HJB characterization for the combined integral–peak control problem

Determine whether a corresponding Hamilton–Jacobi–Bellman partial differential equation formulation can be derived whose solutions directly characterize the optimal-control problem combining an integral payoff with a trajectory-wide supremum (L^\infty) penalty, and investigate whether Filippov-type solution methods can characterize such solutions.

Background

The paper studies finite-horizon deterministic control problems whose objective combines an integral running reward and terminal reward with a penalty given by the maximum of a state-dependent quantity over the entire time horizon. The authors establish existence of optimal relaxed controls and construct smooth approximating problems whose solutions converge to the original problem, but the supremum term creates nonsmooth dynamics and complicates direct PDE analysis.

The paper derives dynamic-programming equations in integral form but leaves unresolved whether these relations yield an HJB partial differential equation whose solutions characterize the original combined problem. The authors specifically identify Filippov-type solution frameworks as a possible direction for addressing the discontinuities induced by the supremum term.