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Journal of Applied Mathematics
VolumeΒ 2012, Article IDΒ 672947, 25 pages
http://dx.doi.org/10.1155/2012/672947
Research Article

Optimality Conditions for Infinite Order Distributed Parabolic Systems with Multiple Time Delays Given in Integral Form

1Department of Mathematics, Faculty of Science, Taibah University, Al-Madinah Al-Munawwarah, Saudi Arabia
2Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt

Received 1 March 2012; Accepted 29 April 2012

Academic Editor: WeihaiΒ Zhang

Copyright Β© 2012 Bahaa G. M.. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The optimal boundary control problem for (𝑛×𝑛) infinite order distributed parabolic systems with multiple time delays given in the integral form both in the state equations and in the Neumann boundary conditions is considered. Constraints on controls are imposed. Necessary and suffacient optimality conditions for the Neumann problem with the quadratic performance functional are derived.

1. Introduction

Distributed parameters systems with delays can be used to describe many phenomena in the real world. As is well known, heat conduction, properties of elastic-plastic material, fluid dynamics, diffusion-reaction processes, the transmission of the signals at a certain distance by using electric long lines, and so forth, all lie within this area. The object that we are studying (temperature, displacement, concentration, velocity, etc.) is usually referred to as the state.

During the last twenty years, equations with deviating argument have been applied not only in applied mathematics, physics, and automatic control, but also in some problems of economy and biology. Currently, the theory of equations with deviating arguments constitutes a very important subfield of mathematical control theory.

Consequently, equations with deviating arguments are widely applied in optimal control problems of distributed parameter system with time delays [1].

The optimal control problems of distributed parabolic systems with time-delayed boundary conditions have been widely discussed in many papers and monographs. A fundamental study of such problems is given by [2] and was next developed by [3, 4]. It was also intensively investigated by [1, 5–16] in which linear quadratic problem for parabolic systems with time delays given in the different form (constant, time delays, time-varying delays, time delays given in the integral form, etc.) was presented.

The necessary and sufficient conditions of optimality for systems consist of only one equation and for (𝑛×𝑛) systems governed by different types of partial differential equations defined on spaces of functions of infinitely many variables and also for infinite order systems are discussed for example in [9, 11, 15–18] in which the argument of [19, 20] was used.

Making use of the Dubovitskii-Milyutin Theorem in [13, 21–28] the necessary and sufficient conditions of optimality for similar systems governed by second order operator with an infinite number of variables and also for infinite order systems were investigated. The interest in the study of this class of operators is stimulated by problems in quantum field theory.

In particular, the papers of [1, 8] present necessary and sufficient optimality conditions for the Neumann problem with quadratic performance functionals, applied to a single one equation of second-order parabolic system with fixed time delay and with multiple time delays given in the integral form both in the state equations and in the Neumann boundary conditions, respectively. Such systems constitute a more complex case of distributed parameter systems with time delays given in the integral form.

Also in [9, 11] time-optimal boundary control for a single one equation distributed infinite order parabolic and hyperbolic systems in which constant time lags appear in the integral form both in the state equation and in the Neumann boundary condition is present. Some specific properties of the optimal control are discussed.

In this paper we recall the problem in a more general formulation. A distributed parameter for infinite order parabolic (𝑛×𝑛) systems with multiple time delays given in the integral form both in the state equations and in the Neumann boundary conditions is considered. Such an infinite order parabolic system can be treated as a generalization of the mathematical model for a plasma control process. The quadratic performance functionals defined over a fixed time horizon are taken and some constraints are imposed on the initial state and the boundary control. Such a system may be viewed as a linear representation of many diffusion processes, in which time-delayed signals are introduced at a spatial boundary, and there is a freedom in choosing the controlled process initial state. Following a line of the Lions scheme, necessary and sufficient optimality conditions for the Neumann problem applied to the above system were derived. The optimal control is characterized by the adjoint equations.

This paper is organized as follows. In Section 1, we introduce spaces of functions of infinite order. In Section 2, we formulate the mixed Neumann problem for infinite order parabolic operator with multiple time delays given in the integral form. In Section 3, the boundary optimal control problem for this case is formulated, then we give the necessary and sufficient conditions for the control to be an optimal. In Section 4, we generalized the discussion to two cases, the first case: the optimal control for (2Γ—2) coupled infinite order parabolic systems is studied. The second case: the optimal control for (𝑛×𝑛) coupled infinite order parabolic systems was to be formulated.

2. Sobolev Spaces with Infinite Order

The object of this section is to give the definition of some function spaces of infinite order and the chains of the constructed spaces which will be used later.

Let Ξ© be a bounded open set of ℝ𝑛 with a smooth boundary Ξ“, which is a 𝐢∞ manifold of dimension (π‘›βˆ’1). Locally, Ξ© is totally on one side of Ξ“. We define the infinite order Sobolev space π‘Šβˆž{π‘Žπ›Ό,2}(Ξ©) of infinite order of periodic functions πœ™(π‘₯) defined on Ξ© [29–31] as follows: π‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)=βŽ§βŽ¨βŽ©πœ™(π‘₯)∈C∞(Ξ©)βˆΆβˆžξ“|𝛼|=0π‘Žπ›Όβ€–Dπ›Όπœ™β€–22<∞⎫⎬⎭,(2.1) where 𝐢∞(Ξ©) is the space of infinitely differentiable functions, π‘Žπ›Όβ‰₯0 is a numerical sequence, and β€–β‹…β€–2 is the canonical norm in the space 𝐿2(Ξ©), and 𝐷𝛼=πœ•|𝛼|ξ€·πœ•π‘₯1𝛼1β‹―ξ€·πœ•π‘₯𝑛𝛼𝑛,(2.2)𝛼=(𝛼1,…,𝛼𝑛) being a multi-index for differentiation, |𝛼|=βˆ‘π‘›π‘–=1𝛼𝑖.

The space π‘Šβˆ’βˆž{π‘Žπ›Ό,2}(Ξ©) is defined as the formal conjugate space to the space π‘Šβˆž{π‘Žπ›Ό,2}(Ξ©), namely: π‘Šβˆ’βˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)=βŽ§βŽ¨βŽ©πœ“(π‘₯)βˆΆπœ“(π‘₯)=βˆžξ“|𝛼|=0(βˆ’1)|𝛼|π‘Žπ›Όπ·π›Όπœ“π›Ό(π‘₯)⎫⎬⎭,(2.3) where πœ“π›ΌβˆˆπΏ2(Ξ©) and βˆ‘βˆž|𝛼|=0π‘Žπ›Όβ€–πœ“π›Όβ€–22<∞.

The duality pairing of the spaces π‘Šβˆž{π‘Žπ›Ό,2}(Ξ©) and π‘Šβˆ’βˆž{π‘Žπ›Ό,2}(Ξ©) is postulated by the formula: (πœ™,πœ“)=βˆžξ“|𝛼|=0π‘Žπ›Όξ€œΞ©πœ“π›Ό(π‘₯)π·π›Όπœ™(π‘₯)𝑑π‘₯,(2.4) where πœ™βˆˆπ‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©),πœ“βˆˆπ‘Šβˆ’βˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©).(2.5)

From above, π‘Šβˆž{π‘Žπ›Ό,2}(Ξ©) is everywhere dense in 𝐿2(Ξ©) with topological inclusions and π‘Šβˆ’βˆž{π‘Žπ›Ό,2}(Ξ©) denotes the topological dual space with respect to 𝐿2(Ξ©), so we have the following chain of inclusions: π‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)βŠ†πΏ2(Ξ©)βŠ†π‘Šβˆ’βˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©).(2.6) We now introduce 𝐿2(0,𝑇;𝐿2(Ξ©)) which we will denoted by 𝐿2(𝑄), where 𝑄=Ω×]0,𝑇[ denotes the space of measurable functions π‘‘β†’πœ™(𝑑) such that β€–πœ™β€–πΏ2(𝑄)=ξ‚΅ξ€œπ‘‡0β€–πœ™(𝑑)β€–22𝑑𝑑1/2<∞,(2.7) endowed with the scalar product (𝑓,𝑔)=βˆ«π‘‡0(𝑓(𝑑),𝑔(𝑑))𝐿2(Ξ©)𝑑𝑑, 𝐿2(𝑄) is a Hilbert space.

In the same manner we define the spaces 𝐿2(0,𝑇;π‘Šβˆž{π‘Žπ›Ό,2}(Ξ©)), and 𝐿2(0,𝑇;π‘Šβˆ’βˆž{π‘Žπ›Ό,2}(Ξ©)), as its formal conjugate.

Also, we have the following chain of inclusions: 𝐿2ξ€·0,𝑇;π‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)ξ€ΈβŠ†πΏ2(𝑄)βŠ†πΏ2ξ€·0,𝑇;π‘Šβˆ’βˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)ξ€Έ.(2.8)

The construction of the Cartesian product of 𝑛-times to the above Hilbert spaces can be constructed, for example ξ€·π‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)𝑛=π‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)Γ—π‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)Γ—β‹―Γ—π‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)ξ„Ώξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…ƒξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…€ξ…Œπ‘›-times=𝑛𝑖=1ξ€·π‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)𝑖,(2.9) with norm defined by: β€–πœ™β€–(π‘Šβˆž{π‘Žπ›Ό,2}(Ξ©))𝑛=𝑛𝑖=1β€–β€–πœ™π‘–β€–β€–π‘Šβˆž{π‘Žπ›Ό,2}(Ξ©),(2.10) where πœ™=(πœ™1,πœ™2,…,πœ™π‘›)=(πœ™π‘–)𝑛𝑖=1 is a vector function and πœ™π‘–βˆˆπ‘Šβˆž{π‘Žπ›Ό,2}(Ξ©).

Finally, we have the following chain of inclusions: 𝐿2ξ€·0,𝑇;π‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)ξ€Έξ€Έπ‘›βŠ†ξ€·πΏ2(𝑄)ξ€Έπ‘›βŠ†ξ€·πΏ2ξ€·0,𝑇;π‘Šβˆ’βˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)𝑛,(2.11) where (𝐿2(0,𝑇;π‘Šβˆ’βˆž{π‘Žπ›Ό,2}(Ξ©)))𝑛 are the dual spaces of (𝐿2(0,𝑇;π‘Šβˆž{π‘Žπ›Ό,2}(Ξ©)))𝑛. The spaces considered in this paper are assumed to be real.

3. Mixed Neumann Problem for Infinite Order Parabolic System with Multiple Time Lags

The object of this section is to formulate the following mixed initial boundary value Neumann problem for infinite order parabolic system with multiple time delays which defines the state of the system model [1, 5–11, 18, 24, 26].πœ•π‘¦πœ•π‘‘+π’œ(𝑑)𝑦(π‘₯,𝑑)+π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–π‘π‘–(π‘₯,𝑑)𝑦π‘₯,π‘‘βˆ’β„Žπ‘–ξ€Έπ‘‘β„Žπ‘–=𝑒,(π‘₯,𝑑)βˆˆΞ©Γ—(0,𝑇),β„Žπ‘–βˆˆξ€·π‘Žπ‘–,𝑏𝑖,(3.1)𝑦π‘₯,π‘‘ξ…žξ€Έ=Ξ¦0ξ€·π‘₯,π‘‘ξ…žξ€Έ,ξ€·π‘₯,π‘‘ξ…žξ€ΈβˆˆΞ©Γ—(βˆ’Ξ”,0),(3.2)𝑦(π‘₯,0)=𝑦0(π‘₯),π‘₯∈Ω,(3.3)πœ•π‘¦πœ•πœˆπ’œ(π‘₯,𝑑)=𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ π‘π‘ (π‘₯,𝑑)𝑦π‘₯,π‘‘βˆ’π‘˜π‘ ξ€Έπ‘‘π‘˜π‘ +𝑣,(π‘₯,𝑑)βˆˆΞ“Γ—(0,𝑇),π‘˜π‘ βˆˆξ€·π‘π‘ ,𝑑𝑠,(3.4)𝑦π‘₯,π‘‘ξ…žξ€Έ=Ξ¨0ξ€·π‘₯,π‘‘ξ…žξ€Έ,ξ€·π‘₯,π‘‘ξ…žξ€ΈβˆˆΞ“Γ—(βˆ’Ξ”,0),(3.5) where Ξ©βŠ‚π‘…π‘› has the same properties as in Section 1. We have 𝑦≑𝑦(π‘₯,𝑑;𝑒),𝑦(0)≑𝑦(π‘₯,0;𝑒),𝑦(𝑇)≑𝑦(π‘₯,𝑇;𝑒),𝑒≑𝑒(π‘₯,𝑑),𝑣≑𝑣(π‘₯,𝑑),𝑄=Ω×(0,𝑇),𝑄=Ω×[0,𝑇],𝑄0=Ω×[βˆ’Ξ”,0),Ξ£=Γ×(0,𝑇),Ξ£0=Γ×[βˆ’Ξ”,0),(3.6)(i)𝑇is a specified positive number representing a finite time horizon,(ii)β„Žπ‘–,π‘˜π‘  are time delays, such that β„Žπ‘–βˆˆ(π‘Žπ‘–,𝑏𝑖) and π‘˜π‘ βˆˆ(𝑐𝑠,𝑑𝑠) where 0<π‘Ž1<π‘Ž2<β‹―<π‘Žπ‘š, 0<𝑏1<𝑏2<β‹―<π‘π‘š, for 𝑖=1,2,…,π‘š and 0<𝑐1<𝑐2<β‹―<𝑐𝑙, 0<𝑑1<𝑑2<β‹―<𝑑𝑙, for 𝑠=1,2,…,𝑙,(iii)𝑏𝑖(𝑑),𝑖=1,2,…,π‘š are given real 𝐢∞ functions defined on 𝑄,(iv)𝑐𝑠(π‘₯,𝑑),𝑠=1,2,…,𝑙 are given real 𝐢∞ functions defined on Ξ£,(v)Ξ”=max{π‘π‘š,𝑑𝑙},(vi)𝑦 is a function defined on 𝑄 such that Ω×(0,𝑇)βˆ‹(π‘₯,𝑑)→𝑦(π‘₯,𝑑)βˆˆπ‘…,(vii)𝑒,𝑣 are functions defined on 𝑄 and Ξ£ such that Ω×(0,𝑇)βˆ‹(π‘₯,𝑑)→𝑒(π‘₯,𝑑)βˆˆπ‘… and Γ×(0,𝑇)βˆ‹(π‘₯,𝑑)→𝑣(π‘₯,𝑑)βˆˆπ‘…,(viii)Ξ¦0,Ξ¨0 are initial functions defined on 𝑄0 and Ξ£0, respectively, such that Ω×[βˆ’Ξ”,0)βˆ‹(π‘₯,π‘‘ξ…ž)β†’Ξ¦0(π‘₯,π‘‘ξ…ž)βˆˆπ‘…. Γ×[βˆ’Ξ”,0)βˆ‹(π‘₯,π‘‘ξ…ž)β†’Ξ¨0(π‘₯,π‘‘ξ…ž)βˆˆπ‘….

The parabolic operator (πœ•/πœ•π‘‘)+π’œ(𝑑) in the state equation (3.1) is an infinite order parabolic operator and π’œ(𝑑) [17, 21, 29–31] is given by: π’œπ‘¦=βˆžξ“|𝛼|=0(βˆ’1)|𝛼|π‘Žπ›Όπ·2𝛼𝑦(π‘₯,𝑑),π’œ=βˆžξ“|𝛼|=0(βˆ’1)|𝛼|π‘Žπ›Όπ·2𝛼(3.7) is an infinite order self-adjoint elliptic partial differential operator maps π‘Šβˆž{π‘Žπ›Ό,2}(Ξ©) onto π‘Šβˆ’βˆž{π‘Žπ›Ό,2}(Ξ©).

For this operator we define the bilinear form as follows.

Definition 3.1. For each π‘‘βˆˆ(0,𝑇), we define a family of bilinear forms on π‘Šβˆž{π‘Žπ›Ό,2}(Ξ©) by: πœ‹(𝑑;𝑦,πœ™)=(π’œ(𝑑)𝑦,πœ™)𝐿2(Ξ©),𝑦,πœ™βˆˆπ‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©),(3.8) where π’œ(𝑑) maps π‘Šβˆž{π‘Žπ›Ό,2}(Ξ©) onto π‘Šβˆ’βˆž{π‘Žπ›Ό,2}(Ξ©) and takes the above form. Then πœ‹(𝑑;𝑦,πœ™)=(π’œ(𝑑)𝑦,πœ™)𝐿2(Ξ©)=βŽ›βŽœβŽβˆžξ“|𝛼|=0(βˆ’1)|𝛼|π‘Žπ›Όπ·2𝛼𝑦(π‘₯,𝑑),πœ™(π‘₯)⎞⎟⎠𝐿2(Ξ©)=ξ€œΞ©βˆžξ“|𝛼|=0π‘Žπ›Όπ·π›Όπ‘¦(π‘₯)π·π›Όπœ™(π‘₯)𝑑π‘₯.(3.9)

Lemma 3.2. The bilinear form πœ‹(𝑑;𝑦,πœ™) is coercive on π‘Šβˆž{π‘Žπ›Ό,2}(Ξ©), that is, πœ‹(𝑑;𝑦,𝑦)β‰₯πœ†β€–π‘¦β€–2π‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©),πœ†>0.(3.10)

Proof. It is well known that the ellipticity of π’œ(𝑑) is sufficient for the coerciveness of πœ‹(𝑑;𝑦,πœ™) on π‘Šβˆž{π‘Žπ›Ό,2}(Ξ©): πœ‹(𝑑;πœ™,πœ“)=ξ€œΞ©βˆžξ“|𝛼|=0π‘Žπ›Όπ·π›Όπœ™π·π›Όπœ“π‘‘π‘₯.(3.11) Then πœ‹(𝑑;𝑦,𝑦)=ξ€œΞ©βˆžξ“|𝛼|=0π‘Žπ›Όπ·π›Όπ‘¦π·π›Όπ‘¦π‘‘π‘₯β‰₯βˆžξ“|𝛼|=0π‘Žπ›Όβ€–β€–π·2𝛼𝑦(π‘₯)β€–β€–2𝐿2(Ξ©)β‰₯πœ†β€–π‘¦β€–2π‘Šβˆž{π‘Žπ›Ό,2}(Ξ©),πœ†>0.(3.12) Also we have βˆ€π‘¦,πœ™βˆˆπ‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)thefunctionπ‘‘βŸΆπœ‹(𝑑;𝑦,πœ™)iscontinuouslydifferentiablein(0,𝑇)andπœ‹(𝑑;𝑦,πœ™)=πœ‹(𝑑;πœ™,𝑦).(3.13)
Equations (3.1)–(3.5) constitute a Neumann problem. Then the left-hand side of the boundary condition (3.4) may be written in the following form: πœ•π‘¦(π‘₯,𝑑)πœ•πœˆπ’œ=βˆžξ“|πœ”|=0(π·πœ”π‘¦(π‘₯,𝑑))cos𝑛,π‘₯π‘˜ξ€Έ=π‘ž(π‘₯,𝑑),π‘₯βˆˆΞ“,π‘‘βˆˆ(0,𝑇),(3.14) where πœ•/πœ•πœˆπ’œ is a normal derivative at Ξ“, directed towards the exterior of Ξ©, and cos(𝑛,π‘₯π‘˜) is the π‘˜th direction cosine of 𝑛, with 𝑛 being the normal at Ξ“ exterior to Ξ©.
Then (3.4) can be written as: π‘ž(π‘₯,𝑑)=𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ π‘π‘ (π‘₯,𝑑)𝑦π‘₯,π‘‘βˆ’π‘˜π‘ ξ€Έπ‘‘π‘˜π‘ +𝑣(π‘₯,𝑑),π‘₯βˆˆΞ“,π‘‘βˆˆ(0,𝑇).(3.15)

Remark 3.3. We will apply the indication π‘ž(π‘₯,𝑑) appearing in (3.14) to prove the existence of a unique solution for (3.1)–(3.5).

We will formulate sufficient conditions for the existence of a unique solution of the mixed boundary value problem (3.1)–(3.5) for the cases where the boundary control π‘£βˆˆπΏ2(Ξ£).

For this purpose, we introduce the Sobolev space π‘Šβˆž,1(𝑄) [20, Vol. 2, page 6] defined by: π‘Šβˆž,1(𝑄)=𝐿2ξ€·0,𝑇;π‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)ξ€Έβˆ©π‘Š1ξ€·0,𝑇;𝐿2(Ξ©)ξ€Έ,(3.16) which is a Hilbert space normed by β€–π‘¦β€–π‘Šβˆž,1(𝑄)=ξ‚Έξ€œπ‘‡0ξ€œβ€–π‘¦β€–2π‘Šβˆžξ€½π‘Žπ›Ό,2ξ€Ύ(Ξ©)𝑑𝑑+‖𝑦‖2π‘Š1ξ€·0,𝑇;𝐿2(Ξ©)ξ€Έξ‚Ή1/2=βŽ‘βŽ’βŽ£ξ€œπ‘„βŽ›βŽœβŽβˆžξ“|𝛼|=0π‘Žπ›Ό||𝐷𝛼𝑦||2+|||πœ•π‘¦πœ•π‘‘|||2βŽžβŽŸβŽ π‘‘π‘₯π‘‘π‘‘βŽ€βŽ₯⎦1/2=βŽ‘βŽ’βŽ£ξ€œπ‘„βŽ›βŽœβŽπ‘Ž0||𝑦||2+βˆžξ“|𝛼|=1π‘Žπ›Ό||𝐷𝛼𝑦||2+|||πœ•π‘¦πœ•π‘‘|||2βŽžβŽŸβŽ π‘‘π‘₯π‘‘π‘‘βŽ€βŽ₯⎦1/2,π‘Ž0>0,(3.17) where the space π‘Š1(0,𝑇;𝐿2(Ξ©)) denotes the Sobolev space of order 1 of functions defined on (0,𝑇) and taking values in 𝐿2(Ξ©) [20, Vol. 1].

The existence of a unique solution for the mixed initial-boundary value problem (3.1)–(3.5) on the cylinder 𝑄 can be proved using a constructive method, that is, solving at first equations (3.1)–(3.5) on the subcylinder 𝑄1 and in turn on 𝑄2 and so forth, until the procedure covers the whole cylinder 𝑄. In this way, the solution in the previous step determines the next one.

For simplicity, we introduce the following notation: 𝐸𝑗=((π‘—βˆ’1)πœ†,π‘—πœ†),𝑄𝑗=Ω×𝐸𝑗,Σ𝑗=Γ×𝐸𝑗for𝑗=1,…,𝐾,πœ†=minξ€½π‘Ž1,𝑐1ξ€Ύ.(3.18)

Making use of the results of [7, 20] we can prove that the following result holds.

Theorem 3.4. Let 𝑦0, Ξ¦0, Ξ¨0, 𝑣 and 𝑒 be given with 𝑦0βˆˆπ‘Šβˆž{π‘Žπ›Ό,2}(Ξ©), Ξ¦0βˆˆπ‘Šβˆž,1(𝑄0), Ξ¨0∈𝐿2(Ξ£0), π‘£βˆˆπΏ2(Ξ£) and π‘’βˆˆπ‘Šβˆ’βˆž,βˆ’1(𝑄). Then, there exists a unique solution π‘¦βˆˆπ‘Šβˆž,1(𝑄) for the mixed initial-boundary value problem (3.1)–(3.5). Moreover, 𝑦(β‹…,π‘—πœ†)βˆˆπ‘Šβˆž{π‘Žπ›Ό,2}(Ξ©)for𝑗=1,…,𝐾.

4. Problem Formulation-Optimization Theorems

Now, we formulate the optimal control problem for (3.1)–(3.5) in the context of the Theorem 3.4, that is π‘£βˆˆπΏ2(Ξ£).

Let us denote by π‘ˆ=𝐿2(Ξ£) the space of controls. The time horizon 𝑇 is fixed in our problem.

The performance functional is given by 𝐼(𝑣)=πœ†1ξ€œπ‘„ξ€Ίπ‘¦(π‘₯,𝑑;𝑣)βˆ’π‘§π‘‘ξ€»2𝑑π‘₯𝑑𝑑+πœ†2ξ€œΞ£(𝑁𝑣)𝑣𝑑Γ𝑑𝑑,(4.1) where πœ†π‘–β‰₯0, and πœ†1+πœ†2>0,𝑧𝑑 is a given element in 𝐿2(𝑄); 𝑁 is a positive linear operator on 𝐿2(Ξ£) into 𝐿2(Ξ£).

Control Contraints
We define the set of admissible controls π‘ˆad such that π‘ˆadisclosed,convexsubsetofπ‘ˆ=𝐿2(Ξ£).(4.2)

Let 𝑦(π‘₯,𝑑;𝑣) denote the solution of the mixed initial-boundary value problem (3.1)–(3.5) at (π‘₯,𝑑) corresponding to a given control π‘£βˆˆπ‘ˆad. We note from Theorem 3.4 that for any π‘£βˆˆπ‘ˆad the performance functional (4.1) is well-defined since (𝑣)βˆˆπ‘Šβˆž,1(𝑄)βŠ‚πΏ2(𝑄).

Making use of the Loins's scheme we will derive the necessary and sufficient conditions of optimality for the optimization problem (3.1)–(3.5), (4.1), (4.2). The solving of the formulated optimal control problem is equivalent to seeking a π‘£βˆ—βˆˆπ‘ˆad such that πΌξ€·π‘£βˆ—ξ€Έβ‰€πΌ(𝑣),βˆ€π‘£βˆˆπ‘ˆad.(4.3)

From the Lion's scheme [19, Theorem  1.3, page 10], it follows that for πœ†2>0 a unique optimal control π‘£βˆ— exists. Moreover, π‘£βˆ— is characterized by the following condition: πΌξ…žξ€·π‘£βˆ—ξ€Έξ€·π‘£βˆ’π‘£βˆ—ξ€Έβ‰₯0,βˆ€π‘£βˆˆπ‘ˆad.(4.4) For the performance functional of form (4.1) the relation (4.4) can be expressed as πœ†1ξ€œπ‘„ξ€·π‘¦ξ€·π‘£βˆ—ξ€Έβˆ’π‘§π‘‘ξ€Έξ€Ίπ‘¦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘π‘₯𝑑𝑑+πœ†2ξ€œΞ£π‘π‘£βˆ—ξ€·π‘£βˆ’π‘£βˆ—ξ€Έπ‘‘Ξ“π‘‘π‘‘β‰₯0,βˆ€π‘£βˆˆπ‘ˆad.(4.5)

In order to simplify (4.5), we introduce the adjoint equation, and for every π‘£βˆˆπ‘ˆad, we define the adjoint variable 𝑝=𝑝(𝑣)≑𝑝(π‘₯,𝑑;𝑣) as the solution of the equations: βˆ’πœ•π‘(𝑣)πœ•π‘‘+π’œβˆ—(𝑑)𝑝(𝑣)+π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–π‘π‘–ξ€·π‘₯,𝑑+β„Žπ‘–ξ€Έπ‘ξ€·π‘₯,𝑑+β„Žπ‘–;π‘£ξ€Έπ‘‘β„Žπ‘–=πœ†1𝑦(𝑣)βˆ’π‘§π‘‘ξ€Έ,(π‘₯,𝑑)βˆˆΞ©Γ—(0,π‘‡βˆ’Ξ”),β„Žπ‘–βˆˆξ€·π‘Žπ‘–,𝑏𝑖,(4.6)βˆ’πœ•π‘(𝑣)πœ•π‘‘+π’œβˆ—(𝑑)𝑝(𝑣)=πœ†1𝑦(𝑣)βˆ’π‘§π‘‘ξ€Έ,(π‘₯,𝑑)βˆˆΞ©Γ—(π‘‡βˆ’Ξ”,𝑇),(4.7)𝑝(π‘₯,𝑇;𝑣)=0,π‘₯∈Ω,(4.8)𝑝(π‘₯,𝑑;𝑣)=0,(π‘₯,𝑑)βˆˆΞ©Γ—[π‘‡βˆ’Ξ”+πœ†,𝑇),(4.9)πœ•π‘(𝑣)πœ•πœˆπ’œβˆ—(π‘₯,𝑑)=𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ π‘π‘ ξ€·π‘₯,𝑑+π‘˜π‘ ξ€Έπ‘ξ€·π‘₯,𝑑+π‘˜π‘ ;π‘£ξ€Έπ‘‘π‘˜π‘ ,(π‘₯,𝑑)βˆˆΞ“Γ—(0,π‘‡βˆ’Ξ”(𝑇)),π‘˜π‘ βˆˆξ€·π‘π‘ ,𝑑𝑠,(4.10)πœ•π‘(𝑣)πœ•πœˆπ’œβˆ—(π‘₯,𝑑)=0,(π‘₯,𝑑)βˆˆΞ“Γ—(π‘‡βˆ’Ξ”(𝑇),𝑇),(4.11) where πœ•π‘(𝑣)πœ•πœˆπ’œβˆ—(π‘₯,𝑑)=βˆžξ“|πœ”|=0(π·πœ”π‘(𝑣))cos𝑛,π‘₯πœ”ξ€Έ(π‘₯,𝑑),π’œβˆ—(𝑑)𝑝(𝑣)=βˆžξ“|𝛼|=0(βˆ’1)|𝛼|π‘Žπ›Όπ·2𝛼𝑝(π‘₯,𝑑).(4.12)

As in the above section with change of variables, that is, with reversed sense of time. that is, π‘‘ξ…ž=π‘‡βˆ’π‘‘, for given π‘§π‘‘βˆˆπΏ2(𝑄) and any π‘£βˆˆπΏ2(Ξ£), there exists a unique solution 𝑝(𝑣)βˆˆπ‘Šβˆž,1(𝑄) for problem (4.6)–(4.11).

The existence of a unique solution for the problem (4.6)–(4.11) on the cylinder Ω×(0,𝑇) can be proved using a constructive method. It is easy to notice that for given 𝑧𝑑 and 𝑒, the problem (4.6)–(4.11) can be solved backwards in time starting from 𝑑=𝑇, that is, first solving (4.6)–(4.11) on the subcylinder 𝑄𝐾 and in turn on π‘„πΎβˆ’1, and so forth until the procedure covers the whole cylinder Ω×(0,𝑇). For this purpose, we may apply Theorem 3.4 (with an obvious change of variables).

Hence, using Theorem 3.4, the following result can be proved.

Lemma 4.1. Let the hypothesis of Theorem 3.4 be satisfied. Then for given π‘§π‘‘βˆˆπΏ2(Ξ©,π‘…βˆž) and any π‘£βˆˆπΏ2(Ξ£), there exists a unique solution 𝑝(𝑣)βˆˆπ‘Šβˆž,1(𝑄) for the adjoint problem (4.6)–(4.11).

We simplify (4.5) using the adjoint equation (4.6)–(4.11). For this purpose denoting by 𝑝(0)≑𝑝(π‘₯,0;𝑣) and 𝑝(𝑇)≑𝑝(π‘₯,𝑇;𝑣), respectively, setting 𝑣=π‘£βˆ— in (4.6)–(4.11), multiplying both sides of (4.6) and (4.7) by 𝑦(𝑣)βˆ’π‘¦(π‘£βˆ—), then integrating over Ω×(0,π‘‡βˆ’Ξ”) and Ω×(π‘‡βˆ’Ξ”,𝑇), respectively and then adding both sides of (4.6), (4.11), we get πœ†1ξ€œπ‘„ξ€·π‘¦ξ€·π‘£βˆ—ξ€Έβˆ’π‘§π‘‘ξ€Έξ€Ίπ‘¦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘π‘₯𝑑𝑑=ξ€œπ‘„βŽ›βŽœβŽβˆ’πœ•π‘ξ€·π‘£βˆ—ξ€Έπœ•π‘‘+π’œβˆ—(𝑑)π‘ξ€·π‘£βˆ—ξ€ΈβŽžβŽŸβŽ Γ—ξ€Ίπ‘¦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘π‘₯𝑑𝑑+ξ€œπ‘‡βˆ’Ξ”0ξ€œΞ©βŽ›βŽœβŽπ‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–π‘π‘–ξ€·π‘₯,𝑑+β„Žπ‘–ξ€Έπ‘ξ€·π‘₯,𝑑+β„Žπ‘–;π‘£βˆ—ξ€Έπ‘‘β„Žπ‘–βŽžβŽŸβŽ Γ—ξ€Ίπ‘¦(π‘₯,𝑑;𝑣)βˆ’π‘¦ξ€·π‘₯,𝑑;π‘£βˆ—ξ€Έξ€»π‘‘π‘₯𝑑𝑑=ξ€œπ‘‡0ξ€œΞ©π‘ξ€·π‘£βˆ—ξ€Έπœ•πœ•π‘‘ξ€Ίπ‘¦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘π‘₯𝑑𝑑+ξ€œπ‘‡0ξ€œΞ©π’œβˆ—(𝑑)π‘ξ€·π‘£βˆ—ξ€Έξ€Ίπ‘¦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘π‘₯𝑑𝑑+π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–ξ€œΞ©ξ€œπ‘‡βˆ’Ξ”0𝑏𝑖π‘₯,𝑑+β„Žπ‘–ξ€Έπ‘ξ€·π‘₯,𝑑+β„Žπ‘–;π‘£βˆ—ξ€Έξ€ΈΓ—ξ€Ίπ‘¦(π‘₯,𝑑;𝑣)βˆ’π‘¦ξ€·π‘₯,𝑑;π‘£βˆ—ξ€Έξ€»π‘‘π‘₯π‘‘π‘‘π‘‘β„Žπ‘–.(4.13) Using (3.1), the first integral on the right-hand side of (4.13) can be written as: ξ€œπ‘‡0ξ€œΞ©π‘ξ€·π‘£βˆ—ξ€Έπœ•πœ•π‘‘ξ€Ίπ‘¦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘π‘₯𝑑𝑑=βˆ’ξ€œπ‘„π‘ξ€·π‘£βˆ—ξ€Έπ’œ(𝑑)𝑦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€Έπ‘‘π‘₯π‘‘π‘‘βˆ’π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–ξ€œΞ©ξ€œπ‘‡0𝑝π‘₯,𝑑;π‘£βˆ—ξ€Έπ‘π‘–(π‘₯,𝑑)×𝑦π‘₯,π‘‘βˆ’β„Žπ‘–;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘βˆ’β„Žπ‘–;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘π‘‘π‘₯π‘‘β„Žπ‘–=βˆ’ξ€œπ‘„π‘ξ€·π‘£βˆ—ξ€Έπ’œ(𝑑)𝑦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€Έπ‘‘π‘₯π‘‘π‘‘βˆ’π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–ξ€œΞ©ξ€œπ‘‡βˆ’β„Žπ‘–βˆ’β„Žπ‘–π‘ξ€·π‘₯,π‘‘ξ…ž+β„Žπ‘–;π‘£βˆ—ξ€Έπ‘π‘–ξ€·π‘₯,π‘‘ξ…ž+β„Žπ‘–ξ€ΈΓ—ξ€Ίπ‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘β€²π‘‘π‘₯π‘‘β„Žπ‘–=βˆ’ξ€œΞ©π‘ξ€·π‘£βˆ—ξ€Έπ’œ(𝑑)𝑦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€Έπ‘‘π‘₯π‘‘π‘‘βˆ’π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–ξ€œΞ©ξ€œ0βˆ’β„Žπ‘–π‘ξ€·π‘₯,π‘‘ξ…ž+β„Žπ‘–;π‘£βˆ—ξ€Έπ‘π‘–ξ€·π‘₯,π‘‘ξ…ž+β„Žπ‘–ξ€ΈΓ—ξ€Ίπ‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘β€²π‘‘π‘₯π‘‘β„Žπ‘–βˆ’π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–ξ€œΞ©ξ€œπ‘‡βˆ’Ξ”0𝑝π‘₯,π‘‘ξ…ž+β„Žπ‘–;π‘£βˆ—ξ€Έπ‘π‘–ξ€·π‘₯,π‘‘ξ…ž+β„Žπ‘–ξ€ΈΓ—ξ€Ίπ‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘β€²π‘‘π‘₯π‘‘β„Žπ‘–βˆ’π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–ξ€œΞ©ξ€œπ‘‡βˆ’β„Žπ‘–π‘‡βˆ’Ξ”π‘ξ€·π‘₯,π‘‘ξ…ž+β„Žπ‘–;π‘£βˆ—ξ€Έπ‘π‘–ξ€·π‘₯,π‘‘ξ…ž+β„Žπ‘–ξ€ΈΓ—ξ€Ίπ‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘β€²π‘‘π‘₯π‘‘β„Žπ‘–=βˆ’ξ€œπ‘„π‘ξ€·π‘£βˆ—ξ€Έπ’œ(𝑑)𝑦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€Έπ‘‘π‘₯π‘‘π‘‘βˆ’π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–ξ€œΞ©ξ€œ0βˆ’β„Žπ‘–π‘ξ€·π‘₯,π‘‘ξ…ž+β„Žπ‘–;π‘£βˆ—ξ€Έπ‘π‘–ξ€·π‘₯,π‘‘ξ…ž+β„Žπ‘–ξ€ΈΓ—ξ€Ίπ‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘β€²π‘‘π‘₯π‘‘β„Žπ‘–βˆ’π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–ξ€œΞ©ξ€œπ‘‡βˆ’Ξ”0𝑝π‘₯,π‘‘ξ…ž+β„Žπ‘–;π‘£βˆ—ξ€Έπ‘π‘–ξ€·π‘₯,π‘‘ξ…ž+β„Žπ‘–ξ€ΈΓ—ξ€Ίπ‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘β€²π‘‘π‘₯π‘‘β„Žπ‘–βˆ’π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–ξ€œΞ©ξ€œπ‘‡π‘‡βˆ’Ξ”+β„Žπ‘–π‘ξ€·π‘₯,𝑑;π‘£βˆ—ξ€Έπ‘π‘–(π‘₯,𝑑)×𝑦π‘₯,π‘‘βˆ’β„Žπ‘–;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘βˆ’β„Žπ‘–;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘π‘‘π‘₯π‘‘β„Žπ‘–.(4.14) Using Green’s formula, the second integral on the right-hand side of (4.13) can be written as: ξ€œπ‘‡0ξ€œΞ©π’œβˆ—(𝑑)π‘ξ€·π‘£βˆ—ξ€Έξ€Ίπ‘¦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘π‘₯𝑑𝑑=ξ€œπ‘‡0ξ€œΞ©π‘ξ€·π‘£βˆ—ξ€Έπ’œ(𝑑)𝑦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘π‘₯𝑑𝑑+ξ€œπ‘‡0ξ€œΞ“π‘ξ€·π‘£βˆ—ξ€ΈβŽ›βŽœβŽπœ•π‘¦(𝑣)πœ•πœˆπ’œβˆ’πœ•π‘¦ξ€·π‘£βˆ—ξ€Έπœ•πœˆπ’œβŽžβŽŸβŽ π‘‘Ξ“π‘‘π‘‘βˆ’ξ€œπ‘‡0ξ€œΞ“πœ•π‘ξ€·π‘£βˆ—ξ€Έπœ•πœˆπ’œβˆ—ξ€Ίπ‘¦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘Ξ“π‘‘π‘‘.(4.15) Using the boundary condition (3.2), one can transform the second integral on the right-hand side of (4.15) into the form: ξ€œπ‘‡0ξ€œΞ“π‘ξ€·π‘£βˆ—ξ€ΈβŽ›βŽœβŽπœ•π‘¦(𝑣)πœ•πœˆπ’œβˆ’πœ•π‘¦ξ€·π‘£βˆ—ξ€Έπœ•πœˆπ’œβŽžβŽŸβŽ π‘‘Ξ“π‘‘π‘‘=𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ ξ€œΞ“ξ€œπ‘‡0𝑝π‘₯,𝑑;π‘£βˆ—ξ€Έπ‘π‘ (π‘₯,𝑑)×𝑦π‘₯,π‘‘βˆ’π‘˜π‘ ;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘βˆ’π‘˜π‘ ;π‘£βˆ—ξ€Έξ€»π‘‘Ξ“π‘‘π‘‘π‘‘π‘˜π‘ +ξ€œπ‘‡0ξ€œΞ“π‘ξ€·π‘£βˆ—ξ€Έξ€·π‘£βˆ’π‘£βˆ—ξ€Έπ‘‘Ξ“π‘‘π‘‘=𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ ξ€œΞ“ξ€œπ‘‡βˆ’π‘˜π‘ βˆ’π‘˜π‘ π‘ξ€·π‘₯,π‘‘ξ…ž+π‘˜π‘ ;π‘£βˆ—ξ€Έπ‘π‘ ξ€·π‘₯,π‘‘ξ…ž+π‘˜π‘ ξ€ΈΓ—ξ€Ίπ‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘ξ…žπ‘‘Ξ“π‘‘π‘˜π‘ +ξ€œπ‘‡0ξ€œΞ“π‘ξ€·π‘£βˆ—ξ€Έξ€·π‘£βˆ’π‘£βˆ—ξ€Έπ‘‘Ξ“π‘‘π‘‘=𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ ξ€œΞ“ξ€œ0βˆ’π‘˜π‘ π‘ξ€·π‘₯,π‘‘ξ…ž+π‘˜π‘ ;π‘£βˆ—ξ€Έπ‘π‘ ξ€·π‘₯,π‘‘ξ…ž+π‘˜π‘ ξ€ΈΓ—ξ€Ίπ‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘ξ…žπ‘‘Ξ“π‘‘π‘˜π‘ +𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ ξ€œΞ“ξ€œπ‘‡βˆ’Ξ”0𝑝π‘₯,π‘‘ξ…ž+π‘˜π‘ ;π‘£βˆ—ξ€Έπ‘π‘ ξ€·π‘₯,π‘‘ξ…ž+π‘˜π‘ ξ€ΈΓ—ξ€Ίπ‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘ξ…žπ‘‘Ξ“π‘‘π‘˜π‘ +𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ ξ€œΞ“ξ€œπ‘‡βˆ’π‘˜π‘ π‘‡βˆ’Ξ”π‘ξ€·π‘₯,π‘‘ξ…ž+π‘˜π‘ ;π‘£βˆ—ξ€Έπ‘π‘ ξ€·π‘₯,π‘‘ξ…ž+π‘˜π‘ ξ€ΈΓ—ξ€Ίπ‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘ξ…žπ‘‘Ξ“π‘‘π‘˜π‘ +ξ€œπ‘‡0ξ€œΞ“π‘ξ€·π‘£βˆ—ξ€Έξ€·π‘£βˆ’π‘£βˆ—ξ€Έπ‘‘Ξ“π‘‘π‘‘=𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ ξ€œΞ“ξ€œ0βˆ’π‘˜π‘ π‘ξ€·π‘₯,π‘‘ξ…ž+π‘˜π‘ ;π‘£βˆ—ξ€Έπ‘π‘ ξ€·π‘₯,π‘‘ξ…ž+π‘˜π‘ ξ€ΈΓ—ξ€Ίπ‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘ξ…žπ‘‘Ξ“π‘‘π‘˜π‘ +𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ ξ€œΞ“ξ€œπ‘‡βˆ’Ξ”0𝑝π‘₯,π‘‘ξ…ž+π‘˜π‘ ;π‘£βˆ—ξ€Έπ‘π‘ ξ€·π‘₯,π‘‘ξ…ž+π‘˜π‘ ξ€ΈΓ—ξ€Ίπ‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘ξ…ž;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘ξ…žπ‘‘Ξ“π‘‘π‘˜π‘ +𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ ξ€œΞ“ξ€œπ‘‡π‘‡βˆ’Ξ”+π‘˜π‘ π‘ξ€·π‘₯,𝑑;π‘£βˆ—ξ€Έπ‘π‘ (π‘₯,𝑑)×𝑦π‘₯,π‘‘βˆ’π‘˜π‘ ;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘βˆ’π‘˜π‘ ;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘π‘‘Ξ“π‘‘π‘˜π‘ +ξ€œπ‘‡0ξ€œΞ“π‘ξ€·π‘£βˆ—ξ€Έξ€·π‘£βˆ’π‘£βˆ—ξ€Έπ‘‘Ξ“π‘‘π‘‘.(4.16)

The last component in (4.15) can be rewritten as ξ€œπ‘‡0ξ€œΞ“πœ•π‘ξ€·π‘£βˆ—ξ€Έπœ•πœˆπ’œβˆ—ξ€Ίπ‘¦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘Ξ“π‘‘π‘‘=ξ€œπ‘‡βˆ’Ξ”0ξ€œΞ“πœ•π‘ξ€·π‘£βˆ—ξ€Έπœ•πœˆπ’œβˆ—ξ€Ίπ‘¦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘Ξ“π‘‘π‘‘+ξ€œπ‘‡π‘‡βˆ’Ξ”ξ€œΞ“πœ•π‘ξ€·π‘£βˆ—ξ€Έπœ•πœˆπ’œβˆ—ξ€Ίπ‘¦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘Ξ“π‘‘π‘‘.(4.17) Substituting (4.16) and (4.17) into (4.15) and then the results into (4.13), we obtain πœ†1ξ€œπ‘„ξ€·π‘¦ξ€·π‘£βˆ—ξ€Έβˆ’π‘§π‘‘ξ€Έξ€Ίπ‘¦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘π‘₯𝑑𝑑=ξ€œπ‘‡0ξ€œΞ“π‘ξ€·π‘£βˆ—ξ€Έξ€·π‘£βˆ’π‘£βˆ—ξ€Έπ‘‘Ξ“π‘‘π‘‘βˆ’ξ€œπ‘„π‘ξ€·π‘£βˆ—ξ€Έπ’œ(𝑑)𝑦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘π‘₯π‘‘π‘‘βˆ’π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–ξ€œΞ©ξ€œ0βˆ’β„Žπ‘–π‘π‘–ξ€·π‘₯,𝑑+β„Žπ‘–ξ€Έπ‘ξ€·π‘₯,𝑑+β„Žπ‘–;π‘£βˆ—ξ€ΈΓ—ξ€Ίπ‘¦(π‘₯,𝑑;𝑣)βˆ’π‘¦ξ€·π‘₯,𝑑;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘π‘‘π‘₯π‘‘β„Žπ‘–βˆ’π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–ξ€œΞ©ξ€œπ‘‡βˆ’Ξ”0𝑏𝑖π‘₯,𝑑+β„Žπ‘–ξ€Έπ‘ξ€·π‘₯,𝑑+β„Žπ‘–;π‘£βˆ—ξ€ΈΓ—ξ€Ίπ‘¦(π‘₯,𝑑;𝑣)βˆ’π‘¦ξ€·π‘₯,𝑑;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘π‘‘π‘₯π‘‘β„Žπ‘–+ξ€œπ‘„π‘ξ€·π‘£βˆ—ξ€Έπ’œ(𝑑)𝑦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€»π‘‘π‘₯π‘‘π‘‘βˆ’π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–ξ€œΞ©ξ€œπ‘‡π‘‡βˆ’Ξ”+β„Žπ‘–π‘ξ€·π‘₯,𝑑;π‘£βˆ—ξ€Έπ‘π‘–(π‘₯,𝑑)×𝑦π‘₯,π‘‘βˆ’β„Žπ‘–;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘βˆ’β„Žπ‘–;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘π‘‘π‘₯π‘‘β„Žπ‘–+𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ ξ€œΞ“ξ€œ0βˆ’π‘˜π‘ π‘π‘ ξ€·π‘₯,𝑑+π‘˜π‘ ξ€Έπ‘ξ€·π‘₯,𝑑+π‘˜π‘ ;π‘£βˆ—ξ€ΈΓ—ξ€Ίπ‘¦(π‘₯,𝑑;𝑣)βˆ’π‘¦ξ€·π‘₯,𝑑;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘π‘‘Ξ“π‘‘π‘˜π‘ βˆ’π‘™ξ“π‘ =1ξ€œπ‘‘π‘ π‘π‘ ξ€œΞ“ξ€œπ‘‡βˆ’Ξ”0𝑐𝑠π‘₯,𝑑+π‘˜π‘ ξ€Έπ‘ξ€·π‘₯,𝑑+π‘˜π‘ ;π‘£βˆ—ξ€ΈΓ—ξ€Ίπ‘¦(π‘₯,𝑑;𝑣)βˆ’π‘¦ξ€·π‘₯,𝑑;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘π‘‘Ξ“π‘‘π‘˜π‘ βˆ’π‘™ξ“π‘ =1ξ€œπ‘‘π‘ π‘π‘ ξ€œΞ“ξ€œπ‘‡π‘‡βˆ’Ξ”+π‘˜π‘ π‘π‘ (π‘₯,𝑑)𝑝π‘₯,𝑑;π‘£βˆ—ξ€ΈΓ—ξ€Ίπ‘¦ξ€·π‘₯,π‘‘βˆ’π‘˜π‘ ;π‘£ξ€Έβˆ’π‘¦ξ€·π‘₯,π‘‘βˆ’π‘˜π‘ ;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘π‘‘Ξ“π‘‘π‘˜π‘ βˆ’ξ€œΞ“ξ€œπ‘‡βˆ’Ξ”0πœ•π‘ξ€·π‘£βˆ—ξ€Έπœ•πœˆπ’œβˆ—Γ—ξ€Ίπ‘¦(π‘₯,𝑑;𝑣)βˆ’π‘¦ξ€·π‘₯,𝑑;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘π‘‘Ξ“βˆ’ξ€œΞ“ξ€œπ‘‡π‘‡βˆ’Ξ”πœ•π‘ξ€·π‘£βˆ—ξ€Έπœ•πœˆπ’œβˆ—Γ—ξ€Ίπ‘¦(π‘₯,𝑑;𝑣)βˆ’π‘¦ξ€·π‘₯,𝑑;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘π‘‘Ξ“+π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–ξ€œΞ©ξ€œπ‘‡βˆ’Ξ”0𝑏𝑖π‘₯,𝑑+β„Žπ‘–ξ€Έπ‘ξ€·π‘₯,𝑑+β„Žπ‘–;π‘£βˆ—ξ€ΈΓ—ξ€Ίπ‘¦(π‘₯,𝑑;𝑣)βˆ’π‘¦ξ€·π‘₯,𝑑;π‘£βˆ—ξ€Έξ€»π‘‘π‘‘π‘‘π‘₯π‘‘β„Žπ‘–.(4.18)

Afterwards, using the facts that 𝑦(π‘₯,𝑑,𝑣)=𝑦(π‘₯,𝑑,π‘£βˆ—)=Ξ¦0(π‘₯,𝑑) for π‘₯∈Ω and π‘‘βˆˆ[βˆ’Ξ”,0) and 𝑦(π‘₯,𝑑,𝑣)=𝑦(π‘₯,𝑑,π‘£βˆ—)=Ξ¨0(π‘₯,𝑑) for π‘₯βˆˆΞ“ and π‘‘βˆˆ[βˆ’Ξ”,0), 𝑝|Ξ©(π‘₯,𝑑;π‘£βˆ—)=0 and consequently 𝑝|Ξ“(π‘₯,𝑑;π‘£βˆ—)=0 for π‘‘βˆˆ[π‘‡βˆ’Ξ”+πœ†,𝑇), we obtain πœ†1ξ€œπ‘„ξ€Ίπ‘¦ξ€·π‘£βˆ—ξ€Έβˆ’π‘§π‘‘ξ€»Γ—ξ€·π‘¦(𝑣)βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έξ€Έπ‘‘π‘₯𝑑𝑑=ξ€œπ‘‡0ξ€œΞ“π‘ξ€·π‘£βˆ—ξ€Έξ€·π‘£βˆ’π‘£βˆ—ξ€Έπ‘‘Ξ“π‘‘π‘‘.(4.19)

Substituting (4.19) into (4.5) gives ξ€œπ‘‡0ξ€œΞ“ξ€·π‘ξ€·π‘£βˆ—ξ€Έ+πœ†2π‘π‘£βˆ—ξ€Έξ€·π‘£βˆ’π‘£βˆ—ξ€Έπ‘‘Ξ“π‘‘π‘‘β‰₯0,βˆ€π‘£βˆˆπ‘ˆad.(4.20)

The foregoing result is now summarized.

Theorem 4.2. For the problem (3.1)–(3.5), with the performance functional (4.1) with π‘§π‘‘βˆˆπΏ2(𝑄) and πœ†2>0 and with constraints on controls (4.2), there exists a unique optimal control π‘£βˆ— which satisfies the maximum condition (4.20).

4.1. Mathematical Examples

Example 4.3. Consider now the particular case where π‘ˆad=π‘ˆ=𝐿2(Ξ£) (no constraints case). Thus the maximum condition (4.20) is satisfied when π‘£βˆ—=βˆ’πœ†2π‘βˆ’1π‘ξ€·π‘£βˆ—ξ€Έ.(4.21) If 𝑁 is the identity operator on 𝐿2(Ξ£), then from Lemma 4.1 it follows that π‘£βˆ—βˆˆπ‘Šβˆž,1(𝑄).

Example 4.4. We can also consider an analogous optimal control problem where the performance functional is given by: 𝐼(𝑣)=πœ†1ξ€œΞ£ξ€Ίπ‘¦(π‘₯,𝑑;𝑣)βˆ£Ξ£βˆ’π‘§π‘‘ξ€»2𝑑Γ𝑑𝑑+πœ†2ξ€œΞ£(𝑁𝑣)𝑣𝑑Γ𝑑𝑑,(4.22) where π‘§π‘‘βˆˆπΏ2(Ξ£).

From Theorem 3.4 and the Trace Theorem [20, Vol. 2, page 9], for each π‘£βˆˆπΏ2(Ξ£), there exists a unique solution 𝑦(𝑣)βˆˆπ‘Šβˆž,1(𝑄) with π‘¦βˆ£Ξ£βˆˆπΏ2(Ξ£). Thus, 𝐼(𝑣) is well defined. Then, the optimal control π‘£βˆ— is characterized by: πœ†1ξ€œΞ£ξ€·π‘¦ξ€·π‘£βˆ—ξ€Έβˆ£Ξ£βˆ’π‘§π‘‘ξ€Έξ€Ίπ‘¦(𝑣)βˆ£Ξ£βˆ’π‘¦ξ€·π‘£βˆ—ξ€Έβˆ£Ξ£ξ€»π‘‘Ξ“π‘‘π‘‘+πœ†2ξ€œΞ£π‘π‘£βˆ—ξ€·π‘£βˆ’π‘£βˆ—ξ€Έπ‘‘Ξ“π‘‘π‘‘β‰₯0,βˆ€π‘£βˆˆπ‘ˆad.(4.23) We define the adjoint variable 𝑝=𝑝(π‘£βˆ—)=𝑝(π‘₯,𝑑;π‘£βˆ—) as the solution of the equations: βˆ’πœ•π‘ξ€·π‘£βˆ—ξ€Έπœ•π‘‘+π’œβˆ—(𝑑)π‘ξ€·π‘£βˆ—ξ€Έ+π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–π‘π‘–ξ€·π‘₯,𝑑+β„Žπ‘–ξ€Έπ‘ξ€·π‘₯,𝑑+β„Žπ‘–;π‘£βˆ—ξ€Έπ‘‘β„Žπ‘–=0,(π‘₯,𝑑)βˆˆΞ©Γ—(0,π‘‡βˆ’Ξ”),β„Žπ‘–βˆˆξ€·π‘Žπ‘–,𝑏𝑖,βˆ’πœ•π‘ξ€·π‘£βˆ—ξ€Έπœ•π‘‘+π’œβˆ—(𝑑)π‘ξ€·π‘£βˆ—ξ€Έ=0,(π‘₯,𝑑)βˆˆΞ©Γ—(π‘‡βˆ’Ξ”,𝑇),𝑝π‘₯,𝑇;π‘£βˆ—ξ€Έ=0,π‘₯∈Ω,𝑝π‘₯,𝑑;π‘£βˆ—ξ€Έ=0,(π‘₯,𝑑)βˆˆΞ©Γ—[π‘‡βˆ’Ξ”+πœ†,𝑇),πœ•π‘ξ€·π‘£βˆ—ξ€Έπœ•πœˆπ’œβˆ—(π‘₯,𝑑)=𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ π‘π‘ ξ€·π‘₯,𝑑+π‘˜π‘ ξ€Έπ‘ξ€·π‘₯,𝑑+π‘˜π‘ ;π‘£βˆ—ξ€Έπ‘‘π‘˜π‘ +πœ†1ξ€·π‘¦π‘£βˆ—βˆ£Ξ£βˆ’π‘§Ξ£π‘‘ξ€Έ,(π‘₯,𝑑)βˆˆΞ“Γ—(0,π‘‡βˆ’Ξ”(𝑇)),π‘˜π‘ βˆˆξ€·π‘π‘ ,𝑑𝑠,πœ•π‘ξ€·π‘£βˆ—ξ€Έπœ•πœˆπ’œβˆ—(π‘₯,𝑑)=πœ†1ξ€·π‘¦ξ€·π‘£βˆ—ξ€Έβˆ£Ξ£βˆ’π‘§Ξ£π‘‘ξ€Έ,(π‘₯,𝑑)βˆˆΞ“Γ—(π‘‡βˆ’Ξ”(𝑇),𝑇).(4.24) As in the above section, we have the following result.

Lemma 4.5. Let the hypothesis of Theorem 3.4 be satisfied. Then, for given π‘§Ξ£π‘‘βˆˆπΏ2(Ξ£) and any π‘£βˆˆπΏ2(Ξ£), there exists a unique solution 𝑝(π‘£βˆ—)βˆˆπ‘Šβˆž,1(𝑄) to the adjoint problem (4.24).

Using the adjoint equations (4.24)in this case, the condition (4.23) can also be written in the following form: ξ€œπ‘‡0ξ€œΞ“ξ€·π‘ξ€·π‘£βˆ—ξ€Έ+πœ†2π‘π‘£βˆ—ξ€Έξ€·π‘£βˆ’π‘£βˆ—ξ€Έπ‘‘Ξ“π‘‘π‘‘β‰₯0,βˆ€π‘£βˆˆπ‘ˆad.(4.25) The following result is now summarized.

Theorem 4.6. For the problem (3.1)–(3.5) with the performance function (4.22) with π‘§Ξ£π‘‘βˆˆπΏ2(Ξ£) and πœ†2>0, and with constraint (4.2), and with adjoint equations (4.24), there exists a unique optimal control π‘£βˆ— which satisfies the maximum condition (4.25).

Example 4.7 (π‘’βˆˆπΏ2(𝑄)). We can also consider an analogous optimal control problem where the performance functional is given by: 𝐼(𝑒)=πœ†1ξ€œπ‘„ξ€Ίπ‘¦(π‘₯,𝑑;𝑒)βˆ’π‘§π‘‘ξ€»2𝑑π‘₯𝑑𝑑+πœ†2ξ€œπ‘„(𝑁𝑒)𝑒𝑑π‘₯𝑑𝑑,(4.26) where π‘§π‘‘βˆˆπΏ2(𝑄).

From Theorem 3.4 and the Trace Theorem [20, Vol. 2, page 9], for each π‘’βˆˆπΏ2(𝑄), there exists a unique solution 𝑦(𝑒)βˆˆπ‘Šβˆž,1(𝑄). Thus, 𝐼 is well defined. Then, the optimal control π‘’βˆ— is characterized by: πœ†1ξ€œπ‘„ξ€·π‘¦ξ€·π‘’βˆ—ξ€Έβˆ’π‘§π‘‘ξ€Έξ€Ίπ‘¦(𝑒)βˆ’π‘¦ξ€·π‘’βˆ—ξ€Έξ€»π‘‘π‘₯𝑑𝑑+πœ†2ξ€œπ‘„π‘π‘’βˆ—ξ€·π‘’βˆ’π‘’βˆ—ξ€Έπ‘‘π‘₯𝑑𝑑β‰₯0,βˆ€π‘’βˆˆπ‘ˆad.(4.27) We define the adjoint variable 𝑝=𝑝(π‘’βˆ—)=𝑝(π‘₯,𝑑;π‘’βˆ—) as the solution of the equations: βˆ’πœ•π‘ξ€·π‘’βˆ—ξ€Έπœ•π‘‘+π’œβˆ—(𝑑)π‘ξ€·π‘’βˆ—ξ€Έ+π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–π‘π‘–ξ€·π‘₯,𝑑+β„Žπ‘–ξ€Έπ‘ξ€·π‘₯,𝑑+β„Žπ‘–;π‘’βˆ—ξ€Έπ‘‘β„Žπ‘–=πœ†1ξ€·π‘¦ξ€·π‘’βˆ—ξ€Έβˆ’π‘§π‘‘ξ€Έ,(π‘₯,𝑑)βˆˆΞ©Γ—(0,π‘‡βˆ’Ξ”),β„Žπ‘–βˆˆξ€·π‘Žπ‘–,𝑏𝑖,βˆ’πœ•π‘ξ€·π‘’βˆ—ξ€Έπœ•π‘‘+π’œβˆ—(𝑑)π‘ξ€·π‘’βˆ—ξ€Έ=πœ†1ξ€·π‘¦ξ€·π‘’βˆ—ξ€Έβˆ’π‘§π‘‘ξ€Έ,(π‘₯,𝑑)βˆˆΞ©Γ—(π‘‡βˆ’Ξ”,𝑇),𝑝π‘₯,𝑇;π‘’βˆ—ξ€Έ=0,π‘₯∈Ω,𝑝π‘₯,𝑑;π‘’βˆ—ξ€Έ=0,(π‘₯,𝑑)βˆˆΞ©Γ—[π‘‡βˆ’Ξ”+πœ†,𝑇),πœ•π‘ξ€·π‘’βˆ—ξ€Έπœ•πœˆπ’œβˆ—(π‘₯,𝑑)=𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ π‘π‘ ξ€·π‘₯,𝑑+π‘˜π‘ ξ€Έπ‘ξ€·π‘₯,𝑑+π‘˜π‘ ;π‘’βˆ—ξ€Έπ‘‘π‘˜π‘ ,(π‘₯,𝑑)βˆˆΞ“Γ—(0,π‘‡βˆ’Ξ”(𝑇)),π‘˜π‘ βˆˆξ€·π‘π‘ ,𝑑𝑠,πœ•π‘ξ€·π‘’βˆ—ξ€Έπœ•πœˆπ’œβˆ—(π‘₯,𝑑)=0,(π‘₯,𝑑)βˆˆΞ“Γ—(π‘‡βˆ’Ξ”(𝑇),𝑇).(4.28) As in the above section, we have the following result.

Lemma 4.8. Let the hypothesis of Theorem 3.4 be satisfied. Then, for given π‘§π‘‘βˆˆπΏ2(𝑄) and any π‘’βˆˆπΏ2(𝑄), there exists a unique solution 𝑝(π‘’βˆ—)βˆˆπ‘Šβˆž,1(𝑄) to the adjoint problem (4.28).
Using the adjoint equations (4.28) in this case, the condition (4.27) can also be written in the following form: ξ€œπ‘„ξ€·π‘ξ€·π‘’βˆ—ξ€Έ+πœ†2π‘π‘’βˆ—ξ€Έξ€·π‘’βˆ’π‘’βˆ—ξ€Έπ‘‘π‘₯𝑑𝑑β‰₯0,βˆ€π‘’βˆˆπ‘ˆπ‘Žπ‘‘.(4.29) The following result is now summarized.

Theorem 4.9. For the problem (3.1)–(3.5) with the performance function (4.26) with π‘§π‘‘βˆˆπΏ2(𝑄) and πœ†2>0, and with constraint (4.2), and with adjoint equations (4.28), there exists a unique optimal control π‘’βˆ— which satisfies the maximum condition (4.29).

Example 4.10. We can also consider an analogous optimal control problem where the performance functional is given by: 𝐼(𝑒)=πœ†1ξ€œΞ£ξ€Ίπ‘¦βˆ£Ξ£(π‘₯,𝑑;𝑒)βˆ’π‘§Ξ£π‘‘ξ€»2𝑑Γ𝑑𝑑+πœ†2ξ€œπ‘„(𝑁𝑒)𝑒𝑑π‘₯𝑑𝑑,(4.30) where π‘§Ξ£π‘‘βˆˆπΏ2(Ξ£).

From Theorem 3.4 and the Trace Theorem [20, Vol. 2, page 9], for each π‘’βˆˆπΏ2(𝑄), there exists a unique solution 𝑦(𝑒)βˆˆπ‘Šβˆž,1(𝑄) with π‘¦βˆ£Ξ£βˆˆπΏ2(Ξ£). Thus, 𝐼 is well defined. Then, the optimal control π‘’βˆ— is characterized by: πœ†1ξ€œΞ£ξ€·π‘¦ξ€·π‘’βˆ—ξ€Έβˆ’π‘§Ξ£π‘‘ξ€Έξ€Ίπ‘¦(𝑒)βˆ’π‘¦ξ€·π‘’βˆ—ξ€Έξ€»π‘‘Ξ“π‘‘π‘‘+πœ†2ξ€œπ‘„π‘π‘’βˆ—ξ€·π‘’βˆ’π‘’βˆ—ξ€Έπ‘‘π‘₯𝑑𝑑β‰₯0,βˆ€π‘’βˆˆπ‘ˆad.(4.31)

The above inequality can be simplified by introducing an adjoint equation, the form of which is identical to (4.24). Then using Theorem 3.4 we can establish the existence of a unique solution 𝑝=𝑝(π‘’βˆ—)=𝑝(π‘₯,𝑑;π‘’βˆ—)βˆˆπ‘Šβˆž,1(𝑄) for (4.24).

As in the above section, we have the following result.

Lemma 4.11. Let the hypothesis of Theorem 3.4 be satisfied. Then, for given π‘§Ξ£π‘‘βˆˆπΏ2(Ξ£) and any π‘’βˆˆπΏ2(𝑄), there exists a unique solution 𝑝(π‘’βˆ—)βˆˆπ‘Šβˆž,1(𝑄) to the adjoint problem (4.24)–(37).
Using the adjoint equations (4.24)–(37) in this case, the condition (4.31) can also be written in the following form: ξ€œπ‘„ξ€·π‘ξ€·π‘’βˆ—ξ€Έ+πœ†2π‘π‘’βˆ—ξ€Έξ€·π‘’βˆ’π‘’βˆ—ξ€Έπ‘‘π‘₯𝑑𝑑β‰₯0,βˆ€π‘’βˆˆπ‘ˆπ‘Žπ‘‘.(4.32) The following result is now summarized.

Theorem 4.12. For the problem (3.1)–(3.5) with the performance function (4.30) with π‘§Ξ£π‘‘βˆˆπΏ2(Ξ£) and πœ†2>0, and with constraint (4.2), and with adjoint equations (4.24), there exists a unique optimal control π‘’βˆ— which satisfies the maximum condition (4.32).

5. Generalization

The optimal control problems presented here can be extended to certain different two cases. Case 1: optimal control for 2Γ—2 coupled infinite order parabolic systems with multiple time delays. Case 2: optimal control for 𝑛×𝑛 coupled infinite order parabolic systems with multiple time delays. Such extension can be applied to solving many control problems in mechanical engineering.

Case 1 (optimal control for 2Γ—2 coupled infinite order parabolic systems with multiple time delays). We can extend the discussions to study the optimal control for 2Γ—2 coupled infinite order parabolic systems with multiple time delays. We consider the case where 𝑣=(𝑣1,𝑣2)∈𝐿2(Ξ£)×𝐿2(Ξ£), the performance functional is given by [15, 16]: 𝐼(𝑣)=𝐼1(𝑣)+𝐼2(𝑣)=2𝑖=1ξ‚΅πœ†1ξ€œπ‘„ξ€Ίπ‘¦π‘–(π‘₯,𝑑;𝑣)βˆ’π‘§π‘–π‘‘ξ€»2𝑑π‘₯𝑑𝑑+πœ†2ξ€œΞ£ξ€·π‘π‘–π‘£π‘–ξ€Έπ‘£π‘–π‘‘π‘₯𝑑𝑑,(5.1) where 𝑧𝑑=(𝑧1𝑑,𝑧2𝑑)∈(𝐿2(𝑄))2.

The following results can now be proved.

Theorem 5.1. Let 𝑦0, Ξ¦0,Ξ¨0, 𝑣, and 𝑒 be given with 𝑦0=(𝑦0,1,𝑦0,2)∈(π‘Šβˆž{𝛼𝛼,2}(Ξ©))2, Ξ¨0=(Ξ¨0,1,Ξ¨0,2)∈(𝐿2(Ξ£0))2,Ξ¦0=(Ξ¦0,1,Ξ¦0,2)∈(π‘Šβˆž,1(𝑄0))2,𝑣=(𝑣1,𝑣2)∈(𝐿2(Ξ£))2, and 𝑒=(𝑒1,𝑒2)∈(π‘Šβˆ’βˆž,βˆ’1(𝑄))2. Then, there exists a unique solution 𝑦=(𝑦1,𝑦2)∈(π‘Šβˆ’βˆž,1(𝑄))2 for the following mixed initial-boundary value problem: πœ•π‘¦1πœ•π‘‘+βŽ›βŽœβŽβˆžξ“|𝛼|=0(βˆ’1)|𝛼|π‘Žπ›Όπ·2𝛼+1βŽžβŽŸβŽ π‘¦1+π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–π‘π‘–(π‘₯,𝑑)𝑦1ξ€·π‘₯,π‘‘βˆ’β„Žπ‘–ξ€Έπ‘‘β„Žπ‘–βˆ’π‘¦2=𝑒1,in𝑄,β„Žπ‘–βˆˆξ€·π‘Žπ‘–,𝑏𝑖,πœ•π‘¦2πœ•π‘‘+βŽ›βŽœβŽβˆžξ“|𝛼|=0(βˆ’1)|𝛼|π‘Žπ›Όπ·2𝛼+1βŽžβŽŸβŽ π‘¦2+π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–π‘π‘–(π‘₯,𝑑)𝑦2ξ€·π‘₯,π‘‘βˆ’β„Žπ‘–ξ€Έπ‘‘β„Žπ‘–+𝑦1=𝑒2,in𝑄,β„Žπ‘–βˆˆξ€·π‘Žπ‘–,𝑏𝑖,𝑦1ξ€·π‘₯,π‘‘ξ…ž;𝑒=Ξ¦0,1ξ€·π‘₯,π‘‘ξ…žξ€Έ,ξ€·π‘₯,π‘‘ξ…žξ€ΈβˆˆΞ©Γ—[βˆ’Ξ”,0),𝑦2ξ€·π‘₯,π‘‘ξ…ž;𝑒=Ξ¦0,2ξ€·π‘₯,π‘‘ξ…žξ€Έ,ξ€·π‘₯,π‘‘ξ…žξ€ΈβˆˆΞ©Γ—[βˆ’Ξ”,0),𝑦1(π‘₯,0;𝑒)=𝑦0,1,π‘₯∈Ω,𝑦2(π‘₯,0;𝑒)=𝑦0,2,π‘₯∈Ω,πœ•π‘¦1πœ•πœˆπ’œ(π‘₯,𝑑)=𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ π‘π‘ 1(π‘₯,𝑑)𝑦1ξ€·π‘₯,π‘‘βˆ’π‘˜π‘ ξ€Έπ‘‘π‘˜π‘ +𝑣1,onΞ£,π‘˜π‘ βˆˆξ€·π‘π‘ ,𝑑𝑠,πœ•π‘¦2πœ•πœˆπ’œ=𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ π‘π‘ 2(π‘₯,𝑑)𝑦2ξ€·π‘₯,π‘‘βˆ’π‘˜π‘ ξ€Έπ‘‘π‘˜π‘ +𝑣2,onΞ£,π‘˜π‘ βˆˆξ€·π‘π‘ ,𝑑𝑠,𝑦1ξ€·π‘₯,π‘‘ξ…ž;𝑒=Ξ¨0,1ξ€·π‘₯,π‘‘ξ…žξ€Έ,ξ€·π‘₯,π‘‘ξ…žξ€ΈβˆˆΞ“Γ—[βˆ’Ξ”,0),𝑦2ξ€·π‘₯,π‘‘ξ…ž;𝑒=Ξ¨0,2ξ€·π‘₯,π‘‘ξ…žξ€Έ,(π‘₯,𝑑′)βˆˆΞ“Γ—[βˆ’Ξ”,0),(5.2) where 𝑦≑𝑦(π‘₯,𝑑;𝑒)=𝑦1(π‘₯,𝑑;𝑒),𝑦2(π‘₯,𝑑;𝑒)ξ€Έβˆˆξ€·π‘Šβˆž,1(𝑄)ξ€Έ2,𝑒≑𝑒(π‘₯,𝑑)=𝑒1(π‘₯,𝑑),𝑒2(π‘₯,𝑑)ξ€Έβˆˆξ€·ξ€·π‘Šβˆž,1(𝑄)ξ€Έξ…žξ€Έ2,𝑣≑𝑣(π‘₯,𝑑)=𝑣1(π‘₯,𝑑),𝑣2(π‘₯,𝑑)ξ€Έβˆˆξ€·πΏ2(Ξ£)ξ€Έ2.(5.3)

Lemma 5.2. Let the hypothesis of Theorem 5.1 be satisfied. Then for given 𝑧𝑑=(𝑧1𝑑,𝑧2𝑑)∈(𝐿2(𝑄))2 and any 𝑣=(𝑣1,𝑣2)∈(𝐿2(Ξ£))2, there exists a unique solution 𝑝(𝑣)=(𝑝1(𝑣),𝑝2(𝑣))∈(π‘Šβˆž,1(𝑄))2 for the adjoint problem: βˆ’πœ•π‘1(𝑣)πœ•π‘‘+βŽ›βŽœβŽβˆžξ“|𝛼|=0(βˆ’1)|𝛼|π‘Žπ›Όπ·2𝛼+1βŽžβŽŸβŽ π‘1(𝑣)+π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–π‘π‘–ξ€·π‘₯,𝑑+β„Žπ‘–ξ€Έπ‘1ξ€·π‘₯,𝑑+β„Žπ‘–;π‘£ξ€Έπ‘‘β„Žπ‘–+𝑝2(𝑣)=πœ†1𝑦1(𝑣)βˆ’π‘§1𝑑,(π‘₯,𝑑)βˆˆΞ©Γ—(0,π‘‡βˆ’Ξ”),β„Žπ‘–βˆˆξ€·π‘Žπ‘–,𝑏𝑖,βˆ’πœ•π‘2(𝑣)πœ•π‘‘+βŽ›βŽœβŽβˆžξ“|𝛼|=0(βˆ’1)|𝛼|π‘Žπ›Όπ·2𝛼+1βŽžβŽŸβŽ π‘2(𝑣)+π‘šξ“π‘–=1ξ€œπ‘π‘–π‘Žπ‘–π‘π‘–ξ€·π‘₯,𝑑+β„Žπ‘–ξ€Έπ‘2ξ€·π‘₯,𝑑+β„Žπ‘–;π‘£ξ€Έπ‘‘β„Žπ‘–βˆ’π‘1(𝑣)=πœ†1𝑦2(𝑣)βˆ’π‘§2𝑑,(π‘₯,𝑑)βˆˆΞ©Γ—(0,π‘‡βˆ’Ξ”),β„Žπ‘–βˆˆξ€·π‘Žπ‘–,𝑏𝑖,πœ•π‘1(𝑣)πœ•π‘‘+βŽ›βŽœβŽβˆžξ“|𝛼|=0(βˆ’1)|𝛼|π‘Žπ›Όπ·2𝛼+1βŽžβŽŸβŽ π‘1(𝑣)=πœ†1𝑦1(𝑣)βˆ’π‘§1𝑑,(π‘₯,𝑑)βˆˆΞ©Γ—(π‘‡βˆ’Ξ”,𝑇),πœ•π‘2(𝑣)πœ•π‘‘+βŽ›βŽœβŽβˆžξ“|𝛼|=0(βˆ’1)|𝛼|π‘Žπ›Όπ·2𝛼+1βŽžβŽŸβŽ π‘2(𝑣)=πœ†1𝑦2(𝑣)βˆ’π‘§2𝑑,(π‘₯,𝑑)βˆˆΞ©Γ—(π‘‡βˆ’Ξ”,𝑇),𝑝1(π‘₯,𝑇;𝑣)=0,𝑝2(π‘₯,𝑇;𝑣)=0,π‘₯∈Ω,𝑝1(π‘₯,𝑑;𝑣)=0,𝑝2(π‘₯,𝑑;𝑣)=0,(π‘₯,𝑑)βˆˆΞ©Γ—[π‘‡βˆ’Ξ”+πœ†,𝑇),πœ•π‘1(π‘₯,𝑑;𝑣)πœ•πœˆπ’œβˆ—=𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ π‘π‘ 1ξ€·π‘₯,𝑑+π‘˜π‘ ξ€Έπ‘1ξ€·π‘₯,𝑑+π‘˜π‘ ;π‘£ξ€Έπ‘‘π‘˜π‘ ,(π‘₯,𝑑)βˆˆΞ“Γ—(0,π‘‡βˆ’Ξ”),π‘˜π‘ βˆˆξ€·π‘π‘ ,𝑑𝑠,πœ•π‘2(π‘₯,𝑑;𝑣)πœ•πœˆπ’œβˆ—=𝑙𝑠=1ξ€œπ‘‘π‘ π‘π‘ π‘π‘ 2ξ€·π‘₯,𝑑+π‘˜π‘ ξ€Έπ‘2ξ€·π‘₯,𝑑+π‘˜π‘ ;π‘£ξ€Έπ‘‘π‘˜π‘ ,(π‘₯,𝑑)βˆˆΞ“Γ—(0,π‘‡βˆ’Ξ”),π‘˜π‘ βˆˆξ€·π‘π‘ ,𝑑𝑠,πœ•π‘1(π‘₯,𝑑)πœ•πœˆπ’œβˆ—=0,πœ•π‘2(π‘₯,𝑑)πœ•