Deterministic and Non - Blocking Hybrid

Consider the ordinary diierential equation _ xt = F , x t ; x 0 = x 0 : 1 A solution 1 on 0; T , T T 0, to 1 is a continuously diierentiable function x : : 0 ; T !R n satisfying xt = x 0 + Z t 0 F , x s ds: We m a y ask for which functions F there exist a solution to 1 and, if so, if the solution is unique. Is it, for instance, suucient that F is a continuous function? The answer is no concerning the uniqueness issue, as illustrated by the following example: Example 1. Let F z = p z and consider the diierential equation 1 for x 0 = 0. Then both xt = t 2 = 4 and xt = 0 are solutions. 2 If we, however, not only assume continuity but also restrict the rate of change of F , w e can show some nice properties. The following deenition is needed, where k k denotes the Euclidean norm kzk 2 = z 2 1 + + z 2 n : a function F is locally Lipschitz continuous if there exist r; L 0 such that kFz , F yk L k z , y k for all kzk; kyk r. If there exists L L 0 independent o f r such that the condition holds for any r r 0, then F is globally Lipschitz continuous. Theorem 1 Existence and Uniqueness. If F is locally Lipschitz continuous , then there exists 0 such that 1 has a unique solution on 0; ;. Moreover , if F is globally Lipschitz continuous, then 1 c an be chosen arbitrarily large. 1 This solution is called a solution in the sense of Caratheodory. 2 Actually, for any t0 0, xt = 0 for t 2 0; t 0 and xt = t , t 0 2 = 4 for t 2 t0; 1 is a solution.