On stability of solutions to linear systems with periodic coefficients (Q2729334)

The authors consider the following system of differential equations with periodic coefficients: NEWLINE\[NEWLINE \frac{dy}{dt} = A(t)y,\quad t \geq 0, NEWLINE\]NEWLINE where \(A(t)\) is an \(n\times n\)-matrix with continuous \(T\)-periodic coefficients, i.e., \(A(t + T) = A(t)\). The authors study the Lyapunov differential equations NEWLINE\[NEWLINE \frac{d}{dt}H + HA(t) + A^*(t)H = - C. NEWLINE\]NEWLINE With the help of this equation, a criterion for the asymptotic stability is formulated; moreover, the authors obtain a uniform estimate on the matriciant to the system which makes it possible to indicate the rate of decaying for the solutions as \(t\to +\infty\). Finally, the influence of periodic perturbations is discussed.