Effective Computability of Solutions of Differential Inclusions The Ten Thousand Monkeys Approach

In this paper we consider the computability of the solution of the initialvalue problem for differential inclusions with semicontinuous right-hand side. We present algorithms for the computation of the solution using the “ten thousand monkeys” approach, in which we generate all possible solution tubes, and then check which are valid. In this way, we show that the solution of an upper-semicontinuous differential inclusion is upper-semicomputable, and the solution of a differential inclusion defined by a one-sided locally Lipschitz function is lower-semicomputable computable. We show that the solution of a locally Lipschitz differential equation is computable even if the function is not effectively locally Lipschitz. We also recover a result of Ruohonen, in which it is shown that if the solution is unique, then it is computable, even if the right-hand side is not locally Lipschitz. We also prove that the maximal interval of existence for the solution must be effectively enumerable open, and give an example of a computable locally Lipschitz function which is not effectively locally Lipschitz.