The Cayley transform of the generator of a polynomially stable $$C_0$$-semigroup

In this paper, we study the decay rate of the Cayley transform of the generator of a polynomially stable $$C_0$$ C 0 -semigroup. To estimate the decay rate of the Cayley transform, we develop an integral condition on resolvents for polynomial stability. Using this integral condition, we relate polynomial stability to Lyapunov equations. We also study robustness of polynomial stability for a certain class of structured perturbations.

Closed Form Solutions of Lyapunov Equations Using the Vech and Veck Operators

New closed forms are presented of the solutions of the continuous and discrete Lyapunov equations using the vech and veck operators. The proposed solutions are faster than the classical solutions derived using the vec operator. The solutions via veck operator are faster than the solutions via vech operator

The extended nonsymmetric block Lanczos methods for solving large-scale differential Lyapunov equations

In this paper, we present a new approach for solving large-scale differential Lyapunov equations. The proposed approach is based on projection of the initial problem onto an extended block Krylov subspace by using extended nonsymmetric block Lanczos algorithm then, we get a low-dimensional differential Lyapunov matrix equation. The latter differential matrix equation is solved by the Backward Differentiation Formula method (BDF) or Rosenbrock method (ROS), the obtained solution allows to build a low-rank approximate solution of the original problem. Moreover, we also give some theoretical results. The numerical results demonstrate the performance of our approach.