# American Institute of Mathematical Sciences

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

## Optimal control of dynamical systems with polynomial impulses

 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

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.
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.

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##### 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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