Constrained optimal control of stochastic switched affine systems using randomization

We consider a finite-horizon optimal control problem for a switched affine system with controlled switches, affected by uncertainty and subject to input and/or state constraints. We show how the logical statements that govern the underlying switching mechanism can be transformed into robust mixed-integer inequalities, leading to an infinite dimensional linear program with robust constraints. Following a randomized methodology, based on enforcing the constraints only on a finite number of uncertainty instances/scenarios, we relax the infinite dimensional program to a mixed-integer linear program, which is amenable to existing numerical tools. We establish a probabilistic link between the infinite dimensional robust program and its scenario-based relaxation, showing that the optimal solution of the latter is feasible, in a probabilistic sense, for the former