April  2011, 29(2): 403-415. doi: 10.3934/dcds.2011.29.403

On a generalization of the impulsive control concept: Controlling system jumps

1. 

People's Friendship University of Russia, 117198, Moscow, Miklukho-Maklaya str. 6, Russian Federation

2. 

Faculdade de Engenharia da Universidade do Porto, DEEC, Rua Dr. Roberto Frias, s/n 4200-465 Porto, Portugal, Portugal

Received  August 2009 Revised  March 2010 Published  October 2010

This paper concerns the investigation of a general impulsive control problem. The considered impulsive processes are of non-standard type: control processes admit ordinary type controls as the impulse develops. New necessary conditions of optimality in the form of Pontryagin Maximum Principle are obtained. These conditions are applied to a model problem and are shown to yield useful information about optimal control modes.
Citation: Aram Arutyunov, Dmitry Karamzin, Fernando L. Pereira. On a generalization of the impulsive control concept: Controlling system jumps. Discrete and Continuous Dynamical Systems, 2011, 29 (2) : 403-415. doi: 10.3934/dcds.2011.29.403
References:
[1]

A. V. Arutyunov, "Optimality Conditions: Abnormal and Degenerate Problems," Math. Appl., Kluwer Academic Publisher, 2000.

[2]

A. V. Arutyunov, D. Yu. Karamzin and F. L. Pereira, A nondegenerate maximum principle for the impulse control problem with state constraints, SIAM J. Control Optim., 43 (2005), 1812-1843. doi: doi:10.1137/S0363012903430068.

[3]

A. V. Arutyunov and D. Yu. Karamzin, Necessary conditions for minimum in impulsive control problems, Nonlinear Dynamics and Control, Moscow, Fizmatlit, (2004), 205-240, (in Russian).

[4]

A. V. Arutyunov, D. Yu. Karamzin, and F. L. Pereira, On constrained impulsive control problems, Sovremennaya Matematika i Ee Prilozheniya, 65 (2009) (in Russian, the English translation in Journal of Mathematical Sciences, 165 (2010), 654-687).

[5]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Matematica Italiana B, 2 (1988), 641-656.

[6]

A. Bressan and F. Rampazzo, Impulsive control systems with commutative vector fields, J. Optim. Theory and Appl., 71 (1991), 67-83. doi: doi:10.1007/BF00940040.

[7]

V. A. Dykhta and O. N. Samsonyuk, "Optimal Impulse Control and Applications," Fizmatlit, Moscow, 2000, (in Russian).

[8]

N. N. Krasovski, "The Theory of Motion Control," Nauka, Moscow, 1968, (in Russian)

[9]

A. B. Kurzhanski., Optimal systems with impulse controls, in "Differential Games and Control Problems," UNC AN SSSR. Sverdlovsk, 1975, (in Russian).

[10]

A. B. Kurzhanski and A. N. Daryin, Dynamic programming for impulse controls, Annual Reviews in Control, 32 (2008), 213-227. doi: doi:10.1016/j.arcontrol.2008.08.001.

[11]

F. L. Pereira and G. N. Silva, Necessary conditions of optimality for vector-valued impulsive control problems, Systems and Control Letters, 40 (2000), 205-215. doi: doi:10.1016/S0167-6911(00)00027-X.

[12]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Gordon and Beach, New York, 1986.

[13]

R. W. Rishel, An extended Pontryagin principle for control systems, whose control laws contain measures, J. SIAM. Ser. A. Control, 3 (1965), 191-205.

[14]

G. N. Silva and R. B. Vinter, Measure differential inclusions, J. Math. Anal. Appl., 202 (1996), 727-746. doi: doi:10.1006/jmaa.1996.0344.

[15]

R. B. Vinter and F. L. Pereira, A maximum principle for optimal processes with discontinuous trajectories, SIAM J. Control Optim., 26 (1988), 205-229. doi: doi:10.1137/0326013.

show all references

References:
[1]

A. V. Arutyunov, "Optimality Conditions: Abnormal and Degenerate Problems," Math. Appl., Kluwer Academic Publisher, 2000.

[2]

A. V. Arutyunov, D. Yu. Karamzin and F. L. Pereira, A nondegenerate maximum principle for the impulse control problem with state constraints, SIAM J. Control Optim., 43 (2005), 1812-1843. doi: doi:10.1137/S0363012903430068.

[3]

A. V. Arutyunov and D. Yu. Karamzin, Necessary conditions for minimum in impulsive control problems, Nonlinear Dynamics and Control, Moscow, Fizmatlit, (2004), 205-240, (in Russian).

[4]

A. V. Arutyunov, D. Yu. Karamzin, and F. L. Pereira, On constrained impulsive control problems, Sovremennaya Matematika i Ee Prilozheniya, 65 (2009) (in Russian, the English translation in Journal of Mathematical Sciences, 165 (2010), 654-687).

[5]

A. Bressan and F. Rampazzo, On differential systems with vector-valued impulsive controls, Boll. Un. Matematica Italiana B, 2 (1988), 641-656.

[6]

A. Bressan and F. Rampazzo, Impulsive control systems with commutative vector fields, J. Optim. Theory and Appl., 71 (1991), 67-83. doi: doi:10.1007/BF00940040.

[7]

V. A. Dykhta and O. N. Samsonyuk, "Optimal Impulse Control and Applications," Fizmatlit, Moscow, 2000, (in Russian).

[8]

N. N. Krasovski, "The Theory of Motion Control," Nauka, Moscow, 1968, (in Russian)

[9]

A. B. Kurzhanski., Optimal systems with impulse controls, in "Differential Games and Control Problems," UNC AN SSSR. Sverdlovsk, 1975, (in Russian).

[10]

A. B. Kurzhanski and A. N. Daryin, Dynamic programming for impulse controls, Annual Reviews in Control, 32 (2008), 213-227. doi: doi:10.1016/j.arcontrol.2008.08.001.

[11]

F. L. Pereira and G. N. Silva, Necessary conditions of optimality for vector-valued impulsive control problems, Systems and Control Letters, 40 (2000), 205-215. doi: doi:10.1016/S0167-6911(00)00027-X.

[12]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes," Gordon and Beach, New York, 1986.

[13]

R. W. Rishel, An extended Pontryagin principle for control systems, whose control laws contain measures, J. SIAM. Ser. A. Control, 3 (1965), 191-205.

[14]

G. N. Silva and R. B. Vinter, Measure differential inclusions, J. Math. Anal. Appl., 202 (1996), 727-746. doi: doi:10.1006/jmaa.1996.0344.

[15]

R. B. Vinter and F. L. Pereira, A maximum principle for optimal processes with discontinuous trajectories, SIAM J. Control Optim., 26 (1988), 205-229. doi: doi:10.1137/0326013.

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