Abstract
We consider the linear system of differential equations Au''(t)+Bu'(t) +Cu(t)=0, where A,B, and C are n x n real symmetric matrices, and study the problem of, given A and C, finding B which optimizes the asymptotic rate of decay of solutions of this equation. The optimal value for this rate of decay is found and we show that it is attained within this class of matrices. We then study some of the properties of optimal matrices. In particular, we show that the solution is not unique in general and that optimal matrices do not have to be positive definite. A sufficient condition for all optimal matrices to be positive definite is given and some examples are discussed.
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