Equivalent conditions on periodic feedback stabilization for linear periodic evolution equations

Abstract

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 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, n 0 T ] (where n 0 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 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.

Characterization by detectability inequality for periodic stabilization of linear time-periodic evolution systems

“…• The similar equivalence results in Theorem 1.1 were obtained in [8] for time-invariant systems. Different kinds of characterizations of the periodic stabilization for time-periodic systems have been studied in [5], [6], [9] and [10]. The characterization (for the system (1.1)), given in Theorem 1.1, seems to be new.…”

“…We will present an equivalence between the detectability inequality (1.6) (or (1.7)) and a qualitative unique continuation property for the adjoint system (1.5), under the setting (H 1 )-(H 4 ). This qualitative unique continuation was introduced in [9] (see also [10]).…”

Attainable Subspaces and the Bang-Bang Property of Time Optimal Controls for Heat Equations

“…The attainable subspaces play important roles in the studies of control problems governed by Equation (1.1) (see, for instance, [29] where the connection of attainable subspaces and the stabilization for some periodic evolution system are provided). To our surprise, the studies on the attainable subspaces of internally controlled heat equations are quite limited from the past publications.…”

“…The method used there is borrowed from [23] and [11], where the elements of a subspace of the controlled wave equation (without the geometric condition imposed on the control region) are explicitly expressed via a Riesz basis. In [29] (see also [30]), the authors presented some properties of attainable subspaces for some T -periodic evolution systems. Those properties gives the connection of the space t>0 A t,∞ and the spaces A kT,∞ , k ∈ N. The observations presented in Theorem 1.1 seem to be new.…”

“…29) By(2.25),(2.27), with ξ = χ ω ϕ, and (2.29), for each admissible control v to (NP ) y T ,p , we see…”