Optimal Control for the Navier–Stokes Equation with Time Delay in the Convection: Analysis and Finite Element Approximations

Abstract

A distributed optimal control problem for the 2D incompressible Navier–Stokes equation with delay in the convection term is studied. The delay corresponds to the non-instantaneous effect of the motion of a fluid parcel on the mass transfer, and can be realized as a regularization or stabilization to the Navier–Stokes equation. The existence of optimal controls is established, and the corresponding first-order necessary optimality system is determined. A semi-implicit discontinuous Galerkin scheme with respect to time and conforming finite elements for space is considered. Error analysis for this numerical scheme is discussed and optimal convergence rates are proved. The fully discrete problem is solved by the Barzilai-Borwein gradient method. Numerical examples for the velocity-tracking and vorticity minimization problems based on the Taylor-Hood elements are presented.

Appendix

In the following, an extension of the Gronwall Lemma that is needed in the analysis of the state and linearized state equations is presented. The reader is reminded on the notation of various time intervals in (2.2).

Lemma 7.1

Suppose that $a \ge 0$, $\phi \in L^\infty (I_r)\cap L^1(I)$, $\varphi \in L^1(J_r)$, $\alpha ,\, \beta ,\, \psi \in L^1(I)$, and $\gamma \in L^\infty (I)$ are nonnegative and for a.e. $t \in I$ it holds that

$$\begin{aligned} \phi (t) + \int _0^t \varphi (s){{\,\mathrm{d\!}\,}}s \le a + \int _0^t \alpha (s)\phi (s) + \beta (s)\phi ^r(s) + \gamma (s)\varphi ^r(s) + \psi (s) {{\,\mathrm{d\!}\,}}s. \end{aligned}$$

(7.1)

Then $\phi \in L^\infty (I)$ and there exists a continuous function ${\mathfrak {c}} > 0$ such that

$$\begin{aligned} \Vert \phi \Vert _{L^\infty (I)} + \Vert \varphi \Vert _{L^1(I)} \le {\mathfrak {c}}_{T, r, \alpha , \beta , \gamma }(a + \Vert \phi \Vert _{L^\infty (I_r)} + \Vert \varphi \Vert _{L^1(I_r)} + \Vert \psi \Vert _{L^1(I)}) \end{aligned}$$

where ${\mathfrak {c}}_{T, r, \alpha , \beta , \gamma } := {\mathfrak {c}}(T, r, \Vert \alpha \Vert _{L^1(I)}, \Vert \beta \Vert _{L^1(I)}, \Vert \gamma \Vert _{L^\infty (I)}).$

Proof

Let N be the largest positive integer such that $(N-1)r < T \le Nr$, and set $I_n := [0, nr]$ for $n=1,\ldots , N$. For each n, we shall demonstrate by induction that

$$\begin{aligned} \Vert \phi \Vert _{L^\infty (I_n)} + \Vert \phi \Vert _{L^1(I_n)} \le {\mathfrak {c}}_{T, r, \alpha , \beta , \gamma }(a + \Vert \phi \Vert _{L^\infty (I_r)} + \Vert \varphi \Vert _{L^1(I_r)} + \Vert \psi \Vert _{L^1(I_n)}). \end{aligned}$$

(7.2)

Let us verify this for $n = 1$. Using the assumption (7.1) restricted to $t \in I_1$, we can apply the usual Gronwall Lemma so that

$$\begin{aligned} \Vert \phi \Vert _{L^\infty (I_1)} \le (a + \Vert \beta \Vert _{L^1(I)}\Vert \phi \Vert _{L^\infty (I_r)} + \Vert \gamma \Vert _{L^\infty (I)}\Vert \varphi \Vert _{L^1(I_r)} + \Vert \psi \Vert _{L^1(I_1)})e^{\Vert \alpha \Vert _{L^1(I)}}. \end{aligned}$$

(7.3)

On the other hand, (7.1) also yields the following estimate

$$\begin{aligned} \Vert \varphi \Vert _{L^1(I_1)} \le a&+ \Vert \alpha \Vert _{L^1(I)}\Vert \phi \Vert _{L^\infty (I_1)} + \Vert \beta \Vert _{L^1(I)}\Vert \phi \Vert _{L^\infty (I_r)} \nonumber \\&+ \Vert \gamma \Vert _{L^\infty (I)}\Vert \varphi \Vert _{L^1(I_r)} + \Vert \psi \Vert _{L^1(I_1)}. \end{aligned}$$

(7.4)

Substituting (7.3) in the second term of the right hand side in (7.4) and then adding the resulting inequality with (7.3) prove (7.2) for $n = 1$.

Now, suppose that (7.2) holds for $n = k$. For $t \in I_{k+1}$, we obtain from (7.1) that

$$\begin{aligned} \phi (t)&+ \int _0^t \varphi (s){{\,\mathrm{d\!}\,}}s \le a + \Vert \beta \Vert _{L^1(I)}\max \{\Vert \phi \Vert _{L^\infty (I_k)}, \Vert \phi \Vert _{L^\infty (I_r)}\} \\&+ \Vert \gamma \Vert _{L^\infty (I)}(\Vert \varphi \Vert _{L^1(I_k)} + \Vert \varphi \Vert _{L^1(I_r)}) + \int _0^t \alpha (s)\phi (s) + \psi (s) {{\,\mathrm{d\!}\,}}s. \end{aligned}$$

Thus, applying the Gronwall Lemma once more, one has the estimate

$$\begin{aligned} \Vert \phi \Vert _{L^\infty (I_{k+1})} \le (a&+ \Vert \beta \Vert _{L^1(I)}\max \{\Vert \phi \Vert _{L^\infty (I_k)}, \Vert \phi \Vert _{L^\infty (I_r)}\} \\&+ \Vert \gamma \Vert _{L^\infty (I)}(\Vert \varphi \Vert _{L^1(I_k)} + \Vert \varphi \Vert _{L^1(I_r)}) + \Vert \psi \Vert _{L^1(I_{k+1})})e^{\Vert \alpha \Vert _{L^1(I)}} \end{aligned}$$

and as a consequence it follows that

$$\begin{aligned} \Vert \varphi \Vert _{L^1(I_{k+1})} \le a&+ \Vert \beta \Vert _{L^1(I)}\max \{\Vert \phi \Vert _{L^\infty (I_k)}, \Vert \phi \Vert _{L^\infty (I_r)}\} \\&+ \Vert \gamma \Vert _{L^\infty (I)}(\Vert \varphi \Vert _{L^1(I_k)} + \Vert \varphi \Vert _{L^1(I_r)}) + \Vert \alpha \Vert _{L^1(I)}\Vert \phi \Vert _{L^\infty (I_{k+1})} + \Vert \psi \Vert _{L^1(I_{k+1})}. \end{aligned}$$

The last two inequalities along with the induction hypothesis imply (7.2) for $n = k+1$. This completes the proof of the induction step. $\square$

Next, we recall the following discrete version of the Gronwall Lemma, see [34] for instance. This is utilized in the error analysis of the fully-discrete optimal control problem.

Lemma 7.2

Let $n \in {\mathbb {N}}$, $a \ge 0$, $\{a_k\}_{k=1}^n$, $\{b_k\}_{k=1}^n$, and $\{c_k\}_{k=1}^{n-1}$ be nonnegative sequences with

$$\begin{aligned} a_j + \sum _{k=1}^j b_k \le a + \sum _{k=1}^{j-1} c_k a_k \quad \text {for all } j=1, \ldots , n. \end{aligned}$$

Then it holds that

$$\begin{aligned} \max _{1 \le k \le l}a_k + \sum _{k=1}^l b_k \le a \exp \Bigl (\, \sum _{k=1}^{l-1} c_k\Bigl ) \quad \text {for all } l=1,\ldots ,n. \end{aligned}$$