Relaxation of infinite dimensional control systems

Abstract

IT IS well known that convexity is crucial in the solution of optimal control problems. Namely, if the orientor field determined from the deparametrization operation is convex valued and satisfies some kind of set valued upper semicontinuity hypothesis, then one can prove the existence of optimal controls. The first such existence result was proved by Filippov [ 141 for a time-optimal control problem in IR”, with a control vector field that is continuously differentiable in the state variable [ 14, theorem 11. In that same paper, Filippov presents an example of a time optimal control problem that does not satisfy the convexity hypothesis and so does not have an optimal solution (see Section V of [14]). What happens in that example, is that the minimizing sequence of optimal controls (u,( -)I, z , , rapidly oscillates between + 1 and 1, remaining at each value for the same length of time. Then the Dirac measures 6u,(-), converge weakly (narrowly in the Bourbaki terminology), to the probability distribution &I, + $3_, . This suggests that we have to consider an extension of the original variational problem with measured valued controls, which satisfies the convexity requirement. This extension of control problems, was introduced independently by Gamkrelidze [15], Ghouila-Houri [ 171 and Warga [27]. Gamkrelidze called the controls of the extended problem “sliding regimes” or “chattering controls”, while Warga called them “relaxed controls”. Here we follow the terminology of Warga, which appears to be more widely used, at least in the western literature. We should also mention that other existence results like theorem 1 of Filippov [14] were also proved by Roxin [24] and Wazewski [29]. Later Berkovitz, Cesari, Olech and others improved those results. A very lucid account of the existing theory with complete bibliographical information, can be found in the books of Berkovitz [2] (in particular Chapters III and IV) and Cesari [12]. A detailed study of the relaxed problem, can also be found in the books of Gamkrelidze [ 161 and Warga [28], In the late sixties, Cesari in a series of important papers (see [8-111 and the references therein), addressed the question of existence of optimal controls for distributed parameter systems. Throughout his existence results, Cesari had a convexity hypothesis in the form of the well known by now, property (Q). Our work in this paper can be viewed to a certain extend, as a continuation of the work of Cesari 18-l 11. Namely we examine what happens when the con_ _ vexity hypothesis is no longer satisfied. If this is the case, then we can always convexify the orientor field and this in a control theoretic level corresponds to the introduction of time