Decay rate estimations for linear quadratic optimal regulators

Let $u(t)=-Fx(t)$ be the optimal control of the open-loop system $x'(t)=Ax(t)+Bu(t)$ 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 $x'(t)=(A-BF)x(t)$. 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 $(A, B_j)$, $j=1, \dots, N$, 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.