Linear Quadratic Optimal Control of Continuous-Time LTI Systems With Random Input Gains

This note studies the linear quadratic (LQ) optimal control of continuous-time linear time-invariant systems with random gains imposed on the input channels. We start from the indefinite LQ problem, in which the cost weighting matrix can be indefinite. The definite LQ problem is discussed as a special case. The main novelty originates from the point of view that in networked control, designing the channels and controller jointly often leads to an easier problem and achieves better performance than designing them separately. Specifically, we formulate the LQ problem as a channel/controller co-design problem assuming that the channel capacities can be allocated among the input channels subject to an overall capacity constraint. Necessary and sufficient conditions are obtained for the well-posedness and the attainability of the indefinite LQ problem under a given channel capacity allocation satisfying the stabilization requirement. The optimal controller is given by a linear state feedback associated with the mean-square stabilizing solution of a modified algebraic Riccati equation.