The Time Optimal Control of Two Dimensional Convective Brinkman–Forchheimer Equations

Abstract

In this work, we discuss the time optimal control of two dimensional convective Brinkman–Forchheimer (2D CBF) equations, which describe the motion of incompressible viscous fluid through a rigid, homogeneous, isotropic, porous medium. We establish Pontryagin’s maximum principle for the time optimal control of the 2D CBF equations.

Appendix A. Global Solvability of the Linearized System

In this section, we discuss the existence of a global strong solution to the linearized system (4.26). Let ${\mathbf {u}}(\cdot )$ be the unique strong solution to the system (A.3) with the control $\mathrm {U}$ satisfying ${\mathbf {u}}\in \mathrm {C}([0,T];{\mathbb {V}})\cap \mathrm {L}^2(0,T;\mathrm {D}(\mathrm {A}))$. Then, we have the following result.

Theorem A.1

For $\mathrm {U}\in \mathrm {L}^2(0,T;{\mathbb {H}})$, there exists a unique strong solution to the system

$$\begin{aligned} \left\{ \begin{aligned}&\partial _t{\mathbf {z}}(t)+\mu \mathrm {A}{\mathbf {z}}(t)+\mathrm {B}'({\mathbf {u}}(t)){\mathbf {z}}(t)+\alpha {\mathbf {z}}(t)+\beta {\mathcal {C}}'({\mathbf {u}}(t)){\mathbf {z}}(t)=\mathrm {U}(t),\ \text { in }\ {\mathbb {H}},\\&\quad {\mathbf {z}}(0)={\mathbf {0}}, \end{aligned} \right. \end{aligned}$$

(A.1)

for a.e. $t\in (0,T)$, satisfying

$$\begin{aligned} {\mathbf {z}}\in \mathrm {C}([0,T];{\mathbb {V}})\cap \mathrm {L}^2(0,T;\mathrm {D}(\mathrm {A})). \end{aligned}$$

Proof

Using the standard Faedo-Galerkin technique, one can obtain the global solvability results. We only provide a-priori energy estimates satisfied by the system (A.1) and one can use Faedo-Galerkin approximation to make the calculations rigorous. Taking the inner product with ${\mathbf {z}}(\cdot )$ to the first equation in (A.1), we obtain

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {H}}}^2+\mu \Vert {\mathbf {z}}(t)\Vert _{{\mathbb {V}}}^2+\alpha \Vert {\mathbf {z}}(t)\Vert _{{\mathbb {H}}}^2\nonumber \\&\qquad +\beta \Vert |{\mathbf {u}}(t)|^{\frac{r-1}{2}}{\mathbf {z}}(t)\Vert _{{\mathbb {H}}}^2\mathrm {d}t+(r-1)\beta \Vert |{\mathbf {u}}(t)|^{r-3}({\mathbf {u}}(t)\cdot {\mathbf {z}}(t))\Vert _{{\mathbb {H}}}^2\nonumber \\&\quad =-(\mathrm {B}({\mathbf {z}}(t),{\mathbf {u}}(t)),{\mathbf {z}}(t))+(\mathrm {U}(t),{\mathbf {z}}(t))\le \sqrt{2}\Vert {\mathbf {u}}(t)\Vert _{{\mathbb {V}}}\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {H}}}\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {V}}}\nonumber \\&\qquad +\frac{1}{\sqrt{\lambda _1}}\Vert \mathrm {U}(t)\Vert _{{\mathbb {H}}}\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {V}}}\nonumber \\&\quad \le \frac{\mu }{2}\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {V}}}^2+\frac{2}{\mu }\Vert {\mathbf {u}}(t)\Vert _{{\mathbb {V}}}^2\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {H}}}^2+\frac{1}{\mu \lambda _1}\Vert \mathrm {U}(t)\Vert _{{\mathbb {H}}}^2. \end{aligned}$$

(A.2)

Integrating the above inequality from 0 to t, we find

$$\begin{aligned}&\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {H}}}^2+\mu \int _0^t\Vert {\mathbf {z}}(s)\Vert _{{\mathbb {V}}}^2\mathrm {d}s+2\alpha \int _0^t\Vert {\mathbf {z}}(s)\Vert _{{\mathbb {H}}}^2\mathrm {d}s+2\beta \int _0^t\Vert |{\mathbf {u}}(s)|^{\frac{r-1}{2}}{\mathbf {z}}(s)\Vert _{{\mathbb {H}}}^2\mathrm {d}s\nonumber \\&\qquad +2(r-1)\beta \int _0^t\Vert |{\mathbf {u}}(s)|^{r-3}({\mathbf {u}}(s)\cdot {\mathbf {z}}(s))\Vert _{{\mathbb {H}}}^2\mathrm {d}s\nonumber \\&\quad \le \Vert {\mathbf {z}}_0\Vert _{{\mathbb {H}}}^2+\frac{4}{\mu }\int _0^t\Vert {\mathbf {u}}(s)\Vert _{{\mathbb {V}}}^2\Vert {\mathbf {z}}(s)\Vert _{{\mathbb {H}}}^2\mathrm {d}s+\frac{2}{\mu \lambda _1}\int _0^t\Vert \mathrm {U}(s)\Vert _{{\mathbb {H}}}^2\mathrm {d}s, \end{aligned}$$

(A.3)

for all $t\in (0,T)$. An application of Gornwall’s inequality in (A.3) yields

$$\begin{aligned}&\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {H}}}^2\le \left\{ \Vert {\mathbf {z}}_0\Vert _{{\mathbb {H}}}^2+\frac{2}{\mu \lambda _1}\int _0^T\Vert \mathrm {U}(t)\Vert _{{\mathbb {H}}}^2\mathrm {d}t\right\} \exp \left( \frac{4}{\mu }\int _0^T\Vert {\mathbf {u}}(t)\Vert _{{\mathbb {V}}}^2\mathrm {d}t\right) , \end{aligned}$$

(A.4)

for all $t\in (0,T)$. Using (A.4) in (A.3), we get

$$\begin{aligned}&\sup _{t\in [0,T]}\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {H}}}^2+\mu \int _0^T\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {V}}}^2\mathrm {d}t\nonumber \\&\quad \le \left\{ \Vert {\mathbf {z}}_0\Vert _{{\mathbb {H}}}^2+\frac{2}{\mu }\int _0^T\Vert \mathrm {U}(t)\Vert _{{\mathbb {H}}}^2\mathrm {d}t\right\} \exp \left( \frac{8}{\mu }\int _0^T\Vert {\mathbf {u}}(t)\Vert _{{\mathbb {V}}}^2\mathrm {d}t\right) . \end{aligned}$$

(A.5)

Taking inner product with $\mathrm {A}{\mathbf {z}}(\cdot )$ to the first equation in (A.1), we obtain

$$\begin{aligned}&\frac{1}{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {V}}}^2+\mu \Vert \mathrm {A}{\mathbf {z}}(t)\Vert _{{\mathbb {H}}}^2+\alpha \Vert {\mathbf {z}}(t)\Vert _{{\mathbb {V}}}^2\nonumber \\&\quad =-(\mathrm {B}({\mathbf {u}}(t),{\mathbf {z}}(t)),\mathrm {A}{\mathbf {z}}(t))-(\mathrm {B}({\mathbf {z}}(t),{\mathbf {u}}(t)),\mathrm {A}{\mathbf {z}}(t))\nonumber \\&\qquad -\beta ({\mathcal {C}}'({\mathbf {u}}(t)){\mathbf {z}}(t),\mathrm {A}{\mathbf {z}}(t))+(\mathrm {U}(t),\mathrm {A}{\mathbf {z}}(t)). \end{aligned}$$

(A.6)

For $r=2$, we estimate terms on the right hand side of the equality (A.6) using Hölder’s, Ladyzhenskaya’s, Agmon’s and Young’s inequalities as

$$\begin{aligned} |(\mathrm {B}({\mathbf {u}},{\mathbf {z}}),\mathrm {A}{\mathbf {z}})|&\le \Vert {\mathbf {u}}\Vert _{{\widetilde{{\mathbb {L}}}}^{\infty }}\Vert {\mathbf {z}}\Vert _{{\mathbb {V}}}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}\le \frac{\mu }{8}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}^2+\frac{C}{\mu }\Vert {\mathbf {u}}\Vert _{{\mathbb {H}}}\Vert \mathrm {A}{\mathbf {u}}\Vert _{{\mathbb {H}}}\Vert {\mathbf {z}}\Vert _{{\mathbb {V}}}^2, \end{aligned}$$

(A.7)

$$\begin{aligned} |(\mathrm {B}({\mathbf {z}},{\mathbf {u}}),\mathrm {A}{\mathbf {z}})|&\le \Vert {\mathbf {z}}\Vert _{{\widetilde{{\mathbb {L}}}}^{\infty }}\Vert {\mathbf {u}}\Vert _{{\mathbb {V}}}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}\le C\Vert {\mathbf {z}}\Vert _{{\mathbb {H}}}^{1/2}\Vert {\mathbf {u}}\Vert _{{\mathbb {V}}}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}^{3/2}\nonumber \\&\le \frac{\mu }{8}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}^2+\frac{C}{\mu ^3}\Vert {\mathbf {u}}\Vert _{{\mathbb {V}}}^4\Vert {\mathbf {z}}\Vert _{{\mathbb {H}}}^2, \end{aligned}$$

(A.8)

$$\begin{aligned} \beta |({\mathcal {C}}'({\mathbf {u}}){\mathbf {z}},\mathrm {A}{\mathbf {z}})|&\le 2\beta \Vert {\mathbf {u}}\Vert _{{\widetilde{{\mathbb {L}}}}^4}\Vert {\mathbf {z}}\Vert _{{\widetilde{{\mathbb {L}}}}^4}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}\le 2\sqrt{2}\beta \Vert {\mathbf {u}}\Vert _{{\widetilde{{\mathbb {L}}}}^4}\Vert {\mathbf {z}}\Vert _{{\mathbb {H}}}^{1/2}\Vert {\mathbf {z}}\Vert _{{\mathbb {V}}}^{1/2}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}\nonumber \\&\le \frac{\mu }{8}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}^2+\frac{16\beta ^2}{\mu }\Vert {\mathbf {u}}\Vert _{{\widetilde{{\mathbb {L}}}}^4}^2\Vert {\mathbf {z}}\Vert _{{\mathbb {H}}}\Vert {\mathbf {z}}\Vert _{{\mathbb {V}}}\nonumber \\&\le \frac{\mu }{8}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}^2+\frac{32\beta ^2}{\mu }\Vert {\mathbf {u}}\Vert _{{\widetilde{{\mathbb {L}}}}^4}^4\Vert {\mathbf {z}}\Vert _{{\mathbb {V}}}^2+\frac{32\beta ^2}{\mu }\Vert {\mathbf {z}}\Vert _{{\mathbb {H}}}^2, \end{aligned}$$

(A.9)

$$\begin{aligned} |(\mathrm {U},\mathrm {A}{\mathbf {z}})|&\le \Vert \mathrm {U}\Vert _{{\mathbb {H}}}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}\le \frac{\mu }{8}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}^2+\frac{2}{\mu }\Vert \mathrm {U}\Vert _{{\mathbb {H}}}^2. \end{aligned}$$

(A.10)

Combining (A.7)-(A.10), substituting it in (A.6) and then integrating from 0 to t, we find

$$\begin{aligned}&\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {V}}}^2+\mu \int _0^t\Vert \mathrm {A}{\mathbf {z}}(s)\Vert _{{\mathbb {H}}}^2\mathrm {d}s+2\alpha \int _0^t\Vert {\mathbf {z}}(s)\Vert _{{\mathbb {V}}}^2\mathrm {d}s\nonumber \\&\quad \le \frac{4}{\mu }\int _0^t\Vert \mathrm {U}(s)\Vert _{{\mathbb {H}}}^2\mathrm {d}s+\frac{64\beta ^2}{\mu }\int _0^t\Vert {\mathbf {z}}(s)\Vert _{{\mathbb {H}}}^2\mathrm {d}s\nonumber \\&\qquad + \int _0^t\left[ \frac{C}{\mu }\left( \Vert {\mathbf {u}}(s)\Vert _{{\mathbb {H}}}^2+\Vert \mathrm {A}{\mathbf {u}}(s)\Vert _{{\mathbb {H}}}^2\right) +\frac{64\beta ^2}{\mu }\Vert {\mathbf {u}}(s)\Vert _{{\widetilde{{\mathbb {L}}}}^4}^4\right] \Vert {\mathbf {z}}(s)\Vert _{{\mathbb {V}}}^2\mathrm {d}s. \end{aligned}$$

(A.11)

Applying Gronwall’s inequality in (A.11), we get

$$\begin{aligned}&\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {V}}}^2+\mu \int _0^t\Vert \mathrm {A}{\mathbf {z}}(s)\Vert _{{\mathbb {H}}}^2\mathrm {d}s+2\alpha \int _0^t\Vert {\mathbf {z}}(s)\Vert _{{\mathbb {V}}}^2\mathrm {d}s\nonumber \\&\quad \le \left\{ \frac{4}{\mu }\int _0^T\Vert \mathrm {U}(t)\Vert _{{\mathbb {H}}}^2\mathrm {d}t+\frac{64\beta ^2}{\mu }\int _0^T\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {H}}}^2\mathrm {d}t\right\} \nonumber \\&\qquad \times \exp \left\{ \frac{C}{\mu }\int _0^T\left[ \Vert {\mathbf {u}}(t)\Vert _{{\mathbb {H}}}^2+\Vert \mathrm {A}{\mathbf {u}}(t)\Vert _{{\mathbb {H}}}^2+\beta ^2\Vert {\mathbf {u}}(t)\Vert _{{\widetilde{{\mathbb {L}}}}^4}^4\right] \mathrm {d}t\right\} , \end{aligned}$$

(A.12)

for all $t\in [0,T]$. For $r=3$, we need to estimate $\beta |({\mathcal {C}}'({\mathbf {u}}){\mathbf {z}},\mathrm {A}{\mathbf {z}})|$ only. Using Hölder’s, Ladyzhenskaya’s, Agmon’s and Young’s inequalities, we estimate it as

$$\begin{aligned} \beta ({\mathcal {C}}'({\mathbf {u}}){\mathbf {z}},\mathrm {A}{\mathbf {z}})&\le 3\beta \Vert {\mathbf {u}}\Vert _{{\widetilde{{\mathbb {L}}}}^8}^2\Vert {\mathbf {z}}\Vert _{{\widetilde{{\mathbb {L}}}}^4}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}\le C\beta \Vert {\mathbf {u}}\Vert _{{\mathbb {V}}}^2\Vert {\mathbf {z}}\Vert _{{\mathbb {H}}}^{1/2}\Vert {\mathbf {z}}\Vert _{{\mathbb {V}}}^{1/2}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}\nonumber \\&\le \frac{\mu }{8}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}^2+\frac{C\beta ^2}{\mu }\Vert {\mathbf {u}}\Vert _{{\mathbb {V}}}^4\Vert {\mathbf {z}}\Vert _{{\mathbb {H}}}\Vert {\mathbf {z}}\Vert _{{\mathbb {V}}}\nonumber \\&\le \frac{\mu }{8}\Vert \mathrm {A}{\mathbf {z}}\Vert _{{\mathbb {H}}}^2+\frac{C\beta ^2}{\mu }\left( \Vert {\mathbf {u}}\Vert _{{\mathbb {V}}}^8\Vert {\mathbf {z}}\Vert _{{\mathbb {V}}}^2+\Vert {\mathbf {z}}\Vert _{{\mathbb {H}}}^2\right) . \end{aligned}$$

(A.13)

In order to obtain time derivative estimates, we take the inner product with $\partial _t{\mathbf {u}}(\cdot )$ with the first equation in (A.1) to find

$$\begin{aligned}&\Vert \partial _t{\mathbf {z}}(t)\Vert _{{\mathbb {H}}}^2+\frac{\mu }{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {V}}}^2+\frac{\alpha }{2}\frac{\mathrm {d}}{\mathrm {d}t}\Vert {\mathbf {z}}(t)\Vert _{{\mathbb {H}}}^2\nonumber \\&\quad = -(\mathrm {B}({\mathbf {u}}(t),{\mathbf {z}}(t)),\partial _t{\mathbf {z}}(t))-(\mathrm {B}({\mathbf {z}}(t),{\mathbf {u}}(t)),\partial _t{\mathbf {z}}(t))\nonumber \\&\qquad -\beta ({\mathcal {C}}'({\mathbf {u}}(t)){\mathbf {z}}(t),\partial _t{\mathbf {u}}(t))+(\mathrm {U}(t),\partial _t{\mathbf {z}}(t)). \end{aligned}$$

(A.14)

We estimate $|(\mathrm {B}({\mathbf {z}},{\mathbf {u}}),\partial _t{\mathbf {z}})|$ using (2.11) and Young inequality as

$$\begin{aligned} |(\mathrm {B}({\mathbf {z}},{\mathbf {u}}),\partial _t{\mathbf {z}})|&\le C\Vert {\mathbf {z}}\Vert _{{\mathbb {H}}}^{1/2}\Vert {\mathbf {z}}\Vert _{{\mathbb {V}}}^{1/2}\Vert {\mathbf {u}}\Vert _{{\mathbb {V}}}^{1/2}\Vert \mathrm {A}{\mathbf {u}}\Vert _{{\mathbb {H}}}^{1/2}\Vert \partial _t{\mathbf {z}}\Vert _{{\mathbb {H}}}\nonumber \\&\le \frac{\mu }{8}\Vert \partial _t{\mathbf {z}}\Vert _{{\mathbb {H}}}^2+\frac{C}{\mu \sqrt{\lambda _1}}\Vert {\mathbf {u}}\Vert _{{\mathbb {V}}}\Vert \mathrm {A}{\mathbf {u}}\Vert _{{\mathbb {H}}}\Vert {\mathbf {z}}\Vert _{{\mathbb {V}}}^2. \end{aligned}$$

(A.15)

One can use the calculations similar to (A.7)-(A.10) and (A.13) to obtain $\int _0^t\Vert \partial _t {\mathbf {u}}(s)\Vert _{{\mathbb {H}}}^2\mathrm {d}s\le C,$ for all $t\in [0,T]$ and hence one can complete the Theorem. $\square$