Effective computability of solutions of differential inclusions --- the ten thousand monkeys approach

In this paper we consider the computability of the solution of the initial-value problem for differential equations and 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 a locally Lipschitz differential equation is computable even if the function is not effectively locally Lipschitz, and recover a result of Ruohonen, in which it is shown that if the solution is unique, then it is computable. We give an example of a computable locally Lipschitz function which is not effectively locally Lipschitz. We also show that the solutions of a convex-valued upper-semicontinuous differential inclusion are upper-semicomputable, and the solutions of a lower-semicontinuous one-sided Lipschitz differential inclusion are lower-semicomputable.