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

Vivek Natarajan, George Weiss

Research output: Chapter in Book/Report/Conference proceedingConference contributionpeer-review


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 t 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 t, 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.

Original languageEnglish
Title of host publication2013 IEEE 52nd Annual Conference on Decision and Control, CDC 2013
PublisherInstitute of Electrical and Electronics Engineers Inc.
Number of pages5
ISBN (Print)9781467357173
StatePublished - 2013
Event52nd IEEE Conference on Decision and Control, CDC 2013 - Florence, Italy
Duration: 10 Dec 201313 Dec 2013

Publication series

NameProceedings of the IEEE Conference on Decision and Control


Conference52nd IEEE Conference on Decision and Control, CDC 2013

All Science Journal Classification (ASJC) codes

  • Control and Systems Engineering
  • Modelling and Simulation
  • Control and Optimization


Dive into the research topics of 'Improving the exponential decay rate by back and forth iterations of the feedback in time'. Together they form a unique fingerprint.

Cite this