Second-order sensitivity relations and regularity of the value function for Mayer's problem in optimal control

TL;DR

This paper explores second-order sensitivity relations for the value function in Mayer optimal control problems, establishing conditions for its regularity and twice differentiability along optimal trajectories.

Contribution

It extends sensitivity analysis to second order, linking sub/superjets with regularity of the value function in differential inclusion-based control problems.

Findings

Value function is twice differentiable along optimal trajectories.

Sufficient conditions for local $C^2$ regularity of the value function.

Exclusion of conjugate points through sensitivity analysis.

Abstract

This paper investigates the value function, $V$, of a Mayer optimal control problem with the state equation given by a differential inclusion. First, we obtain an invariance property for the proximal and Fr\'echet subdifferentials of $V$ along optimal trajectories. Then, we extend the analysis to the sub/superjets of $V$, obtaining new sensitivity relations of second order. By applying sensitivity analysis to exclude the presence of conjugate points, we deduce that the value function is twice differentiable along any optimal trajectory starting at a point at which $V$ is proximally subdifferentiable. We also provide sufficient conditions for the local $C^2$ regularity of $V$ on tubular neighborhoods of optimal trajectories.