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Gramian Optimization with Input-Power Constraints

The optimization of the controllability Gramian of a given linear continuous-time system via proper design of its input matrix is a rather unexplored problem in the control literature. The most well-studied version of this problem is related to the optimal placement of actuators in a large-scale system. In this case, the input matrix is constrained to be a matrix whose columns are elements of the canonical basis. Inspired by this problem, in this paper we investigate two Gramian optimization problems in which the input matrix is subject to a Frobenius norm (or "power") constraint. Specifically, we consider the minimization of the maximum eigenvalue of the controllability Gramian and its dual, namely the maximization of the minimum eigenvalue of the Gramian, subject to a fixed Frobenius norm constraint on the input matrix. Surprisingly, these two problems admit the same optimal value, which is a simple function of the trace of the system state matrix. On the other hand, the properties of the optimal input matrices turn out to be much more difficult to capture. In this paper, we also establish several characterizations of these optimal matrices.