Approximate Controllability Results for Linear Viscoelastic Flows

Lack of null controllability of viscoelastic flows

“…This yields that the solutions of these two systems cannot cancel each other outside O, and hence the original system cannot be null controllable using any localized control in the parabolic equation with support inside O, a proper subset of Ω. The suitable initial condition having discontinuity around a neighbourhood of a point of Ω\O can be chosen from the eigenspace of the linear operator associated with the new system taking advantage of the spectrum which has an accumulation point (see [3], Prop. 2.2).…”

“…For η = 0, the single mode linear Maxwell system is exactly controllable at sufficiently large enough time T , using a localized interior control acting only on velocity equation, under the restriction of the reachable set of the stress variable as outlined in the remark above [3]. For small time T , the system cannot be null controllable (due to the finite speed of propagation property the of Maxwell system), using a localized interior control acting only on velocity.…”

“…Under a geometric condition imposed on the controlled region along with an additional condition that the system is underdamped, an exact controllability result for both velocity and stress in a suitable space is established. In [3], a number of refinements of the results obtained in [1,4] are obtained. The approximate controllability results for the single mode and several mode Jeffreys model for both velocity and stress tensor using an interior localized control acting only in the velocity equation are obtained in [3].…”

“…In [3], a number of refinements of the results obtained in [1,4] are obtained. The approximate controllability results for the single mode and several mode Jeffreys model for both velocity and stress tensor using an interior localized control acting only in the velocity equation are obtained in [3]. The exact controllability result for the single mode Maxwell model and the approximate controllability result for the several mode Maxwell model for both velocity and stress tensor are obtained at sufficiently large time using an interior, localized control acting only in velocity.…”

“…To do that, as mentioned in Section 2, we prove Theorem 2.2. Note that, the wellposedness of (2.1) on V 0 n (Ω) × V 1 0 (Ω) can be derived showing that the linear operator associated to (2.1) denoted by (A, D(A)) as defined in (2.4) in [3] generates a C 0 semigroup on V 0 n (Ω) × V 1 0 (Ω) (see [3], Rem. 2.1).…”