Stochastic control with input and state constraints: A relaxation technique to ensure feasibility

We consider the problem of designing a finite-horizon control policy for a stochastic linear system subject to probabilistic constraints on both input and state variables. When the disturbance has unbounded support, a feasibility issue may arise due to the presence of the state constraint. In this paper, we address this issue by introducing a suitable relaxation of the original problem that ensures feasibility. The relaxation is such that the original state constraint is enforced whenever is possible; otherwise, the control that pushes the state closest to the constraint is chosen. This involves formulating a cascade of two chance-constrained optimization problems, which are tackled through a scenario-based randomized scheme expressly tailored to the problem at hand. The theoretical properties of the obtained solution are investigated and it is shown that randomization allows one to achieve computational tractability. The proposed approach finds immediate application to stochastic model predictive control.