Null controllability of the linearized compressible Navier–Stokes equations using moment method

Self Cite

“…Let us choose the space

{\overset{H}{true}}_{p e r}^{2} (I_{2 π}) \times H_{p e r}^{2} (I_{2 π})

where the approximate controllability of is considered. In

{\overset{H}{true}}_{p e r}^{2} (I_{2 π}) \times H_{p e r}^{2} (I_{2 π})

, the well‐posedness of , without control, follows by differentiating the equations twice with respect to x and then using the well‐posedness of the new system (similar to ) in

{\overset{L}{true}}^{2} (I_{2 π}) \times L^{2} (I_{2 π})

(see Remark 2.5 in . Hence, for any small δ > 0, we obtain a control g 4 ∈ L 2 (0,2 π + ε /2; L 2 ( l , r )) such that the solution ( ϱ 4 , υ 4 ) of belongs to

C ([0, 2 π + ε / 2]; {\overset{H}{true}}_{p e r}^{2} (I_{2 π}) \times H_{p e r}^{2} (I_{2 π}))

and satisfies

∥ (ϱ_{f}^{4}, υ_{f}^{4}) ∥_{{\overset{H}{true}}_{p e r}^{2} (I_{2 π}) \times H_{p e r}^{2} (I_{2 π})} ⩽ δ ∥ (ϱ^{4} (\cdot, 0), υ^{4} (\cdot, 0)) ∥_{{\overset{H}{true}}_{p e r}^{2} (I_{2 π}) \times H_{<...>}}

…”

Section: Outline Of the Iterative Processmentioning

confidence: 95%

{\overset{L}{true}}^{2} (I_{2 π}) \times L^{2} (I_{2 π})

at time T , using an L 2 control in velocity. We note that approximate controllability in

{\overset{L}{true}}^{2} (I_{2 π}) \times L^{2} (I_{2 π})

implies approximate controllability in higher Sobolev spaces: Let ϕ 0 be a mollifier.…”

Section: Outline Of the Iterative Processmentioning

confidence: 99%

Section: Step 3 At This Point We Have Thatmentioning

confidence: 99%

Section: Proofmentioning

confidence: 99%

“…We refer to [2][3][4][5][6] for earlier results on a linearized version of our system. There have been a few results on the null controllability of the nonlinear system.…”

Section: Introductionmentioning

confidence: 99%

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In this paper, we consider the linearized compressible barotropic Navier‐Stokes system in a bounded interval with a time‐varying delay term acting in the Dirichlet boundary or internal feedback of the hyperbolic component. Assuming some suitable conditions on the time‐dependent delay term and the coefficients of feedback (delayed or not), we study the exponential stability of the concerned hyperbolic‐parabolic system. Due to the presence of the time‐varying delay term, the corresponding spatial operator is also time dependent. Using classical semigroup theory with Kato's variable norm approach, we first show the existence and uniqueness of the Navier‐Stokes system with time delay, acting in the boundary or interior. Next, we prove the two stabilization results by means of interior delay feedback and boundary delay. In both cases, we establish the exponential stability results by introducing some suitable functional energy and using the Lyapunov function approach.