Lack of null controllability of viscoelastic flows

Abstract

We consider controllability of linear viscoelastic flow with a localized control in the momentum equation. We show that, for Jeffreys fluids or for Maxwell fluids with more than one relaxation mode, exact null controllability does not hold. This contrasts with known results on approximate controllability.Mathematics Subject Classification. 35Q35, 76A10, 93B05.

Approximate controllability and stabilizability of a linearized system for the interaction between a viscoelastic fluid and a rigid body

“…In order to prove the above result, we have the following results: Proof. The proof is based on Theorem 3.3 and the proof of Theorem 3.1 in [21]. First, from Proposition 2.1, we deduce that…”

“…The proof is based on a localization procedure in the fluid and thus can be deduced from results obtained in [21]. The statement is the following one: Theorem 5.1.…”

“…(5.3) From Theorem 3.5 in [21], there exists u 0 * ∈ H 1 (R 3 ) and v 0 * ∈ H 2 (R 3 ) such that the solution of the above system with ϑ 0 = curl u 0 * and ζ 0 = curl v 0 * satisfies for any t > 0,…”

“…Then, we apply Lemma 3.7 in [21] to deduce the existence of u 0 ∈ H 1 0 (F) and v 0 ∈ H 2 (F) ∩ H 1 0 (F), with div u 0 = div v 0 = 0 such that curl u 0 = ϑ 0 and curl v 0 = ζ 0 in a neighborhood of x0. Extending u 0 and v 0 by 0 in S, we deduce that u 0 ∈ H 1/2 and v 0 ∈ H1.…”

“…Under the usual geometric condition on the controlled domain, they established the exact controllability with interior controls for the single mode and the approximate controllability for multimode Maxwell models. The lack of null controllability of Jeffreys and Maxwell fluids with distributed control is considered in [21]. They established that the solution of single mode and multimode Jeffreys systems cannot reach zero.…”