Monotone Invariant Solutions to Differential Inclusions

Let a given set be endowed with a preference preordering, and consider the problem of finding a solution to the differential inclusion A(OeS(*(O) which remains in the given set and evolves monotonically with respect to the preordering. We give sufficient conditions for the existence of such a trajectory, couched in terms of a notion of tangency developed by Clarke. No smoothness or convexity is involved in the construction, which uses techniques of Filippov. 1. Statement of the problem Let X be a compact subset of R", and S a multifunction (set-valued function) from X into R" with non-empty compact values. We regard X as the state set of a dynamical system and S(x) as the set of feasible velocities of the system when its state is x. We introduce a preordering " y ^ x " (y is better than x) on X (that is, a relation which is both reflexive and transitive). Let [0, T] be any finite interval (T > 0). We say that an absolutely continuous function x from [0, T] into R" is a " monotone invariant trajectory for S starting at x o eX"if (i) dx/dt e S(x(tj) for almost all t in [0, 7 ] , (ii) x(0) = x 0 , W } (1.1) (iii) x(t)eXfora\lt€[0, T], l (iv) if t^ s then x(t) is better than x(s). Our main result, which we state in this section, gives reasonable sufficient conditions implying the existence of at least one monotone invariant trajectory. In §2 we discuss a notion of tangency for arbitrary closed sets [5] which is central in our results, while §3 is devoted to the proof of the main theorem. We define the set p(x) = {yeX such that y^x} of elements y better than x. The multifunction P satisfies the properties: (i) V x e l , xeP(x)) (1-2) (ii) VxeX,VyeP(x), P(y) a P(x). J (Conversely, if P is a multifunction from X to X satisfying (1.2), the relation y ^= x defined by y e P(x) is a preordering on X.) We recall that P is said to be Lipschitz if and only if there exists L > 0 such that P(x) cz P(y) + L\\x—y\\B, where B is the unit ball. We recall also that S is upper (respectively lower) semicontinuous if, for Received 24 January, 1977.