On attainability of optimal controls in coefficients for system of Hammerstein type with anisotropic p-Laplacian

Abstract

In this paper we consider an optimal control problem (OCP) for the coupled system of a nonlinear monotone Dirichlet problem with anisotropic p-Laplacian and matrix-valued $L^\infty (\varOmega ,\mathbb {R}^{N\times N})$-controls in its coefficients and a nonlinear equation of Hammerstein type. Using the direct method in calculus of variations, we prove the existence of an optimal control in considered problem and provide sensitivity analysis for a specific case of considered problem with respect to two-parameter regularization.

Appendix

Proof of Proposition 3

Boundedness. From the assumptions on $\mathcal {F}_k$ and the boundedness of A, we get

$$\begin{aligned} \Vert \mathcal {A}_{{\varepsilon },k}\Vert&=\sup _{\Vert y\Vert _{H^1_0(\varOmega )}\le 1}\Vert \mathcal {A}_{{\varepsilon },k} y\Vert _{H^{-1}(\varOmega )} \\&=\sup _{\Vert y\Vert _{H^1_0(\varOmega )}\le 1} \sup _{\Vert v\Vert _{H^1_0(\varOmega )}\le 1}\left<\mathcal {A}_{{\varepsilon },k} y,v\right>_{H^{-1}(\varOmega );H_0^1(\varOmega )}\\&= \sup _{\Vert y\Vert _{H^1_0(\varOmega )}\le 1} \sup _{\Vert v\Vert _{H^1_0(\varOmega )}\le 1} \int _\varOmega \left[ {\varepsilon }+\mathcal {F}_k\big (|A^\frac{1}{2}\nabla y|^2\big )\right] ^{\frac{p-2}{2}}\left( \nabla v,A\nabla y\right) _{\mathbb {R}^N} dx\\&\le \frac{\Vert \xi _2\Vert ^2_{L^\infty (\varOmega )}}{\left( {\varepsilon }+k^2+1\right) ^{\frac{2-p}{2}}}\sup _{\Vert y\Vert _{H^1_0(\varOmega )}\le 1} \sup _{\Vert v\Vert _{H^1_0(\varOmega )}\le 1} \Vert y\Vert _{H^1_0(\varOmega )}\Vert v\Vert _{H^1_0(\varOmega )}=C_{{\varepsilon },k}. \end{aligned}$$

$\square$

Strict monotonicity We make use of the following algebraic inequality, which is proved in [18, Proposition 4.4]:

$$\begin{aligned} \left( \big ({\varepsilon }+\mathcal {F}_k(|a|^2)\big )^{\frac{p-2}{2}}a- \big ({\varepsilon }+\mathcal {F}_k(|b|^2)\big )^{\frac{p-2}{2}}b,a-b\right) _{\mathbb {R}^N} \ge {\varepsilon }^{\frac{p-2}{2}} |a-b|^2,\quad a,b\!\in \! \mathbb {R}^N. \end{aligned}$$

With this, having put $a:=A^\frac{1}{2}\nabla y$, $b:=A^\frac{1}{2}\nabla v$ we obtain

$$\begin{aligned} \big < \mathcal {A}_{{\varepsilon },k}(A,y)-\mathcal {A}_{{\varepsilon },k}(A,v), y-v\big >_{H^{-1}(\varOmega );H^1_0(\varOmega )}&\ge {\varepsilon }^{\frac{p-2}{2}} \int _\varOmega |A^\frac{1}{2}\nabla y-A^\frac{1}{2}\nabla v|^2 dx \\&\ge \alpha ^2 {\varepsilon }^{\frac{p-2}{2}} \Vert y-v\Vert ^2_{H^1_0(\varOmega )}\ge 0. \end{aligned}$$

Since the relation $\big < \mathcal {A}_{{\varepsilon },k}(A,y)-\mathcal {A}_{{\varepsilon },k}(A,v), y-v\big >_{H^{-1}(\varOmega );H^1_0(\varOmega )}=0$ implies $y=v$, it follows that the strict monotonicity property (13), (14) holds true for each $A\in \mathfrak {A}_{ad}$, $k\in \mathbb {N}$, and ${\varepsilon }>0$.

Coercivity The coercivity property obviously follows from the estimate

$$\begin{aligned} \big <\mathcal {A}_{{\varepsilon },k}(A,y),y\big >_{H^{-1}(\varOmega );H^1_0(\varOmega )} \ge \alpha ^2{\varepsilon }^{\frac{p-2}{2}}\Vert y\Vert ^2_{H_0^1(\varOmega )}. \end{aligned}$$

(90)

Semi-continuity In order to get the equality

$$\begin{aligned} \lim _{t\rightarrow 0} \langle \mathcal {A}_{{\varepsilon },k}(A,y+tw),v\rangle _{H^{-1}(\varOmega );H^1_0(\varOmega )}= \langle \mathcal {A}_{{\varepsilon },k}(A,y),v\rangle _{H^{-1}(\varOmega );H^1_0(\varOmega )}, \end{aligned}$$

it is enough to observe that

$$\begin{aligned} ({\varepsilon }+\mathcal {F}_k(|A^\frac{1}{2}(\nabla y+t\nabla w)|^2))^{\frac{p-2}{2}}A\left( \nabla y+t\nabla w\right) \rightarrow ({\varepsilon }+\mathcal {F}_k(|A^\frac{1}{2}\nabla y|^2))^{\frac{p-2}{2}}A\nabla y, \end{aligned}$$

as $t\rightarrow 0$ almost everywhere in $\varOmega$, and make use of Lebesgue’s dominated convergence theorem.

Proof of Proposition 4

Similarly to the proof of Proposition 3, the boundedness, strict monotonicity, and radial continuity of $F_{{\varepsilon },k}$ can be shown. It remains to prove the compactness property. Let $y_n\rightharpoonup y_0$ in $H_0^1(\varOmega )$. Hence, $y_n\rightarrow y_0$ strongly in $L^2(\varOmega )$ and, up to a subsequence, $y_n\rightarrow y_0$ a.e. in $\varOmega$. We must show that $F_{{\varepsilon },k}(y_n,z)\rightarrow F_{{\varepsilon },k}(y_0,z)$ strongly in $L^2(\varOmega )$, i.e.

$$\begin{aligned}&\int _\varOmega |F_{{\varepsilon },k}(y_n,z)- F_{{\varepsilon },k}(y_0,z)|^2\,dx\nonumber \\&\quad =\int _\varOmega |({\varepsilon }+\mathcal {F}_k(|y_n|^2))^{\frac{p-2}{2}}y_n-({\varepsilon }+\mathcal {F}_k(|y_0|^2))^{\frac{p-2}{2}}y_0|^2dx\rightarrow 0\quad \text{ as } \,n\rightarrow \infty .\nonumber \\ \end{aligned}$$

(91)

Obviously, $|({\varepsilon }+\mathcal {F}_k(|y_n|^2))^{\frac{p-2}{2}}y_n-({\varepsilon }+\mathcal {F}_k(|y_0|^2))^{\frac{p-2}{2}}y_0|^2\rightarrow 0$ a.e. in $\varOmega$. Also, the following estimate implies the equi-integrability property of this function

$$\begin{aligned}&\int _\varOmega |({\varepsilon }+\mathcal {F}_k(|y_n|^2))^{\frac{p-2}{2}}y_n-({\varepsilon }+\mathcal {F}_k(|y_0|^2))^{\frac{p-2}{2}}y_0|^2\,dx\\&\quad \le 2 ({\varepsilon }+k^2+1)^{p-2}\int _\varOmega (|y_n|^2+|y_0|^2)\,dx\le C_{{\varepsilon },k},\;\forall \, n\in \mathbb {N}. \end{aligned}$$

Therefore, due to Lebesgue’s Theorem 1,

$$\begin{aligned} |({\varepsilon }+\mathcal {F}_k(|y_n|^2))^{\frac{p-2}{2}}y_n-({\varepsilon }+\mathcal {F}_k(|y_0|^2))^{\frac{p-2}{2}}y_0|^2\rightarrow 0 \text{ strongly } \text{ in } L^1(\varOmega ), \end{aligned}$$

i.e. (91) holds true.

Proof of Proposition 6

Let us fix an arbitrary element y of $H^1_0(\varOmega )$. We associate with this element the set $\varOmega _k(A,y)$, where $\varOmega _k(A,y):=\{x\in \varOmega : |A^\frac{1}{2}\nabla y(x)|>\sqrt{k^2+1}\}$. Then

$$\begin{aligned} \int _\varOmega g y\,dx&= \Vert g\Vert _{L^2(\varOmega )}\Vert y\Vert _{L^2(\varOmega )}\mathop {\le }\limits ^{\text {by Friedrich's inequality}} C_\varOmega \Vert g\Vert _{L^2(\varOmega )}\Vert \nabla y\Vert _{L^2(\varOmega )^N}\\&\le C_\varOmega \Vert g\Vert _{L^2(\varOmega )}\big [\Vert \nabla y\Vert _{L^2(\varOmega \setminus \varOmega _k(A,y))^N}+ \Vert \nabla y\Vert _{L^2(\varOmega _k(A,y))^N}\big ].\nonumber \end{aligned}$$

(92)

Since $\mathcal {F}_k(|A^\frac{1}{2}\nabla y|^2)=|A^\frac{1}{2}\nabla y|^2$ a.e. in $\varOmega {\setminus }\varOmega _k(A,y)$, and $k^2\le \mathcal {F}_k(|A^\frac{1}{2}\nabla y|^2)\le k^2+ 1$ a.e. in $\varOmega _k(A,y)$ $\forall \,k\in \mathbb {N}$, we get

$$\begin{aligned}&\Vert \nabla y\Vert _{L^2(\varOmega \setminus \varOmega _k(A,y))^N}\le \alpha ^{-1}\left( \int _{\varOmega \setminus \varOmega _k(A,y)}|A^\frac{1}{2}\nabla y|^2\,dx\right) ^\frac{1}{2}\\&\quad \le \alpha ^{-1} |\varOmega \setminus \varOmega _k(A,y)|^{\frac{p-2}{2p}}\left( \int _{\varOmega \setminus \varOmega _k(A,y)} |A^\frac{1}{2}\nabla y|^p\,dx\right) ^{\frac{1}{p}}\\&\quad \le \alpha ^{-1}|\varOmega |^{\frac{p-2}{2p}} \left( \int _{\varOmega \setminus \varOmega _k(A,y)} ({\varepsilon }+|A^\frac{1}{2}\nabla y|^2)^{\frac{p-2}{2}}|A^\frac{1}{2}\nabla y|^2\,dx\right) ^{\frac{1}{p}}\\&\quad =\alpha ^{-1} |\varOmega |^{\frac{p-2}{2p}} \left( \int _{\varOmega \setminus \varOmega _k(A,y)} ({\varepsilon }+\mathcal {F}_k(|A^\frac{1}{2}\nabla y|^2))^{\frac{p-2}{2}}|A^\frac{1}{2}\nabla y|^2\,dx\right) ^{\frac{1}{p}} \\&\quad \le \alpha ^{-1}|\varOmega |^{\frac{p-2}{2p}}\Vert y\Vert _{A,{\varepsilon },k}, \end{aligned}$$

and

$$\begin{aligned}&\Vert \nabla y\Vert _{L^2(\varOmega _k(A,y))^N}\le \alpha ^{-1}\left( \int _{\varOmega _k(A,y)}|A^\frac{1}{2}\nabla y|^2\,dx\right) ^{\frac{1}{2}}\\&\quad \le \displaystyle \frac{\alpha ^{-1}}{k^\frac{p-2}{2}}\left( \int _{\varOmega _k(A,y)} ({\varepsilon }+\mathcal {F}_k(|A^\frac{1}{2}\nabla y|^2))^{\frac{p-2}{2}}|A^\frac{1}{2}\nabla y|^2\,dx\right) ^{\frac{1}{2}}\le \displaystyle \frac{\alpha ^{-1}}{k^\frac{p-2}{2}}\Vert y\Vert ^{\frac{p}{2}}_{A,{\varepsilon },k}. \end{aligned}$$

As a result, inequality (92) finally implies the desired estimate. The proof is complete.

Proof of Proposition 8

Due to strong convergence $\left( {\varepsilon }_n+\mathcal {F}_{k_n}(y_0^2)\right) ^\frac{p-2}{2}y_0\rightarrow |y_0|^{p-2}y_0$ in $L^q(\varOmega )$ and relations

$$\begin{aligned}&\left( {\varepsilon }_n+\mathcal {F}_{k_n}(y_n^2)\right) ^\frac{p-2}{2}y_n-\left( {\varepsilon }_n+\mathcal {F}_{k_n}(y_0^2)\right) ^\frac{p-2}{2}y_0 =\left( {\varepsilon }_n+\mathcal {F}_{k_n}(y_n^2)\right) ^\frac{p-2}{2}(y_n-y_0)\nonumber \\&\quad +\left( \left( {\varepsilon }_n+\mathcal {F}_{k_n}(y_n^2)\right) ^\frac{p-2}{2}- \left( {\varepsilon }_n+\mathcal {F}_{k_n}(y_0^2)\right) ^\frac{p-2}{2}\right) y_0=I_1^k+I_2^k, \end{aligned}$$

(93)

it is enough to prove that $I_i^k\rightarrow 0$ strongly in $L^q(\varOmega )$ as $k\rightarrow \infty$ for $i=1,2$.

Step 1 To prove $\Vert I_1\Vert ^q_{L^q}=\int _\varOmega \left| \left( {\varepsilon }_n+\mathcal {F}_{k_n}(y_n^2)\right) ^\frac{p-2}{2}\right| ^\frac{p}{p-1}|y_n-y_0|^\frac{p}{p-1}\,dx\rightarrow 0$ we use Lemma 2. The initial suppositions imply that sequence $\{\varphi _n=|y_n-y_0|^\frac{p}{p-1}\}_{n\in \mathbb {N}}$ is bounded in $L^1(\varOmega )$ and converges to 0 almost everywhere in $\varOmega$. On this step we are left to prove only the equi-integrability property of the sequence $\left| {\varepsilon }_n+\mathcal {F}_{k_n}(y_n^2)\right| ^\frac{p(p-2)}{2(p-1)}$. Let us notice, that $\displaystyle \frac{p(p-2)}{2(p-1)}< \displaystyle \frac{p-1}{2}<\displaystyle \frac{p}{2}$ and, using Hölder inequality with exponents $1/s+1/s'=1$, where $s=\displaystyle \frac{p}{2}/\displaystyle \frac{p(p-2)}{2(p-1)}=\displaystyle \frac{p-1}{p-2}$, $s'=p-1$, we have

$$\begin{aligned} \int _\varOmega \left| {\varepsilon }_n+\mathcal {F}_{k_n}(y_n^2)\right| ^\frac{p(p-2)}{2(p-1)}\,dx&\le \left( \int _\varOmega \left| {\varepsilon }_n+\mathcal {F}_{k_n}(y_n^2)\right| ^\frac{p}{2}dx\right) ^\frac{p-2}{p-1}|\varOmega |^\frac{1}{p-1}\nonumber \\&\le \left( \int _\varOmega \left| {\varepsilon }_n+\mathcal {F}_{k_n}(y_n^2)\right| ^\frac{p-2}{2}({\varepsilon }_n+y_n^2)dx\right) ^\frac{p-2}{p-1}|\varOmega |^\frac{1}{p-1}\nonumber \\&=\left( \int _\varOmega (J_1+J_2)\,dx\right) ^\frac{p-2}{p-1}|\varOmega |^\frac{1}{p-1}\le C|\varOmega |^\frac{1}{p-1}. \end{aligned}$$

(94)

Indeed, $\int _\varOmega J_1\,dx ={\varepsilon }_n\int _\varOmega \left| {\varepsilon }_n+\mathcal {F}_{k_n}(y_n^2)\right| ^\frac{p-2}{2}\,dx\rightarrow 0$, because ${\varepsilon }_n\rightarrow 0$ and within a subsequence, still denoted by the same index, $\left| {\varepsilon }_n+\mathcal {F}_{k_n}(y_n^2)\right| ^\frac{p-2}{2}\rightarrow |y_0|^{p-2}$ a.e. in $\varOmega$. As for the second integral, we have

$$\begin{aligned} \int _\varOmega J_2\,dx\le \Vert \left( {\varepsilon }_n+\mathcal {F}_{k_n}(y_n^2)\right) ^\frac{p-2}{2}y_n\Vert _{L^2(\varOmega )}\Vert y_n\Vert _{L^2(\varOmega )}\le C\sup _{n\in \mathbb {N}}\Vert y_n\Vert _{L^2(\varOmega )}. \end{aligned}$$

Step 2 Here we prove that $I_2^q\rightarrow 0$ strongly in $L^1(\varOmega )$. Indeed, within a subsequence, $({\varepsilon }_n+\mathcal {F}_{k_n}(y_n^2))^\frac{p-2}{2}- ({\varepsilon }_n+\mathcal {F}_{k_n}(y_0^2))^\frac{p-2}{2}\rightarrow 0$ a.e. in $\varOmega$ and closely following the arguments of the previous step it can be shown that

$$\begin{aligned}&\int _\varOmega \left| ({\varepsilon }_n+\mathcal {F}_{k_n}(y_n^2))^\frac{p-2}{2}-({\varepsilon }_n+\mathcal {F}_{k_n}(y_0^2))^\frac{p-2}{2}\right| ^\frac{p}{p-1}\,dx\\&\quad \le \left( \int _\varOmega |{\varepsilon }_n+\mathcal {F}_{k_n}(y_n^2)|^\frac{p(p-2)}{2(p-1)}\,dx+\int _\varOmega |{\varepsilon }_n+\mathcal {F}_{k_n}(y_0^2)|^\frac{(p-2)p}{2(p-1)}\,dx\right) ^\frac{p}{p-1}\le c. \end{aligned}$$

It remains to apply Lemma 2.