Pontryagin maximum principle for general Caputo fractional optimal control problems with Bolza cost and terminal constraints

Abstract

In this paper we focus on a general optimal control problem involving a dynamical system described by a nonlinear Caputo fractional differential equation of order 0 < α ≤ 1, associated to a general Bolza cost written as the sum of a standard Mayer cost and a Lagrange cost given by a Riemann-Liouville fractional integral of order β ≥ α. In addition the present work handles general control and mixed initial/final state constraints. Adapting the standard Filippov's approach based on appropriate compactness assumptions and on the convexity of the set of augmented velocities, we give an existence result for at least one optimal solution. Then, the major contribution of this paper is the statement of a Pontryagin maximum principle which provides a first-order necessary optimality condition that can be applied to the fractional framework considered here. In particular, Hamiltonian maximization condition and transversality conditions on the adjoint vector are derived. Our proof is based on the sensitivity analysis of the Caputo fractional state equation with respect to needle-like control perturbations and on Ekeland's variational principle. The paper is concluded with two illustrating examples and with a list of several perspectives for forthcoming works.