Not All Feedback Stabilized Hyperbolic Systems are Robust with Respect to Small Time Delays in Their Feedbacks

Stability of the heat and of the wave equations with boundary time-varying delays

“…Since τ (t) ≤ M (see (3)) and in view of the definition (22) of E(t), there exists a constant γ > 0 (depending on γ and δ, namely γ ≤ min(…”

“…Asτ (t) < 1 (see (2)), we obtain (49) d dt E 2 (t) ≤ −2δE 2 (t) + qu x )dx − 2γδE 2 (t) + (u t (π, t), u t (π, t − τ (t)))Φ q (u t (π, t), u t (π, t − τ (t))) , whereΦ q is the matrix defined bỹ , where λ is the greatest negative eigenvalue of Ψ q given by (30),Φ q is negative and therefore Since τ (t) ≤ M (see (3)), in view of the definition of E, there exists a constant γ > 0 (depending on γ and δ: γ ≤ 2γ min 1, 2δe −2δM ) such that d dt E(t) ≤ −γ E(t).…”

“…In the case of distributed parameter systems, even arbitrarily small delays in the feedback may destabilize the system (see e.g. [3,9,15,10]). The stability issue of systems with delay is, therefore, of theoretical and practical importance.…”

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

“…The robustness of delay equations has been studied by many authors (see cf. [Ba1,Ba2,Da,EN,Hu,FN,JGH,Liu]). …”

Stabilization of second-order evolution equations with time delay

“…Time delays are often present in applications and practical problems and it is by now wellknown that even an arbitrarily small delay in the feedback may destabilize a system which is uniformly exponentially stable in absence of delay. For some examples in this sense we refer to [8,9,21,27].…”