Optimal control of dynamical systems with polynomial impulses

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September 2015, 35(9): 4367-4384. doi: 10.3934/dcds.2015.35.4367

Optimal control of dynamical systems with polynomial impulses

Elena Goncharova ^1, and Maxim Staritsyn ^1,

1.	Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russian Federation, Russian Federation

Received April 2014 Revised September 2014 Published April 2015

Abstract
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The paper is devoted to the $BV$-relaxation of a dynamical system, whose right-hand side is a $p$th degree polynomial with rational powers of control under a uniform bound on its $L_p$-norm, and coefficients containing usual measurable bounded control.
Under natural convexity assumptions, we give an explicit representation of generalized solutions to the control system by a measure differential equation. The main results concern an optimal impulsive control problem for the relaxed system: We establish the existence of a minimizer, and give necessary optimality conditions in the form of a Maximum Principle.

Keywords: Maximum Principle., measure differential equations, Optimal control, impulsive control, $BV$-relaxation.

Mathematics Subject Classification: Primary: 49N25, 49K99; Secondary: 49J9.

Citation: Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367

References:

[1]	A. Arutyunov, D. Karamzin and F. Pereira, On constrained impulsive control problems,, J. Math. Sci., 165 (2010), 654. doi: 10.1007/s10958-010-9834-z. Google Scholar
[2]	A. Arutyunov, D. Karamzin and F. Pereira, A nondegenerate Maximum Principle for the impulse control problem with state constraints,, SIAM J. Control Optim., 43 (2005), 1812. doi: 10.1137/S0363012903430068. Google Scholar
[3]	J.-P. Aubin and A. Cellina, Differential Inclusions,, Springer-Verlag, (1984). doi: 10.1007/978-3-642-69512-4. Google Scholar
[4]	A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids,, Discr. Cont. Dynam. Syst., 20 (2008), 1. doi: 10.3934/dcds.2008.20.1. Google Scholar
[5]	A. Bressan and M. Motta, A class of mechanical systems with some coordinates as controls. A reduction of certain optimization problems for them. Solution methods,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem., 2 (1993). Google Scholar
[6]	A. Bressan and F. Rampazzo, Moving constraints as stabilizing controls in classical mechanics,, Arch. Ration. Mech. Anal., 196 (2010), 97. doi: 10.1007/s00205-009-0237-6. Google Scholar
[7]	A. Bressan and F. Rampazzo, On systems with quadratic impulses and their application to Lagrangean mechanics,, SIAM J. Control Optim., 31 (1993), 1205. doi: 10.1137/0331057. Google Scholar
[8]	A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions,, J. Optim. Theory Appl., 81 (1994), 435. doi: 10.1007/BF02193094. Google Scholar
[9]	V. Dykhta, Impulse-trajectory extension of degenerate optimal control problems,, IMACS Ann. Comput. Appl. Math., 8 (1990), 103. Google Scholar
[10]	V. Dykhta and O. Samsonyuk, A maximum principle for smooth optimal impulsive control problems with multipoint state constraints,, Comput. Math. Math. Phys., 49 (2009), 942. doi: 10.1134/S0965542509060050. Google Scholar
[11]	V. Gurman, On optimal processes with unbounded derivatives,, Autom. Remote Control, 17 (1972), 14. Google Scholar
[12]	A. Ioffe and V. Tikhomirov, Theory of Extremal Problems,, North-Holland, (1979). Google Scholar
[13]	D. Karamzin, Necessary conditions of the minimum in an impulse optimal control problem,, J. Math. Sci., 139 (2006), 7087. doi: 10.1007/s10958-006-0408-z. Google Scholar
[14]	B. Miller, The generalized solutions of nonlinear optimization problems with impulse control,, SIAM J. Control Optim., 34 (1996), 1420. doi: 10.1137/S0363012994263214. Google Scholar
[15]	B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete- Continuous Systems,, Kluwer Academic / Plenum Publishers, (2001). Google Scholar
[16]	M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls,, Differential Integral Equations, 8 (1995), 269. Google Scholar
[17]	M. Motta and C. Sartori, Asymptotic problems in optimal control with a vanishing Lagrangian and unbounded data,, (2014) [published online as , (2014). Google Scholar
[18]	P. Pedregal and J. Tiago, Existence results for optimal control problems with some special nonlinear dependence on state and control,, SIAM J. Control Optim., 48 (2009), 415. doi: 10.1137/08071805X. Google Scholar
[19]	F. Rampazzo and C. Sartori, Hamilton-Jacobi-Bbellman equations with fast gradient-dependence,, Indiana Univ. Math. J., 49 (2000), 1043. doi: 10.1512/iumj.2000.49.1736. Google Scholar
[20]	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. doi: 10.1137/0303016. Google Scholar
[21]	R. Vinter and F. Pereira, A maximum principle for optimal processes with discontinuous trajectories,, SIAM J. Control Optim., 26 (1988), 205. doi: 10.1137/0326013. Google Scholar
[22]	J. Warga, Relaxed variational problems,, J. Math. Anal. Appl., 4 (1962), 111. doi: 10.1016/0022-247X(62)90033-1. Google Scholar
[23]	J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972). Google Scholar
[24]	J. Warga, Variational problems with unbounded controls,, J. SIAM Control Ser. A, 3 (1965), 424. doi: 10.1137/0303028. Google Scholar
[25]	S. Zavalischin and A. Sesekin, Dynamic Impulse Systems: Theory and Applications,, Kluwer Academic Publishers, (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. doi: 10.1007/s10958-010-9834-z. Google Scholar
[2]	A. Arutyunov, D. Karamzin and F. Pereira, A nondegenerate Maximum Principle for the impulse control problem with state constraints,, SIAM J. Control Optim., 43 (2005), 1812. doi: 10.1137/S0363012903430068. Google Scholar
[3]	J.-P. Aubin and A. Cellina, Differential Inclusions,, Springer-Verlag, (1984). doi: 10.1007/978-3-642-69512-4. Google Scholar
[4]	A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids,, Discr. Cont. Dynam. Syst., 20 (2008), 1. doi: 10.3934/dcds.2008.20.1. Google Scholar
[5]	A. Bressan and M. Motta, A class of mechanical systems with some coordinates as controls. A reduction of certain optimization problems for them. Solution methods,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem., 2 (1993). Google Scholar
[6]	A. Bressan and F. Rampazzo, Moving constraints as stabilizing controls in classical mechanics,, Arch. Ration. Mech. Anal., 196 (2010), 97. doi: 10.1007/s00205-009-0237-6. Google Scholar
[7]	A. Bressan and F. Rampazzo, On systems with quadratic impulses and their application to Lagrangean mechanics,, SIAM J. Control Optim., 31 (1993), 1205. doi: 10.1137/0331057. Google Scholar
[8]	A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions,, J. Optim. Theory Appl., 81 (1994), 435. doi: 10.1007/BF02193094. Google Scholar
[9]	V. Dykhta, Impulse-trajectory extension of degenerate optimal control problems,, IMACS Ann. Comput. Appl. Math., 8 (1990), 103. Google Scholar
[10]	V. Dykhta and O. Samsonyuk, A maximum principle for smooth optimal impulsive control problems with multipoint state constraints,, Comput. Math. Math. Phys., 49 (2009), 942. doi: 10.1134/S0965542509060050. Google Scholar
[11]	V. Gurman, On optimal processes with unbounded derivatives,, Autom. Remote Control, 17 (1972), 14. Google Scholar
[12]	A. Ioffe and V. Tikhomirov, Theory of Extremal Problems,, North-Holland, (1979). Google Scholar
[13]	D. Karamzin, Necessary conditions of the minimum in an impulse optimal control problem,, J. Math. Sci., 139 (2006), 7087. doi: 10.1007/s10958-006-0408-z. Google Scholar
[14]	B. Miller, The generalized solutions of nonlinear optimization problems with impulse control,, SIAM J. Control Optim., 34 (1996), 1420. doi: 10.1137/S0363012994263214. Google Scholar
[15]	B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete- Continuous Systems,, Kluwer Academic / Plenum Publishers, (2001). Google Scholar
[16]	M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls,, Differential Integral Equations, 8 (1995), 269. Google Scholar
[17]	M. Motta and C. Sartori, Asymptotic problems in optimal control with a vanishing Lagrangian and unbounded data,, (2014) [published online as , (2014). Google Scholar
[18]	P. Pedregal and J. Tiago, Existence results for optimal control problems with some special nonlinear dependence on state and control,, SIAM J. Control Optim., 48 (2009), 415. doi: 10.1137/08071805X. Google Scholar
[19]	F. Rampazzo and C. Sartori, Hamilton-Jacobi-Bbellman equations with fast gradient-dependence,, Indiana Univ. Math. J., 49 (2000), 1043. doi: 10.1512/iumj.2000.49.1736. Google Scholar
[20]	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. doi: 10.1137/0303016. Google Scholar
[21]	R. Vinter and F. Pereira, A maximum principle for optimal processes with discontinuous trajectories,, SIAM J. Control Optim., 26 (1988), 205. doi: 10.1137/0326013. Google Scholar
[22]	J. Warga, Relaxed variational problems,, J. Math. Anal. Appl., 4 (1962), 111. doi: 10.1016/0022-247X(62)90033-1. Google Scholar
[23]	J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972). Google Scholar
[24]	J. Warga, Variational problems with unbounded controls,, J. SIAM Control Ser. A, 3 (1965), 424. doi: 10.1137/0303028. Google Scholar
[25]	S. Zavalischin and A. Sesekin, Dynamic Impulse Systems: Theory and Applications,, Kluwer Academic Publishers, (1997). doi: 10.1007/978-94-015-8893-5. Google Scholar

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