Optimization Problems with Gradient of Control's in the Coefficients of Elliptic Equations

Abstract

Linear elliptic equations with the coefficients depending on the control function and its gradient are considered, and an optimal control problem for such equations is studied. The existence and uniqueness of the solution of this problem are proved. The results are applied to the optimal control of the boundary of a given domain.

Appendix

Proof of Theorem 2. First of all, demonstrate that the functional J₀(v) is continuous on the set V. Denote by u_k(x; v + Δv) and u_k(x; v) the solutions of problems (1), (7) and (1), (8) that correspond to the controls v + Δv ∈ V and v ∈ V, respectively. Let Δu_k(x) ≡ u_k(x; v + Δv) − u_k(x; v), k = 1, 2. From the equation for u_k(x; v + Δv) subtract the corresponding equation for u_k(x; v). As a result, the function Δu_k(x), k = 1, 2, will satisfy the integral identity

$$\begin{array}{ccccccc}\mathop{\int}\limits_{D}\left[\mathop{\sum }\limits_{i,j=1}^{n}{a}_{ij}(x,v+\Delta v,{v}_{x}+\Delta {v}_{x})\frac{\partial \Delta {u}_{k}}{\partial {x}_{j}}\frac{\partial {\eta }_{k}}{\partial {x}_{i}}\right.\\ \left.+\mathop{\sum }\limits_{i=1}^{n}{b}_{i}(x,v+\Delta v,{v}_{x}+\Delta {v}_{x})\frac{\partial \Delta {u}_{k}}{\partial {x}_{j}}{\eta }_{k}+c(x,v+\Delta v)\Delta {u}_{k}{\eta }_{k}\right]dx\\ =-\mathop{\int}\limits_{D}\left[\mathop{\sum }\limits_{i,j=1}^{n}({a}_{ij}(x,v+\Delta v,{v}_{x}+\Delta {v}_{x})-{a}_{ij}(x,v,{v}_{x}))\frac{\partial {u}_{k}}{\partial {x}_{j}}\frac{\partial {\eta }_{k}}{\partial {x}_{i}}\right.\\ +\left(\mathop{\sum }\limits_{i=1}^{n}({b}_{i}(x,v+\Delta v,{v}_{x}+\Delta {v}_{x})-{b}_{i}(x,v,{v}_{x}))\frac{\partial {u}_{k}}{\partial {x}_{i}}\right.\\ +(c(x,v+\Delta v,{v}_{x}+\Delta {v}_{x})-c(x,v,{v}_{x})){u}_{k}\\ \left.\left.-(f(x,v+\Delta v,{v}_{x}+\Delta {v}_{x})-f(x,v,{v}_{x}))\right){\eta }_{k}(x)\right]dx,\quad k=1,2,\\ \forall {\eta }_{1}={\eta }_{1}(x)\in \mathop {W_2^1}\limits^\circ ({D})\,{\rm{and}}\,\forall {\eta }_{2}={\eta }_{2}(x)\in {W}_{2}^{1}(D).\end{array}$$

(A.1)

In addition,

$$\Delta {u}_{k}(x)\in {W}_{2}^{1}(D),\quad k=1,2.$$

Introduce the auxiliary notations

$$\begin{array}{cccccc}{\bar{A}}_{ij}(x)={a}_{ij}(x,v+\Delta v,{v}_{x}+\Delta {v}_{x}),\quad i,j=\overline{1,n},\\ {\bar{B}}_{i}(x)={b}_{i}(x,v+\Delta v,{v}_{x}+\Delta {v}_{x}),\quad i=\overline{1,n},\\ \bar{C}(x)=c(x,v+\Delta v,{v}_{x}+\Delta {v}_{x}),\\ {F}_{jk}(x)=\mathop{\sum }\limits_{i=1}^{n}\left[{a}_{ij}(x,v+\Delta v,{v}_{x}+\Delta {v}_{x})-{a}_{ij}(x,v,{v}_{x})\right]\frac{\partial {u}_{k}}{\partial {x}_{i}},\\ {F}_{0k}(x)=(f(x,v+\Delta v,{v}_{x}+\Delta {v}_{x})-f(x,v,{v}_{x}))-(c(x,v+\Delta v,{v}_{x}+v)-c(x,v,{v}_{x})){u}_{k}\\ +\mathop{\sum }\limits_{i=1}^{n}\left[{b}_{i}(x,v+\Delta v,{v}_{x}+\Delta {v}_{x})-{b}_{i}(x,v,{v}_{x})\right]\frac{\partial {u}_{k}}{\partial {x}_{i}},\quad j=\overline{1,n},k=1,2.\end{array}$$

Due to the relation (9) (see Definition 1 of a generalized solution from the space ${W}_{2}^{1}(D)$), the functions Δu₁ and Δu₂ represent the generalized solutions of the first and second boundary-value problems, respectively. Then in accordance with [10], these boundary-value problems for the functions Δu₁ and Δu₂ have unique solutions and also satisfy prior upper bounds like (5) and (6). The explicit form of the functions F_jk, $j=\overline{0,n}$, enables rewriting these upper bounds as

$$\begin{array}{cccc}{\left\Vert \Delta {u}_{k}\right\Vert }_{{W}_{2}^{1}(D)}\le {C}_{3}\left\{\mathop{\sum }\limits_{j=1}^{n}\parallel \mathop{\sum }\limits_{i=1}^{n}{[{a}_{ij}(x,v+\Delta v,{v}_{x}+\Delta {v}_{x})-{a}_{ij}(x,v,{v}_{x})\frac{\partial {u}_{k}}{\partial {x}_{i}}]}^{2}{\parallel }_{{L}_{2}(D)}^{1/2}\right.\\ +\mathop{\sum }\limits_{i=1}^{n}\parallel \left({b}_{i}(x,v+\Delta v,{v}_{x}+\Delta {v}_{x})-{b}_{i}(x,v,{v}_{x})\right)\frac{\partial {u}_{k}}{\partial {x}_{i}}{\parallel }_{{L}_{2}(D)}\\ +\parallel f(x,v+\Delta v,{v}_{x}+\Delta {v}_{x})-f(x,v,{v}_{x}){\parallel }_{{L}_{2}(D)}\\ \left.+\parallel (c(x,v+\Delta v,{v}_{x}+\Delta {v}_{x})-c(x,v,{v}_{x})){u}_{k}{\parallel }_{{L}_{2}(D)}\right\},\end{array}$$

(A.2)

where C₃ > 0 is some constant. By assumption 3) the operators A_ij(v) ≡ a_ij(x, v, v_x), $i,j=\overline{1,n}$, B_i(v) ≡ b_i(x, v, v_x), $i=\overline{1,n}$, C(v) ≡ c(x, v, v_x), and F(v) ≡ f(x, v, v_x) continuously act from the functional space ${W}_{\infty }^{1}(D)$ into the functional spaces L_∞(D), L_∞(D), L_∞(D), and L₂(D), respectively. Therefore, the right-hand side of inequality (A.2) can be estimated via Δv and Δv_x in the norm L_∞(D), i.e., in the norm ${\left\Vert \Delta v\right\Vert }_{{W}_{\infty }^{1}}$. This proves the convergence ${\left\Vert \Delta {u}_{k}\right\Vert }_{{W}_{2}^{1}(D)}\to 0$ as ${\left\Vert \Delta v\right\Vert }_{{W}_{\infty }^{1}(D)}\to 0$, k = 1, 2. In other words, the solutions of problems (1), (7) and (1), (8) in the space ${W}_{2}^{1}(D)$ continuously depend on Δv in the norm ${W}_{\infty }^{1}(D)$. Obviously, the increment of the functional J₀(v) can be represented as

$$\begin{array}{ccc}\Delta {J}_{0}(v)={J}_{0}(v+\Delta v)-{J}_{\alpha }(v)\\ ={\left\Vert \omega (v(E)+\Delta v(x))[{u}_{1}(x)+\Delta {u}_{1}(x)-{u}_{2}(x)-\Delta {u}_{2}(x)]\right\Vert }_{{L}_{2}(D)}^{2}\\ -{\left\Vert \omega (v(E))[{u}_{1}(x)-{u}_{2}(x)]\right\Vert }_{{L}_{2}(D)}^{2}.\end{array}$$

(A.3)

Since the function ω(v) is continuously differentiable on $\left[{v}_{1},{v}_{2}\right]$,

$$\omega (v+\Delta v)={\omega }_{v}(v)+0\left({\left\Vert \Delta v\right\Vert }_{{L}_{2}(D)}\right).$$

Then the increment formula (A.3) for the functional ΔJ₀(v), in combination with the convergence ${\left\Vert \Delta {u}_{k}\right\Vert }_{{W}_{2}^{1}(D)}\to 0$, k = 1, 2, for v ∈ V and v + Δv ∈ V as ${\left\Vert \Delta v\right\Vert }_{{W}_{\infty }^{1}(D)}\to 0$, implies the convergence ΔJ₀(v) → 0 as ${\left\Vert \Delta v\right\Vert }_{{W}_{\infty }^{1}(D)}\to 0$. In other words, the functional J₀(v) is continuous on the set V.

Now take advantage of Theorem 1. Under the hypotheses of this theorem, as the space X, the set U, and the functional I₀(v) choose the space L₂(D), the set V, and the functional J₀(v). In accordance with the considerations presented above, the functional J₀(v) is continuous on V. The functional J₀(v) is obviously bounded above. The set V is closed and bounded in L₂(D). The space L₂(D) is uniformly convex. Hence, by Theorem 1 there exists a dense subset K of the space L₂(D) such that for any $\bar{v}\in K$ problem (4) has a unique solution for any α > 0.

The proof of Theorem 2 is complete.

Proof of Theorem. Consider the optimal control problem for the boundary of a given domain. Perform an appropriate transformation of the coordinate system, in order to prove that this problem can be reduced to the problem with controls appearing in the coefficients of an elliptic equation, like (2). For system (13)–(16) introduce the new variables t_i = x_i, i = 1, …, n − 1, ${t}_{n}={x}_{n}/v(x^{\prime} )$, and $t=(t^{\prime},{t}_{n})$. Therefore, $x^{\prime} =t^{\prime}$ and ${x}_{n}={t}_{n}v(t^{\prime} )$. Denote by z(t) the function $z(t^{\prime},{t}_{n})$. As is easily checked, $z(t)=z(x^{\prime},{x}_{n})/v(x^{\prime} )=u(t^{\prime},{t}_{n}v(t^{\prime} ))=u(x^{\prime},{x}_{n})=u(x)$ and the domain D₀ is transformed to the domain D₁ with the boundary Γ₁.

Apply the transformation of the coordinate system to establish that the solutions of the elliptic Eqs. (13) with the boundary conditions (14), which satisfy the integral identities (15), become the identities

$$\begin{array}{ccccc}\mathop{\int}\limits_{{D}_{1}}\left\{\mathop{\sum }\limits_{i,j=1}^{n-1}{a}_{ij}^{0}(t)\left(\frac{\partial {z}_{k}(t)}{\partial {t}_{j}}-\frac{{t}_{n}}{v(t^{\prime} )}\frac{\partial v(t^{\prime} )}{\partial {t}_{i}}\frac{\partial {z}_{k}(t)}{\partial {t}_{n}}\right)\left(\frac{\partial {\varphi }_{k}(t)}{\partial {t}_{i}}-\frac{{y}_{n}}{v(t^{\prime} )}\frac{\partial v(t^{\prime} )}{\partial {t}_{i}}\frac{\partial {\varphi }_{k}(t)}{\partial {t}_{n}}\right)\right.\\ +\mathop{\sum }\limits_{j=1}^{n-1}{a}_{nj}^{0}(t)\left(\frac{\partial {z}_{k}(t)}{\partial {t}_{j}}-\frac{{t}_{n}}{v(t^{\prime} )}\frac{\partial v(t^{\prime} )}{\partial {t}_{j}}\frac{\partial {z}_{k}(t)}{\partial {t}_{n}}\right)\frac{1}{v(y^{\prime} )}\frac{\partial {\varphi }_{k}(t)}{\partial {t}_{n}}\\ +\mathop{\sum }\limits_{i=1}^{n-1}{a}_{in}^{0}(t)\frac{\partial {z}_{k}(t)}{\partial {t}_{n}}\frac{1}{v(t^{\prime} )}\left(\frac{\partial {\varphi }_{k}(t)}{\partial {t}_{i}}-\frac{{t}_{n}}{v(t^{\prime} )}\frac{\partial v(t^{\prime} )}{\partial {t}_{i}}\times \frac{\partial {\varphi }_{k}(t)}{\partial {t}_{n}}\right)\\ +\left[\mathop{\sum }\limits_{i=1}^{n-1}{b}_{i}^{0}(t)\left(\frac{\partial {z}_{k}(t)}{\partial {t}_{i}}-\frac{{t}_{n}}{v(t^{\prime} )}\frac{\partial v(t^{\prime} )}{\partial {t}_{i}}\frac{\partial {z}_{k}(t)}{\partial {t}_{n}}\right)\right.\\ +\left.\left.{b}_{n}^{0}(t)\frac{1}{v(y^{\prime} )}\frac{\partial {z}_{k}(t)}{\partial {t}_{n}}+{c}^{0}(t){z}_{k}(t)-{f}^{0}(t)\right]{\varphi }_{k}(t)\right\}\sqrt{v(t)}dt=0,\quad k=1,2,\end{array}$$

(A.4)

with the additional notations

$$\begin{array}{ccc}{z}_{k}(t)={u}_{k}\left(t^{\prime},{t}_{n}v(t^{\prime} )\right),\quad {\varphi }_{k}(t)={\eta }_{k}\left(t^{\prime},{t}_{n}v(t^{\prime} )\right),\\ {a}_{ij}^{0}(t)={a}_{ij}\left(t^{\prime},{t}_{n}v(t^{\prime} )\right),\quad i,j=\overline{1,n},\quad {b}_{i}^{0}(t)={b}_{i}\left(t^{\prime},{t}_{n}v(t^{\prime} )\right),\quad i=\overline{1,n},\\ {c}^{0}(t)=c\left(t^{\prime},{t}_{n}v(t^{\prime} )\right),\quad {f}^{0}(t)=f\left(t^{\prime},{t}_{n}v(t^{\prime} )\right).\end{array}$$

The performance functional J_α(v) in the new variables takes the form

$${J}_{\alpha }(v)={\left\Vert \omega (v(t))\sqrt{v(t)}\left({z}_{1}(t)-{z}_{2}(t)\right)\right\Vert }_{{L}_{2}({D}_{1})}^{2}+\alpha {\left\Vert v(t)-\bar{v}(t)\right\Vert }_{{L}_{2}(D_0^\prime)}^{2},\quad \alpha \ge 0.$$

(A.5)

Identity (A.4) can be finally written as

$$\begin{array}{cc}\mathop{\int}\limits_{{D}_{1}}\left\{\mathop{\sum }\limits_{i,j=1}^{n}{\overline{a}}_{ij}(t,v(t^{\prime} ),{v}_{t}(t^{\prime} ))\frac{\partial {z}_{k}(t)}{\partial {t}_{j}}\frac{\partial {\varphi }_{k}(t)}{\partial {t}_{i}}+\left[\mathop{\sum }\limits_{i=1}^{n}{\overline{b}}_{i}(t,v(t^{\prime} ),{v}_{t}(t^{\prime} ))\frac{\partial {z}_{k}(t)}{\partial {t}_{i}}\right.\right.\\ \left.\left.+\overline{c}(t,v(t^{\prime} )){z}_{k}(t)-\overline{f}(t,v(t^{\prime} ))\right]{\varphi }_{k}(t)\right\}dt=0,\quad k=1,2,\end{array}$$

for any ${\varphi _1}(t) \in \mathop {W_2^1}\limits^\circ ({D_1})$ and ${\varphi }_{2}(t)\in {W}_{2}^{1}({D}_{1})$, where ${z}_{1}(t)\mathop {W_2^1}\limits^\circ({D}_{1})$, ${z}_{2}(t){W}_{2}^{1}({D}_{1})$ are the solutions of the integral identity (A.4). The other notations are as follows:

$$\begin{array}{ccccccc}{\overline{a}}_{ij}(t,v(t^{\prime} ),{v}_{t}(t^{\prime} ))={a}_{ij}(t^{\prime},{t}_{n}v(t^{\prime} )),\\ {\bar{a}}_{in}^{}(t,v(t^{\prime} ),{v}_{t}(t^{\prime} ))={a}_{in}^{0}(t)\frac{1}{v(t^{\prime} )}-\mathop{\sum }\limits_{j=1}^{n-1}{a}_{ij}^{0}(t)\frac{{t}_{n}}{v(t^{\prime} )}\frac{\partial v(t^{\prime} )}{\partial {t}_{j}},\\ {\bar{a}}_{nn}(t,v(t^{\prime} ),{v}_{t}(t^{\prime} ))=\mathop{\sum }\limits_{i,j=1}^{n-1}{a}_{ij}^{0}(t)\frac{{t}_{n}^{2}}{{v}^{2}(t^{\prime} )}\frac{\partial v(t^{\prime} )}{\partial {t}_{i}}\frac{\partial v(t^{\prime} )}{\partial {t}_{j}}-\mathop{\sum }\limits_{j=1}^{n-1}{a}_{nj}^{0}(t)\frac{{t}_{n}}{{v}^{2}(t^{\prime} )}\frac{\partial v(t^{\prime} )}{\partial {t}_{j}}-\mathop{\sum }\limits_{i=1}^{n-1}{a}_{in}^{0}(t)\frac{{t}_{n}}{{v}^{2}(t^{\prime\prime} )}\frac{\partial v(t^{\prime} )}{\partial {t}_{i}},\\ {\bar{b}}_{i}(t,v(t^{\prime} ),{v}_{t}(t^{\prime} ))={b}_{i}^{0}(t)={b}_{i}(t^{\prime},{t}_{n}v(t^{\prime} )),\\ {\bar{b}}_{n}(t,v(t^{\prime} ),{v}_{t}(t^{\prime} ))={b}_{n}^{0}(t)\frac{1}{v(t^{\prime} )}-\mathop{\sum }\limits_{i=1}^{n-1}{b}_{i}^{0}(t)\frac{{t}_{n}}{v(t^{\prime} )}\frac{\partial v(t^{\prime} )}{\partial {t}_{i}},\\ \bar{c}(t,v(t^{\prime} ),{v}_{y}(t^{\prime\prime} ))={c}^{0}(t)=c(t^{\prime},{t}_{n}v(t^{\prime} )),\\ \bar{f}(t,v(t^{\prime} ),{v}_{t}(t^{\prime} ))={f}^{0}(t)=f(t^{\prime},{t}_{n}v(y^{\prime} )),\quad i=\overline{1,n-1}.\end{array}$$

Clearly, the last expression is an integral identity for solving the boundary-value problems for an elliptic equation with controls appearing in the coefficients of the following form:

$$-\mathop{\sum }\limits_{i,j=1}^{n}\frac{\partial }{\partial {t}_{i}}\left({\overline{a}}_{ij}(t,v(t^{\prime} ),{v}_{t}(t^{\prime} ))\frac{\partial {z}_{k}(t)}{\partial {t}_{j}}\right)$$

(A.6)

$$\begin{array}{cc}+\mathop{\sum }\limits_{i=1}^{n}{\overline{b}}_{i}(t,v(t^{\prime} ),{v}_{t}(t^{\prime} ))\frac{\partial {z}_{k}(t)}{\partial {t}_{i}}+\overline{c}(t,v(t^{\prime} )){z}_{k}(t)=\overline{f}(t,v(t^{\prime} )),\\ {z}_{1}{| }_{{\Gamma }_{1}}={\left.\frac{\partial {z}_{2}}{\partial N}\right|}_{{\Gamma }_{1}}=0,\quad k=1,2.\end{array}$$

(A.7)

Therefore, the above-mentioned transformation of the coordinate system reduces the original optimal control problem for the boundary of a given domain to the optimal control problem with controls appearing in the coefficients of the elliptic Eq. (A.6) in the domain D₁ with the boundary conditions (A.7) and the performance criterion (A.5), which is minimized on the set of admissible controls V₁. This problem satisfies all hypotheses of Theorem 2, and the conclusion follows.

The proof of Theorem 3 is complete.