The Pontryagin type maximum principle for Caputo fractional optimal control problems with terminal and running state constraints

In this paper, we consider the fractional optimal control problem with the terminal and running state constraints. The fractional calculus of derivatives and integrals can be viewed as generalizations of their classical notions to any arbitrary real order. In our problem setup, the dynamical system (or state equation) is captured by the fractional differential equation in the sense of (left) Caputo with order $\alpha \in (0,1)$, and the objective functional is formulated by the Bolza form expressed as the left Riemann-Liouville fractional integral. In addition, there are terminal and running state constraints; while the former is described by initial and final states within a convex set, the latter is given by an explicit instantaneous inequality state constraint. We obtain the Pontryagin maximum principle for the problem of this paper. The proof is based on an application of the Ekeland variational principle and the spike variation, by which we develop fractional variational and duality analysis using fractional calculus and functional analysis techniques, together with the representation results on (RL and Caputo) linear fractional differential equations. In fact, due to the inherent complex nature of the fractional control problem and the presence of the terminal and running state constraints, our maximum principle is new in the optimal control problem, context and its detailed proof must be different from that of the existing literature. As an application, we consider the linear-quadratic fractional optimal control problem with terminal and running state constraints, for which the optimal solution is obtained using the maximum principle of this paper.