Optimization estimates of dynamical system characteristics (Q2739980)

The problem of finding the best estimates for the transition characteristics is reduced to the optimization problem NEWLINE\[NEWLINE H_0= \text{arg inf} _{H \in Z [H ]} \{\phi [ \lambda _{\min} (G[ H]), \lambda_{\max} (G[ H]) ]\}, NEWLINE\]NEWLINE where \(\lambda _{\min} (\cdot), \lambda _{\max}(\cdot)\) are the minimal and maximal characteristic numbers of symmetric positive definite matrices, respectively, \(G[ H]\) is a linear matrix operator, \(\phi[\cdot,\cdot]\) is a continuously differentiable function, \(Z[H]\) is a set of positive definite matrices. The authors prove the existence of a solution of the considered optimization problem, obtain the necessary and sufficient conditions of optimality in a particular case and apply these results to the construction of estimates of the asymptotic stability domain for a system of differential equations with quadratic right sides.