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

Abstract

We consider the control systemẋ = 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 −∞ (deadbeat 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 L 2 . 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.

Some new applications of Russell’s principle to infinite dimensional vibrating systems

“…An abstract version of this principle has been given in [3], in the case of bounded control operators. This version has been further generalized in Rebarber and Weiss [28] and Natarajan and Weiss [24]. More recently, a dual version Russell's principle, asserting that forward and backward detectability implies observability, came to the attention of control theorists.…”