The Regular Linear Systems Associated with the Shift Semigroups and Application to Control Linear Systems with Delay

Abstract

We investigate the infinite dimensional control linear systems with delays in the state and input. We give a new variation of constants formula when the state and control delay operators are unbounded. We prove the existence of mild and classical solutions of such systems. Our approach is based on the theory of abstract and regular linear systems introduced by Salamon (Math Syst Theor 21: [147][148][149][150][151][152][153][154][155][156][157][158][159][160][161][162][163][164] 1989) and Weiss (Isr J Math 65:17-43, 1989). Finally, we apply our abstract framework to an example from population dynamics.

On the Perturbations of Regular Linear Systems and Linear Systems with State and Output Delays

“…We denote such control operator by β X . In order to derive our main theorem in this section, we introduce in X the mass operator L (see [10]) as follows: Obversely, if (A, B, C) generates a regular linear system, (A, B, C) is a regular triple. Using Lemma 5.3, we can obtain the following lemma.…”

“…It is reasonable to consider the conditions L = 0 and D = 0. In fact, as is described in [10], the frequently-used delay operators given by Now we prove the equivalence of system (5.1) and (5.2) under some assumptions. To do this, we first introduce the following lemma:…”

“…Here the result R(λ, (Q X ) −1 )β X = e λ in [10] and Lemma 5.8 are applied. Since X is the classical solution, we obtain…”

“…Since then, Vol. 68 (2010) Perturbations of Regular Systems and Delay systems 359 there have many mathematicians followed the work (see e.g [2,7,9,10,18]). In Sect.…”

Feedback stabilizability of infinite dimensional neutral systems

“…It is shown in [14] and [16] that the triple (A 0 , B 0 , C 0 ) is a regular triple and the identity I : X → X is a feedback operator of its associated regular linear system Σ. Moreover, the operator A coincides with the generator of the closedloop system associated with Σ and the feedback operator I.…”