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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 |
  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.
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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