Robustness with respect to small time-varied delay for linear dynamical systems on Banach spaces

Abstract

Abstract. Under suitable conditions we prove the wellposedness of small time-varied delay equations and then establish the robust stability for such systems on the phase space of continuous vector-valued functions.

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

“…Recently, Bátkai and Farkas [5] obtained a similar result in L p phase space. Li, Gu and Huang [6] also obtained a similar result in the case that B is unbounded and in continuous functions phase space. In addition, many authors have studied the problem of small delays (see, e.g., [7,8,9,10]) by various methods.…”

“…In this paper, we will give a similar result as in [6] in L p phase space when B is unbounded. In order to do this, we suppose B is a closed linear operator satisfying the following assumption:…”

“…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 .…”