Optimal Network Control With Stochastic Link Variations and Indefinite Quadratic Constraints

In this paper, we consider the problem of optimal control of a network with stochastic links modeled as multiplicative noise with given second moments and arbitrary probability distributions. Additional quadratic constraints are imposed, where the quadratic forms may be indefinite and, thus, not necessarily convex. We show that the problem can be transformed to a semidefinite program. The optimization problem is over a certain variable serving as the covariance matrix of the state and the controller. We show that affine controllers are optimal and depend on the optimal covariance matrix. Furthermore, we show that optimal controllers are linear if all of the quadratic forms are convex in the control variable. The solutions are presented for the finite and infinite horizon cases. We give a necessary and sufficient condition for mean square stabilizability of the dynamical system with additive and multiplicative noise. The condition is a Lyapunov-like condition whose solution is again given by the covariance matrix of the state and the control variable. The results are illustrated with an example.