Exact controllability and perturbation analysis for elastic beams (Q1885366)

The author establishes in this very important paper the convergence of the solution of the exact controllability problem for the Rayleigh beam to the corresponding solution of the Bernoulli-Euler beam. Convergence is related to a singular perturbation problem. Here, the main tool in solving the perturbation problem is a weak version of a lower bound for hyperbolic polynomials. In Littman's method, the solution of the control problem is given in the form: \(W= w\psi(t)- v\), where \(\psi(t)\) is a cut-off function: \(\psi(t)= 1\) for \(t\leq t_0\), \(\psi(t)= 0\) for \(t\geq t_1\). In this decomposition, \(w\) (the displacement) satisfies \(L[w]= 0\) with Cauchy data in the \(x\)-axis. On the other hand, \(v\) is of the form: \(v= v_++ v_-\) and satisfies \(L[v]= f\), \(f= L[w\psi]\) (\(f\) is supported in the strip \(t_0\leq t\leq t_1\)). For the Rayleigh equation (the R-beam is a perturbation of the B-E beam) a solution for the exact controllability problem has the form: \(W^I= u^I\psi(t)- v^I\), whereas for the B-E equation one has: \(W^0= u^0\psi(t)- v^0\). Main result: There exists \(I_0> 0\) sufficiently small such that for \(I< I_0\): 1) There exist solutions \(W^I(x, t)\) and \(W^0(x, t)\) of the Cauchy problem \[ \left\{\begin{aligned} &W^I_{tt}- IW^I_{ttxx}+ W^I_{xxxx}= 0 \text{ in }\Omega \times\{t\geq 0\}\\ &W^I(x, 0)= u_0(x),\;W^I_t(x, 0)= u_1(x)\quad \text{in }\Omega\end{aligned}\right. \] and \[ \left\{ \begin{aligned} &W^0_{tt}+ W^0_{xxxx}= 0\quad \text{in }\Omega \times \{t\geq 0\}\\ &W^0(x, 0)= u_0(x),\;W^0_t(x, 0)= u_1(x)\quad \text{in }\Omega\end{aligned}\right. \] both vanishing in \(\overline\Omega \times \{t\geq T\}\) 2) \(W^I\) converges to \(W^0\) when \(I\to 0\) in \(L_\infty(Q\times [0,T])\). 3) There exists \(T_0\), \(0< T_0< T\), such that \(W^I_t\) converges to \(W^0_t\), when \(I\to 0\) in \(L_\infty(Q \times[0, T])\).