Explicit Solutions for Distributed, Boundary and Distributed-Boundary Elliptic Optimal Control Problems

TL;DR

This paper derives explicit solutions for various elliptic optimal control problems involving heat conduction with mixed boundary conditions, providing benchmarks and analyzing convergence as boundary conditions change.

Contribution

It provides the first explicit solutions for several boundary and distributed-boundary elliptic optimal control problems in specific geometries.

Findings

Explicit solutions for control, state, and adjoint in rectangular, annular, and spherical domains.

Convergence of solutions as convective boundary condition parameter alpha approaches infinity.

Order of convergence is 1/alpha, a novel result for these problems.

Abstract

We consider a steady-state heat conduction problem in a multidimensional bounded domain Omega for the Poisson equation with constant internal energy g and mixed boundary conditions given by a constant temperature b in the portion Gamma_1 of the boundary and a constant heat flux q in the remaining portion Gamma_2 of the boundary. Moreover, we consider a family of steady-state heat conduction problems with a convective condition on the boundary Gamma_1 with heat transfer coefficient alpha and external temperature b. We obtain explicitly, for a rectangular domain in R^2, an annulus in R^2 and a spherical shell in R^3, the optimal controls, the system states and adjoint states for the following optimal control problems: a distributed control problem on the internal energy g, a boundary optimal control problem on the heat flux q, a boundary optimal control problem on the external temperature b and a distributed-boundary simultaneous optimal control problem on the source g and the flux q. These explicit solutions can be used for testing new numerical methods as a benchmark test. In agreement with theory, it is proved that the system state, adjoint state, optimal controls and optimal values corresponding to the problem with a convective condition on Gamma_1 converge, when alpha\to\infty, to the corresponding system state, adjoint state, optimal controls and optimal values that arise from the problem with a temperature condition on Gamma_1. Also, we analyze the order of convergence in each case, which turns out to be 1/alpha being new for these kind of elliptic optimal control problems.

Equations

$$\Delta u=g,\quad\text{in }\Omega\qquad\qquad u\big{|}_{\Gamma_{1}}=b,\qquad\qquad-\frac{\partial u}{\partial n}\big{|}_{\Gamma_{2}}=q,\qquad\qquad\frac{\partial u}{\partial n}\big{|}_{\Gamma_{3}}=0,$$

$$-\Delta u_{\alpha}=g\,\ \quad\text{in }\Omega\quad\qquad-\frac{\partial u_{\alpha}}{\partial n}\big{|}_{\Gamma_{1}}=\alpha(u-b),\quad\qquad-\frac{\partial u_{\alpha}}{\partial n}\big{|}_{\Gamma_{2}}=q,\quad\qquad\frac{\partial u_{\alpha}}{\partial n}\big{|}_{\Gamma_{3}}=0,$$

$$\text{ find }{g}_{op}\in \mathbb{R}\text{ such that }{J}_1(g_{op})=\underset{g\in \mathbb{R}}{\min }{J}_1(g)$$

$$\text{find }{g}_{{\alpha }_{op}}\in \mathbb{R}\text{ such that }{J}_{1\alpha }(g_{{\alpha }_{op}})=\underset{g\in \mathbb{R}}{\min }{J}_{1\alpha }(g)$$

$$J_1(g)=\frac{1}{2}{\Vert u_g-z_d\Vert }_H^2+\frac{M_1}{2}{\Vert g\Vert }_H^2\text{and}{J}_{1\alpha }(g)=\frac{1}{2}{\Vert u_{\alpha g}-z_d\Vert }_H^2+\frac{M_1}{2}{\Vert g\Vert }_H^2$$

$$\text{find }{q}_{op}\in \mathbb{R}\text{ such that }{J}_2(q_{op})=\underset{q\in \mathbb{R}}{\min }{J}_2(q)$$

$$\text{find }{q}_{{\alpha }_{op}}\in \mathbb{R}\text{ such that }{J}_{2\alpha }(q_{{\alpha }_{op}})=\underset{q\in \mathbb{R}}{\min }{J}_{2\alpha }(q)$$

$$J_2(q)=\frac{1}{2}{\Vert u_q-z_d\Vert }_H^2+\frac{M_2}{2}{\Vert q\Vert }_Q^2\text{and}{J}_{2\alpha }(q)=\frac{1}{2}{\Vert u_{\alpha q}-z_d\Vert }_H^2+\frac{M_2}{2}{\Vert q\Vert }_Q^2$$

$$\text{find }{b}_{op}\in \mathbb{R}\text{ such that }{J}_3(b_{op})=\underset{b\in \mathbb{R}}{\min }{J}_3(b)$$

$$\text{find }{b}_{{\alpha }_{op}}\in \mathbb{R}\text{ such that }{J}_{3\alpha }(b_{{\alpha }_{op}})=\underset{b\in \mathbb{R}}{\min }{J}_{3\alpha }(q)$$

$$J_3(b)=\frac{1}{2}{\Vert u_b-z_d\Vert }_H^2+\frac{M_3}{2}{\Vert b\Vert }_B^2$$

$$J_{3\alpha }(b)=\frac{1}{2}{\Vert u_{\alpha b}-z_d\Vert }_H^2+\frac{M_3}{2}{\Vert b\Vert }_B^2$$

$$\text{find }(g,q{)}_{op}\in \mathbb{R}\times \mathbb{R}\text{ such that }{J}_4((g,q{)}_{op})=\underset{g\in \mathbb{R},q\in \mathbb{R}}{\min }{J}_4(g,q)$$

$$\text{find }(g,q{)}_{{\alpha }_{op}}\in \mathbb{R}\times \mathbb{R}\text{ such that}{J}_{4\alpha }((g,q{)}_{{\alpha }_{op}})=\underset{g\in \mathbb{R},q\in \mathbb{R}}{\min }{J}_{4\alpha }(g,q)$$

$$J_4(g,q)=\frac{1}{2}{\Vert u_{(g,q)}-z_d\Vert }_H^2+\frac{M_4}{2}{\Vert g\Vert }_H^2+\frac{M_5}{2}{\Vert q\Vert }_Q^2$$

$$J_{4\alpha }(g,q)=\frac{1}{2}{\Vert u_{\alpha (g,q)}-z_d\Vert }_H^2+\frac{M_4}{2}{\Vert g\Vert }_H^2+\frac{M_5}{2}{\Vert q\Vert }_Q^2$$

$$-\Delta p=u-z_{d},\quad\text{in }\Omega\quad\qquad p\big{|}_{\Gamma_{1}}=0,\qquad\quad\frac{\partial p}{\partial n}\big{|}_{\Gamma_{2}}=0,\quad\qquad\frac{\partial p}{\partial n}\big{|}_{\Gamma_{3}}=0,$$

$$-\Delta p_{\alpha}=u_{\alpha}-z_{d},\quad\text{in }\Omega\qquad\quad-\frac{\partial p_{\alpha}}{\partial n}\big{|}_{\Gamma_{1}}=\alpha p_{\alpha},\quad\qquad\frac{\partial p_{\alpha}}{\partial n}\big{|}_{\Gamma_{2}}=0,\qquad\qquad\frac{\partial p_{\alpha}}{\partial n}\big{|}_{\Gamma_{3}}=0$$

$${\Omega }_1=\{(x,y)\in {\mathbb{R}}^2:0<x<x_0,0<y<y_0\}$$

$${\Gamma }_1=\{(x,y)\in {\mathbb{R}}^2:x=0,0\le y\le y_0\}$$

$${\Gamma }_2=\{(x,y)\in {\mathbb{R}}^2:x=x_0,0\le y\le y_0\}$$

$${\Gamma }_3=\{(x,y)\in {\mathbb{R}}^2:y=0,0<x<x_0\}\cup \{(x,y)\in {\mathbb{R}}^2:y=y_0,0<x<x_0\}$$

$$u(x,y)=u(x)=-g\frac{x^2}{2}+(g{x}_0-q)x+b$$

$$p(x,y)=p(x)=g\frac{x^4}{24}-(g{x}_0-q)\frac{x^3}{6}-(b-z_d)\frac{x^2}{2}+Ax$$

$$u_{\alpha }(x,y)=u_{\alpha }(x)=-g\frac{x^2}{2}+(g{x}_0-q)x+\frac{g{x}_0-q}{\alpha }+b$$

$$p_{\alpha }(x,y)=p_{\alpha }(x)=g\frac{x^4}{24}-(g{x}_0-q)\frac{x^3}{6}-\left(\frac{g{x}_0-q}{\alpha }+(b-z_d)\right)\frac{x^2}{2}+A_{\alpha }x+\frac{A_{\alpha }}{\alpha }$$

$$||u_{\alpha }-u|{|}_{H^1({\Omega }_1)}=||u_{\alpha }-u|{|}_{L^2({\Omega }_1)}=\frac{K_1}{\alpha }$$

$$K_1=(x_0{y}_0{)}^{1/2}|q-g{x}_0|.$$

$$\underset{\alpha \to \infty }{\lim }\alpha ||p_{\alpha }-p|{|}_{L^2({\Omega }_1)}=L_1$$

$$L_1$$

$$\displaystyle\left.-60(b-z_{d})(5qx_{0}-4gx_{0}^{2})\Big{|}\right\}^{1/2}.$$

$$\frac{1}{2}\Vert u-z_d{\Vert }_{L^2({\Omega }_1)}^2$$

$$\displaystyle+C_{5}gx_{0}^{3}(b-z_{d})+C_{6}qx_{0}^{2}(b-z_{d})\Big{]}$$

$$C_1=\frac{2}{15},C_2=\frac{1}{3},C_3=1,C_4=-\frac{5}{12},C_5=\frac{2}{3},C_6=-1.$$

Full text

Keywords: Elliptic variational equalities, distributed and boundary optimal control problems, mixed boundary conditions, explicit solutions, optimality conditions.

2000 AMS Subject Classification: 35C05, 35J25, 35J86, 35R35, 49J20, 49K20.

1 Introduction

The goal of this paper is to show the explicit solution for eight elliptic optimal control problems in two and three dimensional cases.

We consider a bounded domain $\Omega$ in ${\mathbb{R}}^{n}$ $(n=2,3)$, whose regular boundary $\Gamma$ consist of the union of three disjoint portions $\Gamma_{1}$, $\Gamma_{2}$ and $\Gamma_{3}$ with $meas(\Gamma_{1})>0$, $meas(\Gamma_{2})>0$ and $meas(\Gamma_{3})\geq 0$. We present the following steady-state heat conduction problems $S$ and $S_{\alpha}$ (for each parameter $\alpha>0)$ respectively, with mixed boundary conditions

[TABLE]

[TABLE]

where $g$ is the internal energy in $\Omega$, $b$ is the temperature on $\Gamma_{1}$ for (1) and the temperature of the external neighborhood of $\Gamma_{1}$ for (2), $q$ is the heat flux on $\Gamma_{2}$ and $\alpha>0$ is the heat transfer coefficient on $\Gamma_{1}$. The above problems can be considered as the steady-state Stefan problems, [10, 24, 25, 26]. Note that mixed boundary conditions play an important role in various applications, e.g. heat conduction and electric potential problems [15]. In general, the solution of a mixed elliptic boundary problems is not so regular [14] but there exist some examples which solutions are regular [1, 19, 23].

Let $u$ and $u_{\alpha}$ the unique solutions of the elliptic problems (1) and (2), respectively. In relation with these state systems, we present the particular eight following optimal control problems [2, 20, 22, 28].

1.1 Distributed optimal control on the constant internal energy $g$

Following [11], we consider the distributed optimal control problems:

[TABLE]

with $J_{1}:\mathbb{R}{\rightarrow}{\mathbb{R}}_{0}^{+}$ and $J_{1\alpha}:\mathbb{R}{\rightarrow}{\mathbb{R}}_{0}^{+}$, given by

[TABLE]

with $H=L^{2}(\Omega)$, and where $u_{g}$ and $u_{\alpha g}$ denote the unique solutions of the problems (1) and (2) respectively, for data $q\in\mathbb{R}$, $b\in\mathbb{R}$, $z_{d}\in\mathbb{R}$ and $M_{1}$ a positive constant.

1.2 Boundary optimal control on the constant heat flux $q$ on $\Gamma_{2}$

Following [12], we formulate the boundary optimal control problems:

[TABLE]

where $J_{2}:\mathbb{R}{\rightarrow}{\mathbb{R}}_{0}^{+}$ and $J_{2\alpha}:\mathbb{R}{\rightarrow}{\mathbb{R}}_{0}^{+}$ given by

[TABLE]

with $Q=L^{2}\left(\Gamma_{2}\right)$ where $u_{q}$ y $u_{\alpha q}$ are the unique solutions of the problems (1) and (2) respectively, for data $g\in\mathbb{R}$, $b\in\mathbb{R}$, $z_{d}\in\mathbb{R}$ and $M_{2}$ a positive constant.

1.3 Boundary optimal control on the constant temperature $b$ in an external neighborhood of $\Gamma_{1}$

Following [3], we consider the boundary optimal control problems:

[TABLE]

with $J_{3}:\mathbb{R}{\rightarrow}{\mathbb{R}}_{0}^{+}$ and $J_{3\alpha}:\mathbb{R}{\rightarrow}{\mathbb{R}}_{0}^{+}$, given by

[TABLE]

[TABLE]

with $B=L^{2}\left(\Gamma_{1}\right)$, where $u_{b}$ y $u_{\alpha b}$ are the unique solutions of the problems (1) and (2) respectively, for data $g\in\mathbb{R}$, $q\in\mathbb{R}$, $z_{d}\in\mathbb{R}$ and $M_{3}$ a positive constant.

1.4 Simultaneous distributed-boundary optimal control on the constant source $g$ and the constant flux $q$

Following [13], we formulate the simultaneous distributed-boundary optimal control problems:

[TABLE]

with the cost functional $J_{4}:\mathbb{R}\times\mathbb{R}{\rightarrow}{\mathbb{R}}_{0}^{+}$ and $J_{4\alpha}:\mathbb{R}\times\mathbb{R}{\rightarrow}{\mathbb{R}}_{0}^{+}$ given by

[TABLE]

where $u_{(g,q)}$ and $u_{\alpha(g,q)}$ are the unique solutions of the problems (1) and (2) respectively, for data $b\in\mathbb{R}$, $z_{d}\in\mathbb{R}$, $M_{4}$ and $M_{5}$ positive constants.

1.5 Adjoint states

We define the adjoint state corresponding to problems $S$ and $S_{\alpha}$ as the unique solution of the following mixed elliptic problems, respectively.

[TABLE]

and

[TABLE]

with $u$ and $u_{\alpha}$ given by the unique solution of (1) and (2), respectively. Other theoretical optimal control problems in the subject was done in [4, 5, 6, 7, 8, 9, 16, 17, 18, 21, 29].

In [3, 11, 12, 13] were obtained results of existence and uniqueness of the optimal controls, as well also convergence results, when the heat transfer coefficient $\alpha$ goes to infinity, of the optimal controls, the system states and the adjoint states, in suitable Sobolev spaces.

In Section 2, we calculate explicitly the optimal controls, the system states and the adjoint states, for the optimal control problems previously formulated, related to $S$ and $S_{\alpha}$ respectively, in a rectangular domain in $\mathbb{R}^{2}$. In Section 3 and Section 4, similar results are obtained in an annulus in $\mathbb{R}^{2}$ and a spherical shell in $\mathbb{R}^{3}$, respectively. In all cases, we obtain, in agreement with theory, the convergence of the optimal controls and values when $\alpha\to\infty$ as it was obtained in [3, 11, 12, 13] and for numerical analysis in [27]. Also, the corresponding rates of convergence are studied, obtaining, in Appendix $A$, that the order of convergence in each case is $1/\alpha$ which is new for these elliptic optimal control problems.

We remark that the expressions for the system states $u$, $u_{\alpha}$, the adjoint states $p$, $p_{\alpha}$, the functional cost $J_{i},J_{i\alpha}$, $i=1,..,4$, and the optimal controls are defined for each particular domain, using the same notation.

2 Optimal solutions for a rectangle in $\mathbb{R}^{2}$

In this Section, we consider a rectangular domain in the plane, that is

[TABLE]

whose boundaries $\Gamma_{1}$, $\Gamma_{2}$ and $\Gamma_{3}$ are given by (see Figure 1):

[TABLE]

[TABLE]

[TABLE]

Figure 1

If we consider constant data $g$, $b$, $\alpha$, $q$ and the desired system state $z_{d}\in\mathbb{R}$, we obtain the following result, which proof is omitted:

Lemma 2.1**.**

i) The system state and adjoint state for the problem (1) and (11) respectively are given by:

[TABLE]

[TABLE]

*where A=x_{0}\Big{[}g\frac{x_{0}^{2}}{3}-q\frac{x_{0}}{2}+(b-z_{d})\Big{]}.

ii) The system state and adjoint state for the problem (2) and (12) respectively take the expressions:*

[TABLE]

[TABLE]

where $A_{\alpha}=x_{0}\left[gx_{0}^{2}\left(\frac{1}{3}+\frac{1}{\alpha x_{0}}\right)-qx_{0}\left(\frac{1}{2}-\frac{1}{\alpha x_{0}}\right)+(b-z_{d})\right]$.

Remark 2.2**.**

It is immediate that $u_{\alpha}$ converges to $u$ and $p_{\alpha}$ to $p$, when $\alpha\to\infty$. Moreover, we can prove that there exists a positive constant $K_{1}=K_{1}(x_{0},y_{0},g,q)$ such that:

[TABLE]

where

[TABLE]

In the same way, a similar estimate can be obtained for the adjoint states $p_{\alpha}$ and $p$. It can be proved that there exists a positive constant $L_{1}=L_{1}(x_{0},y_{0},g,q,b,z_{d})$ such that:

[TABLE]

where

[TABLE]

Next, we present the following lemma that will allow us to find the solution of the optimal control problems:

Lemma 2.3**.**

i) For the problem (1), it can be obtained that:

[TABLE]

with:

[TABLE]

ii) For the problem (2), we have:

[TABLE]

with:

[TABLE]

Remark 2.4**.**

It is clear that $C_{i\alpha}$ converges to $C_{i}$, when $\alpha\to\infty$ for $i=1,2,\dots,6$.

Theorem 2.5**.**

i) For the distributed optimal control problems (3) and (4), the optimal controls are given by:

[TABLE]

[TABLE]

and the optimal values are given by:

[TABLE]

and

[TABLE]

ii) For the boundary optimal control problems (5) and (6), the optimal controls are given by:

[TABLE]

[TABLE]

and the optimal values can be expressed as:

[TABLE]

and

[TABLE]

iii) For the boundary optimal control problems (7) and (8), the optimal controls are given by:

[TABLE]

[TABLE]

and the optimal values are:

[TABLE]

and

[TABLE]

iv) For the distributed-boundary optimal control problem (9) and (10), the optimal solutions are given by:

[TABLE]

where

[TABLE]

and

[TABLE]

with

[TABLE]

obtaining the following optimal values:

[TABLE]

and

[TABLE]

v) When $\alpha\to\infty$ the following convergences and estimates hold:

1. a)

$g_{\alpha_{op}}\to g_{op}$*  with $|g_{\alpha_{op}}-g_{op}|=\mathcal{O}\left(\frac{1}{\alpha}\right)$* 2. b)

$q_{\alpha_{op}}\to q_{op}$*  with $|q_{\alpha_{op}}-q_{op}|=\mathcal{O}\left(\frac{1}{\alpha}\right)$* 3. c)

$b_{\alpha_{op}}\to b_{op}$*  with $|b_{\alpha_{op}}-b_{op}|=\mathcal{O}\left(\frac{1}{\alpha}\right)$* 4. d)

$(g,q)_{\alpha_{op}}\to(g,q)_{op}$*  with $|g_{\alpha}^{op}-g^{op}|=\mathcal{O}\left(\frac{1}{\alpha}\right)\quad\text{and}\quad|q_{\alpha}^{op}-q^{op}|=\mathcal{O}\left(\frac{1}{\alpha}\right)$*

Moreover, when $\alpha\to\infty$, we have:

1. a’)

$J_{1\alpha}\left(g_{\alpha_{op}}\right)\to J_{1}(g_{op})$*  with  $|J_{1\alpha}\left(g_{\alpha_{op}}\right)-J_{1}(g_{op})|=\mathcal{O}\left(\frac{1}{\alpha}\right)$* 2. b’)

$J_{2\alpha}\left(q_{\alpha_{op}}\right)\to J_{2}(q_{op})$* with  $|J_{2\alpha}\left(q_{\alpha_{op}}\right)-J_{2}(q_{op})|=\mathcal{O}\left(\frac{1}{\alpha}\right)$* 3. c’)

$J_{3\alpha}\left(b_{\alpha_{op}}\right)\to J_{3}(b_{op})$* with  $|J_{3\alpha}\left(b_{\alpha_{op}}\right)-J_{3}(b_{op})|=\mathcal{O}\left(\frac{1}{\alpha}\right)$* 4. d’)

$J_{4\alpha}\left((g,q)_{\alpha_{op}}\right)\to J_{4}((g,q)_{op})$* with  $|J_{4\alpha}\left((g,q)_{\alpha_{op}}\right)-J_{4}((g,q)_{op})|=\mathcal{O}\left(\frac{1}{\alpha}\right)$.*

Proof.

i) Taking into account that the functional $J_{1}$ and $J_{1\alpha}$ are given by the following quadratic forms

[TABLE]

and

[TABLE]

we obtain that the optimal solutions $g_{op}$ and $g_{\alpha_{op}}$ for the problems (3) and (4) are given by (13) and (14), respectively since the second derivative is positive in both cases.

In addition, if we evaluate the functional $J_{1}$ at $g_{op}$ it is obtained formula (15). In a similar way, computing $J_{1\alpha}$ at $g_{\alpha_{op}}$ it can be derived the closed form (16).

ii) The functional $J_{2}$ and $J_{2\alpha}$ are given by the expressions:

[TABLE]

and

[TABLE]

and then the corresponding minimum are given by (17) and (18), respectively, since the second derivative is positive in both cases. Evaluating $J_{2}$ and $J_{2\alpha}$ at $q_{op}$ and $q_{\alpha_{op}}$ respectively, and through computations, the formulas (19) and (20) can be obtained.

iii) For the problems (7) and (8), the functional $J_{3}$ and $J_{3\alpha}$ can be expressed as

[TABLE]

and

[TABLE]

and therefore the optimal controls are given by (21) and (22), respectively since the second derivative is positive in both cases. The formulas (23) and (24) are derived from evaluating $J_{3}$ and $J_{3\alpha}$ at $b_{op}$ and $b_{\alpha_{op}}$.

iv) For the distributed-boundary optimal control problems (9) and (10), the functional $J_{4}$ and $J_{4\alpha}$ can be written as:

[TABLE]

and

[TABLE]

Therefore, the optimal solutions of the problems (9) and (10), take the form (25) and (26), respectively, due to the second partial derivative test. In addition, the optimal optimal values given by formulas (27) and (28) are deduced by evaluating $J_{4}$ at $(g,q)_{op}$ and $J_{4\alpha}$ at $(g,q)_{\alpha_{op}}$.

v) The convergences can be easily proved by taking into account Remark 2.4 and the closed forms of the optimal controls and optimal values given by the preceding items (i)-(iv). Moreover, the following limits can be computed for the optimal controls:

[TABLE]

and for the simultaneous control we have:

[TABLE]

with

[TABLE]

In the case of the optimal values, we have:

[TABLE]

∎

3 Optimal solutions for an annulus in $\mathbb{R}^{2}$

We consider the following particular domain

[TABLE]

with boundary $\Gamma_{1}$ and $\Gamma_{2}$ given by (see Figure 2):

[TABLE]

[TABLE]

Figure 2

In similar way to previous Section, if we take constant data $g$, $b$, $\alpha$, $q$ and the desired system state $z_{d}\in\mathbb{R}$, we obtain the following result:

Lemma 3.1**.**

i) The system state and the adjoint state for the problem (1) are given by

[TABLE]

where

[TABLE]

ii) The system state and the adjoint state for the problem (2) are given by

[TABLE]

where

[TABLE]

Remark 3.2**.**

From the formulas given above, it is clear that $u_{\alpha}$ converges to $u$ and $p_{\alpha}$ to $p$, when $\alpha\to\infty$. Furthermore, we can prove that there exists a positive constant $K_{2}=K_{2}(r_{1},r_{2},g,q)$ such that:

[TABLE]

where

[TABLE]

In the same way, a similar estimate can be obtained for the adjoint states $p_{\alpha}$ and $p$. In Appendix A, it is proved that there exists a positive constant $L_{2}=L_{2}(r_{1},r_{2},g,q,b,z_{d})$ such that:

[TABLE]

Now, we present the following lemma that will allow us to obtain the explicit solutions for the optimal control problems on the annulus in $\mathbb{R}^{2}$.

Lemma 3.3**.**

i) For the problem (1), it can be obtained that:

[TABLE]

with:

[TABLE]

ii) For the problem (2), we have:

[TABLE]

with

[TABLE]

Remark 3.4**.**

It is immediate that $E_{i\alpha}$ converges to $E_{i}$, when $\alpha\to\infty$ for $i=1,2,\dots,6$.

Theorem 3.5**.**

i) For the distributed optimal control problems (3) and (4), the optimal solutions are given by:

[TABLE]

and

[TABLE]

and the optimal values can be expressed as:

[TABLE]

and

[TABLE]

ii) For the boundary optimal control problems (5) and (6), the optimal solutions are given by:

[TABLE]

and

[TABLE]

where the optimal values are given by:

[TABLE]

and

[TABLE]

iii) For the boundary optimal control problems (7) and (8), the optimal controls are given by

[TABLE]

and

[TABLE]

respectively. In addition, the optimal values are given by:

[TABLE]

*and

---

[TABLE]

iv) For the distributed-boundary optimal control problem (9) and (10), the optimal solutions are given by

[TABLE]

with

[TABLE]

and

[TABLE]

where

[TABLE]

*Moreover, the optimal values are given by

---

[TABLE]

*and

---

[TABLE]

v) The convergences and estimates obtained in (v) of Theorem 2.5 also hold for the annulus in $\mathbb{R}^{2}$.

Proof.

i) Taking into account that the functional $J_{1}$ and $J_{1\alpha}$ can be expressed in the following quadratic forms:

[TABLE]

and

[TABLE]

it can be obtained that the optimal solutions $g_{op}$ and $g_{\alpha_{op}}$ for the problems (3) and (4) are given by (29) and (30), respectively, since the second derivative is positive in both cases. The optimal values formulas (31) and (32) are deduced by evaluating $J_{1}$ and $J_{1\alpha}$ at $g_{op}$ and $g_{\alpha_{op}}$, respectively.

ii) The functional $J_{2}$ and $J_{2\alpha}$ are given by the expressions:

[TABLE]

and

[TABLE]

Therefore it is immediate that the optimal controls for problems (5) and (6) are given by (33) and (34), respectively, since the second derivative is positive in both cases.

The computation of $J_{2}(q_{op})$ and $J_{2\alpha}(q_{\alpha_{op}})$ leads to the closed formulas (35) and (36) for the optimal values of the control problems.

iii) For the problems (7) and (8), the functional $J_{3}$ and $J_{3\alpha}$ are given by

[TABLE]

and

[TABLE]

Therefore the optimal controls are given by (37) and (38), respectively, since the second derivative is positive in both cases.

The optimal values given by expressions (39) and (40) are obtained by computing $J_{3}$ and $J_{3\alpha}$ at $b_{op}$ and $b_{\alpha_{op}}$, respectively.

iv) For the distributed-boundary optimal control problems (9) and (10), the functional $J_{4}$ can be expressed as

[TABLE]

and the functional $J_{4\alpha}$ is given by:

[TABLE]

from where it can be obtained that the optimal solutions are given by (41) and (42), respectively, due to the second partial derivative test. Formulas (43) and (44) are deduced by evaluating $J_{4}$ at $(g,q)_{op}$ and $J_{4\alpha}$ at $(g,q)_{\alpha_{op}}$.

v) The convergences and estimates of the optimal controls and the optimal values when $\alpha\to\infty$ are obtained by taking into account the closed formulas given in (i)-(iv) and the Remark 3.4. As the computations become cumbersome, they can be found in the Appendix A.

∎

4 Optimal solutions for a spherical shell in $\mathbb{R}^{3}$

We consider the particular domain

[TABLE]

with boundary $\displaystyle\Gamma=\cup_{i=1}^{2}\Gamma_{i}$, where

[TABLE]

[TABLE]

In similar way to previous Sections, if we take constant data $g$, $b$, $\alpha$, $q$ and the desired system state $z_{d}\in\mathbb{R}$, we obtain the following result:

Lemma 4.1**.**

i) The system state and the adjoint state for the problem (1) are given by

[TABLE]

where

[TABLE]

ii) The system state and the adjoint state for the problem (2) are given by

[TABLE]

where

[TABLE]

Remark 4.2**.**

The convergences of $u_{\alpha}$ to $u$, and $p_{\alpha}$ to $p$, when $\alpha\to\infty$ can be immediately verified.

In addition, there exists a positive constant $K_{3}=K_{3}(r_{1},r_{2},g,q)$ such that:

[TABLE]

with

[TABLE]

Analogously, a similar estimate can be proved for the adjoint states $p_{\alpha}$ and $p$ (see Appendix A).

Now, we present the following lemma that will allow us to obtain the explicit solutions for the optimal control problems on the spherical shell in $\mathbb{R}^{3}$.

Lemma 4.3**.**

i) For the problem (1), it can be obtained that:

[TABLE]

with:

[TABLE]

ii) For the problem (2), we have:

[TABLE]

with

[TABLE]

[TABLE]

Remark 4.4**.**

It is clear that $G_{i\alpha}$ converges to $G_{i}$, when $\alpha\to\infty$ for $i=1,2,\dots,6$.

Theorem 4.5**.**

i) For the distributed optimal control problems (3) and (4), the optimal solutions are given by:

[TABLE]

and

[TABLE]

The optimal values corresponding to those optimal controls are given by the following formulas:

[TABLE]

and

[TABLE]

ii) For the boundary optimal control problems (5) and (6), the optimal solutions are given by:

[TABLE]

and

[TABLE]

The corresponding optimal values can be expressed by:

[TABLE]

and

[TABLE]

iii) For the boundary optimal control problems (7) and (8), the optimal controls are given by

[TABLE]

and

[TABLE]

*Moreover, $J_{3}(b_{op})$ and $J_{3\alpha}(b_{\alpha_{op}})$ can be obtained by the following formulas:

---

[TABLE]

*and

---

[TABLE]

iv) For the distributed-boundary optimal control problem (9) and (10), the optimal solutions are given by

[TABLE]

with

[TABLE]

and

[TABLE]

with

[TABLE]

Furthermore, $J_{4}$ at $(g,q)_{op}$ and $J_{4\alpha}$ at $(g,q)_{\alpha_{op}}$ can be computed by the following expressions:

[TABLE]

*and *

[TABLE]

v) The estimates and convergences obtained in (v) of Theorem 2.5 are also verified for the spherical shell in $\mathbb{R}^{3}$.

Proof.

i) Taking into account that the functional $J_{1}$ and $J_{1\alpha}$ can be expressed in the following quadratic forms:

[TABLE]

and

[TABLE]

it can be obtained that the optimal solutions $g_{op}$ and $g_{\alpha_{op}}$ for the problems (3) and (4) are given by (45) and (46), respectively, since the second derivative is positive in both cases. The optimal values formulas (47) and (48) are deduced by evaluating $J_{1}$ and $J_{1\alpha}$ at $g_{op}$ and $g_{\alpha_{op}}$, respectively.

ii) The functional $J_{2}$ and $J_{2\alpha}$ are given by the expressions:

[TABLE]

and

[TABLE]

Therefore it is immediate that the optimal controls for problems (5) and (6) are given by (49) and (50), respectively, since the second derivative is positive in both cases.

The computation of $J_{2}(q_{op})$ and $J_{2\alpha}(q_{\alpha_{op}})$ leads to the closed formulas (51) and (52) for the optimal values of the control problems considered.

iii) For the problems (7) and (8), the functional $J_{3}$ and $J_{3\alpha}$ are given by

[TABLE]

and

[TABLE]

Therefore the optimal controls are given by (53) and (54), respectively, since the second derivative is positive in both cases.

The optimal values given by expressions (55) and (56) are obtained by computing $J_{3}$ and $J_{3\alpha}$ at $b_{op}$ and $b_{\alpha_{op}}$ respectively.

iv) For the distributed-boundary optimal control problems (9) and (10), the functional $J_{4}$ can be expressed as

[TABLE]

and the functional $J_{4\alpha}$ is given by:

[TABLE]

from where it can be obtained that the optimal solutions are given by (57) and (58), respectively, due to the second partial derivative test. Formulas (59) and (60) are deduced by evaluating $J_{4}$ at $(g,q)_{op}$ and $J_{4\alpha}$ at $(g,q)_{\alpha_{op}}$.

v) The convergences and estimates of the optimal controls and the optimal values, when $\alpha\to\infty$ are obtained by taking into account the formulas given in (i)-(iv) and the Remark 4.4. The corresponding computations can be found in Appendix A. They are omitted here due to the fact that they become cumbersome.

∎

5 Conclusions

In this paper, two different steady-state heat conduction problems $S$ and $S_{\alpha}$, for the Poisson equation with constant internal energy $g$ and mixed boundary conditions have been considered. The problem $S$ corresponds to the case when a constant temperature $b$ is prescribed in the portion $\Gamma_{1}$ of the boundary and a constant flux $q$ on $\Gamma_{2}$, while in the problem $S_{\alpha}$, a convective condition is imposed at $\Gamma_{1}$ with a heat transfer coefficient $\alpha$ and external temperature $b$. Different optimal control problems can be also considered: a distributed control problem on the internal energy $g$, a boundary optimal control problem on the heat flux $q$, a boundary optimal control problem on the external temperature $b$ and a distributed-boundary simultaneous optimal control problem on the source $g$ and the flux $q$ have been defined. We have obtained explicitly the optimal values of these optimal control problems, already study theoretically in literature in a general framework, for the particular domains: a rectangle in $\mathbb{R}^{2}$, an annulus in $\mathbb{R}^{2}$ and a spherical shell in $\mathbb{R}^{3}$. We point out that this solutions provide a benchmark for testing the accuracy of numerical methods. Also, the limit behaviour of the system state, adjoint state, optimal controls and optimal values for the optimal control problems defined from $S_{\alpha}$, when $\alpha\to\infty$ have been analysed; concluding that they converge to the corresponding system state, adjoint state, optimal controls and optimal values for the optimal control problems defined from $S$. All these limits have been proved to present an order of convergence of $1/\alpha$ which can be considered as new results for these kind of elliptic optimal control problems. This estimate, obtained for this particular domains, make us to believe that it also holds for a more general domain, encouraging to prove it analytically.

Acknowledgements

The present work has been partially sponsored by the European Union’s Horizon 2020 Research and Innovation Programme under the Marie Sklodowska-Curie grant agreement 823731 CONMECH, and by the Project PIP No. 0275 from CONICET-UA, Rosario, Argentina; by the Project ANPCyT PICTO Austral 2016 No. 0090 for the first and third authors; and by Project PPI No. 18/C468 from SECyT-UNRC, Río Cuarto, Argentina for the second author.

Appendix A Appendix

Explicit solution for the domain $\Omega_{2}$

Order of convergence for $p_{\alpha}$

[TABLE]

Order of convergence for $g_{\alpha_{op}}$

[TABLE]

with

[TABLE]

Order of convergence for $q_{\alpha_{op}}$

[TABLE]

with

[TABLE]

Order of convergence for $b_{\alpha_{op}}$

[TABLE]

Order of convergence for $g_{\alpha}^{op}$ and $q_{\alpha}^{op}$

[TABLE]

and

[TABLE]

with

[TABLE]

Order of convergence for $J_{1{\alpha}}(g_{\alpha_{op}})$

[TABLE]

with

[TABLE]

Order of convergence for $J_{2{\alpha}}(q_{\alpha_{op}})$

[TABLE]

with

[TABLE]

Order of convergence for $J_{3{\alpha}}(b_{\alpha_{op}})$

[TABLE]

Order of convergence for $J_{4{\alpha}}(g_{\alpha}^{op},q_{\alpha}^{op})$

[TABLE]

with

[TABLE]

Explicit solution for the domain $\Omega_{3}$

Order of convergence for $p_{\alpha}$

[TABLE]

Order of convergence for $g_{\alpha_{op}}$

[TABLE]

with

[TABLE]

Order of convergence for $q_{\alpha_{op}}$

[TABLE]

with

[TABLE]

Order of convergence for $b_{\alpha_{op}}$

[TABLE]

Order of convergence for $g_{\alpha}^{op}$ and $q_{\alpha}^{op}$

[TABLE]

and

[TABLE]

with

[TABLE]

Order of convergence for $J_{1{\alpha}}(g_{\alpha_{op}})$

[TABLE]

with

[TABLE]

Order of convergence for $J_{2{\alpha}}(q_{\alpha_{op}})$

[TABLE]

with

[TABLE]

Order of convergence for $J_{3{\alpha}}(b_{\alpha_{op}})$

[TABLE]

Order of convergence for $J_{4{\alpha}}(g_{\alpha}^{op},q_{\alpha}^{op})$

[TABLE]

with

[TABLE]