Well-posedness and stabilizability of a viscoelastic equation in energy space

Abstract:

We consider the well-posedness and exponential stabilizability of the abstract Volterra integrodifferential system \[ \begin {array}{*{20}{c}} {v\prime (t) = - {D^\ast }\sigma (t) + f(t),} \hfill \\ {\sigma (t) = \nu Dv(t) + \int _{ - \infty }^t {a(t - s)Dv(s)ds,\quad t \geq 0,} } \hfill \\ \end {array} \] in ilbeubert space. In a typical viscoelastic interpretation of this equation one lets v represent velocit, $v\prime$ acceleratio $\sigma$, stres, $- {D^ \ast }\sigma$ the divergence of the stres, $v \geq 0$ pure viscosity (usually equal to zero) Dv the time derivative of the strain, and a the linear stress relaxation modulus of the material. The problems that we treat are one-dimensional in the sense that we require a to be scalar. First we prove well-posedness in a new semigroup setting, where the history component of the state space describes the absorbed energy of the system rather than the history of the function v. To get the well-posedness we need extremely weak assumptions on the kernel; it suffices if the system is "passive", i.e., a is of positive type; it may even be a distribution. The system is exponentially stabilizable with a finite dimensional continuous feedback if and only if the essential growth rate of the original system is negative. Under additional assumptions on the kernel we prove that this is indeed the case. The final part of the treatment is based on a new class of kernels. These kernels are of positive type, but they need not be completely monotone. Still, they have many properties similar to those of completely monotone kernels, and a number of results that have been proved earlier for completely monotone kernels can be extended to the new class.