Journal of Convex Analysis 27 (2020), No. 2, 565--581
Copyright Heldermann Verlag 2020
Optimality Conditions for Nonconvex Variational Problems with Integral Constraints in Banach Spaces
Faculty of Economics, Hosei University, Machida, Tokyo 194-0298, Japan
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.
Keywords: Lyapunov convexity theorem, saturated measure space, Bochner integral, Gelfand integral, value function, maximum principle, subdifferential, normal cone.
MSC: 28B20, 49J27, 49K27; 28B05, 46G10, 93C25.
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