On mean square boundedness of stochastic linear systems with bounded controls

We start by summarizing the state of the art in stabilization of stochastic linear systems with bounded inputs and highlight remaining open problems. We then report two new results concerning mean-square boundedness of a linear system with additive stochastic noise. The first states that, given any nonzero bound on the controls, it is possible to construct a policy with bounded memory requirements that renders a marginally stable stabilizable system mean-square bounded in closed-loop. The second states that it is not possible to ensure mean-square boundedness in closed-loop with a bounded control policy for systems affected by unbounded noise and having at least one eigenvalue outside the unit circle.