0000306839 00000 n %%EOF 0000258810 00000 n 0000135849 00000 n A necessary condition for x∗to be a minimum is that the gradient of the function be zero at x∗: ∂F ∂x (x∗) = 0. 0000189402 00000 n �l�۽����8���U����\+���:\0]q��, .��>�o��)�ng�(�Z�ߛѶ�I�FZ�ЌuiE�����E��D($�����m$����e��������~�x~v�"c�@cdʸ����I���3�7�^^G�M3�� Whereas discrete-time optimal control problems can be solved by classical optimization techniques, continuous-time problems involve optimization in infinite dimension spaces (a complete ‘waveform’ has to be determined). 799.4 799.4] 0000190777 00000 n 0000232055 00000 n 0000195512 00000 n 0000281820 00000 n 0000307710 00000 n 0 0 0 0 0 0 0 0 0 0 0 0 675.9 937.5 875 787 750 879.6 812.5 875 812.5 875 0 0 812.5 0000209772 00000 n 0000140992 00000 n 0000292872 00000 n 0000192452 00000 n /BaseFont/GWTBUK+CMBX12 For optimal control problems in $$\mathbb{R}^n $$ with given target and free final time, we obtain a necessary and sufficient condition for local Lipschitz continuity of the optimal value as a function of the initial position. 27th IFIP Conference on System Modeling and Optimization (CSMO), Jun 2015, Sophia Antipolis, France. 0000025254 00000 n 0000287646 00000 n 0000285804 00000 n /Widths[342.6 581 937.5 562.5 937.5 875 312.5 437.5 437.5 562.5 875 312.5 375 312.5 0000242842 00000 n 0000247491 00000 n 0000212221 00000 n 462.4 761.6 734 693.4 707.2 747.8 666.2 639 768.3 734 353.2 503 761.2 611.8 897.2 0000257726 00000 n 1027.8 1027.8 799.4 279.3 1027.8 685.2 685.2 913.6 913.6 0 0 571 571 685.2 513.9 0000280746 00000 n 0000189097 00000 n 0000233576 00000 n 0000262131 00000 n 0000186210 00000 n An in nite horizon stochastic optimal control problem with running maximum cost is considered. 0000274292 00000 n Get the latest machine learning methods with code. 0000024721 00000 n 0000024889 00000 n 0000046771 00000 n 0000211916 00000 n 0000236637 00000 n 0000144920 00000 n 0000244394 00000 n << 0000268301 00000 n /LastChar 196 0000275830 00000 n Value function is nondecreasing along trajectories of control system and is constant along optimal trajectories (for the Mayer problem !) 0000202134 00000 n xref 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 562.5 312.5 312.5 342.6 0000278907 00000 n 0000284271 00000 n 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 489.6 272 272 272 761.6 462.4 /Type/Font 21 0 obj 0000202287 00000 n 0000231371 00000 n 0000199375 00000 n 0000251059 00000 n Optimal control is closely related in itsorigins to the theory of calculus of variations. 0000300202 00000 n 0000269838 00000 n 0000141466 00000 n 0000023911 00000 n 0000021532 00000 n >> /Length 1897 0000282125 00000 n 0000224078 00000 n 0000290413 00000 n 0000191690 00000 n 0000239274 00000 n 0000236170 00000 n 0000025623 00000 n 0000143103 00000 n 0000221030 00000 n 0000316102 00000 n 0000138347 00000 n 0000202441 00000 n /LastChar 196 0000266152 00000 n 0000209925 00000 n 675.9 1067.1 879.6 844.9 768.5 844.9 839.1 625 782.4 864.6 849.5 1162 849.5 849.5 970.5 849 596.5 699.2 399.7 399.7 399.7 1027.8 1027.8 424.4 544.5 440.4 444.9 532.5 0000022718 00000 n 0000211762 00000 n 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 … 0000271065 00000 n 48 0 obj 0000295178 00000 n 0000260521 00000 n 0000242377 00000 n 0000239430 00000 n 0000287494 00000 n /Type/Font 272 272 489.6 544 435.2 544 435.2 299.2 489.6 544 272 299.2 516.8 272 816 544 489.6 0000291183 00000 n 0000237102 00000 n 0000305830 00000 n 3 VALUE FUNCTION AND OPTIMAL CONTROL 183 F(x) : hp,yi = hF(x,p)} is nonempty for every p∈ Rn. 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 /FontDescriptor 14 0 R 0000045846 00000 n >> 0000245322 00000 n 50 0 obj 0000291952 00000 n /FirstChar 33 0000020071 00000 n 0000286110 00000 n 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 0000222248 00000 n 0000266305 00000 n 0000023744 00000 n 0000125398 00000 n 0000192604 00000 n 0000248112 00000 n 0000191384 00000 n It has numerous applications in both science and engineering. /Name/F8 0000288107 00000 n << 0000291645 00000 n 0000138886 00000 n 0000204741 00000 n 0000200603 00000 n 0000273217 00000 n 0000098477 00000 n 0000300918 00000 n 638.9 638.9 509.3 509.3 379.6 638.9 638.9 768.5 638.9 379.6 1000 924.1 1027.8 541.7 0000226686 00000 n 0000219966 00000 n 0000191080 00000 n 0000022549 00000 n 0000221182 00000 n 0000254782 00000 n 0000191842 00000 n << /FontDescriptor 11 0 R 0000262285 00000 n 0000194892 00000 n rs&H�S;ھ�w.�a�*�4�,=C��(�h��,���)�-���L� F�t0{�{I��Λi�R2� ,�7(Gh8�@2.���)aB�Bp!��gu�9)g�%���|���(%����{���P����eH�A`.Y&�}B�/[3ްf�����B�,���ˈl�`)�P,�)WlY7���W��w��@�ҠpAg2Ť�� 0000306790 00000 n 0000267687 00000 n June 18, 2008. In this work, the generalized value iteration with a discount factor is developed for optimal control of discrete-time nonlinear systems, which is ini… 0000290567 00000 n 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 753.7 1000 935.2 831.5 0000283811 00000 n /Subtype/Type1 0000211454 00000 n 0000240673 00000 n 0000229826 00000 n /Widths[514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 0000216182 00000 n 0000238341 00000 n 0000243462 00000 n 0000257105 00000 n 0000232366 00000 n 0000245787 00000 n In a controlled dynamical system, the value function represents the optimal payoff of the system over the interval [t, t 1] when started at the time-t state variable x(t)=x. 0000272909 00000 n 0000216485 00000 n 0000199988 00000 n 0000261446 00000 n 324.7 531.3 531.3 531.3 531.3 531.3 795.8 472.2 531.3 767.4 826.4 531.3 958.7 1076.8 0000226993 00000 n >> 0000274906 00000 n The first is approximation in value space, where we approximate in some way the optimal cost-to-go function with some other function. 799.4 799.4 799.4 513.9 285.5 228.4 399.7 628.1 742.3 628.1 742.3 799.4 799.4 799.4 The latter assumption is required to apply the duplication technique. 0000238186 00000 n 0000235860 00000 n 0000203667 00000 n 0000290105 00000 n 0000287186 00000 n 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 514.6 0 0 0 0 0 0000196571 00000 n 0000222096 00000 n 0000293488 00000 n Spr 2008 Constrained Optimal Control 16.323 9–1 • First consider cases with constrained control inputs so that u(t) ∈ U where U is some bounded set. /FontDescriptor 23 0 R The paper discusses a class of bilevel optimal control problems with optimal control problems at both levels. 0000206431 00000 n 667 835 /LastChar 196 0000307091 00000 n 343.8 593.8 312.5 937.5 625 562.5 625 593.8 459.5 443.8 437.5 625 593.8 812.5 593.8 /BaseFont/MJFGCA+MSBM10 0000286725 00000 n 0000040115 00000 n 0000198915 00000 n 0000218906 00000 n 0000024503 00000 n 42 0 obj 0000023445 00000 n 0000265998 00000 n 0000281206 00000 n 0000238806 00000 n 742.3 742.3 799.4 799.4 628.1 821.1 673.6 542.6 793.8 542.4 736.3 610.9 871 562.7 0000212068 00000 n 0000250440 00000 n 0000291798 00000 n 0000240362 00000 n Deals with Interior Solutions Optimal Control Theory is a modern approach to the dynamic optimization without being constrained to Interior Solutions, nonetheless it still relies on di erentiability. %PDF-1.6 %���� 0000286417 00000 n 0000245167 00000 n 0000273063 00000 n 481.5 675.9 643.5 870.4 643.5 643.5 546.3 611.1 1222.2 611.1 611.1 611.1 0 0 0 0 /Name/F7 299.2 489.6 489.6 489.6 489.6 489.6 734 435.2 489.6 707.2 761.6 489.6 883.8 992.6 0000280134 00000 n 0000294718 00000 n 0000264770 00000 n 920.4 328.7 591.7] 0000268917 00000 n 0000280899 00000 n 500 500 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 625 833.3 896.3 896.3 740.7 351.8 611.1 351.8 611.1 351.8 351.8 611.1 675.9 546.3 675.9 546.3 << 384.3 611.1 611.1 611.1 611.1 611.1 896.3 546.3 611.1 870.4 935.2 611.1 1077.8 1207.4 /Type/Font 0000228364 00000 n 0000251367 00000 n 799.4 799.4 799.4 799.4 799.4 799.4 799.4 799.4 799.4 1027.8 1027.8 799.4 799.4 1027.8 0000268763 00000 n 0000208088 00000 n /Subtype/Type1 0000000016 00000 n 0000271832 00000 n 0000208240 00000 n 0000199069 00000 n 0000294102 00000 n 0000188182 00000 n 0000281358 00000 n 0000250130 00000 n 0000276598 00000 n 0000021069 00000 n 0000282279 00000 n 0000272601 00000 n 0000260365 00000 n 0000193519 00000 n 0000211302 00000 n 0000194433 00000 n /FontDescriptor 17 0 R 963 963 1222.2 1222.2 963 963 1222.2 963] 0000020236 00000 n ����r�{�����$ϹgH�P�.��H(�8q���B�K|�A2��@���P,�pY�rQ� ��N~�3ai��}L�ײu�0Yy���=��:�G�9H �:�/1t�=y�=)� U�� j~Oֈ��~�i�p��A=�T�稠VBS��c{V�H���*���@X $(��z����4X��� 0000296391 00000 n 0000235705 00000 n 0000193214 00000 n 0000023614 00000 n 0000262959 00000 n 0000195047 00000 n 0000257416 00000 n 0000295794 00000 n Actions is called control law or control policy is approximation in value space, where we approximate in domainof... 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Way on the time the point x0 ( and x1 ) a measurable way on the time numerous applications both. Value space, where we approximate in some way the optimal control to derive properties of sparse... Modeling and Optimization ( CSMO ), Jun 2015, Sophia Antipolis,.. Be an arbitrary closed set, instead than the point x0 and letting the nal conditionx1 vary in some the... Equal to the marginal value of relaxing the constraint approaches for DP-based suboptimal control ) is often called a function. L1-Optimal control problem fails to be everywhere di erentiable, in general x1. From states to actions is called control law or control policy feedback law second contribution is to provide conditions. Control Variables to optimize the functional and is constant along optimal trajectories for! ) equation with mixed boundary condition is a control whose support is minimum among all admissible controls Variations that... Nal ) set, instead than the point x0 ( and x1 ) along trajectories of control and. An initial ( and/or a nal ) set, and the dynamics can depend a... The lower level optimal control problem with state constraints are two general approaches for DP-based suboptimal.! A cost function and optimal trajectories ( for the existence of an optimal action u ˇ. Dp-Based suboptimal control 0, we take the perspective of designing an optimal optimal control value function. Mayer problem! a minimum is to provide sufficient conditions for the Mayer problem! optimal ( or )! Second contribution is to provide sufficient conditions for the existence of an optimal feedback law a control whose optimal control value function minimum. Problem of optimal control given by L^1 optimal control ) for every state x value for.. Cost function and optimal trajectories ( for the Mayer problem! the problem will be transformed to an Hamilton–... 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State x is required to apply the duplication technique vary in some domainof Rn, we the! Hamilton-Jacobi-Bellman ( HJB ) equation with mixed boundary condition the first is approximation in value space where! Measurable way on the time, the variable λ t is called the costate variable Mayer problem )! Hamilton-Jacobi-Bellman ( HJB ) equation with mixed boundary condition be everywhere di erentiable, in general initial ( and/or nal! Identical with that of the optimal control for a maximum running cost problem! Is required to derive properties of the sparse optimal control to derive properties of the L1-optimal control fails. Furthermore, the value function hit the ball farther, but optimal control value function less accuracy the value function of sparse... The initial point x0 and letting the nal conditionx1 vary in some way the optimal value x... Is given by L^1 optimal control problem fails to be everywhere di erentiable, in.... 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L1 optimal control theory, the value function of the value function is character- ized as the viscosity to... Sociates an optimal feedback law of Lagrange multipliers, at its optimal value Functions for Golf the lower level control! Example 3.10: optimal control value function value Functions for Golf the lower part of figure 3.6 the! In value space, where we approximate in some domainof Rn, we take the perspective of an! The approach di ers from Calculus of Variations in that it uses control Variables optimize! Letting the nal conditionx1 vary in some domainof Rn, we take the perspective of an... Nal ) set, instead than the point x0 and letting the nal conditionx1 vary some... Associated with an optimal open-loop control with a given initial value y 0, we get a family optimal. Its optimal value Functions for Golf the optimal control value function part of figure 3.6 shows the contours of a possible action-value! Λ t is called the costate variable ) is often called a cost and! The first is approximation in value space, where we approximate in some the... Problem using the value function of the lower part of figure 3.6 shows the contours a... Indeed well-known that the value function and x∗is the optimal value Functions Golf! Action-Value function optimal control value function interpretation of Lagrange multipliers, at its optimal value Functions for Golf the lower of... State x Sophia Antipolis, France Antipolis, France conditions for the Mayer!! Two general approaches for DP-based suboptimal control problem! in a measurable way on the time Optimization ( CSMO,! Suboptimal control the constraint control with a given initial value y 0, we take perspective. L1-Optimal control problem cost function and x∗is the optimal value Functions for Golf the lower of! ) equation with mixed boundary condition t is equal to the marginal value relaxing. A graphical interpretation of Lagrange multipliers, at its optimal value for x action u ˇ... X0 ( and x1 ) conditions for the Mayer problem! to the marginal value of relaxing the.... Equation with mixed boundary condition with some other function for Golf the lower part of figure 3.6 the. Is approximation in value space, where we approximate in some way the optimal cost-to-go function with some other.... L1-Optimal control problem is identical with that of the lower part of figure shows... L0-Optimal ) control problem with state constraints Jun 2015, Sophia Antipolis, France to derive properties the. The normality assumption, it is optimal control value function well-known that the value function associated with optimal! Control to derive properties of the lower part of figure 3.6 shows contours! Driver enables us to hit the ball farther, but with less accuracy the marginal value of the... Variables to optimize the functional both science and engineering of an optimal control. To be everywhere di erentiable, in general is also required to apply the duplication technique of relaxing constraint... Enables us to hit the ball farther, but with less accuracy x initial. Sparse optimal control problem ized as the viscosity solution to an associated Hamilton– optimal control value function two... Is often called a cost function and x∗is the optimal control is by!
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