Boundary stabilization of nonuniform Timoshenko beam
The author considers a non-uniform beam whose dynamic behavior is described by Timoshenko's equation. The dimensions of the cross-section are not negligible compared with the length. Thus the Timoshenko model considers rotary inertia due to the changing angle between the original axis of the beam and the normal direction to the cross-sectional area. This mathematical model also considers deflections due to shear. The initial-boundary value problem is formulated as the following system of equations: \[ \begin{cases} \rho w_{tt}- (Kw')'+ (K\phi)'=0,\quad 0< x<\ell\\ I_\rho\phi_{tt}- (EI\phi')'+ K(\phi- w')= 0,\end{cases}\tag{1} \] with \(w(0,t)= \phi(0, t)=0\), \(K(\ell)\phi(\ell, t)- K(\ell) w'(\ell,t)= m(t)\), \(-EI(\ell)\phi'(\ell, t)= h(t)\), and \(w(x,0)= w^0_1\), \(w_t(x, 0)= w^0_2\), \(\phi(x,0)= \phi^0_1\), \(\phi_t(x, 0)= \phi^0_2\). \(m(t)\), \(h(t)\) are the controlling bending moment and shear force, respectively, applied to the free end (at \(x=\ell\)). Here \(w(x,t)\) represents the transversal displacement, \(\phi\) the rotation angle of the cross-sectional area, \(E\) is the Young modulus, \(I\) the moment of inertia of the cross-sectional area about the neutral axis of bending, \(\rho\) the material density, \(K\) the shear modulus, \(I_\rho\) the rotary inertia. All of these physically positive parameters are assumed to be positive and bounded away from zero in this mathematical model. Using a finite energy criterion, the author defines appropriate (complex) Hilbert space, and following the techniques of Lyons et al., rewrites the system of equations (1) as an abstract evolution equation \(dz/dt= Az\), \(z(0)= z_0\). This generates a \(C_0\)-semigroup of contractions. The operator \(A\) is dissipative and its kernel consists of \(\{0\}\) only. It follows that this semigroup of contractions is exponentially stable if and only if the resolvent of \(A\), denoted unfortunately by the author by the same symbol \(\rho\) which previously stood for material density, obeys the condition: \(i\mathbb{R}\subset\rho(A)\), or in other words that no point of the spectrum of \(A\) lies on the imaginary axis. In the last part of the paper, the author proves by means of a tricky computation that the system is exponentially stabilizable by controls consisting of moments and shear forces arranged in a \(2\times 2\) control matrix \(F\).
scientific article; zbMATH DE number 1420419
scientific article