Linear control systems

Abstract

This chapter starts with basic results on semigroups of linear operators on Banach spaces. Characterizations of the generators of the semigroups due to Hille and Yosida and to Lions are given. Abstract material is illustrated by self-adjoint and differential operators. Non-homogeneous differential equations in Banach spaces, which are the mathematical models of infinite-dimensional control systems, are studied. Their weak and strong solutions are examined. The final sections are devoted to controlled delay equations: existence of solutions and the semigroup representations.

Bibliographical notes

Our presentation of the material on semigroups was often based on elegant sources like A. Pazy [76] and J. Kisynski [57]. The general result on weak solutions is due to J. Ball [5]. The semigroup approach to control theory of infinite-dimensional systems is the subject of the monographs by R.F. Curtain and A.J. Pritchard [22], R.F. Curtain and H. Zwart [23], see also the author’s paper of survey character [109]. This approach was initiated by A.V. Balakrishnan [4] and M.O. Fattorini [33]. The semigroup approach to delay systems, in the product space $\mathcal H$, was initiated by M.C. Delfour and S.K. Mitter in [27] and developed by many authors, see, e.g. R.F. Curtain and H. Zwart [23] and references therein. Important contributions are due to A. Manitius and R. Triggiani, see, e.g. [68]. In the author’s paper J. Zabczyk [111] some delay equations were treated using the concept of decomposable generators.