Boundary stabilization of a hybrid system (Q2712211)

In this very important and useful paper the boundary stabilization of a degenerate hybrid system composed of an Euler-Bernoulli beam with a tip mass is considered. The hybrid system is governed by the equations NEWLINE\[NEWLINE\rho w_{tt}+ pw''''= 0,\quad (x,t)\in (0,L)\times (0,\infty),NEWLINE\]NEWLINE NEWLINE\[NEWLINEw(0, t)= w'(0, t)= 0,\;Jw_{tt}'(L, t)+ pw''(L, t)= g(t),\;Mw_{tt}(L, t)- pw'''(L, t)= h(t)NEWLINE\]NEWLINE \(p,\rho>0\) are the elasticity modulus and mass density, respectively, \(J\) is the rotatory inertia of the tip mass \(M\), \(g(t)\), \(h(t)\) are controls applied at the end \(x= L\).NEWLINENEWLINENEWLINEMain result: The authors prove that the system is exponentially stabilizable when the usual velocity feedback controls are applied at the end with the tip mass. Furthermore, the time reversibility and spectral completeness of the closed-loop system is established.NEWLINENEWLINEFor the entire collection see [Zbl 0958.00050].