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

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 (2×2) 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 Ω be a bounded open set of ℝ^𝑛 with a smooth boundary Γ, which is a 𝐶^∞ manifold of dimension (𝑛−1). Locally, Ω is totally on one side of Γ. We define the infinite order Sobolev space 𝑊^∞{𝑎_𝛼,2}(Ω) of infinite order of periodic functions 𝜙(𝑥) defined on Ω [29–31] as follows: 𝑊^∞𝑎_𝛼,2(Ω)=⎧⎨⎩𝜙(𝑥)∈C^∞(Ω)∶^∞_|𝛼|=0𝑎_𝛼‖D^𝛼𝜙‖²₂<∞⎫⎬⎭,(2.1) where 𝐶^∞(Ω) is the space of infinitely differentiable functions, 𝑎_𝛼≥0 is a numerical sequence, and ‖⋅‖₂ is the canonical norm in the space 𝐿²(Ω), and 𝐷^𝛼=𝜕^|𝛼|𝜕𝑥₁^𝛼1⋯𝜕𝑥_𝑛^𝛼𝑛,(2.2)𝛼=(𝛼₁,…,𝛼_𝑛) being a multi-index for differentiation, |𝛼|=∑^𝑛_𝑖=1𝛼_𝑖.

The space 𝑊^−∞{𝑎_𝛼,2}(Ω) is defined as the formal conjugate space to the space 𝑊^∞{𝑎_𝛼,2}(Ω), namely: 𝑊^−∞𝑎_𝛼,2(Ω)=⎧⎨⎩𝜓(𝑥)∶𝜓(𝑥)=^∞_|𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^𝛼𝜓_𝛼(𝑥)⎫⎬⎭,(2.3) where 𝜓_𝛼∈𝐿²(Ω) and ∑^∞_|𝛼|=0𝑎_𝛼‖𝜓_𝛼‖²₂<∞.

The duality pairing of the spaces 𝑊^∞{𝑎_𝛼,2}(Ω) and 𝑊^−∞{𝑎_𝛼,2}(Ω) is postulated by the formula: (𝜙,𝜓)=^∞_|_𝛼|=0𝑎_𝛼_Ω𝜓_𝛼(𝑥)𝐷^𝛼𝜙(𝑥)𝑑𝑥,(2.4) where 𝜙∈𝑊^∞𝑎_𝛼,2(Ω),𝜓∈𝑊^−∞𝑎_𝛼,2(Ω).(2.5)

From above, 𝑊^∞{𝑎_𝛼,2}(Ω) is everywhere dense in 𝐿²(Ω) with topological inclusions and 𝑊^−∞{𝑎_𝛼,2}(Ω) denotes the topological dual space with respect to 𝐿²(Ω), so we have the following chain of inclusions: 𝑊^∞𝑎_𝛼,2(Ω)⊆𝐿²(Ω)⊆𝑊^−∞𝑎_𝛼,2(Ω).(2.6) We now introduce 𝐿²(0,𝑇;𝐿²(Ω)) which we will denoted by 𝐿²(𝑄), where 𝑄=Ω×]0,𝑇[ denotes the space of measurable functions 𝑡→𝜙(𝑡) such that ‖𝜙‖_𝐿2(𝑄)=^𝑇₀‖𝜙(𝑡)‖²₂𝑑𝑡^1/2<∞,(2.7) endowed with the scalar product (𝑓,𝑔)=∫^𝑇₀(𝑓(𝑡),𝑔(𝑡))_𝐿2(Ω)𝑑𝑡, 𝐿²(𝑄) is a Hilbert space.

In the same manner we define the spaces 𝐿²(0,𝑇;𝑊^∞{𝑎_𝛼,2}(Ω)), and 𝐿²(0,𝑇;𝑊^−∞{𝑎_𝛼,2}(Ω)), as its formal conjugate.

Also, we have the following chain of inclusions: 𝐿²0,𝑇;𝑊^∞𝑎_𝛼,2(Ω)⊆𝐿²(𝑄)⊆𝐿²0,𝑇;𝑊^−∞𝑎_𝛼,2(Ω).(2.8)

The construction of the Cartesian product of 𝑛-times to the above Hilbert spaces can be constructed, for example 𝑊^∞𝑎_𝛼,2(Ω)^𝑛=𝑊^∞𝑎_𝛼,2(Ω)×𝑊^∞𝑎_𝛼,2(Ω)×⋯×𝑊^∞𝑎_𝛼,2(Ω)_{𝑛-times}=^𝑛_𝑖=1𝑊^∞𝑎_𝛼,2(Ω)^𝑖,(2.9) with norm defined by: ‖𝜙‖_{(𝑊∞{𝑎𝛼,2}(Ω))𝑛}=^𝑛_𝑖=1‖‖𝜙_𝑖‖‖_{𝑊∞{𝑎𝛼,2}(Ω)},(2.10) where 𝜙=(𝜙₁,𝜙₂,…,𝜙_𝑛)=(𝜙_𝑖)^𝑛_𝑖=1 is a vector function and 𝜙_𝑖∈𝑊^∞{𝑎_𝛼,2}(Ω).

Finally, we have the following chain of inclusions: 𝐿²0,𝑇;𝑊^∞𝑎_𝛼,2(Ω)^𝑛⊆𝐿²(𝑄)^𝑛⊆𝐿²0,𝑇;𝑊^−∞𝑎_𝛼,2(Ω)^𝑛,(2.11) where (𝐿²(0,𝑇;𝑊^−∞{𝑎_𝛼,2}(Ω)))^𝑛 are the dual spaces of (𝐿²(0,𝑇;𝑊^∞{𝑎_𝛼,2}(Ω)))^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].𝜕𝑦𝜕𝑡+𝒜(𝑡)𝑦(𝑥,𝑡)+^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖𝑏_𝑖(𝑥,𝑡)𝑦𝑥,𝑡−ℎ_𝑖𝑑ℎ_𝑖=𝑢,(𝑥,𝑡)∈Ω×(0,𝑇),ℎ_𝑖∈𝑎_𝑖,𝑏_𝑖,(3.1)𝑦𝑥,𝑡^=Φ₀𝑥,𝑡^,𝑥,𝑡^∈Ω×(−Δ,0),(3.2)𝑦(𝑥,0)=𝑦₀(𝑥),𝑥∈Ω,(3.3)𝜕𝑦𝜕𝜈_𝒜(𝑥,𝑡)=^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠𝑐_𝑠(𝑥,𝑡)𝑦𝑥,𝑡−𝑘_𝑠𝑑𝑘_𝑠+𝑣,(𝑥,𝑡)∈Γ×(0,𝑇),𝑘_𝑠∈𝑐_𝑠,𝑑_𝑠,(3.4)𝑦𝑥,𝑡^=Ψ₀𝑥,𝑡^,𝑥,𝑡^∈Γ×(−Δ,0),(3.5) where Ω⊂𝑅^𝑛 has the same properties as in Section 1. We have 𝑦≡𝑦(𝑥,𝑡;𝑢),𝑦(0)≡𝑦(𝑥,0;𝑢),𝑦(𝑇)≡𝑦(𝑥,𝑇;𝑢),𝑢≡𝑢(𝑥,𝑡),𝑣≡𝑣(𝑥,𝑡),𝑄=Ω×(0,𝑇),𝑄=Ω×[0,𝑇],𝑄₀=Ω×[−Δ,0),Σ=Γ×(0,𝑇),Σ₀=Γ×[−Δ,0),(3.6)(i)𝑇is a specified positive number representing a finite time horizon,(ii)ℎ_𝑖,𝑘_𝑠 are time delays, such that ℎ_𝑖∈(𝑎_𝑖,𝑏_𝑖) and 𝑘_𝑠∈(𝑐_𝑠,𝑑_𝑠) where 0<𝑎₁<𝑎₂<⋯<𝑎_𝑚, 0<𝑏₁<𝑏₂<⋯<𝑏_𝑚, for 𝑖=1,2,…,𝑚 and 0<𝑐₁<𝑐₂<⋯<𝑐_𝑙, 0<𝑑₁<𝑑₂<⋯<𝑑_𝑙, for 𝑠=1,2,…,𝑙,(iii)𝑏_𝑖(𝑡),𝑖=1,2,…,𝑚 are given real 𝐶^∞ functions defined on 𝑄,(iv)𝑐_𝑠(𝑥,𝑡),𝑠=1,2,…,𝑙 are given real 𝐶^∞ functions defined on Σ,(v)Δ=max{𝑏_𝑚,𝑑_𝑙},(vi)𝑦 is a function defined on 𝑄 such that Ω×(0,𝑇)∋(𝑥,𝑡)→𝑦(𝑥,𝑡)∈𝑅,(vii)𝑢,𝑣 are functions defined on 𝑄 and Σ such that Ω×(0,𝑇)∋(𝑥,𝑡)→𝑢(𝑥,𝑡)∈𝑅 and Γ×(0,𝑇)∋(𝑥,𝑡)→𝑣(𝑥,𝑡)∈𝑅,(viii)Φ₀,Ψ₀ are initial functions defined on 𝑄₀ and Σ₀, respectively, such that Ω×[−Δ,0)∋(𝑥,𝑡^)→Φ₀(𝑥,𝑡^)∈𝑅. Γ×[−Δ,0)∋(𝑥,𝑡^)→Ψ₀(𝑥,𝑡^)∈𝑅.

The parabolic operator (𝜕/𝜕𝑡)+𝒜(𝑡) in the state equation (3.1) is an infinite order parabolic operator and 𝒜(𝑡) [17, 21, 29–31] is given by: 𝒜𝑦=^∞_|𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼𝑦(𝑥,𝑡),𝒜=^∞_|𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼(3.7) is an infinite order self-adjoint elliptic partial differential operator maps 𝑊^∞{𝑎_𝛼,2}(Ω) onto 𝑊^−∞{𝑎_𝛼,2}(Ω).

For this operator we define the bilinear form as follows.

Definition 3.1. For each 𝑡∈(0,𝑇), we define a family of bilinear forms on 𝑊^∞{𝑎_𝛼,2}(Ω) by: 𝜋(𝑡;𝑦,𝜙)=(𝒜(𝑡)𝑦,𝜙)_𝐿2(Ω),𝑦,𝜙∈𝑊^∞𝑎_𝛼,2(Ω),(3.8) where 𝒜(𝑡) maps 𝑊^∞{𝑎_𝛼,2}(Ω) onto 𝑊^−∞{𝑎_𝛼,2}(Ω) and takes the above form. Then 𝜋(𝑡;𝑦,𝜙)=(𝒜(𝑡)𝑦,𝜙)_𝐿2(Ω)=⎛⎜⎝^∞_|_𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼𝑦(𝑥,𝑡),𝜙(𝑥)⎞⎟⎠_𝐿²_(Ω)=_Ω^∞_|𝛼|=0𝑎_𝛼𝐷^𝛼𝑦(𝑥)𝐷^𝛼𝜙(𝑥)𝑑𝑥.(3.9)

Lemma 3.2. The bilinear form 𝜋(𝑡;𝑦,𝜙) is coercive on 𝑊^∞{𝑎_𝛼,2}(Ω), that is, 𝜋(𝑡;𝑦,𝑦)≥𝜆‖𝑦‖²_{𝑊∞𝑎𝛼,2(Ω)},𝜆>0.(3.10)

Proof. It is well known that the ellipticity of 𝒜(𝑡) is sufficient for the coerciveness of 𝜋(𝑡;𝑦,𝜙) on 𝑊^∞{𝑎_𝛼,2}(Ω): 𝜋(𝑡;𝜙,𝜓)=_Ω^∞_|𝛼|=0𝑎_𝛼𝐷^𝛼𝜙𝐷^𝛼𝜓𝑑𝑥.(3.11) Then 𝜋(𝑡;𝑦,𝑦)=_Ω^∞_|𝛼|=0𝑎_𝛼𝐷^𝛼𝑦𝐷^𝛼𝑦𝑑𝑥≥^∞_|𝛼|=0𝑎_𝛼‖‖𝐷^2𝛼𝑦(𝑥)‖‖²_𝐿2(Ω)≥𝜆‖𝑦‖²_{𝑊∞{𝑎𝛼,2}(Ω)},𝜆>0.(3.12) Also we have ∀𝑦,𝜙∈𝑊^∞𝑎_𝛼,2(Ω)thefunction𝑡⟶𝜋(𝑡;𝑦,𝜙)iscontinuouslydifferentiablein(0,𝑇)and𝜋(𝑡;𝑦,𝜙)=𝜋(𝑡;𝜙,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: 𝜕𝑦(𝑥,𝑡)𝜕𝜈_𝒜=^∞_|𝜔|=0(𝐷^𝜔𝑦(𝑥,𝑡))cos𝑛,𝑥_𝑘=𝑞(𝑥,𝑡),𝑥∈Γ,𝑡∈(0,𝑇),(3.14) where 𝜕/𝜕𝜈_𝒜 is a normal derivative at Γ, directed towards the exterior of Ω, and cos(𝑛,𝑥_𝑘) is the 𝑘th direction cosine of 𝑛, with 𝑛 being the normal at Γ exterior to Ω.
Then (3.4) can be written as: 𝑞(𝑥,𝑡)=^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠𝑐_𝑠(𝑥,𝑡)𝑦𝑥,𝑡−𝑘_𝑠𝑑𝑘_𝑠+𝑣(𝑥,𝑡),𝑥∈Γ,𝑡∈(0,𝑇).(3.15)

Remark 3.3. We will apply the indication 𝑞(𝑥,𝑡) 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 𝑣∈𝐿²(Σ).

For this purpose, we introduce the Sobolev space 𝑊^∞,1(𝑄) [20, Vol. 2, page 6] defined by: 𝑊^∞,1(𝑄)=𝐿²0,𝑇;𝑊^∞𝑎_𝛼,2(Ω)∩𝑊¹0,𝑇;𝐿²(Ω),(3.16) which is a Hilbert space normed by ‖𝑦‖_{𝑊∞,1(𝑄)}=^𝑇₀‖𝑦‖²_{𝑊∞𝑎𝛼,2(Ω)}𝑑𝑡+‖𝑦‖²_{𝑊10,𝑇;𝐿2(Ω)}^1/2=⎡⎢⎣_𝑄⎛⎜⎝^∞_|_𝛼|=0𝑎_𝛼||𝐷^𝛼𝑦||²+|||𝜕𝑦𝜕𝑡|||²⎞⎟⎠𝑑𝑥𝑑𝑡⎤⎥⎦^1/2=⎡⎢⎣_𝑄⎛⎜⎝𝑎₀||𝑦||²+^∞_|𝛼|=1𝑎_𝛼||𝐷^𝛼𝑦||²+|||𝜕𝑦𝜕𝑡|||²⎞⎟⎠𝑑𝑥𝑑𝑡⎤⎥⎦^1/2,𝑎₀>0,(3.17) where the space 𝑊¹(0,𝑇;𝐿²(Ω)) denotes the Sobolev space of order 1 of functions defined on (0,𝑇) and taking values in 𝐿²(Ω) [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: 𝐸_𝑗=((𝑗−1)𝜆,𝑗𝜆),𝑄_𝑗=Ω×𝐸_𝑗,Σ_𝑗=Γ×𝐸_𝑗for𝑗=1,…,𝐾,𝜆=min𝑎₁,𝑐₁.(3.18)

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

Theorem 3.4. Let 𝑦₀, Φ₀, Ψ₀, 𝑣 and 𝑢 be given with 𝑦₀∈𝑊^∞{𝑎_𝛼,2}(Ω), Φ₀∈𝑊^∞,1(𝑄₀), Ψ₀∈𝐿²(Σ₀), 𝑣∈𝐿²(Σ) and 𝑢∈𝑊^−∞,−1(𝑄). Then, there exists a unique solution 𝑦∈𝑊^∞,1(𝑄) for the mixed initial-boundary value problem (3.1)–(3.5). Moreover, 𝑦(⋅,𝑗𝜆)∈𝑊^∞{𝑎_𝛼,2}(Ω)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 𝑣∈𝐿²(Σ).

Let us denote by 𝑈=𝐿²(Σ) the space of controls. The time horizon 𝑇 is fixed in our problem.

The performance functional is given by 𝐼(𝑣)=𝜆₁_𝑄𝑦(𝑥,𝑡;𝑣)−𝑧_𝑑²𝑑𝑥𝑑𝑡+𝜆₂_Σ(𝑁𝑣)𝑣𝑑Γ𝑑𝑡,(4.1) where 𝜆_𝑖≥0, and 𝜆₁+𝜆₂>0,𝑧_𝑑 is a given element in 𝐿²(𝑄); 𝑁 is a positive linear operator on 𝐿²(Σ) into 𝐿²(Σ).

Control Contraints
We define the set of admissible controls 𝑈_ad such that 𝑈_adisclosed,convexsubsetof𝑈=𝐿²(Σ).(4.2)

Let 𝑦(𝑥,𝑡;𝑣) denote the solution of the mixed initial-boundary value problem (3.1)–(3.5) at (𝑥,𝑡) corresponding to a given control 𝑣∈𝑈_ad. We note from Theorem 3.4 that for any 𝑣∈𝑈_ad the performance functional (4.1) is well-defined since (𝑣)∈𝑊^∞,1(𝑄)⊂𝐿²(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 𝑣^∗∈𝑈_ad such that 𝐼𝑣^∗≤𝐼(𝑣),∀𝑣∈𝑈_ad.(4.3)

From the Lion's scheme [19, Theorem  1.3, page 10], it follows that for 𝜆₂>0 a unique optimal control 𝑣^∗ exists. Moreover, 𝑣^∗ is characterized by the following condition: 𝐼^𝑣^∗𝑣−𝑣^∗≥0,∀𝑣∈𝑈_ad.(4.4) For the performance functional of form (4.1) the relation (4.4) can be expressed as 𝜆₁_𝑄𝑦𝑣^∗−𝑧_𝑑𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡+𝜆₂_Σ𝑁𝑣^∗𝑣−𝑣^∗𝑑Γ𝑑𝑡≥0,∀𝑣∈𝑈_ad.(4.5)

In order to simplify (4.5), we introduce the adjoint equation, and for every 𝑣∈𝑈_ad, we define the adjoint variable 𝑝=𝑝(𝑣)≡𝑝(𝑥,𝑡;𝑣) as the solution of the equations: −𝜕𝑝(𝑣)𝜕𝑡+𝒜^∗(𝑡)𝑝(𝑣)+^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖𝑏_𝑖𝑥,𝑡+ℎ_𝑖𝑝𝑥,𝑡+ℎ_𝑖;𝑣𝑑ℎ_𝑖=𝜆₁𝑦(𝑣)−𝑧_𝑑,(𝑥,𝑡)∈Ω×(0,𝑇−Δ),ℎ_𝑖∈𝑎_𝑖,𝑏_𝑖,(4.6)−𝜕𝑝(𝑣)𝜕𝑡+𝒜^∗(𝑡)𝑝(𝑣)=𝜆₁𝑦(𝑣)−𝑧_𝑑,(𝑥,𝑡)∈Ω×(𝑇−Δ,𝑇),(4.7)𝑝(𝑥,𝑇;𝑣)=0,𝑥∈Ω,(4.8)𝑝(𝑥,𝑡;𝑣)=0,(𝑥,𝑡)∈Ω×[𝑇−Δ+𝜆,𝑇),(4.9)𝜕𝑝(𝑣)𝜕𝜈_𝒜∗(𝑥,𝑡)=^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠𝑐_𝑠𝑥,𝑡+𝑘_𝑠𝑝𝑥,𝑡+𝑘_𝑠;𝑣𝑑𝑘_𝑠,(𝑥,𝑡)∈Γ×(0,𝑇−Δ(𝑇)),𝑘_𝑠∈𝑐_𝑠,𝑑_𝑠,(4.10)𝜕𝑝(𝑣)𝜕𝜈_𝒜∗(𝑥,𝑡)=0,(𝑥,𝑡)∈Γ×(𝑇−Δ(𝑇),𝑇),(4.11) where 𝜕𝑝(𝑣)𝜕𝜈_𝒜∗(𝑥,𝑡)=^∞_|𝜔_|=0(𝐷^𝜔𝑝(𝑣))cos𝑛,𝑥_𝜔(𝑥,𝑡),𝒜^∗(𝑡)𝑝(𝑣)=^∞_|𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼𝑝(𝑥,𝑡).(4.12)

As in the above section with change of variables, that is, with reversed sense of time. that is, 𝑡^=𝑇−𝑡, for given 𝑧_𝑑∈𝐿²(𝑄) and any 𝑣∈𝐿²(Σ), there exists a unique solution 𝑝(𝑣)∈𝑊^∞,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 𝑧_𝑑 and 𝑢, the problem (4.6)–(4.11) can be solved backwards in time starting from 𝑡=𝑇, that is, first solving (4.6)–(4.11) on the subcylinder 𝑄_𝐾 and in turn on 𝑄_𝐾−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 𝑧_𝑑∈𝐿²(Ω,𝑅^∞) and any 𝑣∈𝐿²(Σ), there exists a unique solution 𝑝(𝑣)∈𝑊^∞,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(𝑥,0;𝑣) and 𝑝(𝑇)≡𝑝(𝑥,𝑇;𝑣), respectively, setting 𝑣=𝑣^∗ in (4.6)–(4.11), multiplying both sides of (4.6) and (4.7) by 𝑦(𝑣)−𝑦(𝑣^∗), then integrating over Ω×(0,𝑇−Δ) and Ω×(𝑇−Δ,𝑇), respectively and then adding both sides of (4.6), (4.11), we get 𝜆₁_𝑄𝑦𝑣^∗−𝑧_𝑑𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡=_𝑄⎛⎜⎝−𝜕𝑝𝑣^∗𝜕𝑡+𝒜^∗(𝑡)𝑝𝑣^∗⎞⎟⎠×𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡+^𝑇−Δ₀_Ω⎛⎜⎝^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖𝑏_𝑖𝑥,𝑡+ℎ_𝑖𝑝𝑥,𝑡+ℎ_𝑖;𝑣^∗𝑑ℎ_𝑖⎞⎟⎠×𝑦(𝑥,𝑡;𝑣)−𝑦𝑥,𝑡;𝑣^∗𝑑𝑥𝑑𝑡=^𝑇₀_Ω𝑝𝑣^∗𝜕𝜕𝑡𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡+^𝑇₀_Ω𝒜^∗(𝑡)𝑝𝑣^∗𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡+^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖_Ω^𝑇−Δ₀𝑏_𝑖𝑥,𝑡+ℎ_𝑖𝑝𝑥,𝑡+ℎ_𝑖;𝑣^∗×𝑦(𝑥,𝑡;𝑣)−𝑦𝑥,𝑡;𝑣^∗𝑑𝑥𝑑𝑡𝑑ℎ_𝑖.(4.13) Using (3.1), the first integral on the right-hand side of (4.13) can be written as: ^𝑇₀_Ω𝑝𝑣^∗𝜕𝜕𝑡𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡=−_𝑄𝑝𝑣^∗𝒜(𝑡)𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡−^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖_Ω^𝑇₀𝑝𝑥,𝑡;𝑣^∗𝑏_𝑖(𝑥,𝑡)×𝑦𝑥,𝑡−ℎ_𝑖;𝑣−𝑦𝑥,𝑡−ℎ_𝑖;𝑣^∗𝑑𝑡𝑑𝑥𝑑ℎ_𝑖=−_𝑄𝑝𝑣^∗𝒜(𝑡)𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡−^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖_Ω^{𝑇−ℎ𝑖}_−ℎ𝑖𝑝𝑥,𝑡^+ℎ_𝑖;𝑣^∗𝑏_𝑖𝑥,𝑡^+ℎ_𝑖×𝑦𝑥,𝑡^;𝑣−𝑦𝑥,𝑡^;𝑣^∗𝑑𝑡′𝑑𝑥𝑑ℎ_𝑖=−_Ω𝑝𝑣^∗𝒜(𝑡)𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡−^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖_Ω⁰_−ℎ𝑖𝑝𝑥,𝑡^+ℎ_𝑖;𝑣^∗𝑏_𝑖𝑥,𝑡^+ℎ_𝑖×𝑦𝑥,𝑡^;𝑣−𝑦𝑥,𝑡^;𝑣^∗𝑑𝑡′𝑑𝑥𝑑ℎ_𝑖−^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖_Ω^𝑇−Δ₀𝑝𝑥,𝑡^+ℎ_𝑖;𝑣^∗𝑏_𝑖𝑥,𝑡^+ℎ_𝑖×𝑦𝑥,𝑡^;𝑣−𝑦𝑥,𝑡^;𝑣^∗𝑑𝑡′𝑑𝑥𝑑ℎ_𝑖−^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖_Ω^{𝑇−ℎ𝑖}_𝑇−Δ𝑝𝑥,𝑡^+ℎ_𝑖;𝑣^∗𝑏_𝑖𝑥,𝑡^+ℎ_𝑖×𝑦𝑥,𝑡^;𝑣−𝑦𝑥,𝑡^;𝑣^∗𝑑𝑡′𝑑𝑥𝑑ℎ_𝑖=−_𝑄𝑝𝑣^∗𝒜(𝑡)𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡−^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖_Ω⁰_−ℎ𝑖𝑝𝑥,𝑡^+ℎ_𝑖;𝑣^∗𝑏_𝑖𝑥,𝑡^+ℎ_𝑖×𝑦𝑥,𝑡^;𝑣−𝑦𝑥,𝑡^;𝑣^∗𝑑𝑡′𝑑𝑥𝑑ℎ_𝑖−^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖_Ω^𝑇−Δ₀𝑝𝑥,𝑡^+ℎ_𝑖;𝑣^∗𝑏_𝑖𝑥,𝑡^+ℎ_𝑖×𝑦𝑥,𝑡^;𝑣−𝑦𝑥,𝑡^;𝑣^∗𝑑𝑡′𝑑𝑥𝑑ℎ_𝑖−^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖_Ω^𝑇_{𝑇−Δ+ℎ𝑖}𝑝𝑥,𝑡;𝑣^∗𝑏_𝑖(𝑥,𝑡)×𝑦𝑥,𝑡−ℎ_𝑖;𝑣−𝑦𝑥,𝑡−ℎ_𝑖;𝑣^∗𝑑𝑡𝑑𝑥𝑑ℎ_𝑖.(4.14) Using Green’s formula, the second integral on the right-hand side of (4.13) can be written as: ^𝑇₀_Ω𝒜^∗(𝑡)𝑝𝑣^∗𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡=^𝑇₀_Ω𝑝𝑣^∗𝒜(𝑡)𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡+^𝑇₀_Γ𝑝𝑣^∗⎛⎜⎝𝜕𝑦(𝑣)𝜕𝜈_𝒜−𝜕𝑦𝑣^∗𝜕𝜈_𝒜⎞⎟⎠𝑑Γ𝑑𝑡−^𝑇₀_Γ𝜕𝑝𝑣^∗𝜕𝜈_𝒜∗𝑦(𝑣)−𝑦𝑣^∗𝑑Γ𝑑𝑡.(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: ^𝑇₀_Γ𝑝𝑣^∗⎛⎜⎝𝜕𝑦(𝑣)𝜕𝜈_𝒜−𝜕𝑦𝑣^∗𝜕𝜈_𝒜⎞⎟⎠𝑑Γ𝑑𝑡=^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠_Γ^𝑇₀𝑝𝑥,𝑡;𝑣^∗𝑐_𝑠(𝑥,𝑡)×𝑦𝑥,𝑡−𝑘_𝑠;𝑣−𝑦𝑥,𝑡−𝑘_𝑠;𝑣^∗𝑑Γ𝑑𝑡𝑑𝑘_𝑠+^𝑇₀_Γ𝑝𝑣^∗𝑣−𝑣^∗𝑑Γ𝑑𝑡=^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠_Γ^{𝑇−𝑘𝑠}_−𝑘𝑠𝑝𝑥,𝑡^+𝑘_𝑠;𝑣^∗𝑐_𝑠𝑥,𝑡^+𝑘_𝑠×𝑦𝑥,𝑡^;𝑣−𝑦𝑥,𝑡^;𝑣^∗𝑑𝑡^𝑑Γ𝑑𝑘_𝑠+^𝑇₀_Γ𝑝𝑣^∗𝑣−𝑣^∗𝑑Γ𝑑𝑡=^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠_Γ⁰_−𝑘𝑠𝑝𝑥,𝑡^+𝑘_𝑠;𝑣^∗𝑐_𝑠𝑥,𝑡^+𝑘_𝑠×𝑦𝑥,𝑡^;𝑣−𝑦𝑥,𝑡^;𝑣^∗𝑑𝑡^𝑑Γ𝑑𝑘_𝑠+^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠_Γ^𝑇−Δ₀𝑝𝑥,𝑡^+𝑘_𝑠;𝑣^∗𝑐_𝑠𝑥,𝑡^+𝑘_𝑠×𝑦𝑥,𝑡^;𝑣−𝑦𝑥,𝑡^;𝑣^∗𝑑𝑡^𝑑Γ𝑑𝑘_𝑠+^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠_Γ^{𝑇−𝑘𝑠}_𝑇−Δ𝑝𝑥,𝑡^+𝑘_𝑠;𝑣^∗𝑐_𝑠𝑥,𝑡^+𝑘_𝑠×𝑦𝑥,𝑡^;𝑣−𝑦𝑥,𝑡^;𝑣^∗𝑑𝑡^𝑑Γ𝑑𝑘_𝑠+^𝑇₀_Γ𝑝𝑣^∗𝑣−𝑣^∗𝑑Γ𝑑𝑡=^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠_Γ⁰_−𝑘𝑠𝑝𝑥,𝑡^+𝑘_𝑠;𝑣^∗𝑐_𝑠𝑥,𝑡^+𝑘_𝑠×𝑦𝑥,𝑡^;𝑣−𝑦𝑥,𝑡^;𝑣^∗𝑑𝑡^𝑑Γ𝑑𝑘_𝑠+^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠_Γ^𝑇−Δ₀𝑝𝑥,𝑡^+𝑘_𝑠;𝑣^∗𝑐_𝑠𝑥,𝑡^+𝑘_𝑠×𝑦𝑥,𝑡^;𝑣−𝑦𝑥,𝑡^;𝑣^∗𝑑𝑡^𝑑Γ𝑑𝑘_𝑠+^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠_Γ^𝑇_{𝑇−Δ+𝑘𝑠}𝑝𝑥,𝑡;𝑣^∗𝑐_𝑠(𝑥,𝑡)×𝑦𝑥,𝑡−𝑘_𝑠;𝑣−𝑦𝑥,𝑡−𝑘_𝑠;𝑣^∗𝑑𝑡𝑑Γ𝑑𝑘_𝑠+^𝑇₀_Γ𝑝𝑣^∗𝑣−𝑣^∗𝑑Γ𝑑𝑡.(4.16)

The last component in (4.15) can be rewritten as ^𝑇₀_Γ𝜕𝑝𝑣^∗𝜕𝜈_𝒜∗𝑦(𝑣)−𝑦𝑣^∗𝑑Γ𝑑𝑡=^𝑇−Δ₀_Γ𝜕𝑝𝑣^∗𝜕𝜈_𝒜∗𝑦(𝑣)−𝑦𝑣^∗𝑑Γ𝑑𝑡+^𝑇_𝑇−Δ_Γ𝜕𝑝𝑣^∗𝜕𝜈_𝒜∗𝑦(𝑣)−𝑦𝑣^∗𝑑Γ𝑑𝑡.(4.17) Substituting (4.16) and (4.17) into (4.15) and then the results into (4.13), we obtain 𝜆₁_𝑄𝑦𝑣^∗−𝑧_𝑑𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡=^𝑇₀_Γ𝑝𝑣^∗𝑣−𝑣^∗𝑑Γ𝑑𝑡−_𝑄𝑝𝑣^∗𝒜(𝑡)𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡−^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖_Ω⁰_−ℎ𝑖𝑏_𝑖𝑥,𝑡+ℎ_𝑖𝑝𝑥,𝑡+ℎ_𝑖;𝑣^∗×𝑦(𝑥,𝑡;𝑣)−𝑦𝑥,𝑡;𝑣^∗𝑑𝑡𝑑𝑥𝑑ℎ_𝑖−^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖_Ω^𝑇−Δ₀𝑏_𝑖𝑥,𝑡+ℎ_𝑖𝑝𝑥,𝑡+ℎ_𝑖;𝑣^∗×𝑦(𝑥,𝑡;𝑣)−𝑦𝑥,𝑡;𝑣^∗𝑑𝑡𝑑𝑥𝑑ℎ_𝑖+_𝑄𝑝𝑣^∗𝒜(𝑡)𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡−^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖_Ω^𝑇_{𝑇−Δ+ℎ𝑖}𝑝𝑥,𝑡;𝑣^∗𝑏_𝑖(𝑥,𝑡)×𝑦𝑥,𝑡−ℎ_𝑖;𝑣−𝑦𝑥,𝑡−ℎ_𝑖;𝑣^∗𝑑𝑡𝑑𝑥𝑑ℎ_𝑖+^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠_Γ⁰_−𝑘𝑠𝑐_𝑠𝑥,𝑡+𝑘_𝑠𝑝𝑥,𝑡+𝑘_𝑠;𝑣^∗×𝑦(𝑥,𝑡;𝑣)−𝑦𝑥,𝑡;𝑣^∗𝑑𝑡𝑑Γ𝑑𝑘_𝑠−^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠_Γ^𝑇−Δ₀𝑐_𝑠𝑥,𝑡+𝑘_𝑠𝑝𝑥,𝑡+𝑘_𝑠;𝑣^∗×𝑦(𝑥,𝑡;𝑣)−𝑦𝑥,𝑡;𝑣^∗𝑑𝑡𝑑Γ𝑑𝑘_𝑠−^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠_Γ^𝑇_{𝑇−Δ+𝑘𝑠}𝑐_𝑠(𝑥,𝑡)𝑝𝑥,𝑡;𝑣^∗×𝑦𝑥,𝑡−𝑘_𝑠;𝑣−𝑦𝑥,𝑡−𝑘_𝑠;𝑣^∗𝑑𝑡𝑑Γ𝑑𝑘_𝑠−_Γ^𝑇−Δ₀𝜕𝑝𝑣^∗𝜕𝜈_𝒜∗×𝑦(𝑥,𝑡;𝑣)−𝑦𝑥,𝑡;𝑣^∗𝑑𝑡𝑑Γ−_Γ^𝑇_𝑇−Δ𝜕𝑝𝑣^∗𝜕𝜈_𝒜∗×𝑦(𝑥,𝑡;𝑣)−𝑦𝑥,𝑡;𝑣^∗𝑑𝑡𝑑Γ+^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖_Ω^𝑇−Δ₀𝑏_𝑖𝑥,𝑡+ℎ_𝑖𝑝𝑥,𝑡+ℎ_𝑖;𝑣^∗×𝑦(𝑥,𝑡;𝑣)−𝑦𝑥,𝑡;𝑣^∗𝑑𝑡𝑑𝑥𝑑ℎ_𝑖.(4.18)

Afterwards, using the facts that 𝑦(𝑥,𝑡,𝑣)=𝑦(𝑥,𝑡,𝑣^∗)=Φ₀(𝑥,𝑡) for 𝑥∈Ω and 𝑡∈[−Δ,0) and 𝑦(𝑥,𝑡,𝑣)=𝑦(𝑥,𝑡,𝑣^∗)=Ψ₀(𝑥,𝑡) for 𝑥∈Γ and 𝑡∈[−Δ,0), 𝑝|_Ω(𝑥,𝑡;𝑣^∗)=0 and consequently 𝑝|_Γ(𝑥,𝑡;𝑣^∗)=0 for 𝑡∈[𝑇−Δ+𝜆,𝑇), we obtain 𝜆₁_𝑄𝑦𝑣^∗−𝑧_𝑑×𝑦(𝑣)−𝑦𝑣^∗𝑑𝑥𝑑𝑡=^𝑇₀_Γ𝑝𝑣^∗𝑣−𝑣^∗𝑑Γ𝑑𝑡.(4.19)

Substituting (4.19) into (4.5) gives ^𝑇₀_Γ𝑝𝑣^∗+𝜆₂𝑁𝑣^∗𝑣−𝑣^∗𝑑Γ𝑑𝑡≥0,∀𝑣∈𝑈_ad.(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 𝑧_𝑑∈𝐿²(𝑄) and 𝜆₂>0 and with constraints on controls (4.2), there exists a unique optimal control 𝑣^∗ which satisfies the maximum condition (4.20).

4.1. Mathematical Examples

Example 4.3. Consider now the particular case where 𝑈_ad=𝑈=𝐿²(Σ) (no constraints case). Thus the maximum condition (4.20) is satisfied when 𝑣^∗=−𝜆₂𝑁⁻¹𝑝𝑣^∗.(4.21) If 𝑁 is the identity operator on 𝐿²(Σ), then from Lemma 4.1 it follows that 𝑣^∗∈𝑊^∞,1(𝑄).

Example 4.4. We can also consider an analogous optimal control problem where the performance functional is given by: 𝐼(𝑣)=𝜆₁_Σ𝑦(𝑥,𝑡;𝑣)∣_Σ−𝑧_𝑑²𝑑Γ𝑑𝑡+𝜆₂_Σ(𝑁𝑣)𝑣𝑑Γ𝑑𝑡,(4.22) where 𝑧_𝑑∈𝐿²(Σ).

From Theorem 3.4 and the Trace Theorem [20, Vol. 2, page 9], for each 𝑣∈𝐿²(Σ), there exists a unique solution 𝑦(𝑣)∈𝑊^∞,1(𝑄) with 𝑦∣_Σ∈𝐿²(Σ). Thus, 𝐼(𝑣) is well defined. Then, the optimal control 𝑣^∗ is characterized by: 𝜆₁_Σ𝑦𝑣^∗∣_Σ−𝑧_𝑑𝑦(𝑣)∣_Σ−𝑦𝑣^∗∣_Σ𝑑Γ𝑑𝑡+𝜆₂_Σ𝑁𝑣^∗𝑣−𝑣^∗𝑑Γ𝑑𝑡≥0,∀𝑣∈𝑈_ad.(4.23) We define the adjoint variable 𝑝=𝑝(𝑣^∗)=𝑝(𝑥,𝑡;𝑣^∗) as the solution of the equations: −𝜕𝑝𝑣^∗𝜕𝑡+𝒜^∗(𝑡)𝑝𝑣^∗+^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖𝑏_𝑖𝑥,𝑡+ℎ_𝑖𝑝𝑥,𝑡+ℎ_𝑖;𝑣^∗𝑑ℎ_𝑖=0,(𝑥,𝑡)∈Ω×(0,𝑇−Δ),ℎ_𝑖∈𝑎_𝑖,𝑏_𝑖,−𝜕𝑝𝑣^∗𝜕𝑡+𝒜^∗(𝑡)𝑝𝑣^∗=0,(𝑥,𝑡)∈Ω×(𝑇−Δ,𝑇),𝑝𝑥,𝑇;𝑣^∗=0,𝑥∈Ω,𝑝𝑥,𝑡;𝑣^∗=0,(𝑥,𝑡)∈Ω×[𝑇−Δ+𝜆,𝑇),𝜕𝑝𝑣^∗𝜕𝜈_𝒜∗(𝑥,𝑡)=^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠𝑐_𝑠𝑥,𝑡+𝑘_𝑠𝑝𝑥,𝑡+𝑘_𝑠;𝑣^∗𝑑𝑘_𝑠+𝜆₁𝑦𝑣^∗∣_Σ−𝑧_Σ𝑑,(𝑥,𝑡)∈Γ×(0,𝑇−Δ(𝑇)),𝑘_𝑠∈𝑐_𝑠,𝑑_𝑠,𝜕𝑝𝑣^∗𝜕𝜈_𝒜∗(𝑥,𝑡)=𝜆₁𝑦𝑣^∗∣_Σ−𝑧_Σ𝑑,(𝑥,𝑡)∈Γ×(𝑇−Δ(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 𝑧_Σ𝑑∈𝐿²(Σ) and any 𝑣∈𝐿²(Σ), there exists a unique solution 𝑝(𝑣^∗)∈𝑊^∞,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: ^𝑇₀_Γ𝑝𝑣^∗+𝜆₂𝑁𝑣^∗𝑣−𝑣^∗𝑑Γ𝑑𝑡≥0,∀𝑣∈𝑈_ad.(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 𝑧_Σ𝑑∈𝐿²(Σ) and 𝜆₂>0, and with constraint (4.2), and with adjoint equations (4.24), there exists a unique optimal control 𝑣^∗ which satisfies the maximum condition (4.25).

Example 4.7 (𝑢∈𝐿²(𝑄)). We can also consider an analogous optimal control problem where the performance functional is given by: 𝐼(𝑢)=𝜆₁_𝑄𝑦(𝑥,𝑡;𝑢)−𝑧_𝑑²𝑑𝑥𝑑𝑡+𝜆₂_𝑄(𝑁𝑢)𝑢𝑑𝑥𝑑𝑡,(4.26) where 𝑧_𝑑∈𝐿²(𝑄).

From Theorem 3.4 and the Trace Theorem [20, Vol. 2, page 9], for each 𝑢∈𝐿²(𝑄), there exists a unique solution 𝑦(𝑢)∈𝑊^∞,1(𝑄). Thus, 𝐼 is well defined. Then, the optimal control 𝑢^∗ is characterized by: 𝜆₁_𝑄𝑦𝑢^∗−𝑧_𝑑𝑦(𝑢)−𝑦𝑢^∗𝑑𝑥𝑑𝑡+𝜆₂_𝑄𝑁𝑢^∗𝑢−𝑢^∗𝑑𝑥𝑑𝑡≥0,∀𝑢∈𝑈_ad.(4.27) We define the adjoint variable 𝑝=𝑝(𝑢^∗)=𝑝(𝑥,𝑡;𝑢^∗) as the solution of the equations: −𝜕𝑝𝑢^∗𝜕𝑡+𝒜^∗(𝑡)𝑝𝑢^∗+^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖𝑏_𝑖𝑥,𝑡+ℎ_𝑖𝑝𝑥,𝑡+ℎ_𝑖;𝑢^∗𝑑ℎ_𝑖=𝜆₁𝑦𝑢^∗−𝑧_𝑑,(𝑥,𝑡)∈Ω×(0,𝑇−Δ),ℎ_𝑖∈𝑎_𝑖,𝑏_𝑖,−𝜕𝑝𝑢^∗𝜕𝑡+𝒜^∗(𝑡)𝑝𝑢^∗=𝜆₁𝑦𝑢^∗−𝑧_𝑑,(𝑥,𝑡)∈Ω×(𝑇−Δ,𝑇),𝑝𝑥,𝑇;𝑢^∗=0,𝑥∈Ω,𝑝𝑥,𝑡;𝑢^∗=0,(𝑥,𝑡)∈Ω×[𝑇−Δ+𝜆,𝑇),𝜕𝑝𝑢^∗𝜕𝜈_𝒜∗(𝑥,𝑡)=^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠𝑐_𝑠𝑥,𝑡+𝑘_𝑠𝑝𝑥,𝑡+𝑘_𝑠;𝑢^∗𝑑𝑘_𝑠,(𝑥,𝑡)∈Γ×(0,𝑇−Δ(𝑇)),𝑘_𝑠∈𝑐_𝑠,𝑑_𝑠,𝜕𝑝𝑢^∗𝜕𝜈_𝒜∗(𝑥,𝑡)=0,(𝑥,𝑡)∈Γ×(𝑇−Δ(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 𝑧_𝑑∈𝐿²(𝑄) and any 𝑢∈𝐿²(𝑄), there exists a unique solution 𝑝(𝑢^∗)∈𝑊^∞,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: _𝑄𝑝𝑢^∗+𝜆₂𝑁𝑢^∗𝑢−𝑢^∗𝑑𝑥𝑑𝑡≥0,∀𝑢∈𝑈_𝑎𝑑.(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 𝑧_𝑑∈𝐿²(𝑄) and 𝜆₂>0, and with constraint (4.2), and with adjoint equations (4.28), there exists a unique optimal control 𝑢^∗ 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(𝑢)=𝜆₁_Σ𝑦∣_Σ(𝑥,𝑡;𝑢)−𝑧_Σ𝑑²𝑑Γ𝑑𝑡+𝜆₂_𝑄(𝑁𝑢)𝑢𝑑𝑥𝑑𝑡,(4.30) where 𝑧_Σ𝑑∈𝐿²(Σ).

From Theorem 3.4 and the Trace Theorem [20, Vol. 2, page 9], for each 𝑢∈𝐿²(𝑄), there exists a unique solution 𝑦(𝑢)∈𝑊^∞,1(𝑄) with 𝑦∣_Σ∈𝐿²(Σ). Thus, 𝐼 is well defined. Then, the optimal control 𝑢^∗ is characterized by: 𝜆₁_Σ𝑦𝑢^∗−𝑧_Σ𝑑𝑦(𝑢)−𝑦𝑢^∗𝑑Γ𝑑𝑡+𝜆₂_𝑄𝑁𝑢^∗𝑢−𝑢^∗𝑑𝑥𝑑𝑡≥0,∀𝑢∈𝑈_ad.(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(𝑢^∗)=𝑝(𝑥,𝑡;𝑢^∗)∈𝑊^∞,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 𝑧_Σ𝑑∈𝐿²(Σ) and any 𝑢∈𝐿²(𝑄), there exists a unique solution 𝑝(𝑢^∗)∈𝑊^∞,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: _𝑄𝑝𝑢^∗+𝜆₂𝑁𝑢^∗𝑢−𝑢^∗𝑑𝑥𝑑𝑡≥0,∀𝑢∈𝑈_𝑎𝑑.(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 𝑧_Σ𝑑∈𝐿²(Σ) and 𝜆₂>0, and with constraint (4.2), and with adjoint equations (4.24), there exists a unique optimal control 𝑢^∗ 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 2×2 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 2×2 coupled infinite order parabolic systems with multiple time delays). We can extend the discussions to study the optimal control for 2×2 coupled infinite order parabolic systems with multiple time delays. We consider the case where 𝑣=(𝑣₁,𝑣₂)∈𝐿²(Σ)×𝐿²(Σ), the performance functional is given by [15, 16]: 𝐼(𝑣)=𝐼₁(𝑣)+𝐼₂(𝑣)=²_𝑖=1𝜆₁_𝑄𝑦_𝑖(𝑥,𝑡;𝑣)−𝑧_𝑖𝑑²𝑑𝑥𝑑𝑡+𝜆₂_Σ𝑁_𝑖𝑣_𝑖𝑣_𝑖𝑑𝑥𝑑𝑡,(5.1) where 𝑧_𝑑=(𝑧_1𝑑,𝑧_2𝑑)∈(𝐿²(𝑄))².

The following results can now be proved.

Theorem 5.1. Let 𝑦₀, Φ₀,Ψ₀, 𝑣, and 𝑢 be given with 𝑦₀=(𝑦_0,1,𝑦_0,2)∈(𝑊^∞{𝛼_𝛼,2}(Ω))², Ψ₀=(Ψ_0,1,Ψ_0,2)∈(𝐿²(Σ₀))²,Φ₀=(Φ_0,1,Φ_0,2)∈(𝑊^∞,1(𝑄₀))²,𝑣=(𝑣₁,𝑣₂)∈(𝐿²(Σ))², and 𝑢=(𝑢₁,𝑢₂)∈(𝑊^−∞,−1(𝑄))². Then, there exists a unique solution 𝑦=(𝑦₁,𝑦₂)∈(𝑊^−∞,1(𝑄))² for the following mixed initial-boundary value problem: 𝜕𝑦₁𝜕𝑡+⎛⎜⎝^∞_|_𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼+1⎞⎟⎠𝑦₁+^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖𝑏_𝑖(𝑥,𝑡)𝑦₁𝑥,𝑡−ℎ_𝑖𝑑ℎ_𝑖−𝑦₂=𝑢₁,in𝑄,ℎ_𝑖∈𝑎_𝑖,𝑏_𝑖,𝜕𝑦₂𝜕𝑡+⎛⎜⎝^∞_|_𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼+1⎞⎟⎠𝑦₂+^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖𝑏_𝑖(𝑥,𝑡)𝑦₂𝑥,𝑡−ℎ_𝑖𝑑ℎ_𝑖+𝑦₁=𝑢₂,in𝑄,ℎ_𝑖∈𝑎_𝑖,𝑏_𝑖,𝑦₁𝑥,𝑡^;𝑢=Φ_0,1𝑥,𝑡^,𝑥,𝑡^∈Ω×[−Δ,0),𝑦₂𝑥,𝑡^;𝑢=Φ_0,2𝑥,𝑡^,𝑥,𝑡^∈Ω×[−Δ,0),𝑦₁(𝑥,0;𝑢)=𝑦_0,1,𝑥∈Ω,𝑦₂(𝑥,0;𝑢)=𝑦_0,2,𝑥∈Ω,𝜕𝑦₁𝜕𝜈_𝒜(𝑥,𝑡)=^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠𝑐_𝑠1(𝑥,𝑡)𝑦₁𝑥,𝑡−𝑘_𝑠𝑑𝑘_𝑠+𝑣₁,onΣ,𝑘_𝑠∈𝑐_𝑠,𝑑_𝑠,𝜕𝑦₂𝜕𝜈_𝒜=^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠𝑐_𝑠2(𝑥,𝑡)𝑦₂𝑥,𝑡−𝑘_𝑠𝑑𝑘_𝑠+𝑣₂,onΣ,𝑘_𝑠∈𝑐_𝑠,𝑑_𝑠,𝑦₁𝑥,𝑡^;𝑢=Ψ_0,1𝑥,𝑡^,𝑥,𝑡^∈Γ×[−Δ,0),𝑦₂𝑥,𝑡^;𝑢=Ψ_0,2𝑥,𝑡^,(𝑥,𝑡′)∈Γ×[−Δ,0),(5.2) where 𝑦≡𝑦(𝑥,𝑡;𝑢)=𝑦₁(𝑥,𝑡;𝑢),𝑦₂(𝑥,𝑡;𝑢)∈𝑊^∞,1(𝑄)²,𝑢≡𝑢(𝑥,𝑡)=𝑢₁(𝑥,𝑡),𝑢₂(𝑥,𝑡)∈𝑊^∞,1(𝑄)^²,𝑣≡𝑣(𝑥,𝑡)=𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡)∈𝐿²(Σ)².(5.3)

Lemma 5.2. Let the hypothesis of Theorem 5.1 be satisfied. Then for given 𝑧_𝑑=(𝑧_1𝑑,𝑧_2𝑑)∈(𝐿²(𝑄))² and any 𝑣=(𝑣₁,𝑣₂)∈(𝐿²(Σ))², there exists a unique solution 𝑝(𝑣)=(𝑝₁(𝑣),𝑝₂(𝑣))∈(𝑊^∞,1(𝑄))² for the adjoint problem: −𝜕𝑝₁(𝑣)𝜕𝑡+⎛⎜⎝^∞_|_𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼+1⎞⎟⎠𝑝₁(𝑣)+^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖𝑏_𝑖𝑥,𝑡+ℎ_𝑖𝑝₁𝑥,𝑡+ℎ_𝑖;𝑣𝑑ℎ_𝑖+𝑝₂(𝑣)=𝜆₁𝑦₁(𝑣)−𝑧_1𝑑,(𝑥,𝑡)∈Ω×(0,𝑇−Δ),ℎ_𝑖∈𝑎_𝑖,𝑏_𝑖,−𝜕𝑝₂(𝑣)𝜕𝑡+⎛⎜⎝^∞_|_𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼+1⎞⎟⎠𝑝₂(𝑣)+^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖𝑏_𝑖𝑥,𝑡+ℎ_𝑖𝑝₂𝑥,𝑡+ℎ_𝑖;𝑣𝑑ℎ_𝑖−𝑝₁(𝑣)=𝜆₁𝑦₂(𝑣)−𝑧_2𝑑,(𝑥,𝑡)∈Ω×(0,𝑇−Δ),ℎ_𝑖∈𝑎_𝑖,𝑏_𝑖,𝜕𝑝₁(𝑣)𝜕𝑡+⎛⎜⎝^∞_|_𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼+1⎞⎟⎠𝑝₁(𝑣)=𝜆₁𝑦₁(𝑣)−𝑧_1𝑑,(𝑥,𝑡)∈Ω×(𝑇−Δ,𝑇),𝜕𝑝₂(𝑣)𝜕𝑡+⎛⎜⎝^∞_|_𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼+1⎞⎟⎠𝑝₂(𝑣)=𝜆₁𝑦₂(𝑣)−𝑧_2𝑑,(𝑥,𝑡)∈Ω×(𝑇−Δ,𝑇),𝑝₁(𝑥,𝑇;𝑣)=0,𝑝₂(𝑥,𝑇;𝑣)=0,𝑥∈Ω,𝑝₁(𝑥,𝑡;𝑣)=0,𝑝₂(𝑥,𝑡;𝑣)=0,(𝑥,𝑡)∈Ω×[𝑇−Δ+𝜆,𝑇),𝜕𝑝₁(𝑥,𝑡;𝑣)𝜕𝜈_𝒜∗=^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠𝑐_𝑠1𝑥,𝑡+𝑘_𝑠𝑝₁𝑥,𝑡+𝑘_𝑠;𝑣𝑑𝑘_𝑠,(𝑥,𝑡)∈Γ×(0,𝑇−Δ),𝑘_𝑠∈𝑐_𝑠,𝑑_𝑠,𝜕𝑝₂(𝑥,𝑡;𝑣)𝜕𝜈_𝒜∗=^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠𝑐_𝑠2𝑥,𝑡+𝑘_𝑠𝑝₂𝑥,𝑡+𝑘_𝑠;𝑣𝑑𝑘_𝑠,(𝑥,𝑡)∈Γ×(0,𝑇−Δ),𝑘_𝑠∈𝑐_𝑠,𝑑_𝑠,𝜕𝑝₁(𝑥,𝑡)𝜕𝜈_𝒜∗=0,𝜕𝑝₂(𝑥,𝑡)𝜕𝜈_𝒜∗=0,(𝑥,𝑡)∈Γ×(𝑇−Δ(𝑇),𝑇).(5.4)

Theorem 5.3. The optimal control 𝑣^∗≡𝑣^∗(𝑥,𝑡)=(𝑣^∗₁(𝑥,𝑡),𝑣^∗₂(𝑥,𝑡))∈(𝐿²(Σ))² is characterized by the following maximum condition: ^𝑇₀_Γ𝑝₁𝑣^∗+𝜆₂𝑁₁𝑣^∗₁𝑣₁−𝑣^∗₁+𝑝₂𝑣^∗+𝜆₂𝑁₂𝑣^∗₂𝑣₂−𝑣^∗₂𝑑Γ𝑑𝑡≥0,∀𝑣=𝑣₁,𝑣₂∈𝐿²(Σ)²,(5.5) where 𝑝≡𝑝(𝑥,𝑡;𝑣)=(𝑝₁(𝑥,𝑡;𝑣),𝑝₂(𝑥,𝑡;𝑣))∈(𝑊^∞,1(𝑄))² is the adjoint state.

The foregoing result is now summarized.

Theorem 5.4. For the problem (5.2) with the performance function (5.1) with 𝑧_𝑑=(𝑧_1𝑑,𝑧_2𝑑)∈(𝐿²(𝑄))² and 𝜆₂>0, and with constraint: 𝑈_𝑎𝑑 is closed, convex subset of (𝐿²(Σ))², and with adjoint problem (5.4), then there exists a unique optimal control 𝑣^∗≡𝑣^∗(𝑥,𝑡)=(𝑣^∗₁(𝑥,𝑡),𝑣^∗₂(𝑥,𝑡))∈(𝐿²(Σ))² which satisfies the maximum condition (5.5).

Case 2 (optimal control for (𝑛×𝑛) coupled infinite order parabolic systems with multiple time delays). We will extend the discussion to (𝑛×𝑛) coupled infinite order parabolic systems. We consider the case where 𝑣=(𝑣₁,𝑣₂,…,𝑣_𝑛)∈((𝐿²(Σ))^𝑛, the performance functional is given by [15, 16]: 𝐼(𝑣)=^𝑛_𝑗=1𝜆₁_𝑄𝑦_𝑗(𝑥,𝑡;𝑣)−𝑧_𝑗𝑑²𝑑𝑥𝑑𝑡+𝜆₂_Σ𝑁_𝑗𝑣_𝑗𝑣_𝑗𝑑𝑥𝑑𝑡,(5.6) where 𝑧_𝑑=(𝑧_1𝑑,𝑧_2𝑑,…,𝑧_𝑛𝑑)∈(𝐿²(𝑄))^𝑛.

The following results can now be proved.

Theorem 5.5. Let 𝑦₀, Φ₀, Ψ₀, 𝑣, and 𝑢 be given with 𝑦_𝑝=(𝑦_𝑝,1,𝑦_𝑝,2,…,𝑦_𝑝,𝑛)∈(𝑊^∞{𝑎_𝛼,2}(Ω))^𝑛, Φ₀=(Φ_0,1,Φ_0,2,…,Φ_0,𝑛)∈(𝑊^∞,1(𝑄₀))^𝑛, Ψ₀=(Ψ_0,1,Ψ_0,2,…,Ψ_0,𝑛)∈(𝐿²(Σ₀))^𝑛, 𝑣=(𝑣₁,𝑣₂,…,𝑣_𝑛)∈(𝐿²(Σ₀))^𝑛, and 𝑢=(𝑢₁,𝑢₂,…,𝑢_𝑛)∈(𝑊^−∞,−1(𝑄))^𝑛. Then, there exists a unique solution 𝑦=(𝑦₁,𝑦₂,…,𝑦_𝑛)∈(𝑊^∞,1(𝑄₀))^𝑛 for the following mixed initial-boundary value problem: forall𝑗,𝑗=1,2,…,𝑛 one has 𝜕𝑦_𝑗𝜕𝑡+𝒮(𝑡)𝑦_𝑗(𝑥,𝑡)+^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖𝑏_𝑖(𝑥,𝑡)𝑦_𝑗𝑥,𝑡−ℎ_𝑖𝑑ℎ_𝑖=𝑢_𝑗,(𝑥,𝑡)∈Ω×(0,𝑇),ℎ_𝑖∈𝑎_𝑖,𝑏_𝑖,𝑦_𝑗(𝑥,𝑡′)=Φ_0,𝑗𝑥,𝑡^𝑥,𝑡^∈Ω×[−Δ,0),𝑦_𝑗(𝑥,0)=𝑦_0,𝑗(𝑥),𝑥∈Ω,𝜕𝑦_𝑗𝜕𝜈_𝒮(𝑥,𝑡)=^𝑙_𝒮₌₁^𝑑𝑠_𝑐𝑠𝑐_𝑠(𝑥,𝑡)𝑦_𝑗𝑥,𝑡−𝑘_𝑠𝑑𝑘_𝑠+𝑣_𝑗,(𝑥,𝑡)∈Γ×(0,𝑇),𝑘_𝑠∈𝑐_𝑠,𝑑_𝑠,𝑦_𝑗𝑥,𝑡^=Ψ_0,𝑗𝑥,𝑡^,𝑥,𝑡^∈Γ×[−Δ,0),(5.7) where 𝑦≡𝑦(𝑥,𝑡;𝑢)=𝑦₁(𝑥,𝑡;𝑢),𝑦₂(𝑥,𝑡;𝑢),…,𝑦_𝑛(𝑥,𝑡;𝑢)∈𝑊^∞,1(𝑄)^𝑛,𝑢≡𝑢(𝑥,𝑡)=𝑢₁(𝑥,𝑡),𝑢₂(𝑥,𝑡),…,𝑢_𝑛(𝑥,𝑡)∈𝑊^−∞,−1(𝑄)^𝑛,𝑣≡𝑣(𝑥,𝑡)=𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡),…,𝑣_𝑛(𝑥,𝑡)∈𝐿²(Σ)^𝑛.(5.8) The operator 𝒮(𝑡) is an 𝑛×𝑛 matrix takes the form [15, 16, 18, 22]: 𝒮(𝑡)=⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝^∞_|𝛼∑_|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼+1−1⋅⋅−11∑^∞_|𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼+1⋅⋅−1⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅11⋅⋅∑^∞_|𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼+1⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠_𝑛×𝑛,(5.9) that is 𝒮(𝑡)𝑦_𝑗(𝑥)=^∞_|𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼𝑦_𝑗(𝑥)+^𝑛_𝑟=1𝐵_𝑗𝑟𝑦_𝑗(𝑥),∀𝑗,𝑗=1,2,…,𝑛,(5.10) where 𝐵_𝑗𝑟=1,if𝑗≥𝑟,−1,if𝑗<𝑟.(5.11)

Lemma 5.6. Let the hypothesis of Theorem 5.5 be satisfied. Then for given 𝑧_𝑑=(𝑧_1𝑑,𝑧_2𝑑,…,𝑧_𝑛𝑑)∈(𝐿²(𝑄))^𝑛 and any 𝑣(𝑥,𝑡)=(𝑣₁(𝑥,𝑡),𝑣₂(𝑥,𝑡),…,𝑣_𝑛(𝑥,𝑡))∈(𝐿²(Σ))^𝑛, there exists a unique solution 𝑝(𝑣)≡𝑝(𝑥,𝑡;𝑣)=𝑝₁(𝑥,𝑡;𝑣),𝑝₁(𝑥,𝑡;𝑣),…,𝑝_𝑛(𝑥,𝑡;𝑣)∈𝑊^∞,1(𝑄)^𝑛,(5.12) for the adjoint problem: forall𝑗,𝑗=1,2,…,𝑛, one has −𝜕𝑝_𝑗(𝑣)𝜕𝑡+𝒮^∗(𝑡)𝑝_𝑗(𝑣)+^𝑚_𝑖=1^𝑏𝑖_𝑎𝑖𝑏_𝑖𝑥,𝑡+ℎ_𝑖𝑝_𝑗𝑥,𝑡+ℎ_𝑖;𝑣𝑑ℎ_𝑖=𝜆₁𝑦_𝑗(𝑣)−𝑧_𝑗𝑑,(𝑥,𝑡)∈Ω×(0,𝑇−Δ),ℎ_𝑖∈𝑎_𝑖,𝑏_𝑖,𝜕𝑝_𝑗(𝑣)𝜕𝑡+𝒮^∗(𝑡)𝑝_𝑗(𝑣)=𝜆_𝑗𝑦_𝑗(𝑣)−𝑧_𝑗𝑑,(𝑥,𝑡)∈Ω×(𝑇−Δ,𝑇),𝑝_𝑗(𝑥,𝑇,𝑣)=0,𝑥∈Ω,𝑝(𝑥,𝑡;𝑣)=0,(𝑥,𝑡)∈Ω×[𝑇−Δ+𝜆,𝑇),𝜕𝑝_𝑗(𝑣)𝜕𝜈_𝒮∗(𝑥,𝑡)=^𝑙_𝑠=1^𝑑𝑠_𝑐𝑠𝑐_𝑠𝑥,𝑡+𝑘_𝑠𝑝_𝑗𝑥,𝑡+𝑘_𝑠;𝑣𝑑𝑘_𝑠,(𝑥,𝑡)∈Γ×(0,𝑇−Δ),𝑘_𝑠∈𝑐_𝑠,𝑑_𝑠,𝜕𝑝_𝑗(𝑣)𝜕𝜈_𝒮∗(𝑥,𝑡)=0,(𝑥,𝑡)∈Γ×(𝑇−Δ,𝑇),(5.13) where 𝒮^∗(𝑡)𝑝_𝑗(𝑥)=^∞_|𝛼|=0(−1)^|𝛼|𝑎_𝛼𝐷^2𝛼𝑝_𝑗(𝑥)+^𝑛_𝑟=1𝐵_𝑟𝑗𝑝_𝑗(𝑥),∀𝑗,𝑗=1,2,…,𝑛,(5.14)𝐵_𝑟𝑗 are the transpose of 𝐵_𝑗𝑟.

Theorem 5.7. The optimal control 𝑣^∗≡𝑣^∗(𝑥,𝑡)=(𝑣^∗₁(𝑥,𝑡),𝑣^∗₂(𝑥,𝑡),…,𝑣^∗_𝑛(𝑥,𝑡))∈(𝐿²(Σ))^𝑛 is characterized by the following maximum condition: ^𝑛_𝑗=1_Σ𝑝_𝑗𝑣^∗+𝜆₂𝑁_𝑗𝑣^∗_𝑗𝑣_𝑗−𝑣^∗_𝑗𝑑Γ𝑑𝑡≥0,∀𝑣=𝑣₁,𝑣₂,…,𝑣_𝑛∈𝑈_𝑎𝑑^𝑛,(5.15) where 𝑝𝑣^∗≡𝑝𝑥,𝑡;𝑣^∗=𝑝₁𝑥,𝑡;𝑣^∗,𝑝₁𝑥,𝑡;𝑣^∗,…,𝑝_𝑛𝑥,𝑡;𝑣^∗∈𝑊^∞,1(𝑄)^𝑛(5.16) is the adjoint state.

The foregoing result is now summarized.

Theorem 5.8. For the problem (5.7) with the performance function (5.6) with 𝑧_𝑑=(𝑧_1𝑑,𝑧_2𝑑,…,𝑧_𝑛𝑑)∈(𝐿²(𝑄))^𝑛 and 𝜆₂>0, and with constraint: 𝑈_𝑎𝑑 is closed, convex subset of (𝐿²(Σ))^𝑛, and with adjoint equations (5.13), then there exists a unique optimal control 𝑣^∗≡𝑣^∗(𝑥,𝑡)=(𝑣^∗₁(𝑥,𝑡),𝑣^∗₂(𝑥,𝑡),…,𝑣^∗_𝑛(𝑥,𝑡))∈(𝐿²(Σ))^𝑛 which satisfies the maximum condition (5.15).

In the case of performance functionals (4.1), (4.22), (4.26), (4.30), (5.1), and (5.6) with 𝜆₁>0 and 𝜆₂=0, the optimal control problem reduces to minimization of the functional on a closed and convex subset in a Hilbert space. Then, the optimization problem is equivalent to a quadratic programming one, which can be solved by the use of the well-known Gilbert algorithm.

6. Conclusions

The optimization problem presented in the paper constitutes a generalization of the optimal boundary control problem for second-order parabolic systems with Neumann boundary condition involving constant time lag appearing in the state and in the boundary conditions considered in [1, 5–9, 14–16, 18, 21, 32].

Moreover, the results obtained in this paper (Theorems 4.2, 4.6, 5.4, and 5.8) can be treated as a generalization of the optimization theorems proved by [8–10]. Also the main result of the paper contains necessary and sufficient conditions of optimality for (𝑛×𝑛) infinite order parabolic systems with multiple time delays given in integral form both in the state equation and in the Neumann boundary condition that give characterization of optimal control (Theorem 5.8). But it is easily seen that obtaining analytical formulas for optimal control are very difficult. This results from the fact that state equations (5.7), adjoint equations (5.13), and maximum condition (5.15) are mutually connected that cause that the usage of derived conditions is difficult. Therefore we must resign from the exact determination of the optimal control and therefore we are forced to use approximation methods.

Also it is evident that by modifying:(i)the boundary conditions, (Dirichlet, Neumann, mixed, etc.),(ii)the nature of the control (distributed, boundary, etc.),(iii)the nature of the observation (distributed, boundary, etc.),(iv)the initial differential system,(v)the time delays (constant time delays, time-varying delays, multiple time-varying delays, time delays given in the integral form, etc.),(vi)the number of variables (finite number of variables, infinite number of variables systems, etc.),(vii)the type of equation (elliptic, parabolic, hyperbolic, etc.),(viii)the order of equation (second order, Schrödinger, infinite order, etc.),(ix)the type of control ( optimal control problem, time-optimal control problem, etc.),

an infinity of variations on the above problems are possible to study with the help of [19] and Dubovitskii-Milyutin formalisms [22–27]. Those problems need further investigations and form tasks for future research. These ideas mentioned above will be developed in forthcoming papers.

Acknowledgments

The research presented here was carried out within the research programme of the Taibah University-Dean of Scientific Research under Project no. 806/1433. The author would like to express his gratitude to the anonymous reviewers for their very valuable remarks.