Optimal control of fractional quantum systems governed by the Caputo time-fractional Schrödinger equation

This paper investigates an optimal control framework for time-fractional quantum systems governed by the Caputo time-fractional Schrödinger equation. The model incorporates a bilinear control term and accommodates general bounded potentials, allowing for a broad range of quantum applications. We establish well-posedness of the state equation using a Galerkin approximation and energy estimates adapted to the fractional setting. Existence of an optimal control is proven via the direct method in the calculus of variations, and first-order necessary optimality conditions are derived through a variational approach involving a suitably defined adjoint system with a right-sided Caputo derivative. The theoretical results are illustrated with several numerical examples, including a one-dimensional harmonic potential, a Gaussian wave packet in a square well, and a two-dimensional harmonic oscillator. In each case, the proposed algorithm effectively steers the quantum state toward the desired target while minimizing the control cost, demonstrating both accuracy and efficiency of the method.