Nash Equilibrium of Bi-objective Optimal Control of Fractional Space-Time Parabolic PDE

TL;DR

This paper studies the existence, uniqueness, and numerical computation of Nash equilibria in a bi-objective optimal control problem governed by fractional space-time parabolic PDEs involving Caputo derivatives and fractional Laplacians.

Contribution

It establishes theoretical conditions for Nash equilibrium existence and uniqueness in fractional PDE control problems and proposes an efficient iterative numerical scheme.

Findings

Theoretical proof of Nash equilibrium existence and uniqueness.

Development of conjugate gradient algorithms for fractional PDE control.

Numerical experiments confirm theoretical results and demonstrate computational efficiency.

Abstract

This work investigates the existence and uniqueness of the Nash equilibrium (solutions to competitive problems in which individual controls aim at separate desired states) for a bi-objective optimal control problem governed by a fractional space-time parabolic partial differential equation. The governing equation involves a Caputo fractional derivative with respect to time of order $\gamma$ in (0,1) and a fractional Laplacian in the spatial variables of order $s$ in (0,1). The system is associated with two independent controls, each aiming at different targets. The problem is formulated as a distributed optimal control system with quadratic cost functionals. Existence and uniqueness of the Nash equilibrium are established under convexity and coercivity assumptions. The solution is computed using conjugate gradient algorithms applied iteratively to the discretized optimal control problems. The numerical experiments agree with the theoretical estimates and demonstrate the efficiency of the proposed scheme.