On an Invariant Set in the Heat Conductivity Problem with Time Lag

Abstract

The problems of weak and strong invariance of a constant multivalued mapping with respect to the heat conductivity equation with time lag are studied. Sufficient conditions of weak and strong invariance of a given multivalued mapping are obtained.

1. Introduction

1.1. Related Works

There are many theoretical and practical problems in control problems with distributed parameters where known methods do not work to solve them. The typical examples of such problems are conservation of temperature of a volume within admissible bounds and deviation from undesirable states.

Note that the works [1–7] were devoted to the problems of invariance of given sets for the controlled systems. In these works, some results on construction of core of liveness, the maximal weak invariant subset of the given set was obtained for the control system.

In the paper [3], a family of trajectories, which is kept to be within the given set until a certain time (viability), is analytically described for control systems given by differential inclusions. The paper [8] deals with the problem of diffusion process control by boundary control.

However, all the works mentioned above relate to control systems with concentrated parameters. In the papers [9, 10], weak and strong invariance of the given set with respect to a system with distributed parameters were studied. Alimov [11], Albeverio and Alimov [12] studied interesting applied control problems on heat distribution by convectors in a volume.

The present paper deals with the problems of weak and strong invariance of given multivalued mapping for the 3rd heat conductivity boundary value problem with time lag. In the equation of this problem, the control parameter appears on the right hand side. We obtain conditions which can be easily checked to determine the invariance of the given constant multivalued mapping.

1.2. Preliminaries

First of all, we recall some definitions. A bounded region Ω(⊂ℝ^𝑛) is referred to as the region with piecewise smooth boundary if its boundary Γ=Ω⧵Ω can be represented as follows: Γ=∑^𝑁_𝑗=1Γ_𝑗, where Γ_𝑗⊂Γ is an open set with respect to the topology on Γ induced by the topology in ℝ^𝑛. Moreover, each Γ_𝑗 is a connected surface of class 𝐶¹; that is, for any point 𝑥₀∈Γ_𝑗 there exists a ball 𝑈_𝜖(𝑥₀) of radius 𝜖>0 such that the piece Γ_𝑗∩𝑈_𝜖(𝑥₀) of the surface is given by the equation of the form 𝑥_𝑘=𝑓_𝑘(𝑥₁,…,𝑥_𝑘−1,𝑥_𝑘+1,…,𝑥_𝑛), where 𝑓_𝑘(⋅)∈𝐶¹ and 1≤𝑘≤𝑛.

Let Ω be a region in ℝ^𝑛 with piecewise smooth boundary. We will use the letter 𝐴 to denote the following differential operator [13, 14]: 𝐴(𝑥,𝐷)𝜑=−^𝑛_𝑖,𝑗=1𝜕𝜕𝑥_𝑖𝑎_𝑖𝑗(𝑥)𝜕𝜑𝜕𝑥_𝑗, (1) where the functions 𝑎_𝑖𝑗(𝑥)∈𝐶¹(Ω)∩𝐶(Ω) satisfy the conditions 𝑎_𝑖𝑗(𝑥)=𝑎_𝑗𝑖(𝑥),𝑥∈Ω, and ^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗(𝑥)𝜉_𝑖𝜉_𝑗≥𝛾^𝑛_𝑖=1𝜉²_𝑖, (2) for all (𝜉₁,𝜉₂,…,𝜉_𝑛)∈𝑅^𝑛. The inequality (2) is called the condition of uniform ellipticity of the operator 𝐴 defined by (1). The domain of the operator 𝐴 is 𝐶²(Ω), which is the space of functions that are twice continuously differentiable in Ω and continuous on Ω∪𝜕Ω.

Define the operator 𝑃 by the equation 𝑃𝜑=^𝑛_𝑖,𝑗=1𝑎_𝑖𝑗(𝑥)𝜕𝜑𝜕𝑥_𝑗cos𝑙,𝑥_𝑖+𝑘(𝑥)𝜑,𝑥∈𝜕Ω, (3) where 𝑙 denotes the upward unit normal vector on 𝜕Ω and 𝑘(𝑥) is a given positive continuous function defined on 𝜕Ω.

Definition 1. The number 𝜆 for which the following boundary value problem: 𝐴(𝑥,𝐷)𝜑=𝜆𝜑,𝑥∈Ω,𝑃𝜑=0,𝑥∈𝜕Ω, (4) has nonzero solution 𝜑_𝜆(𝑥), 𝑥∈Ω is called eigenvalue of this boundary value problem, and the solution 𝜑_𝜆(𝑥) is called eigenfunction of the boundary value problem.

Since problem (4) is homogeneous, we assume that _Ω𝜑²_𝜆𝑑𝑥=1. (5) As the operator (1) is self-adjoint, then it has a discrete spectrum [13, 14]; that is,(i)there exist a countable set of eigenvalues 𝜆_𝑘 of the problem (4) such that 0<𝜆₁≤𝜆₂≤⋯≤𝜆_𝑛≤⋯,𝜆_𝑛⟶∞,𝑛⟶∞, (6)(ii)for each eigenvalue 𝜆_𝑘, there are a finite number of eigenfunctions 𝜑_𝑘 corresponding to it such that ∫_Ω𝜑_𝑖(𝑥)𝜑_𝑗(𝑥)𝑑𝑥=𝛿_𝑖𝑗, where 𝛿_𝑖𝑗=1,if𝑖=𝑗,0,if𝑖≠𝑗, (7)(iii)the set of all eigenvalues {𝜑_𝑘} is complete (closed) in the space 𝐿₂(Ω); that is, any function 𝑓 from the space 𝐿₂(Ω) can be uniquely represented in the form 𝑓(𝑥)=^∞_𝑘=1𝑓_𝑘𝜑_𝑘(𝑥), (8) where the equality is understood in the following sense: _Ω𝑓(𝑥)−^𝑁_𝑘=1𝑓_𝑘𝜑_𝑘(𝑥)²𝑑𝑥⟶0,𝑁⟶∞. (9)

The Fourier coefficients 𝑓_𝑘 in the Fourier series (8) of 𝑓(𝑥) are defined by formula 𝑓_𝑘=_Ω𝑓(𝑥)𝜑_𝑘(𝑥)𝑑𝑥. (10) Since the operator 𝐴 is self-adjoint, in certain conditions on the functions 𝑎_𝑖𝑗, 𝜑, and 𝜓, the Green formula is written in the form _Ω()()𝜓=𝜓𝐴𝑥,𝐷𝜑−𝜑𝐴𝑥,𝐷𝑑𝑥_𝜕Ω𝜓𝑃𝜑−𝜑𝑃𝜓𝑑𝑆. (11) Using these, we construct the following spaces depending on a parameter. Let 𝑟 be any nonnegative number. Denote ℓ_𝑟=𝛼=𝛼₁,𝛼₂,…∣^∞_𝑖=1𝜆^𝑟_𝑖𝛼²_𝑖,𝐻<∞_𝑟(Ω)=𝑓∈𝐿₂(Ω)∣𝑓=^∞_𝑖=1𝛼_𝑖𝜑_𝑖,𝛼∈ℓ_𝑟. (12)

We now define inner product and norm in the spaces ℓ_𝑟,𝐻_𝑟=𝐻_𝑟(Ω). Let 𝛼,𝛽∈ℓ_𝑟,𝑓=^∞_𝑖=1𝛼_𝑖𝜑_𝑖,𝑔=^∞_𝑖=1𝛽_𝑖𝜑_𝑖. (13) Set 𝛼,𝛽_𝑟=^∞_𝑖=1𝜆^𝑟_𝑖𝛼_𝑖𝛽_𝑖,‖𝛼‖=(𝛼,𝛼)_𝑟^1/2,𝑓,𝑔_𝑟=𝛼,𝛽_𝑟,𝑓=‖𝛼‖. (14) It should be noted that 𝐻₀(Ω)=𝐿₂(Ω) and 𝐻_𝑟(Ω)⊂𝐻_𝑠(Ω) for all 0≤𝑠≤𝑟.

Denote by 𝐶(0,𝑇;𝐻_𝑟(Ω)) and 𝐿₂(0,𝑇;𝐻_𝑟(Ω)) the spaces of continuous functions and summable quadratically measurable functions defined on [0,𝑇] with the values in 𝐻_𝑟(Ω), respectively, where 𝑇 is a positive number.

1.3. Auxiliary Statements

Consider the following heat exchange control problem with lag: 𝜕𝑧(𝑥,𝑡)()𝑧()𝜕𝑡+𝐴𝑥,𝐷𝑥,𝑡=𝑧₀(𝑥,𝑡−ℎ)+𝑢(𝑥,𝑡),0<𝑡≤𝑇, (15) with boundary 𝑃𝑧(𝑥,𝑡)=0,0≤𝑡≤𝑇,𝑥∈𝜕Ω, (16) and initial conditions 𝑧(𝑥,0)=𝑧₀(𝑥,0), (17) where 𝑧₀(𝑥,⋅)∈𝑋; 𝑋={𝑧(𝑥,⋅)∣𝑧(𝑥,𝑡)∈𝐻_𝑟,−ℎ≤𝑡≤0}. Here, 𝑧(⋅)=𝑧(𝑥,⋅) and 𝑢(⋅)=𝑢(𝑥,⋅) are abstract functions whose values at each 𝑡>0 are unique elements of the space 𝐻_𝑟(Ω); ℎ is a positive fixed number and 𝑇 is a positive number.

Further, we use the same letter 𝐴 to denote the bounded operator that maps 𝐻_𝑟 into 𝐻_𝑟+2 and is defined by the following formula 𝐴𝜑=𝐴^∞_𝑘=1𝑐_𝑘𝜑_𝑘=^∞_𝑘=1𝜆_𝑘𝑐_𝑘𝜑_𝑘, (18) where 𝜑=∑^∞_𝑘=1𝑐_𝑘𝜑_𝑘. Clearly, we obtain from this that ‖𝐴𝑓‖_𝐻𝑟=‖𝑓‖_𝐻𝑟+2.

The problem (15)–(17) is understood in the sense of the theory of generalized functions (the theory of distributions) with the values in 𝐻_𝑟. We will look for the solution of the problem (15)–(17), which is continuous with respect to 𝑡 and its values belong to one of the spaces 𝐻_𝑟, and the initial and boundary conditions are considered as equality of elements of these spaces.

In the control problem (15)–(17), the control function 𝑢(⋅) is subjected to either constraint ‖𝑢(𝑡)‖_𝐻𝑟≤𝜌,0≤𝑡≤𝑇, (19) or 𝑢(⋅)∈𝐿₂[0,ℎ];𝐻_𝑟,‖𝑢(⋅)‖_{𝐿2([0,ℎ];𝐻𝑟)}≤𝜌. (20) Accordingly, in this paper, we consider both of the problems (15)–(19) and (15)–(17), (20). The control function 𝑢(⋅) that satisfies either (19) or (20) is called admissible.

Assume that the problem (15)–(17) has a solution 𝑧(𝑡), 0≤𝑡≤ℎ, at some admissible control 𝑢(⋅). Then we have 𝑧(𝑡)=^∞_𝑘=1𝑧_𝑘(𝑡)𝜑_𝑘(𝑥);𝑧_𝑘(𝑡)=_Ω𝑧(𝑡)𝜑_𝑘(𝑥)𝑑𝑥. (21) It follows from (15) that 𝜕𝑧(_Ω𝑡)(𝑥)𝜕𝑡+𝐴𝑧−𝑧₀(𝑡−ℎ)−𝑢(𝑡)𝜑_𝑘(𝑥)𝑑𝑥≡0,𝑘=1,2,…. (22) Also, by definition of the functions 𝑧(𝑡), 𝜑_𝑘(𝑥) we obtain 𝑃𝑧(𝑡)=0,0≤𝑡≤ℎ,𝑥∈𝜕Ω,𝑃𝜑_𝑘=0,𝐴𝜑_𝑘=𝜆_𝑘𝜑_𝑘. (23) Therefore, in view of (11) we have _Ω𝜑_𝑘(𝑥)𝐴𝑧(𝑡)𝑑𝑥=𝜆_𝑘𝑧_𝑘(𝑡). (24) Denoting 𝑧_0𝑘(𝑡−ℎ)=_Ω𝑧₀(𝑡−ℎ)𝜑_𝑘(𝑥)𝑢𝑑𝑥,_𝑘(𝑡)=_Ω𝑢(𝑡)𝜑_𝑘(𝑥)𝑑𝑥 (25) and using the fact that 𝜕𝑧(_Ω𝑡)𝜑𝜕𝑡^𝑘(𝑥)𝑑𝑥=𝑑𝑧_𝑘(𝑡),𝑑𝑡 (26) we deduce from formulas (22) and (24) that 𝑧̇_𝑘(𝑡)+𝜆_𝑘𝑧_𝑘(𝑡)=𝑧_0𝑘(𝑡−ℎ)+𝑢_𝑘(𝑡),0≤𝑡≤ℎ,𝑘=1,2,…. (27) Since the function (21) must satisfy the initial condition 𝑧(0)=𝑧₀(0), we have 𝑧_𝑘(0)=𝑧⁰_0𝑘=_Ω𝑧₀(0)𝜑_𝑘(𝑥)𝑑𝑥,𝑘=1,2,…. (28) From this and (27) we see that 𝑧_𝑘(𝑡)=𝑒^{−𝜆𝑘𝑡}𝑧⁰_0𝑘+^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑧_0𝑘(𝜏−ℎ)+𝑢_𝑘(𝜏)𝑑𝜏,𝑘=1,2,…. (29) Thus, (21) and (29) define the formal solution of the problem (15)–(17) on [0,ℎ] which can be written as follows: 𝑧(𝑡)=^∞_𝑘=1𝑒^{−𝜆𝑘𝑡}𝑧⁰_0𝑘+^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑧_0𝑘(𝜏−ℎ)+𝑢_𝑘(𝜏)𝑑𝜏×𝜑_𝑘(𝑥). (30)

The following assumption will be needed throughout the paper.

Assumption 2. Let functions 𝑧(⋅) of the set 𝑋 satisfy the following conditions: ^∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑧_0𝑘(𝜏−ℎ)²𝑑𝜏≤𝑏²,_∞𝑘=1𝜆^𝑟_𝑘𝑧_0𝑘(0)²≤𝑏²,_∞𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑧_0𝑘(𝜏−ℎ)²𝑑𝜏≤𝑏^2𝑡₀𝑒^{−𝜆1(𝑡−𝜏)}𝑑𝜏,0≤𝑡≤ℎ. (31)(32)

Proposition 3. For any function 𝑓(𝑡),‖𝑓(𝑡)‖_𝐻𝑟≤𝑀, 0≤𝑡≤𝑇, the following inequality holds: ^∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑓_𝑘(𝜏)𝑑𝜏²≤𝑎²₁(𝑡)𝑀²;𝑎₁(𝑡)=1−𝑒^−𝜆1𝑡𝜆₁,0≤𝑡≤𝑇, (33) where 𝜆₁,𝜆₂,… are the eigenvalues of the operator 𝐴 arranged in decreasing order, 𝑀 is a positive number, and 𝑓_𝑘(⋅) is the Fourier coefficient of the function 𝑓(𝑡).

Proof. Indeed, for 0≤𝑡≤𝑇, using the Cauchy-Schwartz inequality and the fact that 𝑓(𝜏)²_𝐻𝑟=^∞_𝑘=1𝜆^𝑟_𝑘𝑓²_𝑘(𝜏)≤𝑀²,0≤𝜏≤𝑡≤𝑇, (34) we obtain ^∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑓_𝑘(𝜏)𝑑𝜏²=^∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−(𝜆𝑘/2)(𝑡−𝜏)}⋅𝑒^{−(𝜆𝑘/2)(𝑡−𝜏)}𝑓_𝑘(𝜏)𝑑𝜏²≤^∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−𝜆1(𝑡−𝜏)}𝑑𝜏^𝑡₀𝑒^{−𝜆1(𝑡−𝜏)}𝑓²_𝑘(𝜏)≤𝑑𝜏_𝑡0𝑒^{−𝜆1(𝑡−𝜏)}𝑑𝜏^𝑡₀𝑒^{−𝜆1(𝑡−𝜏)}⋅^∞_𝑘=1𝜆^𝑟_𝑘𝑓²_𝑘(𝜏)𝑑𝜏≤𝑎₁(𝑡)^𝑡₀𝑒^{−𝜆1(𝑡−𝜏)}𝑀²𝑑𝜏=𝑎²₁(𝑡)𝑀². (35)

Theorem 4. Let 𝑢(𝑡), 0≤𝑡≤ℎ be an admissible control that satisfies the condition (19), and let Assumption 2 be satisfied. Then there exists a unique solution of the problem (15)–(17) in the space 𝐶([0,ℎ];𝐻_𝑟(Ω)).

Proof. To prove the theorem we use the formal representation (30) of the solution of the problem (15)–(17). By definition of the norm in 𝐻_𝑟(Ω) for 0≤𝑡≤ℎ, we have ‖𝑧(𝑡)‖²_𝐻𝑟=^∞_𝑘=1𝜆^𝑟_𝑘𝑧_𝑘(𝑡)²=^∞_𝑘=1𝜆^𝑟_𝑘𝑒^{−𝜆𝑘𝑡}𝑧⁰_0𝑘+^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑓_𝑘(𝜏)𝑑𝜏², (36) where 𝑓_𝑘(𝜏)=𝑧_0𝑘(𝜏−ℎ)+𝑢_𝑘(𝜏).
We use the Cauchy-Schwartz inequality to obtain the following chain of relations: ^∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑓_𝑘(𝜏)𝑑𝜏²≤^∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−(𝜆𝑘(𝑡−𝜏))/2}𝑒^{−(𝜆𝑘(𝑡−𝜏))/2}𝑓_𝑘(𝜏)𝑑𝜏²≤^∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑑𝜏^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑓²_𝑘(𝜏)≤𝑑𝜏_∞𝑘=1𝜆^𝑟_𝑘1−𝑒^{−𝜆𝑘𝑡}𝜆_𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑓²_𝑘(𝜏)𝑑𝜏. (37) Therefore, ^∞_𝑘=1𝜆^𝑟_𝑘1−𝑒^{−𝜆𝑘𝑡}𝜆_𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑓²_𝑘(𝜏)𝑑𝜏≤𝑎₁(𝑡)^∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑧_0𝑘(𝜏−ℎ)+𝑢_𝑘(𝜏)²𝑑𝜏. (38) Next, we expand the integrand on the right hand side of the last inequality; then we use the Cauchy-Schwartz inequality and after that we use Assumption 2 to obtain ^∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑓_𝑘(𝜏)𝑑𝜏²≤𝑏+𝜌²𝑎²₁(𝑡). (39) Expanding the square in (36), and then using (39) and Assumption 2, we arrive at ‖𝑧(𝑡)‖²_𝐻𝑟≤𝑒^−𝜆1𝑡𝑏+𝑏+𝜌𝑎₁(𝑡)². (40) Hence, 𝑧(𝑡)∈𝐻_𝑟(Ω) for each 𝑡∈[0,ℎ].
We now show that the function 𝑧(𝑡), 0≤𝑡≤ℎ, is continuous with respect to the norm of the space 𝐻_𝑟. We have ‖𝑧(𝑡+𝛿)−𝑧(𝑡)‖²_𝐻𝑟=^∞_𝑘=1𝜆^𝑟_𝑘𝑧_𝑘(𝑡+𝛿)−𝑧_𝑘(𝑡)²=^𝑁_𝑘=1𝜆^𝑟_𝑘𝑧_𝑘(𝑡+𝛿)−𝑧_𝑘(𝑡)²+^∞_𝑘=𝑁+1𝜆^𝑟_𝑘𝑧_𝑘(𝑡+𝛿)−𝑧_𝑘(𝑡)², (41) where the numbers 𝛿 and 𝑁 will be chosen depending on any 𝜀>0.
By (29), We have 𝑧_𝑘(𝑡+𝛿)−𝑧_𝑘(𝑡)=𝑧⁰_0𝑘𝑒^{−𝜆𝑘𝑡}𝑒^{−𝜆𝑘𝛿}+−1_𝑡0𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑒^{−𝜆𝑘𝛿}−1𝑓_𝑘(𝜏)+𝑑𝜏_{𝑡𝑡+𝛿}𝑒^{−𝜆𝑘(𝑡+𝛿−𝜏)}𝑓_𝑘(𝜏)𝑑𝜏. (42) Combining this with (41), we obtain ‖𝑧(𝑡+𝛿)−𝑧(𝑡)‖²_𝐻𝑟≤𝐼₁+𝐼₂+𝐼₃;𝐼₁=3^𝑁_𝑘=1𝜆^𝑟_𝑘𝑒^{−𝜆𝑘𝛿}−1²𝑧⁰_0𝑘²+^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑓_𝑘(𝜏)𝑑𝜏²𝐼₂=3^∞_𝑘=𝑁+1𝜆^𝑟_𝑘𝑧⁰_0𝑘²+^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑓_𝑘(𝜏)𝑑𝜏²𝐼₃=3^∞_𝑘=1𝜆^𝑟_𝑘𝑡^𝑡+𝛿𝑒^{−𝜆𝑘(𝑡+𝛿−𝜏)}𝑓_𝑘(𝜏)𝑑𝜏². (43) Let 𝜀 be an arbitrary positive number. Since the series in (31) and (39) are convergent, one can choose the number 𝑁 so that 𝐼₂<𝜀²/3. Then, we choose 𝛿 so that 𝐼₁<𝜀²/3. Further, similar to (39) we have ^∞_𝑘=1𝜆^𝑟_𝑘𝑡^𝑡+𝛿𝑒^{−𝜆𝑘(𝑡+𝛿−𝜏)}𝑓_𝑘(𝜏)𝑑𝜏²≤𝑒^{−2𝜆1𝛿∞}_𝑘=1𝜆^𝑟_𝑘𝑡^𝑡+𝛿𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑓_𝑘(𝜏)𝑑𝜏²≤𝑒^{−2𝜆1𝛿}𝑏+𝜌_𝑡^𝑡+𝛿𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑑𝜏². (44) It is clear from these inequalities that we can choose 𝛿 so that 𝐼₃<𝜀²/3. From the estimations for 𝐼₁,𝐼₂,𝐼₃ obtained above we conclude according to (43) that ‖𝑧(𝑡+𝛿)−𝑧(𝑡)‖_𝐻𝑟<𝜀, meaning that the function 𝑧(𝑡), 0≤𝑡≤ℎ, is continuous.
We now show that the problem (15)–(17) has a unique solution. We assume the contrary, that it has two different solutions 𝑧^′(𝑡), 𝑧^′′(𝑡), 0≤𝑡≤ℎ for the same functions 𝑧₀(⋅)∈𝑋 and 𝑢(⋅). Then their difference 𝑍(𝑡)=𝑧^′(𝑡)−𝑧^′′(𝑡), 0≤𝑡≤ℎ, as a solution of corresponding homogeneous problem (15)–(17), can be represented in the form of the Fourier series (21), and the Fourier coefficients 𝑍_𝑘(𝑡), 0≤𝑡≤ℎ, of the function 𝑍(𝑡), 0≤𝑡≤ℎ, are solutions of the following infinite system of differential equations: 𝑧̇_𝑘(𝑡)+𝜆_𝑘𝑧_𝑘(𝑡)=0,𝑧_𝑘(0)=0,0≤𝑡≤ℎ,𝑘=1,2,…. (45) Therefore, from the representation (29) we obtain 𝑍_𝑘(𝑡)=0 on 0≤𝑡≤ℎ. Consequently, 𝑍(𝑡)=0, 0≤𝑡≤ℎ, a contradiction. This proves the theorem.

2. Main results

Consider heat exchange control problem with lag (15)–(17).

Definition 5. A set 𝑊⊂𝑅¹ is referred to as the strong invariant set on the time interval [0,𝑇] with respect to the problem (15)–(19) if for any 𝑧₀(𝑡), ‖𝑧₀(𝑡)‖_𝐻𝑟∈𝑊, and 𝑢(𝑡), ‖𝑢(𝑡)‖_𝐻𝑟≤𝜌, 𝑡∈[0,𝑇], the inclusion ‖𝑧(𝑡)‖_𝐻𝑟∈𝑊 holds true on [0,𝑇].

Definition 6. A set 𝑊⊂𝑅¹ is referred to as the weak invariant set on the time interval [0,𝑇] with respect to the problem (15)–(19) if for any 𝑧₀(𝑡), ‖𝑧₀(𝑡)‖_𝐻𝑟∈𝑊 there exists 𝑢(𝑡), ‖𝑢(𝑡)‖_𝐻𝑟≤𝜌, 𝑡∈[0,𝑇], such that the inclusion ‖𝑧(𝑡)‖_𝐻𝑟∈𝑊 holds true on [0,𝑇].

Further, strong and weak invariant sets of the form 𝑊=[0,𝑏] will be investigated, where 𝑏 is a positive number. We will find relations among the parameters 𝑇, 𝑏, 𝜌, and 𝜆_𝑖 in order to guarantee strong or weak invariance of the set 𝑊 on the time interval [0,𝑇] with respect to the problem (15)–(19) (or (15)–(17), (20)).

Theorem 7. If 𝜆₁≥1, the set 𝑊=[0,𝑏] is strong invariant on [0,𝑇] with respect to the problem (15)–(19) if and only if 𝜌≤𝜆₁−1𝑏. (46)

Proof. Let (46) hold true and let 𝑧₀(𝑡), ‖𝑧₀(𝑡)‖_𝐻𝑟≤𝑏, −ℎ≤𝑡≤0, and 𝑢(𝑡), ‖𝑢(𝑡)‖_𝐻𝑟≤𝜌, 0≤𝑡≤ℎ be any functions. Using the same reasoning as in the proof of Theorem 4, we get from (40) that ‖𝑧(𝑡)‖²_𝐻𝑟≤𝜒²(𝑡);𝜒(𝑡)=𝑒^−𝜆1𝑡𝑏+𝑎₁(𝑡)𝑏+𝜌,0≤𝑡≤ℎ. (47)
Then 𝜒(0)𝜒=𝑏,_′(𝑡)=−𝜆₁𝑒^−𝜆1𝑡𝑏+𝑒^−𝜆1𝑡𝑏+𝜌=𝑒^−𝜆1𝑡−𝜆₁𝑏+𝑏+𝜌≤0. (48) Hence, 𝜒^′(𝑡)≤0 meaning that the function 𝜒(𝑡) is nonincreasing. For this reason, ‖𝑧(𝑡)‖_𝐻𝑟≤𝑏, 0≤𝑡≤ℎ.
The same reasoning can be applied for the time interval [ℎ,2ℎ] to obtain ‖𝑧(𝑡)‖_𝐻𝑟≤𝑏. For any positive number 𝑇>0, we can continue in this fashion to obtain ‖𝑧(𝑡)‖_𝐻𝑟≤𝑏,0≤𝑡≤𝑇; (49) that is, 𝑊 is strong invariant.
We now turn to the proof of the second part of the theorem. Let the set 𝑊 be strong invariant. Suppose, contrary to our claim, that the inequality (46) fails to hold. Then, clearly, the function 𝜒(𝑡) is increasing. Choose the functions 𝑧₀(⋅),𝑢(⋅) as follows: 𝑧₀₁()=𝑏𝑡−ℎ𝜆₁^𝑟/2,𝑢₁(𝑡)=𝜌𝜆₁^𝑟/2,𝑧_0𝑘()𝑢𝑡−ℎ=0,_𝑘(𝑡)=0,𝑘≥2,0≤𝑡≤ℎ. (50) Then ‖𝑧(𝑡)‖²_𝐻𝑟=𝜆^𝑟₁𝑒^−𝜆1𝑡𝑧⁰₀₁+^𝑡₀𝑒^{−𝜆1(𝑡−𝜏)}𝑧₀₁(𝑡−𝜏)+𝑢₁(𝜏)𝑑𝜏²=𝜒²(𝑡). (51) Since the function 𝜒(𝑡) is increasing on [0,ℎ] and 𝜒(0)=𝑏, therefore 𝑡,𝜒(𝑡)>𝑏; that is, ‖𝑧(𝑡)‖_𝐻𝑟>𝑏, which contradicts the strong invariance of the set 𝑊. This completes the proof of Theorem 7.

Theorem 8. Let 𝑇 be any positive number. If 𝜆₁≥1, the set 𝑊=[0,𝑏] is weak invariant on [0,𝑇] with respect to the problem (15)–(19).

Proof. Let 𝜆₁≥1. Show that, under condition 9, the set 𝑊=[0,𝑏] is weak invariant with respect to the problem (15)–(17). Let 𝑧₀(⋅) be an arbitrary function from 𝑋. Set 𝑢(⋅)=0. Then we obtain from the representation (29) of the solution of the problem (27) that ‖𝑧(𝑡)‖²_𝐻𝑟=^∞_𝑘=1𝜆^𝑟_𝑘𝑒^{−𝜆𝑘𝑡}𝑧⁰_0𝑘+^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑧_0𝑘(𝑡−𝜏)𝑑𝜏²≤𝑒^{−2𝜆1𝑡∞}_𝑘=1𝜆^𝑟_𝑘𝑧⁰_0𝑘²+2𝑒^{−𝜆1𝑡∞}_𝑘=1𝜆^𝑟_𝑘𝑧⁰_0𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑧_0𝑘(+𝑡−𝜏)𝑑𝜏_∞𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑧⁰_0𝑘(𝑡−𝜏)𝑑𝜏²≤𝑒^{−2𝜆1𝑡}𝑏²+2𝑒^−𝜆1𝑡𝑏𝑎₁(𝑡)𝑏+𝑎²₁(𝑡)𝑏²=𝑒^−𝜆1𝑡+𝑎₁(𝑡)²𝑏². (52) Here, we used the following inequality: ^∞_𝑘=1𝜆^𝑟_𝑘𝑧⁰_0𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑧_0𝑘(≤𝑡−𝜏)𝑑𝜏_∞𝑘=1𝜆^𝑟_𝑘𝑧⁰_0𝑘^2∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑧_0𝑘(𝑡−𝜏)𝑑𝜏² (53) and Assumption 2. Since the function 𝜉(𝑡)=𝑒^−𝜆1𝑡+1−𝑒^−𝜆1𝑡𝜆₁,0≤𝑡≤ℎ (54) satisfies 𝜉(0)=1,𝜉^′(𝑡)=1−𝜆₁𝑒^−𝜆1𝑡≤0, (55) we have 0<𝜉(𝑡)≤1. Then in view of (52) we conclude that ||𝑧(𝑡)||_𝐻𝑟≤𝑏, 0≤𝑡≤ℎ. Repeated application of this reasoning to the intervals [𝑗ℎ,(𝑗+1)ℎ], 𝑗=1,2,… enables us to conclude that ‖𝑧(𝑡)‖_𝐻𝑟≤𝑏, 0≤𝑡≤𝑇, and this completes the proof.

We now turn to the problem (15)–(17), (20).

Definition 9. The set 𝑊⊂𝑅¹ is called weak invariant on [0,𝑇] with respect to the problem (15)–(17), (20) if for any 𝑧₀(𝑡), ‖𝑧₀(𝑡)‖_𝐻𝑟∈𝑊, 𝑡∈[−ℎ,0] there exists 𝑢(𝑡), ‖𝑢(⋅)‖_{𝐿2([0,𝑇];𝐻𝑟)}≤𝜌, such that the inclusion ||𝑧(𝑡)||_𝐻𝑟∈𝑊 holds for all 𝑡∈[0,𝑇].

Definition 10. The set 𝑊⊂𝑅¹ is called strong invariant on [0,𝑇] with respect to the problem (15)–(17), (20) if for any 𝑧₀(𝑡), ‖𝑧₀(𝑡)‖_𝐻𝑟∈𝑊, 𝑡∈[−ℎ,0] and 𝑢(𝑡), ‖𝑢(⋅)‖_{𝐿2([0,𝑇];𝐻𝑟)}≤𝜌 the inclusion ‖𝑧(𝑡)‖_𝐻𝑟∈𝑊 holds for all 𝑡∈[0,𝑇].

Proposition 11. For any function 𝑔(𝜏), 0≤𝜏≤𝑡≤𝑇, for which ‖𝑔(⋅)‖_{𝐿2([0,𝑇];𝐻𝑟)}≤𝑁, the following inequality holds: ^∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑔_𝑘(𝜏)𝑑𝜏²≤𝑎₂(𝑡)𝑁²,𝑎0≤𝑡≤𝑇;₂(𝑡)=1−𝑒^{−2𝜆1𝑡}2𝜆₁, (56) where 𝑇 a positive number.

The proof follows from the following chain of relations ^∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−𝜆𝑘(𝑡−𝜏)}𝑔_𝑘(𝜏)𝑑𝜏²≤^∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑒^{−2𝜆𝑘(𝑡−𝜏)}𝑑𝜏⋅^𝑡₀𝑔²_𝑘(𝜏)𝑑𝜏≤𝑎₂(𝑡)^∞_𝑘=1𝜆^𝑟_𝑘^𝑡₀𝑔²_𝑘(𝜏)𝑑𝜏=𝑎₂(𝑡)𝑔(⋅)²_{𝐿2][0,𝑡;𝐻𝑟}≤𝑎₂(𝑡)𝑁²; (57) here we used the Cauchy-Schwartz inequality and the relation 𝑒^{−𝜆𝑘𝑡}≤𝑒^−𝜆1𝑡, 𝑡≥0, 𝑘=1,2,….

Theorem 12. If 𝜌>0, then the set 𝑊=[0,𝑏] is not strong invariant with respect to the problem (15)–(17), (20) on [0,𝑇], where 𝑇 is any number.

Proof. Let 𝜌>0. To prove the theorem, we use the fact that modulus of the admissible control function can take any big values on small time interval. Consider the function 𝜂(𝑡)=𝑒^−𝜆1𝑡𝑏+1−𝑒^−𝜆1𝑡𝜆₁𝑏+𝑒^−𝜆1𝑡√𝑡𝜌,𝑡≥0. (58) Note that 𝜂(0)=𝑏 and 𝜂(𝑡)>0 for all 𝑡>0. Assuming 𝑡>0, we have 𝜂^′(𝑡)=𝑒^−𝜆1𝑡𝑏−𝜆₁𝑏−𝜆₁𝜌√𝜌𝑡+2√𝑡. (59) Due to the term 𝑒^−𝜆1𝑡(𝜌/2√𝑡), we obtain that there exists 𝑡₁, 0<𝑡₁<ℎ, such that 𝜂^′(𝑡)>0 whenever 0<𝑡≤𝑡₁.
We now specify the initial function and admissible control function on [0,ℎ] as follows: 𝑧₀()=𝑏𝜏−ℎ𝜆₁^𝑟/2𝜑₁𝑢(𝜏)=𝜌,0≤𝜏≤ℎ,𝜆₁^𝑟/2𝑡₁𝜑₁,0≤𝜏<𝑡₁,0,𝑡₁≤𝜏≤ℎ. (60) We observe that 𝑧₀(𝜏)_𝐻𝑟‖𝑢(⋅≤𝑏,−ℎ≤𝜏≤0,)‖_{𝐿2([0,ℎ];𝐻𝑟)}≤𝜌. (61) Then by the representation (30) we have 𝑧(𝑡)=𝑒^−𝜆1𝑡𝑏𝜆₁^𝑟/2+^𝑡₀𝑒^{−𝜆1(𝑡−𝜏)}𝑏𝜆₁^𝑟/2+𝜌𝜆₁^𝑟/2𝑡₁𝑑𝜏𝜑₁,0≤𝑡≤𝑡₁. (62) Hence ‖𝑧(𝑡)‖²_𝐻𝑟=𝜆^𝑟₁𝑒^−𝜆1𝑡𝑏𝜆₁^𝑟/2+^𝑡₀𝑒^{−𝜆1(𝑡−𝜏)}𝑏𝜆₁^𝑟/2+𝜌𝜆₁^𝑟/2𝑡₁𝑑𝜏²,0≤𝑡≤𝑡₁. (63) This implies that 𝑧𝑡_1𝐻𝑟=𝑒^{−𝜆1𝑡1}𝑏+𝑏^𝑡1₀𝑒^{−𝜆1(𝑡1−𝜏)}+𝑑𝜏_𝑡10𝑒^{−𝜆1(𝑡1−𝜏)}𝜌𝑡₁𝑑𝜏≥𝑒^{−𝜆1𝑡1}𝑏+𝑏^𝑡1₀𝑒^{−𝜆1(𝑡1−𝜏)}+𝑑𝜏_𝑡10𝑒^{−𝜆1𝑡1}𝜌𝑡₁𝑑𝜏=𝜂𝑡₁. (64)
Since 𝜂(0)=𝑏 and 𝜂^′(𝑡)>0 for 0<𝑡≤𝑡₁, then 𝜂(𝑡₁)>𝑏. Therefore ‖𝑧(𝑡₁)‖_𝐻𝑟>𝑏 which shows that the set 𝑊=[0,𝑏] is not strong invariant, and the proof of Theorem 12 is complete.

Theorem 13. If 𝜆₁≥1, then the set 𝑊=[0,𝑏] is weak invariant with respect to the problem (15)–(17), (20) on [0,ℎ].

The proof of the theorem is similar to that of Theorem 8.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The present research was partially supported by the National Fundamental Research Grant Scheme FRGS of Malaysia (no. 01-01-13-1228FR).