Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control

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June 2017, 7(2): 201-210. doi: 10.3934/naco.2017014

Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control

Stepan Sorokin ^, and Maxim Staritsyn

Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of the Russian Academy of Sciences, 134, Lermontov St., 664033, Irkutsk, Russia

^* Corresponding author: Stepan Sorokin

Received December 2016 Revised May 2017 Published June 2017

Fund Project: This paper was prepared at the occasion of The 10th International Conference on Optimization: Techniques and Applications (ICOTA 2016), Ulaanbaatar, Mongolia, July 23-26,2016, with its Associate Editors of Numerical Algebra, Control and Optimization (NACO) being Prof. Dr. Zhiyou Wu, School of Mathematical Sciences, Chongqing Normal University, Chongqing, China, Prof. Dr. Changjun Yu, Department of Mathematics and Statistics, Curtin University, Perth, Australia, and Shanghai University, China, and Prof. Gerhard-Wilhelm Weber, Middle East Technical University, Ankara, Turkey.

We consider a class of rightpoint-constrained state-linear (but non convex) optimal control problems, which takes its origin in the impulsive control framework. The main issue is a strengthening of the Pontryagin Maximum Principle for the addressed problem. Towards this goal, we adapt the approach, based on feedback control variations due to V.A. Dykhta [4,5,6,7]. Our necessary optimality condition, named the feedback maximum principle, is expressed completely in terms of the classical Maximum Principle, but is shown to discard non-optimal extrema. As a connected result, we derive a certain form of duality for the considered problem, and propose the dual version of the proved necessary optimality condition.

Keywords: Optimal control, impulsive control, Maximum Principle, feedback controls, nonstandard duality.

Mathematics Subject Classification: Primary: 49K15, 49K99; Secondary: 93C30.

Citation: Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control and Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014

References:

[1]	A. Arutyunov, D. Karamzin and F. Pereira, On constrained impulsive control problems, J. Math. Sci., 165 (2010), 654-688.
[2]	A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J Optim. Theory Appl., 81 (1994), 435-457. doi: 10.1007/BF02193094.
[3]	F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.
[4]	V. A. Dykhta, Variational necessary optimality conditions with feedback descent controls for optimal control problems, Doklady Mathematics, 91 (2015), 394-396.
[5]	V. A. Dykhta, Positional strengthenings of the maximum principle and sufficient optimality conditions, Proceedings of the Steklov Institute of Mathematics, 293 (2016), S43-S57.
[6]	V. A. Dykhta, Weakly monotone solutions of the Hamilton-Jacobi inequality and optimality conditions with positional controls, Autom. Remote Control, 75 (2014), 829-844. doi: 10.1134/S0005117914050038.
[7]	V. A. Dykhta, Nonstandard duality and nonlocal necessary optimality conditions in nonconvex optimal control problems, Autom. Remote Control, 75 (2014), 1906-1921. doi: 10.1134/S0005117914110022.
[8]	V. Dykhta and O. Samsonyuk, Optimal Impulsive Control with Applications, Fizmathlit, Moscow, (in Russian), 2000.
[9]	A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides: Control System, Kluwer Acad. Publ., 1988. doi: 10.1007/978-94-015-7793-9.
[10]	V. Gurman, Singular Problems in Optimal Control, Nauka, Moscow, (in Russian), 1977.
[11]	N. N. Krasovskii and A. I. Subbotin, Game-theoretical Control Problems, Springer, New York, 1988. doi: 10.1007/978-1-4612-3716-7.
[12]	N. N. Krasovskii and A. I. Subbotin, Positional Differential Games, Fizmatlit, Moscow, 1974.
[13]	V. M. Matrosov, L. U. Anapolskii and S. N. Vasiliev, Comparison Method in Mathematical Control Theory, Nauka, Novosibirsk, (in Russian), 1980.
[14]	B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic / Plenum Publishers, New York, 2001. doi: 10.1007/978-1-4615-0095-7.
[15]	M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288.
[16]	L. S. Pontryagin, V. G. Boltyanskiy, R. V. Gamkrelidze and E. F. Mishenko, Mathematical Theory of Optimal Processes, Fizmatlit, Moscow, (in Russian), 1961.
[17]	R. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures}, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 191-205.
[18]	O. N. Samsonyuk, Invariance sets for the nonlinear impulsive control systems}, Autom. Remote Control, 76 (2015), 405-418. doi: 10.1134/S0005117915030054.
[19]	S. P. Sorokin, Necessary feedback optimality conditions and nonstandard duality in problems of discrete system optimization, Autom. Remote Control, 75 (2014), 1556-1564. doi: 10.1134/S0005117914090021.
[20]	A. I. Subbotin, Generalized Solutions of First Order Partial Derivative Equations. Prospects of Dynamical Optimization, Inst. Komp. Issled. , Izhevsk, (in Russian), 2003.
[21]	J. Warga, Variational problems with unbounded controls}, J. SIAM Control Ser.A, 3 (1987), 424-438.
[22]	S. Zavalischin and A. Sesekin, Dynamic Impulse Systems: Theory and Applications, Kluwer Academic Publishers, Dorderecht, 1997. doi: 10.1007/978-94-015-8893-5.

show all references

References:

[1]	A. Arutyunov, D. Karamzin and F. Pereira, On constrained impulsive control problems, J. Math. Sci., 165 (2010), 654-688.
[2]	A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J Optim. Theory Appl., 81 (1994), 435-457. doi: 10.1007/BF02193094.
[3]	F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.
[4]	V. A. Dykhta, Variational necessary optimality conditions with feedback descent controls for optimal control problems, Doklady Mathematics, 91 (2015), 394-396.
[5]	V. A. Dykhta, Positional strengthenings of the maximum principle and sufficient optimality conditions, Proceedings of the Steklov Institute of Mathematics, 293 (2016), S43-S57.
[6]	V. A. Dykhta, Weakly monotone solutions of the Hamilton-Jacobi inequality and optimality conditions with positional controls, Autom. Remote Control, 75 (2014), 829-844. doi: 10.1134/S0005117914050038.
[7]	V. A. Dykhta, Nonstandard duality and nonlocal necessary optimality conditions in nonconvex optimal control problems, Autom. Remote Control, 75 (2014), 1906-1921. doi: 10.1134/S0005117914110022.
[8]	V. Dykhta and O. Samsonyuk, Optimal Impulsive Control with Applications, Fizmathlit, Moscow, (in Russian), 2000.
[9]	A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides: Control System, Kluwer Acad. Publ., 1988. doi: 10.1007/978-94-015-7793-9.
[10]	V. Gurman, Singular Problems in Optimal Control, Nauka, Moscow, (in Russian), 1977.
[11]	N. N. Krasovskii and A. I. Subbotin, Game-theoretical Control Problems, Springer, New York, 1988. doi: 10.1007/978-1-4612-3716-7.
[12]	N. N. Krasovskii and A. I. Subbotin, Positional Differential Games, Fizmatlit, Moscow, 1974.
[13]	V. M. Matrosov, L. U. Anapolskii and S. N. Vasiliev, Comparison Method in Mathematical Control Theory, Nauka, Novosibirsk, (in Russian), 1980.
[14]	B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems, Kluwer Academic / Plenum Publishers, New York, 2001. doi: 10.1007/978-1-4615-0095-7.
[15]	M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288.
[16]	L. S. Pontryagin, V. G. Boltyanskiy, R. V. Gamkrelidze and E. F. Mishenko, Mathematical Theory of Optimal Processes, Fizmatlit, Moscow, (in Russian), 1961.
[17]	R. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures}, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 191-205.
[18]	O. N. Samsonyuk, Invariance sets for the nonlinear impulsive control systems}, Autom. Remote Control, 76 (2015), 405-418. doi: 10.1134/S0005117915030054.
[19]	S. P. Sorokin, Necessary feedback optimality conditions and nonstandard duality in problems of discrete system optimization, Autom. Remote Control, 75 (2014), 1556-1564. doi: 10.1134/S0005117914090021.
[20]	A. I. Subbotin, Generalized Solutions of First Order Partial Derivative Equations. Prospects of Dynamical Optimization, Inst. Komp. Issled. , Izhevsk, (in Russian), 2003.
[21]	J. Warga, Variational problems with unbounded controls}, J. SIAM Control Ser.A, 3 (1987), 424-438.
[22]	S. Zavalischin and A. Sesekin, Dynamic Impulse Systems: Theory and Applications, Kluwer Academic Publishers, Dorderecht, 1997. doi: 10.1007/978-94-015-8893-5.

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