Filippov's existence theorem and Pontryagin maximum principle for general Caputo fractional optimal control problems

Maïtine Bergounioux (1) , Loïc Bourdin (2)

1 IDP - Institut Denis Poisson
2 XLIM-MATHIS - Mathématiques & Sécurité de l'information

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, our first main result ensures the existence of at least one optimal solution. Secondly, 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. AMS subject classifications. 34K35; 26A33; 34A08; 49J15; 49K15; 49K40; 34H05; 93C15. 1. Introduction. 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. Most of the literature focuses on dynamical systems driven by ordinary differential equations. In that framework, the Filippov's theorem ensures the existence of at least one optimal trajectory under appropriate compactness and convexity assumptions (see [24], and see, e.g., [15, 39] for recent references and [19, Chapter 9] for some extensions). On the other hand, the Pontryagin Maximum Principle (denoted in short PMP), established at the end of the fifties (see [44], and see [26] for the history of this discovery), is the milestone of the classical optimal control theory. It provides a first-order necessary condition for optimality, by asserting that any optimal trajectory must be the projection of an extremal. The PMP then reduces the search of optimal trajectories to a boundary value problem on extremals. Optimal control theory, and in particular the PMP, have a wide field of applications in various domains. We refer the reader to textbooks such as [1, 15, 17, 18, 28, 32, 37, 48, 50, 51, 53] for theoretical results and/or practical applications, essentially for dynamical systems described by ordinary differential equations. Before concluding this paragraph, recall that the classical optimal control theory can be seen as a generalization of the historical calculus of variations from the 18th century (see, e.g., [39, Section 3.2]). From this point of view, the PMP corresponds

Mots clés: Optimal control; fractional calculus; Riemann-Liouville and Caputo operators; Filippov's existence theorem; Pontryagin maximum principle; needle-like variations; Ekeland's variational principle; adjoint vector; Hamiltonian system; Hamiltonian maximization condition; transversality conditions

Domaines: Optimisation et contrôle [math.OC]

Maïtine Bergounioux, Loïc Bourdin. Filippov's existence theorem and Pontryagin maximum principle for general Caputo fractional optimal control problems. 2018. ⟨hal-01796387v1⟩