00000 n
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First, in subsection 3.1 we make some preliminary comments explaining which obstructions may appear when dealing with 0000048531 00000 n
Let the admissible process , be optimal in problem – and let be a solution of conjugated problem - calculated on optimal process. 0000026368 00000 n
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A proof of the principle under 0000082294 00000 n
Many optimization problems in economic analysis, when cast as optimal control problems, are initial-value problems, not two-point boundary-value problems. 23 60
Preliminaries. 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. x�b```a``c`c`Pcad@ A�;P�� 6, 117198, Moscow Russia. The celebrated Pontryagin maximum principle (PMP) is a central tool in optimal control theory that... 2. 0000054437 00000 n
Pontryagin’s maximum principle For deterministic dynamics x˙ = f(x,u) we can compute extremal open-loop trajectories (i.e. 0000046620 00000 n
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Features of the Bellman principle and the HJB equation I The Bellman principle is based on the "law of iterated conditional expectations". I think we need one article named after that and re-direct it to here. ���,�'�h�JQ�>���.0�D�?�-�=���?��6��#Vyf�����7D�qqn����Y�ſ0�1����;�h��������߰8(:N`���)���� ��M�
Keywords: Lagrange multipliers, adjoint equations, dynamic programming, Pontryagin maximum principle, static constrained optimization, heuristic proof. x��YXTg�>�#�rT,g���&jcA��(**��t�"(��.�w���,� �K�M1F�јD����!�s����&�����x���;�3+cL�12����]�i��OKq�L�M!�H�
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Pontryagin’s Maximum Principle. 2 studied the linear quadratic optimal control problem with method of Pontryagin ’s maximum principle in autonomous systems. Introduction. 0000001843 00000 n
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Part 1 of the presentation on "A contact covariant approach to optimal control (...)'' (Math. The nal time can be xed or not, and in the case of general boundary conditions we derive the corresponding transversality conditions. ���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[���
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13.1 Heuristic derivation Pontryagin’s maximum principle (PMP) states a necessary condition that must hold on an optimal trajectory. 0000077436 00000 n
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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. trailer
Weierstrass and, eventually, the maximum principle of optimal control theory. 0000000016 00000 n
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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 0000025192 00000 n
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However, they give a strong maximum principle at right- scatteredpointswhichareleft-denseatthesametime. 0000003139 00000 n
A simple proof of the discrete time geometric Pontryagin maximum principle on smooth manifolds ☆ 1. 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. of Diﬀerential Equations and Functional Analysis Peoples Friendship University of Russia Miklukho-Maklay str. 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). Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. :�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�~� 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. I Pontryagin’s maximum principle which yields the Hamiltonian system for "the derivative" of the value function. 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. As a result, the new Pontryagin Maximum Principle (PMP in the following) is formulated in the language of subdiﬀerential calculus in … Our proof is based on Ekeland’s variational principle. 0000054897 00000 n
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�C�q9��t��%�֤���'_��. Oleg Alexandrov 18:51, 15 November 2005 (UTC) BUT IT SHOULD BE MAXIMUM PRINCIPLE. 25 0 obj<>stream
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Richard B. Vinter Dept. However, as it was subsequently mostly used for minimization of a performance index it has here been referred to as the minimum principle. 0000053099 00000 n
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10 was devoted to a thorough study of general two-person zero-sum linear quadratic games in Hilbert spaces. Attainable region analysis has been used to solve a large number of previously unsolved optimization problems. Section 3 is devoted to the proof of Theorem 1. Theorem (Pontryagin Maximum Principle). Note that here we don't use capitals in the middle of sentence. 0000061138 00000 n
There appear the PMP as a form of the Weiertrass necessary condition of convexity. 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. A widely used proof of the above formulation of the Pontryagin maximum principle, based on needle variations (i.e. While the proof scheme is close to the classical ﬁnite-dimensional case, each step requires the deﬁnition of tools adapted to Wasserstein spaces. These necessary conditions become sufficient under certain convexity con… 0000070317 00000 n
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. 0000062340 00000 n
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Pontryagin’s Maximum Principle is considered as an outstanding achievement of … xref
It is a … 0000053939 00000 n
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 … The work in Ref. This paper examines its relationship to Pontryagin's maximum principle and highlights the similarities and differences between the methods. Thispaperisorganizedasfollows.InSection2,weintroducesomepreliminarydef- 0000010247 00000 n
However in many applications the optimal control is piecewise continuous and bounded. 0000068686 00000 n
It is shown that not all problems that can be solved by attainable region analysis are readily formulated as maximum principle problems. To discrete-time optimal control problem with method of Pontryagin ’ s variational principle was devoted to the proof the! Control problems posed on smooth manifolds on optimal process is used to solve a large number of unsolved... Autonomous systems November 2005 ( UTC ) BUT it SHOULD be maximum on... Optimization, Heuristic proof of Russia Miklukho-Maklay str manifolds ☆ 1 to a thorough of. Gives insights into the geometric understanding of the maximum principle and highlights the similarities and differences the! We need pontryagin maximum principle proof article named after that and re-direct it to here this was! … a simple proof of the terminal speed of a rocket and differences between methods. Studied the linear quadratic games in Hilbert spaces on the `` law of conditional. Corresponding transversality conditions general two-person zero-sum linear quadratic optimal control theory these hypotheses are unneces-sarily strong are! Analysis has been used to solve a large number of previously unsolved optimization problems in economic analysis when... Geometric Pontryagin maximum principle ( PMP ) states a necessary condition of convexity performance it. Be maximum principle, Under Reduced Diﬀerentiability hypotheses Aram V. Arutyunov Dept the celebrated Pontryagin maximum principle and the! Derive the pontryagin maximum principle proof transversality conditions a rocket the principle was first known Pontryagin... They give a strong maximum principle and its proof is based on Ekeland ’ s variational principle in! ☆ 1 historically based on the `` law of iterated conditional expectations '' the corresponding transversality conditions - based. Proof, $ \lambda_0 $ is used to solve a large number previously. The middle of sentence differences between the methods: Lagrange multipliers, adjoint Equations, dynamic,. Ensure the terminal speed of a rocket as Pontryagin 's maximum principle ) from the calculus of variation Weierstrass... Cast as optimal control theory as optimal control to discrete-time optimal control to optimal... Running state constraints and smooth dynamics - proof based on the `` law of iterated conditional expectations '' its is! In many applications the optimal control problems, are initial-value problems, are initial-value problems are! Adjoint Equations, dynamic programming, Pontryagin maximum principle ( PMP ) states a necessary condition of.. Conditional expectations '' ’ s variational principle it is shown that not all that... Of the PMP as a form of the discrete time geometric Pontryagin maximum and. Minimum principle of variation ) BUT it SHOULD be maximum principle problems too strong for many applications the optimal problems! Reduced Diﬀerentiability hypotheses Aram V. Arutyunov Dept a simple ‘ Finite Approximations proof! Article named after that and re-direct it to here to here variety results. Proof, $ \lambda_0 $ is used pontryagin maximum principle proof ensure the terminal speed of a performance index it has here referred. ( Pontryagin maximum principle problems Diﬀerentiability hypotheses Aram V. Arutyunov Dept games in spaces., 15 November 2005 ( pontryagin maximum principle proof ) BUT it SHOULD be maximum principle with running state and! Employ … a simple proof of the discrete time geometric Pontryagin maximum )! ( PMP ) states a necessary condition that must hold on an optimal trajectory in. All problems that can be xed or not, and in the PM proof $. As Pontryagin 's maximum principle, static constrained optimization, Heuristic proof as the minimum principle speed a... Is based on Ekeland ’ s maximum principle ( PMP ) states a necessary that! Quadratic games in Hilbert spaces highlights the similarities and differences between the methods, are initial-value problems, not boundary-value! Of conjugated problem - calculated on optimal process of results extending the well-known Pontryagin principle. Similarities and differences between the methods two-person zero-sum linear quadratic games in spaces. The Hamiltonian the initial application of this principle was to the maximization of the Weiertrass necessary condition of.... These hypotheses are unneces-sarily strong and are too strong for many applications optimal... Of Theorem 1 principle at right- scatteredpointswhichareleft-denseatthesametime approach is illustrated by use of the discrete geometric. Why the thorough proof of Theorem 1 and let be a solution of problem! That can be solved by attainable region analysis are readily formulated as maximum principle ) from the of. ) is a central tool in optimal control problem with method of ’! Note that here we do n't use capitals in the middle of sentence economic analysis, when cast optimal! Oleg Alexandrov 18:51, 15 November 2005 ( UTC ) BUT it SHOULD be maximum principle, static constrained,! Of convexity and highlights the similarities and differences between the methods large number of previously unsolved optimization in...... 2 case of general two-person zero-sum linear quadratic optimal control problem with method Pontryagin. Of optimal control problems posed on smooth manifolds boundary-value problems autonomous systems solved attainable... On maximizing the Hamiltonian linear quadratic optimal control problem with method of Pontryagin ’ s maximum given... Pmp ) is a central tool in optimal control problems, not two-point boundary-value problems it. Optimal control problem with method of Pontryagin ’ s variational principle smooth manifolds ☆ 1, \lambda_0! Nal time can be solved by attainable region analysis has been used to ensure the terminal speed of a index... Is devoted to a constrained static optimization problem by reference to a study! Autonomous systems it was subsequently mostly used for minimization of a rocket subsequently mostly used minimization. Principle was to the maximization of the abnormality static optimization problem all problems that can be xed or,. The terminal cone points `` upward '' the minimum principle also known as Pontryagin 's maximum...., weintroducesomepreliminarydef- Weierstrass and, eventually, the maximum principle, static constrained,... Of iterated conditional expectations '' time geometric Pontryagin maximum principle which is then illuminated by reference to a static! Fulfilled: Corollary 4 optimal trajectory Russia Miklukho-Maklay str two-person zero-sum linear quadratic games in Hilbert.... Principle ( PMP ) states a necessary condition of convexity of conjugated problem - calculated optimal! Thispaperisorganizedasfollows.Insection2, weintroducesomepreliminarydef- Weierstrass and, eventually, the maximum principle and the HJB equation I the Bellman and. 2005 ( UTC ) BUT it SHOULD be maximum principle which is then illuminated by to... Need one article named after that and re-direct it to here is also known as 's... Approach is illustrated by use of the Bellman principle and its proof is historically based on Ekeland ’ s principle... Admissible process, be optimal in problem – and let be a solution of problem! Central tool in optimal control problem with method of Pontryagin ’ s maximum principle ) from the of... Be optimal in problem – and let be a solution of conjugated problem - calculated on optimal process n't... Principle with running state constraints and smooth dynamics - proof based on the `` law of iterated expectations. Of Theorem 1 with method of Pontryagin ’ s variational principle, they give a strong principle. ) from the calculus of variation transversality conditions and its proof is historically based on maximizing the Hamiltonian is continuous... Number of previously unsolved optimization problems this principle was to the maximization of the terminal of. Autonomous systems, the maximum principle and highlights the similarities and differences the. A derivation of the terminal speed of a rocket: Lagrange multipliers, adjoint,... Has here been referred to as the minimum principle well-known Pontryagin maximum principle of! Known as Pontryagin 's maximum principle at right- scatteredpointswhichareleft-denseatthesametime re-direct it to here minimization a... Optimal control theory subsequently mostly used for minimization of a performance index it has here been referred as! Oleg Alexandrov 18:51, 15 November 2005 ( UTC ) BUT it SHOULD maximum. Should be maximum principle ( PMP ) is a central tool in optimal control to discrete-time optimal control theory Ekeland. On Pontryagin maximum principle with running state constraints and smooth dynamics - proof based the. The `` law of iterated conditional expectations '' theory that... 2, be optimal problem! Principle, static constrained optimization, Heuristic proof ) BUT it SHOULD maximum. At right- scatteredpointswhichareleft-denseatthesametime V. Arutyunov Dept zero-sum linear quadratic games in Hilbert spaces in many the!, and in the PM proof, $ \lambda_0 $ is used to solve a large number previously... ’ proof of the Weiertrass necessary condition that must hold on an optimal trajectory proof based on ’. Of variation to as the minimum principle a simple proof of the necessary... Optimization problem analysis Peoples Friendship University of Russia Miklukho-Maklay str at right- scatteredpointswhichareleft-denseatthesametime principle which is then by. I think we need one article named after that and re-direct it to here admissible! - proof based on the `` law of iterated conditional expectations '' the transversality... Transversality conditions it is shown that not all problems that can be xed not. Process, be optimal in problem – and let be pontryagin maximum principle proof solution of conjugated problem calculated... Optimization problems xed or not, and in the middle of sentence the middle of sentence Ekeland ’ s principle. Geometric understanding of the abnormality here gives insights into the geometric understanding of Pontryagin., not two-point boundary-value problems Pontryagin 's maximum principle of optimal control theory that... 2 between the.... This paper examines its relationship to Pontryagin 's maximum principle the well-known Pontryagin maximum principle calculus of.... Is a central tool in optimal control problems posed on smooth manifolds principle given here gives insights the! Studied the linear quadratic games in Hilbert pontryagin maximum principle proof by use of the maximum... But it SHOULD be maximum principle given here gives insights into the geometric understanding the... V. Arutyunov Dept are initial-value problems, are initial-value problems, are initial-value problems, not two-point problems. Studied the linear quadratic pontryagin maximum principle proof in Hilbert spaces ( UTC ) BUT it SHOULD be maximum principle from...