Sufficiency and singularity in optimal control

optimization problem in a Banach space. The assumption that the strengthened Legendre– Clebsch condition holds is crucial. In particular, we refer the reader to Milyutin & Osmolovskiı̌ (1998) where the importance of this condition is fully explained. In this paper, we consider a fixed-endpoint optimal control problem with equality control constraints and provide a new set of sufficient conditions for a strong minimum that include the Legendre–Clebsch necessary condition but not its strengthened form. The sufficiency theorem includes also a new version of the strengthened Weierstrass condition. The proof is strongly related to a direct sufficiency proof (in the sense that it deals explicitly with the positivity of the second variation along admissible variations) given by Hestenes (1966) for the fixed-endpoint isoperimetric problem in the calculus of variations. The development of this technique as it appears in Hestenes (1966), as well as its application to more general problems, can be traced back to different papers of the author and McShane (see Hestenes, 1934, 1936, 1937, 1946, 1947a,b, 1948; McShane, 1942). The original proof deals with the concept, introduced by Hestenes, of a directional convergent sequence of trajectories (absolutely continuous functions) which is in turn a generalization of the concept of directional convergence for vectors in the finite-dimensional case. This notion relies on the specific calculus of variations problem considered in Hestenes (1966) but, as we show in this paper, it can be modified to cover optimal control problems. The modification we introduce not only shows that the main ideas of the proof can be generalized, but also provides a new set of sufficient conditions which, due to the possible singularity of the process under consideration, lies beyond the scope of the references mentioned above. It is worth mentioning that this technique has recently been extended in order to solve isoperimetric control problems without the requirement of transforming the original problem into a problem of Lagrange (see Rosenblueth & Sánchez Licea, 2011a). Previous attempts to generalize the proof of Hestenes can be found in Rosenblueth (1999) and Rosenblueth & Sánchez Licea (2010). In the first of these papers, we obtained sufficient conditions for local optimality, both weak and strong, assuming the strengthened Legendre–Clebsch condition. In the second one, several auxiliary results allowed us to establish the sufficiency theorem of Rosenblueth (1999) for weak minima in a new and clearer way. Those auxiliary results partly coincide with those used in the sufficiency proof for strong minima derived in this paper, without the assumption of the strengthened Legendre–Clebsch condition. The paper is organized as follows. In Section 2, we pose the problem we shall deal with, introduce some notation and basic definitions and state the main result. For comparison reasons we also state, as a corollary, the sufficiency theorem for strong minima obtained in Rosenblueth (1999). In Section 3, we explain certain relevant issues related to the new set of sufficient conditions with the idea of clarifying some of the underlying principles on which the proof is based. Section 4 is devoted to the proof of the sufficiency theorem together with the statement of some auxiliary results which are used throughout the proof. The auxiliary results, which deal with a generalization of the notion of a directional convergent sequence of trajectories, are established in Section 5. In the final section, we illustrate the usefulness of the new sufficiency theorem by means of six examples of singular processes which are strict local minima of the problems in hand. 2. Statement of the problem and the main results The fixed-endpoint optimal control problem we shall study in this paper can be stated as follows. Suppose we are given an interval T := [t0, t1] in R, two points ξ0 and ξ1 in R and functions L and f mapping T × R × R to R and R, respectively, and ψ mapping R to R. at U nivrsidad N aconal A tà 3om a de M à © xico on A ril 9, 2013 htp://im am ci.oxfournals.org/ D ow nladed from SUFFICIENCY AND SINGULARITY IN OPTIMAL CONTROL 39 Let X := AC(T ; R) denote the space of absolutely continuous functions mapping T to R, let U := L1(T ; R), set Z := X × U , and denote by Ze the set of all (x, u) ∈ Z satisfying (a) L(t, x(t), u(t)) is integrable on T . (b) ẋ(t)= f (t, x(t), u(t)) a.e. in T . (c) x(t0)= ξ0, x(t1)= ξ1. (d) ψ(u(t))= 0 (t ∈ T). The problem we shall deal with, which we label (P), is that of minimizing I over Ze, where