Abstract
The Lyapunov matrix differential equation plays an important role in many scientific and engineering fields. In this paper, we first give a class relation between the eigenvalue of functional matrix derivative and the derivative of functional matrix eigenvalue. Using this relation, we convert the Lyapunov matrix differential equation into an eigenvalue differential equation. Further, by the Schur theorem, combining Hölder’s integral inequality with arithmetic–geometric average inequality, we provide several lower and upper bounds on the eigenvalue product for the solution of the Lyapunov matrix differential equation. As an application in control and optimization, we show that our bounds can be used to discuss the stability of a class of time-varying nonlinear systems. Finally, we illustrate the superiority and effectiveness of the derived bounds by a numerical example.
Highlights
Consider the following Lyapunov matrix differential equation [1]:P (t) = AH (t)P(t) + P(t)A(t) + Q(t), P(t0) = P0 = P0H ≥ 0, (1)where A(t) ∈ Cn×n, Q(t) = QH (t) ∈ Cn×n, and Q(t) ≥ 0 are continuous functions of t, and P(t) ∈ Cn×n is the Hermitian positive semidefinite solution of (1).It is well known that the linear matrix differential equation (1) has many express styles, and it usually can be found in time-varying nonlinear and linear systems
(2019) 2019:172 show that the eigenvalue bounds of the Lyapunov matrix differential equation (1) can be used to discuss the stability of the time-varying nonlinear system (2)
3 Lower bounds on eigenvalue product we offer some new lower bounds on the eigenvalue product of the solution of the Lyapunov matrix differential equation (1)
Summary
1 Introduction Consider the following Lyapunov matrix differential equation [1]: P (t) = AH (t)P(t) + P(t)A(t) + Q(t), P(t0) = P0 = P0H ≥ 0, (1) Where x(t), r(t) ∈ Cn, and P(t) is the positive definite solution of the Lyapunov matrix differential equation (1). (2019) 2019:172 show that the eigenvalue bounds of the Lyapunov matrix differential equation (1) can be used to discuss the stability of the time-varying nonlinear system (2).
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