A compact perturbation method for the boundary stabilization of the Rayleigh beam equation
The purpose of this important work is to investigate the stabilization of a clamped Rayleigh beam equation by only one boundary feedback control moment, or control force: \[ y_{tt}- \gamma y_{xxtt}+ y_{xxxx} =0, \quad 0< x< 1, \qquad y(0,t)= y_x (0, t)= 0, \] \[ y_{xx} (1, t)+ \alpha y_{xt} (1, t) =0, \qquad y_{xxx} (1, t)- \gamma y_{xtt} (1, t)- \beta y_t (1, t) =0, \] where \(\gamma>0\) is a physical constant, and where \(\alpha y_{xt} (1, t)\) and \(\beta y_t (1,t)\) designate respectively the boundary feedback control moment and control force. The main result is the following: The Rayleigh beam equation can be uniformly exponentially stabilized by only one control moment. Moreover the strong asymptotic stabilization and the lack of uniform exponential stabilization in the case of only one control force are proved.
scientific article; zbMATH DE number 883410
scientific article