Lower and upper bounds of the solution for the Lyapunov matrix differential equation and an application in input-output finite-time stability of linear systems

In control theory and practical engineering fields, such as the controllability, observability, input–output finite-time stability of the linear systems, it is a significant problem to study the properties of the solution for the Lyapunov matrix differential equation where there are no restrictions on the system matrix. In this paper, by constructing an equivalent form of the Lyapunov matrix differential equation and utilizing some important matrix eigenvalue inequalities, lower and upper bounds for the matrix solution of the Lyapunov matrix differential equation that remove the strict restrictions for the system matrix are proposed. As an application in linear systems, we show that our bounds can be used to discuss the input–output finite-time stability for linear systems. Finally, we give some corresponding numerical examples to illustrate the effectiveness and superiority of our results.