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

Abstract

Abstract. We consider the well-posedness and exponential stabilizability of the abstract Volterra integrodifferential system v'(t) = -D'o(t) + f(t), a(t) = vDv(t) + i a(t-s)Dv{s)ds, t > 0, J-oo in a Hubert space. In a typical viscoelastic interpretation of this equation one lets v represent velocity, v' acceleration, a stress, -D*o the divergence of the stress, v > 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.

Optimal control for stochastic Volterra equations with multiplicative Lévy noise

“…In the past years an important approach to the analysis of equations with memory was provided by Desch and Miller [28] for deterministic Volterra equations arising in linear viscoelasticity, see also [27,52], and further developed by several authors also in the stochastic case, see [32,7,8,10].…”

Introduction to Diffusive Representation

“…The so-called diffusive representation (Montseny [2005]) is a theory (the beginnings of which can be found for example in Montseny [1991], Montseny et al [1993b,a], Staffans [1994]) devoted to exact as well as approximate state realizations of a wide class of integral operators H, that is, when the support of u is in R + , to some formulations of Hu of the following form:…”

Numerical solutions to a BBM‐Burgers model with a nonlocal viscous term

“…The classical methods used in the literature consist in the approximation of these fractional operators using either convolution integrals or the so‐called Gear scheme for fractional operators. Recently, number of authors in the automatic community developed an alternative method, called the diffusive realization, which is devoted to causal pseudodifferential operators . Different applications of this approach can be found in .…”