Nonlinear boundary stabilization of nonuniform Timoshenko beam
The aim is to study the stabilization problem for nonuniform Timoshenko beam by using nonlinear boundary feedback controls. The boundary stabilization problem is characterized by the distributed system of the form \[ \begin{cases} \rho\ddot w+ (K(\varphi- w'))'= 0\quad &0< x< \ell,\;t> 0,\\ I_\rho\ddot\varphi- (EI\varphi')'+ K(\varphi- w')= 0\quad & 0< x< \ell,\;t> 0,\\ w(0, t)= \varphi(0,t)= 0\quad & t> 0,\\ K(\ell)(\varphi(\ell, t)- w'(\ell,t))\in f_1(\dot w(\ell, t))\quad & t> 0,\\ -EI(\ell)\varphi'(\ell, t)\in f_2(\dot\varphi(\ell, t))\quad &t> 0,\end{cases}\tag{1} \] where \(\rho\) is the mass per unit length, \(w(x,t)\) is the deflection of the beam, \(\varphi(x, t)\) is the total rotatory angle of the beam at \(x\), \(I_\rho\) \(EI\) are respectively the mass moment of inertia and rigidity coefficient of the cross-section, and \(K\) is the shear modulus of elasticity. Two functions \(f_1(\dot w(\ell,t))\) and \(f_2(\dot\varphi(\ell, t))\) are the control force and moment applied to the free end of the beam. The main result: Using the nonlinear semigroup theory and exponential multiplier method, it is proved that the energy associated with the solution to system (1) decays as a negative power of time \(t\) and exponentially in some special case as \(t\to\infty\). The precise proofs of all lemmas are presented. Finally, two examples show that a great variety of controls (including saturated ones) can act as boundary controls of the system (1).
scientific article; zbMATH DE number 2101811