Decay rate estimations for linear quadratic optimal regulators

Let u(t)=−Fx(t)u(t)=−Fx(t) be the optimal control of the open-loop system x′(t)=Ax(t)+Bu(t)x′(t)=Ax(t)+Bu(t) in a linear quadratic optimization problem. By using different complex variable arguments, we give several lower and upper estimates of the exponential decay rate of the closed-loop system x′(t)=(A−BF)x(t)x′(t)=(A−BF)x(t). Main attention is given to the case of a skew-Hermitian matrix A. Given an operator A, for a class of cases, we find a matrix B that provides an almost optimal decay rate.We show how our results can be applied to the problem of optimizing the decay rate for a large finite collection of control systems (A,Bj)(A,Bj), j=1,…,Nj=1,…,N, and illustrate this on an example of a concrete mechanical system. At the end of the article, we pose several questions concerning the decay rates in the context of linear quadratic optimization and in a more general context of the pole placement problem.