Zhan-Dong Mei

On robustness of exact controllability and exact observability under cross perturbations of the generator in Banach spaces

This paper is concerned with the exact controllability and exact observability of linear systems in the Banach space setting. It is proved that both the admissibility of control operators and the admissibility of observation operators are invariant to cross perturbations of the generator of a Co-semigroup. Moreover, under the admissibility invariance premise, the robustness of the exact controllability as well as the exact observability to such cross perturbations is verified. An illustrative example is presented.

A characteristic of fractional resolvents

In this paper we define and develop a theory of the Riemann-Liouville fractional semigroups and show that they are equivalent to the fractional resolvents of [K. Li, J. Peng, Applied Mathematics Letters, 25 (2012), 808–812].

Convoluted Fractional

We present the notion of convoluted fractional -semigroup, which is the generalization of convoluted -semigroup in the Banach space setting. We present two equivalent functional equations associated with convoluted fractional -semigroup. Moreover, the well-posedness of the corresponding fractional abstract Cauchy problems is studied.

C

Abstract In this paper, it is proved in general setting that p -admissibilities of control operators and observation operators are invariant to any q -type of perturbations of generator of C 0 -semigroups on Banach space. Moreover, some relations between the Λ -extensions of observation operators with respect to the original generator and the perturbed generator are also characterized, so that the output can be expressed in the mild sense. As an application, the admissibility as well as the mild expressibility of output of a class of observation systems with time delay in state is deduced.

-Semigroups and Fractional Abstract Cauchy Problems

In this paper, we are concerned with a class of abstract fractional relaxation equations. We develop a new notion, named fractional resolvent and derive some of its properties. By virtue of the obtained properties and the properties of general Mittag-Leffler function, we present some sufficient conditions to guarantee that the classical solutions of homogeneous and inhomogeneous fractional relaxation equations exist. An illustrative example is presented.