Optimization of a fractional Mayer problem - existence of solutions, maximum principle, gradient methods

Abstract

Abstract. In the paper, we study a linear-quadratic optimal control problem of Mayer type given by a fractional control system. First, we prove a theorem on the existence of a solution to such a problem. Next, using the local implicit function theorem, we derive a formula for the gradient of a cost functional under constraints given by a control system and prove a maximum principle in the case of a control constraint set. The formula for the gradient is used to implement the gradient methods for the problem under consideration.

On a linear-quadratic problem with Caputo derivative

“…also [11] for the comprehensive study of the classical linear-quadratic problems). The case of the fractional Riemann-Liouville derivative is studied in [10] for J containing only the pointwise term, in the space AC α,2 a+ (defined below) of trajectories and in the space L 2 of controls. The presence of the Caputo derivative means that the problem has to be investigated in quite different (given above) spaces of trajectories and controls.…”