Solution for a System of Hamilton–Jacobi Equations of Special Type and a Link with Nash Equilibrium
The paper is concerned with systems of Hamilton–Jacobi PDEs of the special type. This type of systems of Hamilton–Jacobi PDEs is closely related with a bilevel optimal control problem. The paper aims to construct equilibria in this bilevel optimal control problem using the generalized solution for the system of the Hamilton–Jacobi PDEs. We introduce the definition of the solution for the system of the Hamilton–Jacobi PDEs in a class of multivalued functions. The notion of the generalized solution is based on the notions of minimax solution and M-solution to Hamilton–Jacobi equations proposed by Subbotin. We prove the existence theorem for the solution of the system of the Hamilton–Jacobi PDEs.
This work was supported by the Russian Fond of Fundamental Researches under grant No. 17-01-00074.
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