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.

We consider the terminal state-constrained optimal control problem for Volterra integral equations with singular kernels. A singular kernel introduces abnormal behavior of the state trajectory with respect to the parameter of $\alpha \in (0,1)$. Our state equation covers various state dynamics such as any types of classical Volterra integral equations with nonsingular kernels, (Caputo) fractional differential equations, and ordinary differential state equations. We prove the maximum principle for the corresponding state-constrained optimal control problem. In the proof of the maximum principle, due to the presence of the (terminal) state constraint and the control space being only a separable metric space, we have to employ the Ekeland variational principle and the spike variation technique, together with the intrinsic properties of distance function and the generalized Gronwall's inequality, to obtain the desired necessary conditions for optimality. The maximum principle of this paper is new in the optimal control problem context and its proof requires a different technique, compared with that for classical Volterra integral equations studied in the existing literature.

In this paper, we study the infinite-dimensional endpoint state-constrained optimal control problem for fractional evolution equations. The state equation is modeled by the $\mathsf{X}$-valued left Caputo fractional evolution equation with the analytic semigroup, where $\mathsf{X}$ is a Banach space. The objective functional is formulated by the Bolza form, expressed in terms of the left Riemann-Liouville (RL) fractional integral running and initial/terminal costs. The endpoint state constraint is described by initial and terminal state values within convex subsets of $\mathsf{X}$. Under this setting, we prove the Pontryagin maximum principle. Unlike the existing literature, we do not assume the strict convexity of ${\mathsf{X}}^{\ast }$, the dual space of $\mathsf{X}$. This assumption is particularly important, as it guarantees the differentiability of the distance function of the endpoint state constraint. In the proof, we relax this assumption via a separation argument and constructing a family of spike variations for the Ekeland variational principle. Subsequently, we prove the maximum principle, including nontriviality, adjoint equation, transversality, and Hamiltonian maximization conditions, by establishing variational and duality analysis under the finite codimensionality of initial- and end-point variational sets. Our variational and duality analysis requires new representation results on left Caputo and right RL linear fractional evolution equations in terms of (left and right RL) fractional state transition operators. Indeed, due to the inherent complex nature of the problem of this paper, our maximum principle and its proof technique are new in the optimal control context. As an illustrative example, we consider the state-constrained fractional diffusion PDE control problem, for which the optimality condition is derived by the maximum principle of this paper.