Frankowska, H.
(1989).
*Optimal trajectories associated with a solution of the contingent Hamilton-Jacobi equation.*
Applied Mathematics & Optimization 19 (1) 291-311. 10.1007/BF01448202.

## Abstract

In this paper we study the existence of optimal trajectories associated with a generalized solution to the 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 prove a relationship between superdifferentials of solutions introduced by Crandall et al. [10] and the Pontryagin principle and discuss the link of viscosity solutions with Clarke's approach to the Hamilton-Jacobi equation.

Item Type: | Article |
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Research Programs: | System and Decision Sciences - Core (SDS) |

Depositing User: | Romeo Molina |

Date Deposited: | 19 Apr 2016 09:40 |

Last Modified: | 27 Aug 2021 17:40 |

URI: | https://pure.iiasa.ac.at/12801 |

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