Fractional optimal control problems with trajectory-dependent singular coefficients

Abstract

We investigate a class of fractional optimal control problems governed by semilinear fractional differential equations with trajectory-dependent singular coefficients. The state equation is formulated using the Caputo fractional derivative and contains a singular coefficient of the form (t−s(t))−β\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(t-s(t))^{-\beta }$\end{document}, where the prescribed trajectory s(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$s(t)$\end{document} satisfies a suitable separation condition ensuring integrability of the singular kernel. This framework combines fractional memory effects with time-dependent singular perturbations, leading to several analytical and numerical difficulties. Existence and uniqueness of mild solutions are established by means of fixed-point arguments and fractional semigroup techniques. A fractional optimality system is then derived using variational methods together with a fractional integration-by-parts formula, leading to a coupled state–adjoint system and an explicit characterization of the optimal control. To illustrate the theoretical results, we develop a numerical scheme based on the Grünwald–Letnikov approximation in time combined with finite-difference discretization in space within a forward–backward sweep iterative framework. Numerical simulations demonstrate that the proposed method effectively captures the interaction between singular perturbations and fractional memory effects while steering the system toward the desired target trajectory. The convergence behavior of the numerical approximation is also discussed.