We prove existence of solutions to upper semicontinuous differential inclusions and to time optimal control problems under conditions that are strictly weaker than the usual assumption of convexity. Key words. differential inclusions, time optimal control problems AMS subject classifications. 49J15, 34A34, 34A36 PII. S0363012902408046 1. Introduction. The condition of convexity with respect to the variable gradient has been of universal use in the calculus of variations, in optimal control, and

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... in differential inclusions to prove the existence of solutions. In fact, convexity is the property required in order to pass to a weak limit along a sequence, be it a minimizing sequence or a sequence of successive approximations, preserving the properties that are needed. This approach, however, because of its generality, need not always provide the best results, since it does not take into account possible additional information such as, for instance, the presence of symmetries in the problem. One is led to think that, by suitably exploiting these symmetries, the convexity condition could be substantially reduced. The purpose of the present paper is to show that, for the simplest of such symmetries, the time invariance in the problem of the existence of solutions to upper semicontinuous differential inclusions, convexity can be replaced by a strictly weaker condition, our almost convexity, below. Moreover, we show that, in the case of autonomous control systems of the form for the existence of a time optimal solution, Filippov's classical assumption of convexity of the images of the map F (x) = f (x, U (x)) can be replaced by the weaker assumption of almost convexity of the same images. As will be shown, our assumption does not imply that the set of solutions to the differential inclusion is closed in the space of continuous functions with uniform convergence, as happens in the case of the assumption of convexity, but only that the sections of this set of solutions are closed. This property is sufficient to establish the existence of time optimal solutions. Main results. The following is our assumption of almost convexity. Definition 1. Let X be a vector space. A set K ⊂ X is called almost convex if for every ξ ∈ coK there exist λ 1 and λ 2 , 0 ≤ λ 1 ≤ 1 ≤ λ 2 , such that λ 1 ξ ∈ K, λ 2 ξ ∈