The value functions approach and Hopf-Lax formula for multiobjective costs via set optimization

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS

卷:483

Article

Hamel, A. H.¹  Visetti, D.¹ 

[1] Free Univ Bozen Bolzano, Bolzano, Italy

关键词: Keywords multicriteria calculus of variations;    Value function;    Hopf-Lax formula;    Bellman's principle;    Hamilton-Jacobi equation;    Set relations;   

DOI  :  10.1016/j.jmaa.2019.123605

来源: Elsevier

【 摘 要 】

The complete-lattice approach to optimization problems with a vector- or even set-valued objective already produced a variety of new concepts and results and was successfully applied in finance, statistics and game theory. So far, it has only been applied to set-valued dynamic risk measures within a stochastic framework, but not to deterministic calculus of variations and optimal control problems. In this paper, a multi-objective calculus of variations problem is considered which is turned into a set-valued problem by a straightforward extension. A new set-valued value function is introduced, for which a Bellman's optimality principle holds. Also the classical result of the Hopf-Lax formula holds for the generalized value function. Finally, a derivative with respect to the time and a directional derivative with respect to the state variable of the value function are defined. The value function is proved to be a solution of a corresponding Hamilton-Jacobi equation. (C) 2019 Elsevier Inc. All rights reserved.

【 授权许可】

Free