Sufficient conditions for the absence of relaxation gaps in state-constrained optimal control

TL;DR

This paper addresses the absence of relaxation gaps in state-constrained optimal control by introducing nonconvexity-tolerant sufficient conditions that do not rely on convexity of the Lagrangian or velocity sets. It leverages Filippov--Wa{\u017c}ewski-type results and a Bernard-type superposition to establish when $M_o=M_y=M_c$, and further uses inward pointing conditions and boundary-perturbation stability to extend these equalities to domain closures. The authors also provide practical gap bounds via inner approximations and show how the moment–SOS hierarchy can compute these bounds, enabling SDP-based computation even in nonconvex, noncompact settings. The work broadens the applicability of relaxation-based methods to state-constrained problems, with implications for PDE contexts and numerical optimization tools like OCP08.

Abstract

This work presents new sufficient conditions for the absence of a gap corresponding to Young measure and occupation measure relaxations for constrained optimal control problems. Unlike existing conditions, these sufficient conditions do not rely on convexity of the Lagrangian or the set of admissible velocities. We use these conditions to derive new bounds for the size of the relaxation gap.

Theorems & Definitions (16)

• Lemma 1
• Theorem 2: V93
• Example 3
• Theorem 4: No gap on open domains, and on some closed domains
• Theorem 5: Filippov-Wažewski frankowska2000filippov
• proof : Proof of Theorem \ref{['cor:int']}
• Remark 6: Simpler version of \ref{['IPC:lipschitzity']}
• Remark 7: Simpler versions of \ref{['IPC:ipc']}
• Theorem 8
• Remark 9