Some Applications of Hamilton-Jacobi Inequalities for Classical and Impulsive Optimal Control Problems

This article is devoted to some applications of Hamilton–Jacobi inequalities for control problems of ordinary and impulsive dynamical systems. We focus on the study of necessary and sufficient global optimality conditions. The optimality conditions are obtained by using certain families of Lyapunov type functions, where Lyapunov type functions can be strongly or weakly monotone solutions of the corresponding Hamilton–Jacobi inequalities. For classical problems of the optimal control theory, new variants of the Caratheodory and Krotov types optimality conditions and conditions of the so-called Hamilton–Jacobi canonical optimality theory are proposed and compared. For impulsive dynamical systems with trajectories of bounded variation, conditions for the strong and weak monotonicity of the Lyapunov type functions are obtained. These monotonicity conditions are formulated in terms of generalized Hamilton–Jacobi inequalities and they allow to investigate various questions of the position and optimal control theory for impulsive systems.

[1]  F. Rampazzo,et al.  Dynamic Programming for Nonlinear Systems Driven by Ordinaryand Impulsive Controls , 1996 .

[2]  Monica Motta,et al.  Discontinuous Solutions to Unbounded Differential Inclusions under State Constraints. Applications to Optimal Control Problems , 1999 .

[3]  Andrei I. Subbotin,et al.  Generalized solutions of first-order PDEs - the dynamical optimization perspective , 1994, Systems and control.

[4]  Geraldo Nunes Silva,et al.  Hamilton-Jacobi Conditions for an Impulsive Control Problem , 2001 .

[5]  Geraldo Nunes Silva,et al.  Stability for impulsive control systems , 2002 .

[6]  Alexander N. Daryin,et al.  Dynamic programming for impulse controls , 2008, Annu. Rev. Control..

[7]  D. Yu. Karamzin,et al.  Necessary conditions of the minimum in an impulse optimal control problem , 2006 .

[8]  V. Dykhta.  Lyapunov–Krotov Inequality and Sufficient Conditions in Optimal Control , 2004 .

[9]  J. Danskin.  The Theory of Max-Min and its Application to Weapons Allocation Problems , 1967 .

[10]  Frank H. Clarke,et al.  A proximal characterization of the reachable set , 1996 .

[11]  Fernando M. Lobo Pereira,et al.  A Nondegenerate Maximum Principle for the Impulse Control Problem with State Constraints , 2005, SIAM J. Control. Optim..

[12]  Sérgio Loureiro Fraga,et al.  On the feedback control of impulsive dynamic systems , 2008, 2008 47th IEEE Conference on Decision and Control.

[13]  G. N. Silva,et al.  Necessary Conditions for Optimal Impulsive Control Problems , 1997 .

[14]  Yu. S. Ledyaev,et al.  Nonsmooth analysis and control theory , 1998 .

[15]  R. Vinter.  Convex duality and nonlinear optimal control , 1993 .

[16]  Francis H. Clarke,et al.  Nonconvex Duality in Optimal Control , 2005, SIAM J. Control. Optim..

[17]  A. V. Seleznev,et al.  A Dynamic Programming Approach to the Impulse Control Synthesis Problem , 2005, Proceedings of the 44th IEEE Conference on Decision and Control.

[18]  L. Young,et al.  Lectures on the Calculus of Variations and Optimal Control Theory. , 1971 .

[19]  A. A. Mili︠u︡tin,et al.  Calculus of variations and optimal control , 1998 .

[20]  V. Dykhta,et al.  Optimal control: Nonlocal conditions, computational methods, and the variational principle of maximum , 2009 .

[21]  Yu. S. Ledyaev,et al.  The Synthesis of Universal Feedback Pursuit Strategies in Differential Games , 1997 .

[22]  Richard B. Vinter,et al.  Optimal Control , 2000 .

[23]  M. Bardi,et al.  Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations , 1997 .

[24]  G. N. Silva,et al.  Measure Driven Differential Inclusions , 1996 .

[25]  S. T. Zavalishchin,et al.  Dynamic Impulse Systems: Theory and Applications , 1997 .

[26]  Jerzy Zabczyk.  Dynamic programming for impulse control , 2008 .

[27]  M. Hestenes.  Calculus of variations and optimal control theory , 1966 .

[28]  B. Miller.  The generalized solutions of nonlinear optimization problems with impulse control , 1996 .

[29]  F. Rampazzo,et al.  Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls , 1995, Differential and Integral Equations.

[30]  Geraldo Nunes Silva,et al.  Necessary conditions of optimality for vector-valued impulsive control problems , 2000 .

[31]  A. Ioffe,et al.  Theory of extremal problems , 1979 .

[32]  Valeriano Antunes de Oliveira,et al.  Invariance for impulsive control systems , 2008 .