Equivalent Conditions on Periodic Feedback Stabilization for Linear Periodic Evolution Equations

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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 with 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 Poincare 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 Poincare map). It is also proved that a T-periodic controlled evolution equation is linear T-periodic feedback sabilizable if and only if it is linear T-periodic feedback sabilizable with respect to a finite dimensional subspace. Some applications to heat equations with time-periodic potentials are presented.