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Sufficient optimality conditions for extremal controls based on functional increment formulas
Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control
Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of the Russian Academy of Sciences, 134, Lermontov St., 664033, Irkutsk, Russia |
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 [
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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