Robustness with Respect to Delays for Exponential Stability of Distributed Parameter Systems

In this paper we address the question of whether the open-loop exponential growth rate of a linear system can be improved by a feedback in such a way that this improvement is robust with respect to small delays in the feedback loop. When the input operator is admissible, and the class of possible feedbacks consists of compact operators, we find that if a feedback can improve the exponential growth rate, then it can do so robustly. Furthermore, we find that if the control space is finite dimensional and a bounded feedback cannot be found to improve exponential stability, then a large class of unbounded feedbacks cannot improve the exponential growth rate robustly, even if such feedbacks can improve the exponential growth rate in the absence of delays.

[1]  Scott W. Hansen,et al.  The operator Carleson measure criterion for admissibility of control operators for diagonal semigroups on 1 2 , 1991 .

[2]  George Weiss,et al.  Admissibility of unbounded control operators , 1989 .

[3]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[4]  George Weiss.  Two conjectures on the admissibility of control operators , 1991 .

[5]  R. Datko.  Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks , 1988 .

[6]  Dietmar A. Salamon,et al.  On control and observation of neutral systems , 1982 .

[7]  Michael P. Polis,et al.  An example on the effect of time delays in boundary feedback stabilization of wave equations , 1986 .

[8]  J. Prüss.  On the Spectrum of C 0 -Semigroups , 1984 .

[9]  George Weiss,et al.  Regular linear systems with feedback , 1994, Math. Control. Signals Syst..

[10]  Some remarks on open and closed loop stabilizability for infinite dimensional systems , 1989 .

[11]  C. Jacobson,et al.  Linear state-space systems in infinite-dimensional space: the role and characterization of joint stabilizability/detectability , 1988 .

[12]  R. Triggiani.  On the stabilizability problem in Banach space , 1975 .

[13]  R. Triggiani.  Wave equation on a bounded domain with boundary dissipation: An operator approach☆ , 1989 .

[14]  Hartmut Logemann,et al.  The effect of small time-delays on the closed-loop stability of boundary control systems , 1996, Math. Control. Signals Syst..

[15]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[16]  J. Lagnese,et al.  An example of the effect of time delays in boundary feedback stabilization of wave equations , 1985, 1985 24th IEEE Conference on Decision and Control.

[17]  Stuart Townley,et al.  The effect of small delays in the feedback loop on the stability of neutral systems , 1996 .

[18]  R. Rebarber.  Conditions for the equivalence of internal and external stability for distributed parameter systems , 1993, IEEE Trans. Autom. Control..

[19]  P. Grisvard.  Équations différentielles abstraites , 1969 .

[20]  Yuncheng You,et al.  Some second-order vibrating systems cannot tolerate small time delays in their damping , 1991 .

[21]  Hartmut Logemann,et al.  Conditions for Robustness and Nonrobustness of theStability of Feedback Systems with Respect to Small Delays inthe Feedback Loop , 1996 .

[22]  Richard Rebarber.  Necessary conditions for exponential stabilizability of distributed parameter systems with infinite dimensional unbounded feedback , 1990 .

[23]  George Weiss,et al.  Transfer Functions of Regular Linear Systems. Part I: Characterizations of Regularity , 1994 .