Stabilizability of systems of one-dimensional wave equations by one internal or boundary control force (Q2719144)

The authors' starting point, as they call it, is a coupled system of two second-order partial differential equations \(u''(t)= -Au(t)+Bv'(t)\), and \(v''(t)=-B^* u'(t)-C(v(t)) -Dv'(t)\), where \(u\) and \(v\) are vectors in Hilbert spaces \(H\) and \(G\), respectively. That is not the system they study in this paper, but some approaches including the semigroup theory seem to be relevant to the system they study, which is: NEWLINE\[NEWLINEu_{tt}= u_{xx}+b(x)v_t+f, \quad v_{tt}= \eta^2 v_{xx} -b(x)u_t +g \quad \text{in }(0,\infty) \times(0,1).\tag{1}NEWLINE\]NEWLINE Boundary conditions at \(x= 0\) and \(x=1\) (or \(x=\ell)\) are all set equal to zero, while all relevant conditions at the initial time \(t=0\) are known functions. The nature of the admissible control forces \(f\) and \(g\) is not specified. Since initially \(f\) is assumed to be identically zero, and \(g=-a(x)v_t\), there is no problem. Later they note that Lions' relation in a similar problem states that \(f'=g\;\forall\;t\) in \((0, \infty)\). They choose \(g\) to make the system dissipative and compute the decreasing energy. They eventually prove two ``if and only if'' theorems for the condition of strong stability involving the coefficient \(\eta\). Next, the authors consider the abstract hyperbolic system: NEWLINE\[NEWLINE\partial/\partial t({\mathbf u}^T)= -M(x)\partial/ \partial x({\mathbf u}^T)-N(x) ({\mathbf u}^T)\text{ on }(0,T) \times(0,\ell),\tag{2}NEWLINE\]NEWLINE NEWLINE\[NEWLINE\text{ and }d/dt \bigl[v(t,\ell)- Du(t,\ell)\bigr]= Fu(t, \ell)+ Gv(t,\ell)\text{ with }u(t,0)\text{ given}.NEWLINE\]NEWLINE Here \(({\mathbf u}^T)\) stands for the transpose of \((u,v)\). The authors prove some interesting results. Specifically, they rewrite their equation in an abstract operator form first. They construct the relevant semigroups of operators, and use some known and some that they derive. They use results of spectral theory to prove their theorems, which are stated for the initially discussed system (1). These theorems and lemmas imply that internal damping of only one of the equations in system (1) does not result in stability of the system if the wave speeds differ. If the wave speeds are equal, the authors offer necessary and sufficient conditions for stability.