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(⊂ℝn) is referred to as the region with piecewise smooth boundary if its boundary can be represented as follows: , where Γj ⊂ Γ is an open set with respect to the topology on Γ induced by the topology in ℝn. Moreover, each Γj is a connected surface of class C1; that is, for any point x0 ∈ Γj there exists a ball Uϵ(x0) of radius ϵ > 0 such that the piece Γj∩Uϵ(x0) of the surface is given by the equation of the form xk = fk(x1, …, xk−1, xk+1, …, xn), where fk(·) ∈ C1 and 1 ≤ k ≤ n.
Definition 1. The number λ for which the following boundary value problem:
- (i)
there exist a countable set of eigenvalues λk of the problem (4) such that
() - (ii)
for each eigenvalue λk, there are a finite number of eigenfunctions φk corresponding to it such that ∫Ωφi(x)φj(x)dx = δij, where
() - (iii)
the set of all eigenvalues {φk} 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
() -
where the equality is understood in the following sense:
()
Denote by C(0, T; Hr(Ω)) and L2(0, T; Hr(Ω)) the spaces of continuous functions and summable quadratically measurable functions defined on [0, T] with the values in Hr(Ω), respectively, where T is a positive number.
1.3. Auxiliary Statements
The problem (15)–(17) is understood in the sense of the theory of generalized functions (the theory of distributions) with the values in Hr. 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 Hr, and the initial and boundary conditions are considered as equality of elements of these spaces.
The following assumption will be needed throughout the paper.
Assumption 2. Let functions z(·) of the set X satisfy the following conditions:
Proposition 3. For any function , 0 ≤ t ≤ T, the following inequality holds:
Proof. Indeed, for 0 ≤ t ≤ T, using the Cauchy-Schwartz inequality and the fact that
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]; Hr(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 Hr(Ω) for 0 ≤ t ≤ h, we have
We use the Cauchy-Schwartz inequality to obtain the following chain of relations:
We now show that the function z(t), 0 ≤ t ≤ h, is continuous with respect to the norm of the space Hr. We have
By (29), We have
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 Zk(t), 0 ≤ t ≤ h, of the function Z(t), 0 ≤ t ≤ h, are solutions of the following infinite system of differential equations:
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), , and u(t), , t ∈ [0, T], the inclusion 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), there exists u(t), , t ∈ [0, T], such that the inclusion 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 λi 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
Proof. Let (46) hold true and let z0(t), , −h ≤ t ≤ 0, and u(t), , 0 ≤ t ≤ h be any functions. Using the same reasoning as in the proof of Theorem 4, we get from (40) that
Then
The same reasoning can be applied for the time interval [h, 2h] to obtain . For any positive number T > 0, we can continue in this fashion to obtain
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:
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
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), , t ∈ [−h, 0] there exists u(t), , such that the inclusion 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), , t ∈ [−h, 0] and u(t), the inclusion holds for all t ∈ [0, T].
Proposition 11. For any function g(τ), 0 ≤ τ ≤ t ≤ T, for which , the following inequality holds:
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
We now specify the initial function and admissible control function on [0, h] as follows:
Since η(0) = b and η′(t) > 0 for 0 < t ≤ t1, then η(t1) > b. Therefore 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).