Analysis of an Optimal Control Problem of the Backward Fractional Feynman–Kac Equation with Temporal Approximation

Abstract. In this paper, we discuss an optimal control problem governed by the backward fractional Feynman–Kac equation, which depicts the probability density function of functionals of widespread diffusion phenomena in various multiscale fields. In the constrained equation, the time-space coupled nonlocal operator and its noncommutativity with Laplacian bring significant difficulties. The well-posedness, optimality conditions, and solution’s regularity of the continuous optimal control problem are established. Then, we propose a temporal semidiscrete scheme for the optimal control problem by applying the backward Euler convolution quadrature formula to discretize the Riemann–Liouville fractional substantial derivative in the equation. In addition, the error estimate of the proposed semidiscrete scheme is rigorously established, and almost optimal convergence of [Formula: see text] is obtained depending only on the regularity assumptions on the data and without extra assumptions on the solution. Some numerical examples are illustrated to verify the theoretical results.