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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

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, 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.  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), 1812-1843. doi: 10.1137/S0363012903430068.  Google Scholar

[3]

J.-P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 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), 1-35. 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), 30pp.  Google Scholar

[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.  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), 1205-1220. doi: 10.1137/0331057.  Google Scholar

[8]

A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory Appl., 81 (1994), 435-457. doi: 10.1007/BF02193094.  Google Scholar

[9]

V. Dykhta, Impulse-trajectory extension of degenerate optimal control problems, IMACS Ann. Comput. Appl. Math., 8 (1990), 103-109.  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), 942-957. doi: 10.1134/S0965542509060050.  Google Scholar

[11]

V. Gurman, On optimal processes with unbounded derivatives, Autom. Remote Control, 17 (1972), 14-21. Google Scholar

[12]

A. Ioffe and V. Tikhomirov, Theory of Extremal Problems, North-Holland, Amsterdam, 1979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete- Continuous Systems, Kluwer Academic / Plenum Publishers, New York, 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), 269-288.  Google Scholar

[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]. 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), 415-437. 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), 1043-1077. 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), 191-205. 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), 205-229. doi: 10.1137/0326013.  Google Scholar

[22]

J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111-128. doi: 10.1016/0022-247X(62)90033-1.  Google Scholar

[23]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.  Google Scholar

[24]

J. Warga, Variational problems with unbounded controls, J. SIAM Control Ser. A, 3 (1965), 424-438. doi: 10.1137/0303028.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

A. Arutyunov, D. Karamzin and F. Pereira, On constrained impulsive control problems, J. Math. Sci., 165 (2010), 654-688. 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), 1812-1843. doi: 10.1137/S0363012903430068.  Google Scholar

[3]

J.-P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 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), 1-35. 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), 30pp.  Google Scholar

[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.  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), 1205-1220. doi: 10.1137/0331057.  Google Scholar

[8]

A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory Appl., 81 (1994), 435-457. doi: 10.1007/BF02193094.  Google Scholar

[9]

V. Dykhta, Impulse-trajectory extension of degenerate optimal control problems, IMACS Ann. Comput. Appl. Math., 8 (1990), 103-109.  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), 942-957. doi: 10.1134/S0965542509060050.  Google Scholar

[11]

V. Gurman, On optimal processes with unbounded derivatives, Autom. Remote Control, 17 (1972), 14-21. Google Scholar

[12]

A. Ioffe and V. Tikhomirov, Theory of Extremal Problems, North-Holland, Amsterdam, 1979.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15]

B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete- Continuous Systems, Kluwer Academic / Plenum Publishers, New York, 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), 269-288.  Google Scholar

[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]. 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), 415-437. 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), 1043-1077. 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), 191-205. 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), 205-229. doi: 10.1137/0326013.  Google Scholar

[22]

J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111-128. doi: 10.1016/0022-247X(62)90033-1.  Google Scholar

[23]

J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.  Google Scholar

[24]

J. Warga, Variational problems with unbounded controls, J. SIAM Control Ser. A, 3 (1965), 424-438. doi: 10.1137/0303028.  Google Scholar

[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.  Google Scholar

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