Well-Posedness and Stabilizability of a Viscoelastic Equation in Energy Space.

摘要

The authors consider the well-posedness and exponential stabilizability of the abstract Volterra integrodifferential system in a Hilbert space. In a typical viscoelastic interpretation of the equation one lets upsilon represent velocity, upsilon' acceleration, sigma stress, -D* sigma the divergence of the stress (nu) upsilon = or greater than 0 pure viscosity (usually equal to zero), D(upsilon) the time derivative of the strain, and alpha the linear stress relaxation modulus of the material. The problems that they treat are one-dimensional in the sense that they require alpha to be scalar. First they 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 upsilon. To get the well-posedness they need extremely weak assumptions on the kernel; it suffices if the system is 'passive', i.e., alpha 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 they 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.