Improving the exponential decay rate by back and forth iterations of the feedback in time

We consider the control system ẋ=Ax+Bu, where A generates a strongly continuous semigroup T on the Hilbert space X and the control operator B maps into the dual of D(A*), but it is not necessarily admissible for T. We prove that if the pair (A;B) is both forward and backward optimizable (our definition of this concept is slightly more general than the one in the literature), then the system is exactly controllable. This is a generalization of a well-known result called Russell's principle. Moreover the usual stabilization by state feedback u = Fx, where F is an admissible observation operator for the closed-loop semigroup, can be replaced with a more complicated periodic (but still linear) controller. The period τ of the controller has to be chosen large enough to satisfy an estimate. This controller can improve the exponential decay rate of the system to any desired value, including -∞ (dead-beat control). The corresponding control signal u, generated by alternately solving two exponentially stable homogeneous evolution equations on each interval of length τ, back and forth in time, will still be in L₂. The better the decay rate that we want to achieve, the more iterations the controller needs to perform, but (unless we want to achieve -∞) the number of iterations needed on each period is finite.

I. Introduction

We consider a linear infinite-dimensional system described by

$$\dot{x}(t)=Ax(t)+Bu(t),\qquad x(0)=x_{0},\eqno{\hbox{(1.1)}}$$ where generates a strongly continuous semigroup on the Hilbert space , the signal takes values in another Hilbert space and is a possibly unbounded control operator, which means that it maps into a space that contains densely.