| Contents |
| Systems of interest in viscoelasticity are often described by the equation
\[
w''=\Delta w +\int_0^t M(t-s) \Delta w(s) {\rm d} s\eqno{\bf(A)}
\]
where $w= w(x,t)$ and $x$ belongs to a region $\Omega$ with smooth boundary.
The control problem consists on finding a control $f$ (acting on the boundary of $\Omega$, or on an interior subregion $\Omega_1$) which forces the pair $(w(T),w'(T))$ to hit a prescribed target $(\xi,\eta)\in L^2(\Omega)\times H^{-1}(\Omega)$. Several methods can be used to prove the existence of such control (at a suitable time $T$). The common idea is to see system~ {\bf(A)} as a perturbation of the memoryless wave equation.
It is known that the solution of the wave equation can be represented using cosine operator theory, and this fact suggests the use of cosine operators in the study of controllability of system~ {\bf(A)}. The goal of this talk is both to review existing results and to present new results on the application of cosine operator theory to the study of controllability for system~ {\bf(A)}. |
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