Optimal trajectories associated to a solution of contingent Hamilton-Jacobi equation
Frankowska, Halina (1987), Optimal trajectories associated to a solution of contingent Hamilton-Jacobi equation, 26th IEEE Conference on Decision and Control, 1987. Proceedings, IEEE, p. 727-732. http://dx.doi.org/10.1109/CDC.1987.272464
TypeCommunication / Conférence
Conference title26th IEEE Conference on Decision and Control, 1987
Conference cityLos Angeles
Book title26th IEEE Conference on Decision and Control, 1987. Proceedings
MetadataShow full item record
Abstract (EN)In this paper we study the existence of optimal trajectories associated with a generalized solution to Hamilton-Jacobi-Bellman equation arising in optimal control. In general, we cannot expect such solutions to be differentiable. But, in a way analogous to the use of distributions in PDE, we replace the usual derivatives with "contingent epiderivatives" and the Hamilton-Jacobi equation by two "contingent Hamilton-Jacobi inequalities". We show that the value function of an optimal control problem verifies these "contingent inequalities". Our approach allows the following three results: (a) The upper semicontinuous solutions to contingent inequalities are monotone along the trajectories of the dynamical system. (b) With every continuous solution V of the contingent inequalities, we can associate an optimal trajectory along which V is constant. (c) For such solutions, we can construct optimal trajectories through the corresponding optimal feedback. They are also "viscosity solutions" of a Hamilton-Jacobi equation. Finally we discuss the link of viscosity solutions with Clarke's approach to the Hamilton-Jacobi equation.
Subjects / KeywordsDifferential equations; Viscosity; Optimal control
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Unicité des solutions optimales et absence de chocs pour les équations d'Hamilton-Jacobi-Bellman et de Riccati Frankowska, Halina; Byrnes, Christopher (1992) Article accepté pour publication ou publié