Remarks on stabilization of second-order evolution equations by unbounded dynamic feedbacks
Tóm tắt
In this paper, we consider second-order evolution equations with unbounded dynamic feedbacks. Under a regularity assumption, we show that observability properties for the undamped problem imply decay estimates for the damped problem. We consider both uniform and non-uniform decay properties.
Tài liệu tham khảo
Abbas Z., Nicaise S.: Polynomial decay rate for a wave equation with general acoustic boundary feedback laws. S \({{\vec{\rm e}}}\) MA J., 61, 19–47 (2013)
Ammari K., Henrot A., Tucsnak M.: Asymptotic behaviour of the solutions and optimal location of the actuator for the pointwise stabilization of a string. Asymptot. Anal., 28(3–4), 215–240 (2001)
K. Ammari and S. Nicaise. Stabilization of elastic systems by collocated feedback, volume 2124 of Lecture Notes in Mathematics. Springer, Cham, 2015.
K. Ammari, G. Tenenbaum, and M. Tucsnak. Spectral conditions for the stability of a class of second order systems. Preprint
Ammari K., Tucsnak M.: Stabilization of Bernoulli-Euler beams by means of a pointwise feedback force. SIAM J. Control Optim., 39(4), 1160–1181 (2000)
Ammari K., Tucsnak M.: Stabilization of second order evolution equations by a class of unbounded feedbacks. ESAIM Control Optim. Calc. Var., 6, 361–386 (2001)
Guo B.-Z., Wang J.-M., Yang K.-Y.: Dynamic stabilization of an Euler-Bernoulli beam under boundary control and non-collocated observation. Systems Control Lett., 57(9), 740–749 (2008)
Mercier D., Nicaise S.: Polynomial decay rate for a wave equation with weak dynamic boundary feedback laws. J. Abstr. Differ. Equ. Appl., 2(1), 29–53 (2011)
Miller L.: Controllability cost of conservative systems: resolvent condition and transmutation. J. Funct. Anal., 218(2), 425–444 (2005)
Muñoz Rivera J.E., Racke R.: Large solutions and smoothing properties for nonlinear thermoelastic systems. J. Differential Equations, 127(2), 454–483 (1996)
A. Pazy. Semigroups of linear operators and applications to partial differential equations, volume 44 of Applied Math. Sciences. Springer-Verlag, New York, 1983.
L. Toufayli. Stabilisation polynomiale et contrôlabilité exacte des équations des ondes par des contrôles indirects et dynamiques. PhD thesis, Univ. Strasbourg, 2013.
H. Triebel. Interpolation theory, function spaces, differential operators, volume 18 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 1978.
Wehbe A.: Rational energy decay rate for a wave equation with dynamical control. Appl. Math. Lett., 16(3), 357–364 (2003)
Weiss G., Curtain R.F.: Dynamic stabilization of regular linear systems. IEEE Trans. Automat. Control, 42(1), 4–21 (1997)