On the properness of an impulsive control extension of dynamic optimization problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 3, pp. 857-875.

This article investigates the properness, or well-posedness, of impulsive extension of a conventional optimal control problem. This includes both well-posedness of the solution to impulsive control systems arising as result of an impulsive extension of ordinary differential systems, and existence theorems. Well-posedness in the classic Cauchy sense is proved. Approximation lemmas that guarantee sensitivity to small perturbations in control variables are obtained. Filippov type existence theorems are established. A model example is provided to show the relevance of the impulsive controls problems which are under study.

DOI : 10.1051/cocv/2014053
Classification : 49N25, 49J15
Mots clés : Optimal control extensions, well-posedness of solutions, existence of solutions, impulsive control
Karamzin, D.Y. 1, 2 ; de Oliveira, V.A. 1 ; Pereira, F.L. 3 ; Silva, G.N. 1

1 UNESP – Universidade Estadual Paulista, Department of Applied Mathematics, São José do Rio Preto, SP, Brazil
2 Computing Center of the Russian Academy of Sciences, Moscow, Russia
3 SYSTEC, Faculdade de Engenharia, Universidade do Porto, Rua Dr. Roberto Frias, s/n, 4200-465 Porto, Portugal
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Karamzin, D.Y.; de Oliveira, V.A.; Pereira, F.L.; Silva, G.N. On the properness of an impulsive control extension of dynamic optimization problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 21 (2015) no. 3, pp. 857-875. doi : 10.1051/cocv/2014053. http://www.numdam.org/articles/10.1051/cocv/2014053/

M.A. Aronna and F. Rampazzo, A note on systems with ordinary and impulsive controls. IMA J. Math. Control Inf. (2014) DOI:. | DOI | MR

A.V. Arutyunov, Optimality Conditions: Abnormal and Degenerate Problems. Kluwer Academic Publishers, Dordrecht, Boston, London (2000). | MR | Zbl

A.V. Arutyunov and S.M. Aseev, State constraints in optimal control. The degeneracy phenomenon. Systems Control Lett. 26 (1995) 267–273. | DOI | MR | Zbl

A.V. Arutyunov, V. Dykhta and F.L. Pereira, Necessary conditions for impulsive nonlinear optimal control problems without a priori normality assumptions. J. Optimization Theory Appl. 124 (2005) 55–77. | DOI | MR | Zbl

A.V. Arutyunov, V. Jacimovic and F.L. Pereira, Second order necessary conditions for optimal impulsive control problems. J. Dynam. Control Syst. 9 (2003) 131–153. | DOI | MR | Zbl

A.V. Arutyunov, D. 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 | MR | Zbl

A.V. Arutyunov, D.Yu. Karamzin and F.L. Pereira, On impulsive control problems with constraints: control of jumps of systems. J. Math. Sci. 165 (2010) 654–688. | DOI | MR | Zbl

A.V. Arutyunov, D.Yu. Karamzin and F.L. Pereira, On the extension of classical calculus of variations and optimal control to problems with discontinuous trajectories, in Proc. of the 51st IEEE Conference on Decision and Control, CDC 2012. Maui, Hawaii (2012) 6406–6411.

A.V. Arutyunov, D.Yu. Karamzin and F.L. Pereira, Pontryagin’s maximum principle for constrained impulsive control problems. Nonlin. Anal. 75 (2012) 1045–1057. | DOI | MR | Zbl

A.V. Arutyunov and N.T. Tynyanskiy, The maximum principle in a problem with phase constraints. Soviet J. Comput. Systems Sci. 23 (1985) 28–35. | MR | Zbl

A. Bressan and F. Rampazzo, Impulsive control systems with commutative vector fields. J. Optimization Theory Appl. 71 (1991) 67–83. | DOI | MR | Zbl

A. Bressan, and F. Rampazzo, On differential systems with vector-valued impulsive controls. Boll. Un. Mat. Ital. B 72 (1988) 641–656. | MR | Zbl

W.J. Code and P.D. Loewen, Optimal control of non-convex Measure-driven differential inclusions. Set-Valued Var. Anal. 19 (2011) 203–235. | DOI | MR | Zbl

W.J. Code and G.N. Silva, Closed loop stability of measure-driven impulsive control systems. J. Dyn. Control Syst. 16 (2010) 1–21. | DOI | MR | Zbl

V.A. Dykhta and O.N. Samsonyuk, Optimal Impulse Control with Applications. Fizmatlit, Moscow (2000) [in Russian]. | MR

N. Forcadel, Z. Rao and H. Zidani, State-constrained optimal control problems of impulsive differential equations. Appl. Math. Optim. 68 (2013) 1–19. | DOI | MR | Zbl

A.B. Kurzhanski and A.N. Daryin, Dynamic programming for impulse controls. Ann. Rev. Control 32 (2008) 213–227. | DOI

D.F. Lawden, Optimal Trajectories for Space Navigation. Butterworth, London (1963). | MR | Zbl

B.M. Miller and E.Ya. Rubinovich, Impulsive Control in Continuous and Discrete-Continuous Systems. Kluwer Academic/Plenum Publishers, New York (2003). | MR | Zbl

B. Mordukhovich, Existence of optimal controls. Itogi Nauki Tech. Sovr. Prob. Mat. 6 (1976) 207–261. (Russian); English transl. in J. Soviet Math. 7 (1977) 850–886. | MR | Zbl

B. Mordukhovich, Variational Analysis and Generalized Differentiation. Springer-Verlag, Berlin (2006). | MR | Zbl

F.L. Pereira and G.N. Silva, Necessary conditions of optimality for vector-valued impulsive control problems. Syst. Control Lett. 40 (2000) 205–215. | DOI | MR | Zbl

F.L. Pereira and G.N. Silva, Stability for impulsive control systems. Dyn. Syst. 17 (2002) 421–434. | DOI | MR | Zbl

F.L. Pereira, G.N. Silva and V.A. De Oliveira, Invariance for impulsive control systems. Autom. Remote Control 69 (2008) 788–800. | DOI | MR | Zbl

L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze and E.F. Mishchenko, Mathematical Theory Optimal Processes. Classics Sov. Math. Nauka, Moscow (1983). | MR | Zbl

G.N. Silva and R.B. Vinter, Measure driven differential inclusions. J. Math. Anal. Appl. 202 (1996) 727–746. | DOI | MR | Zbl

G.N. Silva and R.B. Vinter, Necessary conditions for optimal impulsive control problems. SIAM J. Control Optim. 35 (1997) 1829-1846. | DOI | MR | Zbl

R.B. Vinter, Optimal Control. Birkhäuser, Boston (2000). | MR | Zbl

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 | MR | Zbl

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

J. Warga, Variational problems with unbounded controls. SIAM J. Control Ser. A 3 (1965) 424–438. | MR | Zbl

P.R. Wolenski and S. Zabic, A differential solution concept for impulsive systems. Dyn. Contin. Discrete Impuls. Syst., Ser. A Math. Anal. 13B (2006) 199–210. | MR

P.R. Wolenski and S. Zabic, A sampling method and approximation results for impulsive systems. SIAM J. Control Optim. 46 (2007) 983–998. | DOI | MR | Zbl

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