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 and Continuous Dynamical Systems, 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), 654-688. doi: 10.1007/s10958-010-9834-z.
[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), 1812-1843. doi: 10.1137/S0363012903430068.
[3]	J.-P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.
[4]	A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids, Discr. Cont. Dynam. Syst., 20 (2008), 1-35. doi: 10.3934/dcds.2008.20.1.
[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), 30pp.
[6]	A. Bressan and F. Rampazzo, Moving constraints as stabilizing controls in classical mechanics, Arch. Ration. Mech. Anal., 196 (2010), 97-141. doi: 10.1007/s00205-009-0237-6.
[7]	A. Bressan and F. Rampazzo, On systems with quadratic impulses and their application to Lagrangean mechanics, SIAM J. Control Optim., 31 (1993), 1205-1220. doi: 10.1137/0331057.
[8]	A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory Appl., 81 (1994), 435-457. doi: 10.1007/BF02193094.
[9]	V. Dykhta, Impulse-trajectory extension of degenerate optimal control problems, IMACS Ann. Comput. Appl. Math., 8 (1990), 103-109.
[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), 942-957. doi: 10.1134/S0965542509060050.
[11]	V. Gurman, On optimal processes with unbounded derivatives, Autom. Remote Control, 17 (1972), 14-21.
[12]	A. Ioffe and V. Tikhomirov, Theory of Extremal Problems, North-Holland, Amsterdam, 1979.
[13]	D. Karamzin, Necessary conditions of the minimum in an impulse optimal control problem, J. Math. Sci., 139 (2006), 7087-7150. doi: 10.1007/s10958-006-0408-z.
[14]	B. Miller, The generalized solutions of nonlinear optimization problems with impulse control, SIAM J. Control Optim., 34 (1996), 1420-1440. doi: 10.1137/S0363012994263214.
[15]	B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete- Continuous Systems, Kluwer Academic / Plenum Publishers, New York, 2001.
[16]	M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288.
[17]	M. Motta and C. Sartori, Asymptotic problems in optimal control with a vanishing Lagrangian and unbounded data, (2014) [published online as arXiv:1406.7655v1].
[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), 415-437. doi: 10.1137/08071805X.
[19]	F. Rampazzo and C. Sartori, Hamilton-Jacobi-Bbellman equations with fast gradient-dependence, Indiana Univ. Math. J., 49 (2000), 1043-1077. doi: 10.1512/iumj.2000.49.1736.
[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), 191-205. doi: 10.1137/0303016.
[21]	R. Vinter and F. Pereira, A maximum principle for optimal processes with discontinuous trajectories, SIAM J. Control Optim., 26 (1988), 205-229. doi: 10.1137/0326013.
[22]	J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111-128. doi: 10.1016/0022-247X(62)90033-1.
[23]	J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.
[24]	J. Warga, Variational problems with unbounded controls, J. SIAM Control Ser. A, 3 (1965), 424-438. doi: 10.1137/0303028.
[25]	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. doi: 10.1007/s10958-010-9834-z.
[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), 1812-1843. doi: 10.1137/S0363012903430068.
[3]	J.-P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.
[4]	A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids, Discr. Cont. Dynam. Syst., 20 (2008), 1-35. doi: 10.3934/dcds.2008.20.1.
[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), 30pp.
[6]	A. Bressan and F. Rampazzo, Moving constraints as stabilizing controls in classical mechanics, Arch. Ration. Mech. Anal., 196 (2010), 97-141. doi: 10.1007/s00205-009-0237-6.
[7]	A. Bressan and F. Rampazzo, On systems with quadratic impulses and their application to Lagrangean mechanics, SIAM J. Control Optim., 31 (1993), 1205-1220. doi: 10.1137/0331057.
[8]	A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory Appl., 81 (1994), 435-457. doi: 10.1007/BF02193094.
[9]	V. Dykhta, Impulse-trajectory extension of degenerate optimal control problems, IMACS Ann. Comput. Appl. Math., 8 (1990), 103-109.
[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), 942-957. doi: 10.1134/S0965542509060050.
[11]	V. Gurman, On optimal processes with unbounded derivatives, Autom. Remote Control, 17 (1972), 14-21.
[12]	A. Ioffe and V. Tikhomirov, Theory of Extremal Problems, North-Holland, Amsterdam, 1979.
[13]	D. Karamzin, Necessary conditions of the minimum in an impulse optimal control problem, J. Math. Sci., 139 (2006), 7087-7150. doi: 10.1007/s10958-006-0408-z.
[14]	B. Miller, The generalized solutions of nonlinear optimization problems with impulse control, SIAM J. Control Optim., 34 (1996), 1420-1440. doi: 10.1137/S0363012994263214.
[15]	B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete- Continuous Systems, Kluwer Academic / Plenum Publishers, New York, 2001.
[16]	M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288.
[17]	M. Motta and C. Sartori, Asymptotic problems in optimal control with a vanishing Lagrangian and unbounded data, (2014) [published online as arXiv:1406.7655v1].
[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), 415-437. doi: 10.1137/08071805X.
[19]	F. Rampazzo and C. Sartori, Hamilton-Jacobi-Bbellman equations with fast gradient-dependence, Indiana Univ. Math. J., 49 (2000), 1043-1077. doi: 10.1512/iumj.2000.49.1736.
[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), 191-205. doi: 10.1137/0303016.
[21]	R. Vinter and F. Pereira, A maximum principle for optimal processes with discontinuous trajectories, SIAM J. Control Optim., 26 (1988), 205-229. doi: 10.1137/0326013.
[22]	J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111-128. doi: 10.1016/0022-247X(62)90033-1.
[23]	J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.
[24]	J. Warga, Variational problems with unbounded controls, J. SIAM Control Ser. A, 3 (1965), 424-438. doi: 10.1137/0303028.
[25]	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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