Fractional evolution equations: Resolvent families, controllability and observability

Abstract

This thesis deals with the study of fractional evolution systems on an infinite-dimensional Banach space X, in which several notions of fractional resolvent families have been defined. We study the associated functional equations, in particular their limiting behavior and control theoretic aspects of the systems, namely controllability and observability. Examples from partial differential equations are examined.<br /> <br /> In the first part, we study the limiting cases of certain functional equations satisfied by the resolvent families. The involved evolution systems have either the Riemann-Liouville or Caputo fractional derivative with order α ∈ (0, 2). In the case α ∈ (0, 1), these functional equations can be seen as a generalization of the Cauchy functional equation S(t + s) = S(t)S(s) for t, s⩾ 0, with (S(t)_)t⩾0 a semigroup on X and when α ∈ (1, 2) as a generalization of the d'Alembert formula C(t+s)+C(t-s) = 2C(s)C(t) for t⩾s⩾0, with (C(t)_)t⩾0 a cosine function on X. Specifically, it is shown that the semigroup equation and cosine function can be recovered from the aforementioned functional equations via convergence in the sense of scalar/vector-valued distributions respectively, without assuming the existence of the Laplace transform for the resolvent families, which is a more general setting of study compared to some previous works.<br /> <br /> In the second part, in the context of the Riemann-Liouville fractional derivative with order α ∈ (1, 2), the concept of Riemann-Liouville cosine-type resolvent family is considered and we obtain some perturbation results, similar to those already obtained for the Caputo fractional derivative.<br /> <br /> In the third part, we also study evolution systems involving the so-called Hilfer time-fractional derivative (it interpolates between the Riemann-Liouville and the Caputo derivatives) associated with the generator A of an ((α,β)-Hilfer resolvent family, which is denoted by (S_α,β(t))_t>0 . Here, we focus first on the concept of mild solution for these evolution problems and their explicit representation through the aforementioned resolvent families. Assuming that the dual space X* has the Radon-Nikodým property, we prove the strong continuity of the dual family (S*_α,β(t))_t>0 on (0, ∞), obtain an integration by parts formula and hence the corresponding dual evolution system is presented. Finally, the notions of <em>γ</em>-controllability and <em>γ</em>-observability of these systems are studied, from which two main results are proved: a) these concepts are in a duality relationship, b) <em>γ</em>-observability is obtained by applying an adaptation of the Lebeau-Robbiano strategy which is a well-known method in control theory. As an application we consider second-order elliptic operators in divergence form with measurable coefficients, from which important cases such as the classical Laplace operator and the spectral fractional Laplacian are considered.