Stability of the time variable elastic system (Q1354587)

In this important paper, the problem of the stability of the time-varying elastic system is considered. The formulation (the equivalent first-order evolution system) of this problem is described as follows: \[ \begin{aligned} {d\over dt} y(t) &=A(t)y(t) \quad 0<t <\infty \\ y(0) =y_0 & ={u_0 \choose u_1},\;A(t) =\left(\begin{matrix} 0 & I \\ -A(t) & -B(t) \end{matrix} \right),\;y={y_1(t) \choose y_2(t)} \end{aligned} \] with \[ \begin{aligned} A(t) \varphi & ={d^2\over dx^2} \left[p(t,x) {d^2\over dx^2} \varphi(x) \right],\;\varphi\in D\bigl(A(t) \bigr) \\ B(t)\varphi & ={d^2 \over dx^2} \left[\eta (t,x) {d^2\over dx^2} \varphi(x)\right] +{N_0\over v_0} \varphi(x),\;\varphi\in D \bigl(B(t)\bigr) \end{aligned} \] where \[ D\bigl(A(t)\bigr) =\{\varphi \mid \varphi',\varphi'', \varphi''' \text{ are absolutely continuous functions}, \] \[ \varphi''' \in H\;\bigl(\text{the subspace }H= \{1,x\}^\perp \text{ of } L_2(0,\ell)\bigr),\;\varphi''(x) \mid_{x=0,\ell} =\varphi(x) |_{x=0,\ell} =0\}, \] \(D(B(t)) =D(A(t))\), \(p(t,x)\) denotes the bending stiffness, \(\eta(t,x) =\mu(t,x)EI\) the inner structural damping, \(N(t,x)=N_0\) (a constant) the aerodynamic loading distribution for a unit angle of attack and \(\ell\) represents the length of the beam (the slender vehicle). Main result: The authors prove the existence, uniqueness, and stability of the solution of the system of the slender flying vehicle.