Decay rates at low and high frequencies for a plate equation with feedback concentrated in interior curves

In this paper we study the stabilization of plate vibrations by means of piezoelectric actuators. In this situation the geometric control condition of Bardos, Lebeau and Rauch [6] is not satisfied. We prove that we have exponential stability for the low frequencies but not for the high frequencies. We give an explicit decay rate for regular initial data at high frequencies while clarifying the behavior of the constant which intervenes in this estimation there function of the frequency of cut n. The method used is based on some trace regularity which reduces stability to some observability inequalities for the corresponding undamped problem. Moreover, we show numerically at low frequencies, that the optimal location of the actuator is the center of the domain Ω. Research supported by the RIP program of Oberwolfach Institut and by the Tunisian Ministry for Scientific Research and Technology (MRST) under Grant 02/UR/15-01. Research supported by the RIP program of Oberwolfach Institut. (Received: September 17, 2003; revised: February 26, 2004)