Well-posedness of linear partial differential equations with unbounded delay operators

Abstract

In this paper we show the well-posedness of the following constant delay equation:where (A, D(A)) generates a strongly continuous semigroup (T (t)) t 0 on a Banach space X and (B, D(B)) is an unbounded and closed linear operator on X. Our approach is to transform the delay equation into an abstract Cauchy problem in a special phase space and apply semigroup theory. Furthermore, we show that the solution semigroup corresponding to the above delay equation is uniformly exponentially bounded. Finally, we give an example to explain the well-posedness of delay equations.  2004 Elsevier Inc. All rights reserved.

Sufficient Condition of Robustness with Respect to Small Delay for Exponential Stability of Linear Dynamical Systems

“…corresponding to t. The following theorem implies the wellposedness of the delay equation (6) (see [11] for details). Theorem 1: Let A be the infinitesimal generator of C 0 -semigroup (T (·)) t≥0 , and B a closed linear operator satisfying the assumption (H).…”

“…In this section we summarize some results on the wellposedness and the characterization of robust exponential stability of (2) with respect to small delay, mainly from [11,12,13]. In addition, in order to study the problem about robust exponential stability in Banach space, we make use of the theory of operator-valued Fourier multipliers.…”

“…Definition 1: System (6) is said to be robustly exponentially stable with respect to small delays, if there exist some r 0 > 0 and M (r 0 ), ω(r 0 ) > 0 such that for any t ≥ 0, r ∈ [0, (11) where (T r (t)) t≥0 is the solution semigroup of delay system (6) in phase space E r .…”