This article is part of the series Recent Trends on Boundary Value Problems and Related Topics.

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Optimal control problem for stationary quasi-optic equations

Yusuf Koçak1* and Ercan Çelik2

Author Affiliations

1 Department of Mathematics, Ağrı İbrahim Çeçen University Faculty of Science and Art, Ağrı, Turkey

2 Department of Mathematics, Atatürk University Faculty of Science, Erzurum, Turkey

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Boundary Value Problems 2012, 2012:151  doi:10.1186/1687-2770-2012-151


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2012/1/151


Received:1 October 2012
Accepted:30 November 2012
Published:28 December 2012

© 2012 Koçak and Çelik; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, an optimal control problem was taken up for a stationary equation of quasi optic. For the stationary equation of quasi optic, at first judgment relating to the existence and uniqueness of a boundary value problem was given. By using this judgment, the existence and uniqueness of the optimal control problem solutions were proved. Then we state a necessary condition to an optimal solution. We proved differentiability of a functional and obtained a formula for its gradient. By using this formula, the necessary condition for solvability of the problem is stated as the variational principle.

Keywords:
stationary equation of quasi optic; boundary value problem; optimal control problem; variational problem

1 Introduction

Optimal control theory for the quantum mechanic systems described with the Schrödinger equation is one of the important areas of modern optimal control theory. Actually, a stationary quasi-optics equation is a form of the Schrödinger equation with complex potential. Such problems were investigated in [1-5]. Optimal control problem for nonstationary Schrödinger equation of quasi optics was investigated for the first time in [6].

2 Formulation of the problem

We are interested in finding the problem of the minimum of the functional

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M1">View MathML</a>

(1)

in the set

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M2">View MathML</a>

under the condition

(2)

(3)

(4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M6">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M8">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M9">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M11">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M12">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M13">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M14">View MathML</a> are numbers, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M15">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M16">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M17">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M18">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M19">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M20">View MathML</a> are complex valued measurable functions and satisfy the conditions

(5)

(6)

respectively, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M23">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M24">View MathML</a>. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M25">View MathML</a> is a Hilbert space that consists of all functions in <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M26">View MathML</a>, which are measurable and square-integrable. <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M27">View MathML</a> is the well-known Lebesgue space consisting of all functions in Ω, which are measurable and square-integrable.

The problem of finding a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M28">View MathML</a> under the condition (2)-(4) for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M29">View MathML</a>, which is a boundary value problem, is a function for Eq. (2).

Generalized solution of this problem is a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M28">View MathML</a> belonging to the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M31">View MathML</a>, and it satisfies the integral identity

(7)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M33">View MathML</a>.

3 Existence and uniqueness of a solution of the optimal control problem

In this section, we prove the optimal control problem using the Galerkin method and the existence and uniqueness of a solution of the problem (1)-(4).

Theorem 1Suppose that a functionfsatisfies the condition (5). So, for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M29">View MathML</a>, the problem (2)-(4) has a unique solution, and for this solution, the estimate

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M35">View MathML</a>

(8)

is valid for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M36">View MathML</a>. Here, the number<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M37">View MathML</a>is independent ofz.

Proof Proof can be done by processes similar to those given in [7]. □

Theorem 2Let us accept that the conditions of Theorem 1 hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M38">View MathML</a>is a given function. Then there is such a setGdense in<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M39">View MathML</a>that the optimal control problem (1)-(4) has a unique solution<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M40">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M41">View MathML</a>.

Proof Firstly, let us show that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M42">View MathML</a>

(9)

is continuous on the set V. Let us take an arbitrary ∈V, and let <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M43">View MathML</a> be an increment of the v for the <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M44">View MathML</a>. Then the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M45">View MathML</a> of the problem (1)-(4) will have an increment <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M46">View MathML</a>. Here, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M47">View MathML</a> is the solution of (2)-(4). On the basis of the assumptions and conditions (2)-(4), it can be shown that the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M48">View MathML</a> is a solution of the following boundary value problem:

(10)

(11)

(12)

Because the problem (10)-(12) and the problem (2)-(4) are the same type problems, we can write the following estimate the same as (8):

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M52">View MathML</a>

(13)

If we use estimate (13) then we can write the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M53">View MathML</a>

(14)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M54">View MathML</a> is constant that does not depend on Δv.

Now, let us evaluate the increment of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M55">View MathML</a> on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M56">View MathML</a>. Using formula (9) we can write the equality as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M57">View MathML</a>

(15)

Using the Cauchy-Bunyakowski inequality and estimates (8) and (14), we write the inequality as

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M58">View MathML</a>

(16)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M59">View MathML</a> is a constant that does not depend on Δv. This inequality shows that the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M55">View MathML</a> is continuous on the set V. On the other hand, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M61">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M62">View MathML</a>; therefore, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M55">View MathML</a> is bounded on V. The set V is closed, bounded on a Hilbert space H. According to Theorem (Goebel) in [8], there is such a set G dense in H that optimalcontrol problem (1)-(4) has a unique solution for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M41">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M40">View MathML</a>. Theorem 2 is proven. □

3.1 Fréchet diffrentiability of the functional

In this section, we prove the Fréchet differentiability of a given functional. For this purpose, we consider the following adjoint boundary value problem:

(17)

(18)

(19)

Here, the function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M69">View MathML</a> is a solution of (2)-(4) for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M56">View MathML</a>. The solution of the boundary value problem (17)-(19) corresponding to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M56">View MathML</a> is a function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M72">View MathML</a> that belongs to the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M73">View MathML</a> and satisfies the integral identity

(20)

As seen, the problem (17)-(19) is an initial boundary value problem. This can easily be obtained by a transform <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M75">View MathML</a>. Actually, if we do a variable transform <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M75">View MathML</a>, we obtain the boundary problem as

(21)

(22)

(23)

where

If we write the complex conjugate of this boundary value problem, we obtain the following boundary value problem:

(24)

(25)

(26)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M84">View MathML</a>

This problem is a type of (2)-(4) boundary value problem. As the right-hand side is equal to zero, and initial function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M85">View MathML</a> belongs to the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M25">View MathML</a> for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M87">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M38">View MathML</a>. By using Theorem 2, it follows that the solution of the bounded value problem (24)-(26) existing in the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M31">View MathML</a> is unique, and the following estimate is obtained:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M90">View MathML</a>

(27)

If we use the problem (24)-(26) as a type of the conjugate problem (17)-(19), we obtain the initial bounded value problem (17)-(19) has a unique solution belonging to the space <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M31">View MathML</a>, and the following estimate is obtained:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M92">View MathML</a>

Here, the number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M93">View MathML</a> is independent of ψ and y. Now, using estimate (8) in this inequality, we easily write the following estimate:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M94">View MathML</a>

(28)

Here, the number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M95">View MathML</a> is constant.

Theorem 3Let us accept that the conditions of Theorem 2 hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M96">View MathML</a>is given. Then the functional<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M97">View MathML</a>can be Frechet differentiable in the setVand the formula below for a gradient of the functional is valid:

(29)

Proof Let us evaluate the increment of the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M97">View MathML</a> for the element <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M29">View MathML</a>. We can write the following equation for the increment of the functional:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M101">View MathML</a>

(30)

The last formula can be written as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M102','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M102">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M103">View MathML</a> is defined as the formula

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M104">View MathML</a>

(31)

Applying the Cauchy-Bunyakowski inequality, we obtain:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M105">View MathML</a>

If we use estimates (13) and (28) in this inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M106">View MathML</a>

(32)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M107">View MathML</a> is a constant that does not depend on Δv. Hence, we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M108">View MathML</a>

(33)

By using equality (33), the increment of the functional can be written as

(34)

Considering this equality (34), and by using the definition of Fréchet differentiable, we can easily obtain the validity of the rule. Theorem 3 is proved. □

3.2 A necessary condition for an optimal solution

In this section, we prove the continuity of a gradient and state a necessary condition to an optimal solution in the variational inequality form using the gradient.

Theorem 4Accept that the conditions of Theorem 3 hold and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M110">View MathML</a>is an optimal solution of the problem (1)-(4). Then the following inequality is valid for<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M29">View MathML</a>:

(35)

Here, the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M113">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M114">View MathML</a>are solutions of the problems (2)-(4) and a conjugate problem corresponding to<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M110">View MathML</a>, respectively.

Proof Now, we prove that the gradient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M116">View MathML</a> is continuous at V. For this we show

(36)

(37)

(38)

(39)

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M121">View MathML</a>.

In order to show (36), using the formula <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M122">View MathML</a> in (29), we can write the following equation:

(40)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M124">View MathML</a> is the solution of the problem (9)-(11) and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M125">View MathML</a> is the solution of the following problem:

(41)

(42)

(43)

This bounded value problem is a type of a conjugate problem. For this solution, the following estimate is valid:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M129">View MathML</a>

(44)

Here, the number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M130">View MathML</a> is constant.

Using (13) and (28), we write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M131">View MathML</a>

(45)

Here, the number <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M132">View MathML</a> is constant. Using (13) and (45) and applying the Cauchy-Bunyakovski inequality, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M133">View MathML</a>

and then

(46)

If we use estimate (8), we can write the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M135">View MathML</a>

(47)

Using this inequality and estimates (13), (28), and (45), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M136">View MathML</a>

(48)

Here, the number of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M137">View MathML</a> is constant. Similarly, we can prove the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M138">View MathML</a>

(49)

If we use inequalities (48) and (49), we see that the correlations limit (36) and (37) is valid.

Now, we prove (38). To prove this using the formula <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M139">View MathML</a> in (29), we can write the following inequality:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M140">View MathML</a>

(50)

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M141">View MathML</a> is a solution of the problem (41). Estimate (45) is valid for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M36">View MathML</a>. Therefore, the following estimate can be written at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M143','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M143">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M144">View MathML</a>

If this inequality is used in (49), we easily can write

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M145">View MathML</a>

(51)

Similarly, if we use (39), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M146">View MathML</a>

(52)

We can see that (38) and (39) are valid by inequalities (51) and (52). That is, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M147">View MathML</a>. On the other hand, V is a convex set according to the definition. So, the functional <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M97">View MathML</a> holds by the condition of Theorem (Goebel) in [8] at V. Therefore, considering Theorem 3, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M149">View MathML</a>

for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2012/1/151/mathml/M62">View MathML</a>. Here, using (29), it is seen that the statement of Theorem 4 is valid. □

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

YK carried out the optimal control problem studies, participated in the sequence alignment and drafted the manuscript. EÇ conceived of the study and, participated in its design and coordination. All authors read and approved the final manuscript.

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