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.
Highlights
Optimal control theory is concerned with the analysis of controlled dynamical systems, where one aims at steering such a system from a given configuration to some desired target by minimizing or maximizing some criterion
We refer the reader to [3, 4, 26, 28, 32] and references therein for some initiating works. These articles constitute a first step in the field and essentially use fractional variational approaches to derive fractional versions of the weak Pontryagin Maximum Principle (PMP) for smooth and unconstrained fractional optimal control problems
A first attempt to establish a strong version of the PMP in the case of a general Riemann-Liouville fractional optimal control problem with a classical Lagrange cost and with control constraints can be found in Theorem 7 of [35]
Summary
Optimal control theory is concerned with the analysis of controlled dynamical systems, where one aims at steering such a system from a given configuration to some desired target by minimizing or maximizing some criterion. The Pontryagin Maximum Principle (denoted in short by PMP), established at the end of the fifties (see [45], and see [27] for the history of this discovery), is the milestone of the classical optimal control theory. It provides a first-order necessary condition for optimality and reduces the search of optimal trajectories to a boundary value problem. Many other variants exist in the literature (based on an implicit function theorem [1], Hahn-Banach separation theorem [16], or Aubin mini-max theorem [54] for example)
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