Distributed Optimal Control in One Non-Self-Adjoint Boundary Value Problem

Abstract

We prove the solvability of the optimal control problem for elliptic equation with nonlocal boundary conditions in a circular sector with terminal quadratic cost functional in the class of distributed controls.

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1 Introduction

The theory of linear-quadratic optimal control problems for distributed systems is well researched [1, 2]. In many cases the original problem can be decomposed with the help of Fourier method [3–5]. In this chapter we consider the control problem for elliptic equation with non-local boundary conditions in circular sector [6] with terminal quadratic cost functional. This problem does not allow total decomposition and using of $L^2$-theory. To resolve this problem in the class of distributed controls we use apparatus of specially constructed biorthonormal basis systems of functions [7] and then we analyze the solutions of Fredholm matrix equations.

2 Setting of the Problem

In a circular sector $Q=\{(r,\theta ) | r\in (0,1), \ \theta \in (0,\pi )\}$ we consider the optimal control problem

$$\begin{aligned} \left\{ \begin{array}{l} \varDelta y := \frac{1}{r}\frac{\partial }{\partial r}(r \frac{\partial y}{\partial r}) + \frac{1}{r^2}\frac{\partial ^2 y}{\partial \theta ^2} = u(r,\theta ), \ (r,\theta )\in Q, \\ y(1,\theta )=p(\theta ), \ p(0)=0, \\ y(r,0)=0, \ r\in (0,1), \\ \frac{\partial y}{\partial \theta } (r,0)=\frac{\partial y}{\partial \theta } (r,\pi ), \ r\in (0,1), \end{array}\right. \end{aligned}$$

(21.1)

$$\begin{aligned} \begin{array}{c} J(y,u)= \Vert y(\alpha )\Vert ^2_D dr + \int \limits _0^1 \Vert u^{2}(r)\Vert dr \rightarrow \inf , \end{array} \end{aligned}$$

(21.2)

where $p\in C^1([0,\pi ])$ is given function, $\alpha \in (0,1)$ is fixed number, $\Vert \cdot \Vert _D$ is a norm in $L^2(0,\pi )$, which is equivalent to standard one and is given by the equality

$$\begin{aligned} \Vert v\Vert _D = \left( \sum \limits _{n=1}^\infty v_n^2 \right) ^{1/2}, \end{aligned}$$

where $\forall n\ge 1$, $v_n=\int \limits _0^\pi v(\theta )\psi _n(\theta )d\theta$, $\psi _0(\theta )=\frac{2}{\pi ^2}$, $\psi _{2n}(\theta )=\frac{4}{\pi ^2} (\pi -\theta )\sin 2n\theta$, $\psi _{2n-1}(\theta )=\frac{4}{\pi ^2}\cos 2n\theta$.

The aim of this paper is to establish classical solvability of the problem (21.1)–(21.2), that is to find optimal one among admissible processes $\{u,y\}\in C(\bar{Q})\times \left( C(\bar{Q})\bigcap C^2(Q)\right)$. For the application of the spectral method we use biorthonormal and complete in $L^2(0,\pi )$ well-known Samarsky-Ionkin systems of functions [7]

$$\begin{aligned} \varPsi =\{\psi _n\}_{n=1}^\infty \ \ \text{ and } \end{aligned}$$

$$\begin{aligned} \varPhi =\{\varphi _0(\theta )=\theta , \ \varphi _{2n} (\theta )=\sin 2n\theta , \ \varphi _{2n-1} (\theta )=\theta \cos 2n\theta \}_{n=1}^\infty . \end{aligned}$$

(21.3)

Then $\forall u\in L^2(Q)$

$$\begin{aligned} u(r,\theta )=\sum _{n=0}^\infty u_n(r) \cdot \varphi _n(\theta ), \end{aligned}$$

(21.4)

where $u_n(r)=\int _0^\pi u(r,\theta )\psi _n(\theta )d\theta$. So, we will seek for the solution of the problem (21.1) in form

$$\begin{aligned} y(r,\theta )=y_0(r)\theta + \sum _{n=1}^\infty \left( y_{2n-1}(r)\theta \cos 2n\theta + y_{2n}(r) \sin 2n\theta \right) , \end{aligned}$$

(21.5)

where the functions $\{y_k(r)\}_{k=0}^\infty$ are solutions of the system of ordinary differential equations

$$\begin{aligned} \frac{d}{dr}(r \frac{dy_0}{dr})=r\cdot u_0(r), \ y_0(1)=p_0, \end{aligned}$$

(21.6)

$$\begin{aligned} r\cdot \frac{d}{dr}\left( r\cdot \frac{dy_{2k-1}}{dr}\right) - (2k)^2 y_{2k-1}=r^2 \cdot u_{2k-1}(r), \ y_{2k-1}(1) =p_{2k-1}, \end{aligned}$$

(21.7)

$$\begin{aligned} r \frac{d}{dr}\left( r \cdot \frac{dy_{2k}}{dr}\right) - (2k)^2 y_{2k} - 4k\cdot y_{2k-1}=r^2 \cdot u_{2k}(r), \ y_{2k}(1) =p_{2k}, \end{aligned}$$

(21.8)

where $p_k=\int _0^\pi p(\theta ) \cdot \psi _k(\theta )d\theta$.

Thus the original problem (21.1)–(21.2) is reduced to the following one: among admissible pairs $\{u_n(r),y_n(r)\}_{n=0}^\infty$ of the problem (21.6)–(21.8) one should minimize the cost functional

$$\begin{aligned} J(y,u)= \,&y^2_0(\alpha ) + \int _0^1 u_0^2(r) dr + \sum _{k=1}^\infty (y_{2k-1}^2(\alpha ) + y_{2k}^2(\alpha )\, +\nonumber \\&+ \int _0^1 (u_{2k-1}^2(r) + u_{2k}^2(r))dr ) = J_0 + \sum _{k=1}^\infty J_k. \end{aligned}$$

(21.9)

Herewith the optimal process $\{\tilde{u}_n(r),\tilde{y}_n(r)\}_{n=0}^\infty$ should be such that the formula (21.4) defines function from $C(\bar{Q})$, and the formula (21.5) defines function from $C(\bar{Q})\bigcap C^2(Q)$.

3 Main Results

A structure of the problem (21.6)–(21.8) allows to reduce it to sequence of the following problems:

On the solutions of (21.6) one should minimize the cost functional

$$\begin{aligned} J_0 = J_0(u_0), \end{aligned}$$

(21.10)

on the solutions of (21.7), (21.8) one should minimize the cost functional

$$\begin{aligned} \ J_k = J_k (u_{2k-1},u_{2k}), \ k\ge 1. \end{aligned}$$

(21.11)

For fixed $\{u_k(r)\}_{k=0}^\infty \subset C([0,1])$ solutions of the problem (21.6)–(21.8) have form

$$\begin{aligned} y_0(r)=p_0 - \int _r^1\left( \frac{1}{s}\int _0^s \xi u_0(\xi )d\xi \right) ds = p_0 + \int _0^1 G_0(r,s)u_0(s)ds, \end{aligned}$$

(21.12)

where

$$\begin{aligned} G_0(r,s)=\left\{ \begin{array}{l} s\ln r, \ s\in [0,r], \\ s\ln s, \ s\in [r,1], \end{array}\right. \end{aligned}$$

$$\begin{aligned} y_{2k-1}(r)= p_{2k-1}\cdot r^{2k} + \frac{1}{4k}\int _0^1 s\cdot G_k (r,s)u_{2k-1}(s)ds, \end{aligned}$$

(21.13)

where

$$\begin{aligned} G_k(r,s)=\left\{ \begin{array}{l} s^{2k}(r^{2k} - r^{-2k}), \ s\in [0,r], \\ r^{2k}(s^{2k} - s^{-2k}), \ s\in [r,1], \end{array}\right. \end{aligned}$$

$$\begin{aligned} y_{2k}(r)=\,&p_{2k}\cdot r^{2k} + p_{2k-1}\cdot r^{2k}\cdot \ln r + \frac{1}{4k} \int _0^1 s\cdot G_k (r,s)u_{2k}(s)ds \,\,+\nonumber \\&+ \frac{1}{4k} \int _0^1 s\cdot \bar{G}_k (r,s)u_{2k-1}(s)ds, \end{aligned}$$

(21.14)

where

$$\begin{aligned} \bar{G}_k(r,s)&=\int _0^1 p^{-1}\cdot G_k (r,p) G_k(p,s)ds \\&=\left\{ \begin{array}{l} \frac{1}{2k}\left( (\frac{s}{r})^{2k} - (rs)^{-2k}\right) + r^{2k}s^{2k}\ln (rs) - (\frac{s}{r})^{2k}\ln (\frac{s}{r}), \ s\in [0,r], \\ \frac{1}{2k}\left( (\frac{r}{s})^{2k} - (rs)^{-2k}\right) + r^{2k}s^{2k}\ln (rs) - (\frac{r}{s})^{2k}\ln (\frac{r}{s}), \ s\in [r,1]. \end{array}\right. \end{aligned}$$

Lemma 21.1

For any $k\ge 0$ the formulas (21.12)–(21.14) define the solutions of the problem (21.6)–(21.8) $y_k\in C([0,1])\bigcap C^2(0,1)$.

Proof

Since $y_k$ are the solutions of the problem (21.6)–(21.8), then it suffices to show that $\forall k\ge 0$ $y_k\in C([0,1])$. We denote $\prod =[0,1]\times [0,1]$. Then $G_0 \in C(\prod )$, $\max \limits _{\prod }|G_0(r,s)|=e^{-1}$, so, $y_0\in C([0,1])$.

For $k\ge 1$ $G_k\in C(\prod \setminus \{0,0\})$, $\max \limits _{\prod }|G_k(r,s)|\le 1$, so, $y_{2k-1}\in C([0,1])$. Since $x^k\ln x\in C([0,1])$, $\max \limits _{x\in [0,1]}|x^k\ln x|=e^{-1}\cdot k^{-1}$, then for $\bar{G}_k\in C(\prod \setminus \{0,0\})$ we have: $\max \limits _{\prod }|\bar{G}_k(r,s)|\le \frac{1}{k}$, so, $y_{2k}\in C([0,1])$. Lemma is proved.

Theorem 21.1

The problems (21.10), (21.11) have the unique solution $\{\tilde{u}_k\}_{k=0}^\infty$, moreover $\forall k\ge 0$ $\tilde{u}_k\in C([0,1])$.

Proof

From the formulas (21.12)–(21.14) it follows that the functionals $J_0 : L^2(0,1)\mapsto \mathbf {R}$, $J_k : L^2(0,1)\times L^2(0,1)\mapsto \mathbf {R}$ are strictly convex, continuous and coercive, which means under [1] that the problems (21.10), (21.11) have the unique solution in the spaces $L^2(0,1)$ and $L^2(0,1)\times L^2(0,1)$ correspondingly.

Equating to zero Frechet derivatives of $J_0, J_k$, we obtain the following Fredholm integral equations:

$$\begin{aligned} u_0(s)=-\int _0^1 G_0(\alpha ,s)G_0(\alpha ,p)u_0(p)dp - p_0\cdot G_0(\alpha ,s), \end{aligned}$$

(21.15)

$$\begin{aligned} u_{2k-1}(s)=&-\frac{1}{2}\cdot \frac{1}{(4k)^2}\int _0^1 [\bigl (s\cdot G_k(\alpha ,s)p\cdot G_k(\alpha ,p) + s\cdot \bar{G}_k(\alpha ,s)p\cdot \bar{G}_k(\alpha ,p)\bigr )u_{2k-1}(p)\,\, + \nonumber \\&+ 2s\cdot \bar{G}_k(\alpha ,s)p\cdot G_k(\alpha ,p)u_{2k}(p)]dp - p_{2k}\alpha ^{2k}\frac{1}{4k}s\cdot G_k(\alpha ,s) \,\,-\nonumber \\&- \left( p_{2k}\alpha ^{2k} + p_{2k-1}\alpha ^{2k}\ln \alpha \right) \frac{1}{4k}\cdot s\cdot \bar{G}_k(\alpha ,s), \end{aligned}$$

(21.16)

$$\begin{aligned} u_{2k}(s)=&-\frac{1}{2}\cdot \frac{1}{(4k)^2}\int _0^1 [\left( 2s\cdot G_k(\alpha ,s)p\cdot \bar{G}_k(\alpha ,p)\right) u_{2k-1}(p)\,\,+ \nonumber \\&+ s\cdot G_k(\alpha ,s)p\cdot \bar{G}_k(\alpha ,p)u_{2k}(p)]dp \,\,-\nonumber \\&-\left( p_{2k}\alpha ^{2k} + p_{2k-1}\alpha ^{2k}\ln \alpha \right) \frac{1}{4k}\cdot s\cdot G_k(\alpha ,s). \end{aligned}$$

(21.17)

Since $\max \limits _{(p,s)\in \prod }|G_0(\alpha ,p)G_0(\alpha ,s)|\le e^{-2}<1$, then the Eq. (21.15) has the unique solution $\tilde{u}_0\in C([0,1])$.

Put

$$\begin{aligned} A_k(p,s)&=\left( \begin{array}{lr} s\cdot G_k(\alpha ,s)p\cdot G_k(\alpha ,p) + s\cdot \bar{G}_k(\alpha ,s)p\cdot \bar{G}_k(\alpha ,p) &{} 2s\cdot \bar{G}_k(\alpha ,s)p\cdot G_k(\alpha ,p) \\ 2s\cdot G_k(\alpha ,s)p\cdot \bar{G}_k(\alpha ,p) &{} s\cdot G_k(\alpha ,s)p\cdot \bar{G}_k(\alpha ,p) \end{array}\right) , \\ f_k(s)&= \left( \begin{array}{c} - p_{2k}\alpha ^{2k}\frac{1}{4k}s\cdot G_k(\alpha ,s)- \left( p_{2k}\alpha ^{2k} + p_{2k-1}\alpha ^{2k}\ln \alpha \right) \frac{1}{4k}\cdot s\cdot \bar{G}_k(\alpha ,s) \\ - \left( p_{2k}\alpha ^{2k} + p_{2k-1}\alpha ^{2k}\ln \alpha \right) \frac{1}{4k}\cdot s\cdot G_k(\alpha ,s) \end{array}\right) . \end{aligned}$$

Then from the Eqs. (21.16), (21.17) we have that vector

$$\begin{aligned} z_k(s)= \left( \begin{array}{c} u_{2k-1}(s) \\ u_{2k}(s) \end{array}\right) \end{aligned}$$

satisfies the equation

$$\begin{aligned} z_k(s) = -\frac{1}{2}\cdot \frac{1}{(4k)^2}\int _0^1 A_k(p,s)z_k(p)dp + f_k(s). \end{aligned}$$

(21.18)

Under estimates from Lemma 21.1 we obtain

$$\begin{aligned} \max \limits _\prod \Vert A_k(p,s)\Vert \le 4, \ \max \limits _{s\in [0,1]}\Vert f_k(s)\Vert \le \frac{\alpha ^{2k-1}}{2k} \bigl (|p_2k| + |p_{2k-1}|\bigr ). \end{aligned}$$

Then $\forall k\ge 1$ the equation (21.18) has the unique solution

$$\begin{aligned} \tilde{z}_k(s)=\left( \begin{array}{c} u_{2k-1}(s) \\ u_{2k}(s) \end{array}\right) \in C([0,1]), \end{aligned}$$

herewith $\forall r\in [0,1]$

$$\begin{aligned} |u_{2k-1}(r)|\le \frac{\alpha ^{2k-1}}{k}\bigl (|p_{2k-1}| + |p_{2k}|\bigr ), \ |u_{2k}(r)|\le \frac{\alpha ^{2k-1}}{k}\bigl (|p_{2k-1}| + |p_{2k}|\bigr ). \end{aligned}$$

(21.19)

The theorem is proved.

From the estimates (21.19) it follows that the series $\sum \limits _{n=0}^\infty \tilde{u}_n(r)\varphi _n(\theta )$ converges uniformly on $\bar{Q}$ and it defines the function $\tilde{u}(r,\theta )\in C(\bar{Q})$ by the formula (21.4).

Theorem 21.2

Series

$$\begin{aligned} \tilde{y}_0(r)\theta + \sum \limits _{n=1}^\infty \bigl (\tilde{y}_{2n-1}(r)\theta \cdot \cos 2n\theta + \tilde{y}_{2n}(r)\sin 2n\theta \bigr ), \end{aligned}$$

defines the function $\tilde{y}(r,\theta )\in C(\bar{Q})\bigcap C^2(Q)$ by the formula (21.5), where $\{\tilde{y}_n\}_{n=0}^\infty$ are the solutions of the system (21.6)–(21.8) with controls $\{\tilde{u}_n\}_{n=1}^\infty$.

Proof

By the formulas (21.12)–(21.14) desired series has the form

$$\begin{aligned}&p_0\cdot \theta + \theta \cdot \int \limits _0^1 G_0(r,s)u_0(s)ds + \sum \limits _{n=1}^\infty \Bigl (p_{2k-1}\cdot r^{2k}\cdot \theta \cos 2n\theta \,\,+\nonumber \\&+ \bigl (p_{2k}\cdot r^{2k} + p_{2k-1}\cdot r^{2k}\cdot \ln r\bigr )\sin 2n\theta \Bigr ) + \sum \limits _{n=1}^\infty \theta \cos 2n\theta \cdot \frac{1}{4n}\int \limits _0^1 s G_n(r,s)\tilde{u}_{2n-1}(s)ds\,\,+\nonumber \\&+ \sum \limits _{n=1}^\infty \sin 2n\theta \Bigl (\frac{1}{4n}\int \limits _0^1 s G_n(r,s)\tilde{u}_{2n}(s)ds + \frac{1}{4n}\int \limits _0^1 s \bar{G}_n(r,s)\tilde{u}_{2n-1}(s)ds\Bigr ). \end{aligned}$$

(21.20)

The functions $r^{2n}\sin 2n\theta$ and $r^{2n}(\ln r\cdot \sin 2n\theta + \theta \cos 2n\theta )$ are harmonic, $p\in C^1([0,\pi ])$, $p(0)=0$, so, from [6] the first series in 21.20) is the function from the class $C(\bar{Q})\bigcap C^2(Q)$.

From Lemma 21.1 and the estimates (21.19) we have under Weierstrass theorem that $\tilde{y}\in C(\bar{Q})$.

On $\forall [a,b]\times [c,d]\subset (0,1)\times (0,\pi )$ it remains to investigate the uniform convergence of the series from the first and second-order derivatives on $r,\theta$ of functions

$$\begin{aligned} B_n(r,\theta )&=\frac{1}{4n}\int \limits _0^1 s G_n(r,s)\tilde{u}_{2n-1}(s)ds \cdot \theta \cos 2n\theta = b_n(r) \cdot \theta \cos 2n\theta , \\ C_n(r,\theta )&=\frac{1}{4n}\int \limits _0^1 s G_n(r,s)\tilde{u}_{2n}(s)ds \cdot \sin 2n\theta =c_n\cdot \sin 2n\theta , \\ D_n(r,\theta )&=\frac{1}{4n}\int \limits _0^1 s \bar{G}_n(r,s)\tilde{u}_{2n-1}(s)ds\cdot \sin 2n\theta =d_n\cdot \sin 2n\theta . \end{aligned}$$

From the estimates (21.19) we obtain that the series from derivatives $\frac{\partial }{\partial \theta }$, $\frac{\partial ^2}{\partial \theta ^2}$ converge on $\bar{Q}$ uniformly under Weierstrass theorem.

For $\forall r\in [a,b]$, $\forall n>1$

$$\begin{aligned} b_n(r)\,{=}\,&\frac{1}{4n}\Bigl ((r^{2n}-r^{-2n})\int \limits _0^r s^{2n+1}\tilde{u}_{2n-1}(s)ds + r^{2n}\int \limits _r^1 (s^{2n+1}-s^{1-2n})\tilde{u}_{2n-1}(s)ds\Bigr ), \nonumber \\ b'_n(r)\,{=}\,&\frac{1}{2}(r^{2n-1}+r^{-2n-1})\int \limits _0^r s^{2n+1}\tilde{u}_{2n-1}(s)ds \,\,+\\&+\frac{1}{2} r^{2n-1}\int \limits _r^1 (s^{2n+1}-s^{1-2n})\tilde{u}_{2n-1}(s)ds,\nonumber \end{aligned}$$

(21.21)

(summands which do not contain integrals are mutually canceled)

$$\begin{aligned} b''_n(r)\,{=}\,&\frac{1}{2}\Bigl ((2n-1)r^{2n-2} + (-2n-2)r^{-2n-2}\Bigr )\int \limits _0^r s^{2n+1}\tilde{u}_{2n-1}(s)ds \,\,+ \nonumber \\&+ \frac{1}{2} (2n-1) r^{2n-2}\int \limits _r^1 (s^{2n+1}-s^{1-2n})\tilde{u}_{2n-1}(s)ds + \tilde{u}_{2n-1}(r). \end{aligned}$$

(21.22)

Since $\int \limits _0^r s^{2n+1}ds=\frac{r^{2n+1}}{2n+2}$,

$$\begin{aligned} \int \limits _r^1 (s^{2n+1}-s^{1-2n})ds = -\frac{n}{1-n^2} - \frac{r^{2n+1}}{2n+2} + \frac{r^{2-2n}}{2-2n}, \end{aligned}$$

then $\exists C_1>0$ such that

$$\begin{aligned} |b'_n(r)|\le \frac{C}{n}\cdot \frac{\alpha ^{2n-1}}{n}(|p_{2n-1}| + |p_{2n}|), \end{aligned}$$

so, the series $\sum \limits _{n=2}^\infty \frac{\partial }{\partial r}B_n(r,\theta )$, $\sum \limits _{n=2}^\infty \frac{\partial }{\partial r} C_n(r,\theta )$, $\sum \limits _{n=2}^\infty \frac{\partial ^2}{\partial r \partial \theta } B_n(r,\theta )$, $\sum \limits _{n=2}^\infty \frac{\partial ^2}{\partial r \partial \theta }C_n(r,\theta )$ converge uniformly on $[a,b]\times [c,d]$.

From the same estimates $|b''_n(r)|\le C_2\cdot \frac{\alpha ^{2n-1}}{n}(|p_{2n-1}| + |p_{2n}|)$ and, thereby the series $\sum \limits _{n=2}^\infty \frac{\partial ^2}{\partial r^2} B_n(r,\theta )$, $\sum \limits _{n=2}^\infty \frac{\partial ^2}{\partial r^2} C_n(r,\theta )$ converge uniformly on $[a,b]\times [c,d]$.

For the function $d_n(r)$ we have $\forall r\in [a,b]$:

$$\begin{aligned} d_n(r)\,{=}\,&\frac{1}{8n^2}r^{-2n}\cdot \int \limits _0^r s^{2n+1}\tilde{u}_{2n-1}(s)ds - \frac{1}{8n^2}\cdot r^{2n} \int \limits _0^r s^{2n+1}\tilde{u}_{2n-1}(s)ds \,\,+\nonumber \\&+ \frac{1}{4n}r^{2n}\int \limits _0^r s^{2n+1}\ln s\tilde{u}_{2n-1}(s)ds + \frac{1}{4n}r^{2n}\ln r\int \limits _0^r s^{2n+1}\tilde{u}_{2n-1}(s)ds \,\,-\nonumber \\&-\frac{r^{-2n}}{4n}\int \limits _0^r s^{2n+1}\ln s\tilde{u}_{2n-1}(s)ds + \frac{r^{-2n}\ln r}{4n}\int \limits _0^r s^{2n+1}\tilde{u}_{2n-1}(s)ds \,\,+\nonumber \\&+\frac{1}{8n^2}r^{2n}\int \limits _r^1 s^{-2n+1}\tilde{u}_{2n-1}(s)ds - \frac{1}{8n^2}r^{2n}\int \limits _r^1 s^{2n+1} \tilde{u}_{2n-1}(s)ds \,\,+\nonumber \\&+ \frac{1}{4n}r^{2n}\int \limits _r^1 s^{2n+1}\ln s\tilde{u}_{2n-1}(s)ds + \frac{1}{4n}r^{2n}\ln s\int \limits _r^1 s^{2n+1}\tilde{u}_{2n-1}(s)ds \,\,-\nonumber \\&- \frac{1}{4n}r^{2n}\ln s\int \limits _r^1 s^{-2n+1}\tilde{u}_{2n-1}(s)ds + \frac{1}{4n}r^{2n}\int \limits _r^1 s^{-2n+1}\ln s\tilde{u}_{2n-1}(s)ds,\nonumber \\ d'_n(r)=&-\frac{1}{4}\frac{1}{n}r^{-2n-1}\int \limits _0^r s^{2n+1}\tilde{u}_{2n-1}(s)ds - \frac{1}{4n}r^{2n-1}\int \limits _0^r s^{2n+1}\tilde{u}_{2n-1}(s)ds \,\,+\nonumber \\&+ \frac{1}{2}r^{2n-1}\int \limits _0^r s^{2n+1}\ln s\tilde{u}_{2n-1}(s)ds + \frac{1}{4n}\bigl (2n r^{2n-1}\ln r+r^{2n-1} \bigr )\int \limits _0^r s^{2n+1}\tilde{u}_{2n-1}(s)ds \,\,+\nonumber \\&+ \frac{1}{2}r^{-2n-1}\int \limits _0^r s^{2n+1}\ln s\tilde{u}_{2n-1}(s)ds + \frac{1}{4n}\bigl (-2n r^{-2n-1}\ln r+r^{-2n-1} \bigr )\int \limits _0^r s^{2n+1}\tilde{u}_{2n-1}(s)ds \,\,+\nonumber \\&+\frac{1}{4n}r^{2n-1}\int \limits _r^1 s^{-2n+1}\tilde{u}_{2n-1}(s)ds - \frac{1}{4n}r^{2n-1}\int \limits _r^1 s^{2n+1}\tilde{u}_{2n-1}(s)ds \,\,+\nonumber \\&+ \frac{1}{2}r^{2n-1}\int \limits _r^1 s^{2n+1}\ln s\tilde{u}_{2n-1}(s)ds + \frac{1}{4n}\bigl (2n r^{2n-1}\ln r+r^{2n-1} \bigr )\int \limits _r^1 s^{2n+1}\tilde{u}_{2n-1}(s)ds \,\,-\nonumber \\&- \frac{1}{4n}\bigl (2n r^{2n-1}\ln r+r^{2n-1} \bigr )\int \limits _r^1 s^{-2n+1}\tilde{u}_{2n-1}(s)ds + \frac{1}{2}r^{2n-1}\int \limits _r^1 s^{-2n+1}\ln s\tilde{u}_{2n-1}(s)ds. \end{aligned}$$

Since

$$\begin{aligned} \int \limits _0^r s^{2n+1}\ln s ds=\frac{1}{2n+1}r^{2n+1}\ln r - \frac{r^{2n+1}}{(2n+1)^2}, \end{aligned}$$

then $\exists C_2 >0$ such that

$$\begin{aligned} |d'_n(r)|\le \frac{C_2}{n}\cdot \frac{\alpha ^{2n-1}}{n}(|p_{2n-1}| + |p_{2n}|), \end{aligned}$$

so, the series $\sum \limits _{n=2}^\infty \frac{\partial }{\partial r} D_n(r,\theta )$, $\sum \limits _{n=2}^\infty \frac{\partial ^2}{\partial r \partial \theta }D_n(r,\theta )$ converge uniformly on $[a,b]\times [c,d]$.

It is easy to see that $\exists C_3 >0$ such that

$$\begin{aligned} |d''_n(r)|\le C_3\cdot \frac{\alpha ^{2n-1}}{n}(|p_{2n-1}| + |p_{2n}|). \end{aligned}$$

Hence, the series $\sum \limits _{n=2}^\infty \frac{\partial ^2}{\partial r^2} D_n(r,\theta )$ converges uniformly on $[a,b]\times [c,d]$.

Thereby, $\tilde{y}\in C(\bar{Q})\bigcap C^2(Q)$ and Theorem is proved.

Remark 21.1

If $u(r,\theta )\in C(\bar{Q})$ and for some constant $C>0$ $\forall n\ge 1$ $|u_n(r)|\le \frac{C}{n^2}$, then the control $u$ is admissible in the problem (21.1)–(21.2), that is the corresponding function $y(r,\theta )$ from (21.5) defines classical solution of (21.1).

4 Conclusions

In this paper we proved a solvability of the optimal control problem on the classical solutions of elliptic boundary value problem in a circular sector with equality of flows on radiuses and equality of the solution on the one from radiuses to zero in distributed control class for quadratic cost functional.