Volterra Equations And Linear Systems In Abstract Spaces

Volterra equations in abstract spaces are frequently encountered in engineering problems involving viscoelastic material behaviour, such as the transient velocity field in an isotropic viscoelastic fluid, Timoshenko beam, and heat conduction with memory. In this dissertation, asymptotic behaviors of the solutions to Volterra equations in ab-stract spaces, admissibility of the observation and control operators for the correspond-ing Volterra systems together with observability and controllability of the systems are investigated.Chapter 1 provides the research background and the main results of this disserta-tion.Chapter 2 contains some preliminaries including Bochner integral of vector-valued functions, Laplace transform, semigroups of linear operators and their perturbations, and resolvents of linear Volterra equations.Chapter 3 is devoted to discussing uniform exponential stability of the solutions to Volterra equations. Three different methods are used:The idea of the first method is to combine the semigroup approach with spectral analysis of an associated operator matrix. The discussion is made in the setting of Banach spaces, and the kernel function a∈Lp(R+,C) (1< p<∞).The second method is to combine the semigroup approach with a Gearhart type theorem. Thus, it requires the state space to be Hilbertian and the kernel function a∈L2(R+,C).The third method is to apply a representation theorem with regard to Laplace transforms of vector-valued functions. A Tauber type result is obtained.Sufficient conditions are obtained under which the solutions are uniformly expo-nentially stable. Several examples are given to illustrate our results.Chapter 4 is assigned to study time regularities and other asymptotic behaviors of the solutions, namely, strong stability and almost asymptotic periodicity. We deal with the regularities by using the semigroup approach. The strong stability and almost asymptotic periodicity are discussed by combining the Laplace transform tool with some Tauberian theorems.Chapter 5 deals with admissibility of the observation operators and control op-erators for Volterra systems. We mainly study the infinite-time admissibility of the observation operators. Two distinct methods are adopted:The first method is to obtain infinite-time admissibility of the observation operator for the Volterra system from that of an associated observation operator for the Cauchy system in a product space.The second method is to utilize analyticity of the resolvent of the Volterra equa-tion.Chapter 6 is an attempt to explore observability and controllability for general Volterra systems.