Equivalent conditions on periodic feedback stabilization for linear periodic evolution equations

• Published in: Journal of functional analysis Vol. 266; no. 8; pp. 5126 - 5173
• Main Authors: Wang, Gengsheng, Xu, Yashan
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 15.04.2014
• Subjects: Attainable subspaces; Equivalent conditions; Kato projection; Periodic evolution equations; Periodic feedback stabilization; Poincaré map; Unique continuation properties; Unique continuation properties; Poincaré map; Attainable subspaces; Periodic evolution equations; Equivalent conditions; Kato projection; Periodic feedback stabilization
• ISSN: 0022-1236; 1096-0783
• DOI: 10.1016/j.jfa.2014.01.029

This paper studies the periodic feedback stabilization for a class of linear T-periodic evolution equations. Several equivalent conditions on the linear periodic feedback stabilization are obtained. These conditions are related to the following subjects: the attainable subspace of the controlled evolution equation under consideration; the unstable subspace (of the evolution equation with the null control) provided by the Kato projection; the Poincaré map associated with the evolution equation with the null control; and two unique continuation properties for the dual equations on different time horizons [0,T] and [0,n0T] (where n0 is the sum of algebraic multiplicities of distinct unstable eigenvalues of the Poincaré map). It is also proved that a T-periodic controlled evolution equation is linear T-periodic feedback stabilizable if and only if it is linear T-periodic feedback stabilizable with respect to a finite-dimensional subspace. Some applications to heat equations with time-periodic potentials are presented.