International Journal of Mathematics and Mathematical Sciences
Volume 29 (2002), Issue 3, Pages 155-166

On the Lyapunov equation in Banach spaces and applications to control problems

Vu Ngoc Phat1 and Tran Tin Kiet2

1Institute of Mathematics, Box 631, Bo Ho, Hanoi 10.000, Vietnam
2Department of Mathematics, Quinhon Pedagogical University, Binh Dinh, Qui Nhon City, Vietnam

Received 30 August 2000; Revised 5 March 2001

Copyright © 2002 Vu Ngoc Phat and Tran Tin Kiet. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


By extending the Lyapunov equation A*Q+QA=P to an arbitrary infinite-dimensional Banach space, we give stability conditions for a class of linear differential systems. Relationship between stabilizability and exact null-controllability is established. The result is applied to obtain new sufficient conditions for the stabilizability of a class of nonlinear control systems in Banach spaces.