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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 & 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.   Google Scholar

[2]

A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J Optim. Theory Appl., 81 (1994), 435-457.  doi: 10.1007/BF02193094.  Google Scholar

[3]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.  Google Scholar

[4]

V. A. Dykhta, Variational necessary optimality conditions with feedback descent controls for optimal control problems, Doklady Mathematics, 91 (2015), 394-396.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

V. Dykhta and O. Samsonyuk, Optimal Impulsive Control with Applications, Fizmathlit, Moscow, (in Russian), 2000.  Google Scholar

[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.  Google Scholar

[10]

V. Gurman, Singular Problems in Optimal Control, Nauka, Moscow, (in Russian), 1977.  Google Scholar

[11]

N. N. Krasovskii and A. I. Subbotin, Game-theoretical Control Problems, Springer, New York, 1988. doi: 10.1007/978-1-4612-3716-7.  Google Scholar

[12]

N. N. Krasovskii and A. I. Subbotin, Positional Differential Games, Fizmatlit, Moscow, 1974.  Google Scholar

[13]

V. M. Matrosov, L. U. Anapolskii and S. N. Vasiliev, Comparison Method in Mathematical Control Theory, Nauka, Novosibirsk, (in Russian), 1980. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

L. S. Pontryagin, V. G. Boltyanskiy, R. V. Gamkrelidze and E. F. Mishenko, Mathematical Theory of Optimal Processes, Fizmatlit, Moscow, (in Russian), 1961.  Google Scholar

[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.   Google Scholar

[18]

O. N. Samsonyuk, Invariance sets for the nonlinear impulsive control systems}, Autom. Remote Control, 76 (2015), 405-418.  doi: 10.1134/S0005117915030054.  Google Scholar

[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.  Google Scholar

[20]

A. I. Subbotin, Generalized Solutions of First Order Partial Derivative Equations. Prospects of Dynamical Optimization, Inst. Komp. Issled. , Izhevsk, (in Russian), 2003. Google Scholar

[21]

J. Warga, Variational problems with unbounded controls}, J. SIAM Control Ser.A, 3 (1987), 424-438.   Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. Arutyunov, D. Karamzin and F. Pereira, On constrained impulsive control problems, J. Math. Sci., 165 (2010), 654-688.   Google Scholar

[2]

A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J Optim. Theory Appl., 81 (1994), 435-457.  doi: 10.1007/BF02193094.  Google Scholar

[3]

F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer-Verlag, New York, 1998.  Google Scholar

[4]

V. A. Dykhta, Variational necessary optimality conditions with feedback descent controls for optimal control problems, Doklady Mathematics, 91 (2015), 394-396.   Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

V. Dykhta and O. Samsonyuk, Optimal Impulsive Control with Applications, Fizmathlit, Moscow, (in Russian), 2000.  Google Scholar

[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.  Google Scholar

[10]

V. Gurman, Singular Problems in Optimal Control, Nauka, Moscow, (in Russian), 1977.  Google Scholar

[11]

N. N. Krasovskii and A. I. Subbotin, Game-theoretical Control Problems, Springer, New York, 1988. doi: 10.1007/978-1-4612-3716-7.  Google Scholar

[12]

N. N. Krasovskii and A. I. Subbotin, Positional Differential Games, Fizmatlit, Moscow, 1974.  Google Scholar

[13]

V. M. Matrosov, L. U. Anapolskii and S. N. Vasiliev, Comparison Method in Mathematical Control Theory, Nauka, Novosibirsk, (in Russian), 1980. Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[16]

L. S. Pontryagin, V. G. Boltyanskiy, R. V. Gamkrelidze and E. F. Mishenko, Mathematical Theory of Optimal Processes, Fizmatlit, Moscow, (in Russian), 1961.  Google Scholar

[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.   Google Scholar

[18]

O. N. Samsonyuk, Invariance sets for the nonlinear impulsive control systems}, Autom. Remote Control, 76 (2015), 405-418.  doi: 10.1134/S0005117915030054.  Google Scholar

[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.  Google Scholar

[20]

A. I. Subbotin, Generalized Solutions of First Order Partial Derivative Equations. Prospects of Dynamical Optimization, Inst. Komp. Issled. , Izhevsk, (in Russian), 2003. Google Scholar

[21]

J. Warga, Variational problems with unbounded controls}, J. SIAM Control Ser.A, 3 (1987), 424-438.   Google Scholar

[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.  Google Scholar

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