On the convexity of optimal decentralized control problem and sparsity path

This paper is concerned with an important special case of the stochastic optimal decentralized control (SODC) problem, where the objective is to design a static structurally constrained controller for a stable stochastic system. This problem is non-convex and hard to solve in general. We show that if either the measurement noise covariance or the input weighting matrix is not too small, the problem is locally convex. Under such circumstances, the design of a decentralized controller with a bounded norm subject to an arbitrary sparsity pattern is naturally a convex problem. We also study the problem of designing a sparse controller using a regularization technique, where the control structure is not pre-specified but penalized in the objective function. Under some genericity assumptions, we prove that this method is able to design a decentralized controller with any arbitrary sparsity level. Although this paper is focused on stable systems, the results can be generalized to unstable systems as long as an initial stabilizing controller with a desirable structure is known a priori.