Optimality Conditions for Nonconvex Variational Problems with Integral Constraints in Banach Spaces

TL;DR

This paper establishes that saturation of measure spaces is essential for solving nonconvex variational problems with integral constraints in Banach spaces, providing optimality conditions and existence results without relaxed controls.

Contribution

It demonstrates the critical role of measure space saturation in existence and characterization of solutions, extending optimality conditions to infinite-dimensional settings.

Findings

Saturation is necessary for solution existence in nonconvex variational problems.

Optimality characterized via maximum principle for the Hamiltonian.

Existence of solutions is equivalent to measure space saturation.

Abstract

This paper exemplifies that saturation is an indispensable structure on measure spaces to obtain the existence and characterization of solutions to nonconvex variational problems with integral constraints in Banach spaces and their dual spaces. We provide a characterization of optimality via the maximum principle for the Hamiltonian and an existence result without the purification of relaxed controls, in which the Lyapunov convexity theorem in infinite dimensions under the saturation hypothesis on the underlying measure space plays a crucial role. We also demonstrate that the existence of solutions for certain class of primitives is necessary and sufficient for the measure space to be saturated.