STABILIZATION OF SECOND ORDER EVOLUTION EQUATIONS WITH UNBOUNDED FEEDBACK WITH DELAY

We consider abstract second order evolution equations with unbounded feedback with delay. Existence results are obtained under some realistic assumptions. Sufficient and explicit conditions are derived that guarantee the exponential or polynomial stability. Some new examples that enter into our abstract framework are presented.

[1]  K. P. Hadeler,et al.  Delay equations in biology , 1979 .

[2]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[3]  R. Rebarber Exponential Stability of Coupled Beams with Dissipative Joints: A Frequency Domain Approach , 1995 .

[4]  Z. Bien,et al.  Use of time-delay actions in the controller design , 1980 .

[5]  Enrique Zuazua,et al.  Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures (Mathématiques et Applications) , 2005 .

[6]  Stuart Townley,et al.  Robustness with Respect to Delays for Exponential Stability of Distributed Parameter Systems , 1999 .

[7]  C. Baiocchi,et al.  Ingham-Beurling type theorems with weakened gap conditions , 2002 .

[8]  Marius Tucsnak,et al.  How to Get a Conservative Well-Posed Linear System Out of Thin Air. Part II. Controllability and Stability , 2003, SIAM J. Control. Optim..

[9]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[10]  Genqi Xu,et al.  Stabilization of wave systems with input delay in the boundary control , 2006 .

[11]  A. Ingham Some trigonometrical inequalities with applications to the theory of series , 1936 .

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

[13]  Marius Tucsnak,et al.  How to get a conservative well-posed linear system out of thin air. Part I. Well-posedness and energy balance , 2003 .

[14]  Wolfgang Arendt,et al.  Tauberian theorems and stability of one-parameter semigroups , 1988 .

[15]  Kaïs Ammari,et al.  Stabilization of Bernoulli--Euler Beams by Means of a Pointwise Feedback Force , 2000, SIAM J. Control. Optim..

[16]  C. Abdallah,et al.  Delayed Positive Feedback Can Stabilize Oscillatory Systems , 1993, 1993 American Control Conference.

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

[18]  V. Jurdjevic,et al.  Controllability and stability , 1978 .

[19]  Serge Nicaise,et al.  Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks , 2006, SIAM J. Control. Optim..

[20]  Kaïs Ammari,et al.  STABILIZATION OF SECOND ORDER EVOLUTION EQUATIONS BY A CLASS OF UNBOUNDED FEEDBACKS , .

[21]  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.

[22]  Serge Nicaise,et al.  Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks , 2007, Networks Heterog. Media.

[23]  Irena Lasiecka,et al.  Inverse/Observability Estimates for Second-Order Hyperbolic Equations with Variable Coefficients , 1999 .

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

[25]  S. Boulite,et al.  Feedback stabilization of a class of evolution equations with delay , 2009 .

[26]  R. Datko,et al.  Two examples of ill-posedness with respect to time delays revisited , 1997, IEEE Trans. Autom. Control..