Existence of Solutions for Fractional Optimal Control Problems with Superlinear-subcritical Controls

TL;DR

This paper establishes the existence of solutions for fractional elliptic optimal control problems with superlinear-subcritical controls, extending classical control theory to fractional and nonlinear settings.

Contribution

It introduces a new framework for admissible controls in fractional problems, enabling the application of the Mountain Pass Theorem for existence results.

Findings

Existence of nontrivial solutions for fractional control problems.

Construction of admissible control sets satisfying growth and superlinear-subcritical conditions.

Extension of control theory to fractional elliptic equations.

Abstract

This paper gives an existence result for solutions to an elliptic optimal control problem based on a general fractional kernel, where the admissible controls come from a class satisfying both a growth bound and a superlinear-subcritical condition. Each admissible control is known to produce a nontrivial corresponding state by applying the Mountain Pass Theorem to fractional equations. The main theoretical contribution is the construction of a suitable set of admissible controls on which the the standard existence theory for control problems with linear and semi-linear state constraints can be adapted. Extra care is taken to explain what new difficulties arise for these types of control problems, to justify the limitations of this theory. For completeness, the corresponding local elliptic control problem is also studied.