Lyapunov Differential Equation Research Articles

Lyapunov and Riccati equations: Periodic inertia theorems

The Lyapunov and Riccati differential equations with periodically time-varying coefficients are considered. Under the assumption of detectability of the underlying periodic system, two inertia theorems are provided linking the inertia of the solution to the one of the so-called monodromy matrix.

On a numerical method of solving the Lyapunov and Sylvester equations

We present a numerical method of solving the time-varying Lyapunov and Sylvester matrix differential equations. The method is easy to program, fast, and accurate. It is shown that the error of computation is O(Δt 3) where Δt is the interval of computation. Two examples are given.

Stability Analysis of Linear Periodic Systems Via the Lyapunov Equation

It is well known that the asymptotic stability of time-invariant linear systems is connected with the solution of a Lyapunov algebraic equation being positive definite. This result is here extended to periodic linear systems. More precisely, it is shown that the system is asymptotically stable if and only if the associated Lyapunov differential equation admits a periodic solution positive definite at each time instance. The discussion will focus on continuous-time systems. However, parallel results will be shortly outlined for discrete-time systems as well. The case when the discrete-time transition matrix is singular is also discussed.