Boundary Stabilization of Heat Equation with Multi-Point Heat Source

In this paper, we consider boundary stabilization problem of heat equation with multi-point heat source. Firstly, a state feedback controller is designed mainly by backstepping approach. Under the designed state controller, the exponential stability of closed-loop system is guaranteed. Then, an observer-based output feedback controller is proposed. We prove the exponential stability of resulting closed-loop system using operator semigroup theory. Finally, the designed state and output feedback controllers are effective via some numerical simulations.


Introduction
The parabolic partial differential equations (PDEs) have played a crucial role in the realworld practice over the past several decades. At early stages, Masi et al. [1] investigated the microscopic fluctuations of interacting particle systems on a lattice by the nonlinear parabolic PDE. In [2], Price et al. showed the analysis and enhancement on special types of images based on parabolic PDEs. Afterwards, under measuring the temperature changes of ovens' processing area on different points, the heating capability of forced convection reflow ovens and the effect of the ovens' construction on it, were considered in [3]. By the experimental data, the parabolic PDE was used to describe the specific characteristics. Besides, Pardo et al. [4] applied the three-dimensional parabolic PDE to the non-destructive evaluation of minefields. Recently, Gao et al. [5] proposed a method of parabolic equation modeling to address the problem of predicting electromagnetic wave propagation caused by atmospheric dust and rough sea surfaces in the maritime environment. Regarding the study of the parabolic PDEs, there are some other results and references [6][7][8][9][10] therein.
In the control engineering filed, many scholars have been keen on boundary stabilization problems of parabolic PDEs during the past decades. For boundary control problem of a class of unstable scalar parabolic PDEs, ref. [11] employed a gradient-based optimization method to parameterize the feedback kernel as a second-order polynomial, and made the optimized kernel generate closed-loop stability with restricting on the kernel coefficients. Liu [12] proposed the successive approximation to address the boundary stabilization problem of unstable parabolic PDE. Wu et al. [13] made use of fuzzy control approach to solve the stabilization problem of nonlinear parabolic PDEs. Vazquez and Krstic [14] adopted finite-dimension feedback linearization method to transform parabolic PDEs with Volterra nonlinearities into a stable system by the Volterra series nonlinear operators. Actually, the proposed finite-dimension feedback linearization method was infinite dimensional extension of the backstepping approach. It is well known that backstepping transformation is mainstream method to deal with boundary stabilization problems of parabolic PDEs, such as, linear parabolic PDEs [15][16][17][18][19], nonlinear parabolic PDEs [20][21][22], quasi-linear parabolic PDEs [23], coupled parabolic PDEs and ODE [24][25][26][27].
To the best of our knowledge, heat equation is an extremely classic class of parabolic PDEs. The boundary control results of heat equation with unstable term or source term have been proposed and references [19,[28][29][30][31][32][33][34] therein. Krstic first proposed a separation principle result based on the passive and swapping identifiers and combined backstepping method to solve adaptive boundary control problem for heat equation with term λu(x, t) in [33], where λ is unknown parameter. Afterward, based on expressly parametrized control formula, adaptive controllers for parabolic PDEs with the term λu(x, t) were designed in [31], where λ is an unknown constant parameter. Additionally, Baccoli et al. [28] employed backstepping method to represent boundary stabilization result of the coupled reaction-diffusion processes with term ΛQ(x, t), where Λ ∈ R n×n is a real-valued square matrix. In [19], heat equation system is considered by whereÛ(t) is boundary control, Γ 0 is initial value, τ 0 is a constant, ξ 0 ∈ (0, 1). Zhou and Guo designed a state feedback controller based on backstepping approach for problem (1). Motivated by [19], we consider extension of heat system (1) is non-local term. As a matter of convenience, let τ = ∑ Z i=1 τ i . In our paper, owing to the existence of our multiterm τ, normal backstepping transformation is no longer applicable. A new backstepping transformation is constructed in our paper. Unlike the unstable terms mentioned above, can be seen as 1 0 τ i δ(s − ξ i )Γ(s, t)ds (a nice explanation in [19]). When i = 1, system (2) becomes system (1). Actually, [35] adopted backstepping method to solve the output feedback problem of transport equation with non-local term µu(x 0 ). Besides that there are other results about hyperbolic PDEs with integral term by backstepping approach in [36,37].
The rest of paper is organized as follows. In Section 2, we design a state feedback controller by backstepping approach to stabilize system (2). In Section 3, an output feedback controller is constructed. The resulting closed-loop system is proved to be exponentially stable. Section 4 illustrates the effectiveness of the proposed controller on the basis of some simulations. Finally, the concluding remarks are introduced in Section 5.

The State Feedback Controller Design by Backstepping
In this section, we introduce the backstepping transformation as below where a i , b i (i = 1, 2, ..., Z) and c are kernels and to be determined later.
Under transformation (3), we convert system (2) into the following target system where σ ∈ (−∞, 0] is any given number. Now, we prove the kernels are unique. Firstly, define the triangular domain Λ For transformation (3), we take derivative of it with regard to ξ, there is Next, we calculate the derivation of (3) with regard to t According to P t (ξ, t) = P ξξ (ξ, t) + σP(ξ, t) in (4), one has In addition, by P ξ (0, t) = 0 and Γ ξ (0, t) = 0,we get By the (8) and (9), there is For our main result, the following assumption is given by , and c(·, ·) ∈ C 2 (Λ).
Proof of Lemma 1. For convenience, we split (10) into and Firstly, we prove that for (12), there exist classical solutions . According to the first equation of (12), one has where σ = −β 2 . For computing (12) conveniently, we make |τ| = λ 2 . The proof is split the following three steps.

Remark 1.
From the above proof, we cannot explain that there are unique solutions a i (·) (12). However, a i (ξ)b i (s) is unique, which indicates transformation (3) is unique.
Next, we show that transformation (3) is reversible. Assume that inverse backstepping transformation is governed by Analogously, from (6) to (7), we get and Owing Additionally, By (51) and (52) , we get

Proof of Lemma 2.
Similarly, let (53) split into the following and Firstly, we prove that for (54), there exist classical solutions Combining the first equation of (54) and (56), we have We discuss (57) by the following three cases. Case 1: τ > σ By (57), we have Then, it gives Case 2: τ = σ According to (57), we obtain So we have Then, Now, we prove that there exists classical solution k(·, ·) ∈ C 2 (Λ) for (55). We firstly show that kernel k(ξ, s) is independent of h i (s). In fact, we obtain h i ( Then, combining (55), we obtain and Then, Hence, A is any constant in this case, moreover, k s (ξ, 0) = Aτ. By (60), we have By (66) with τ = σ, we can get Let From (72), one has There is a unique solution in (73), which is written by Combining (72) and (74) Then, substituting (75) into (71), we have We can see A is also any constant in this case. Next, we prove that A = 1. Substituting (48) into (3), we make the kernels satisfying and From (77), we have For simplifying calculation, we only give the result under case τ > 0. In fact, case τ > 0 is the same as τ < 0. Substituting b i (s) = ζ i (sinh(λs) − tanh(λξ i ) cosh(λs)) of (28) into (79), one has A = 1.
Finally, we can see that (55) is the same as (13) except for the last equations. However, it can be transformed into form "H(ξ)" for the last of Equation (55). So the remaining steps are similar from (43) to (47).
We design the state feedback controller The closed-loop system (2) corresponding with controller (80) is described by The system (81) is considered in the state space H = L 2 (0, 1). We define system operator A : D(A ) → H for closed-loop system (81) (82) Theorem 1. Under assumption (11), for each initial value Γ 0 (·) ∈ H , the closed-loop system (81) admits a unique solution Γ(·, t) ∈ C([0, +∞); H ). Moreover, the closed-loop system generates an exponentially stable C 0 -semigroup such that where L A and ρ two positive constants.
Proof. Define linear operator A 0 : D(A 0 ) → H for the target system (4) For any ω ∈ D(A 0 ), which means A 0 is dissipative on H . On the other hand, for anyφ ∈ H , A 0 (ϕ) =φ, bŷ ϕ = ϕ + σϕ, we have which shows that A −1 0 ∈ L (H ) is compact on H . By the Lumer Phillips theorem [38], A 0 generates a compressed C 0 -semigroup e A 0 on H . That is to say, where L A 0 are δ two positive constants. Define Lyapunov function Find the derivative of (85), and the calculus of the derivatives gives as the following quantityĖ which means Thus, the target system (4) is exponentially stable. Based on the transformation (3) and reversible transformation (48), we define the following bounded invertible operator So there exists a bounded reversible operator K satisfying A = K −1 A 0 K . Hence, the operator A yields an exponentially stable C 0 -semigroup on H , that is, The Theorem 1 holds by the exponential stability of e A 0 t .
the closed-loop system (90) is equivalent to PDEs as follows Next, we prove that system(93) is exponentially stable. Notice that "Γ-part" of system (93) is special case at σ = 0 for system (4), so "Γ-part" has a unique solution which is exponentially stable. Now, we only need to prove that "Γ-part" of system (93) has a unique solution and is exponentially stable. Similar to transformation (3), we give the following transformation under which "Γ-part" of system (93) is mapped to Now, we are in a position to prove that system (95) admits a unique solution and is exponentially stable. Based on (84), the solution of system (95) can be obtained as Define the following energy function E 1 (t) of system (88) A simple computation of derivative of (97) with respect to t shows thaṫ which means For any t 2 > t 1 > 0, one has Owing to the proof process of Theorem 1, there is Simultaneously, Additionally, from (99), for any i = 1, 2, ..., Z, we obtain(see [39]) Then, it can be obtained that Therefore, system (95) is exponentially stable, which means Theorem 2 holds.

Simulation Results
In this section, some simulation results are presented to explain the effectiveness of proposed controller by the finite element method. For the open-loop system of (2) and the closed-loop system (90), we choose parameters    τ 1 = 1, τ 2 = −1, τ 3 = 3; ξ 1 = 0.3, ξ 2 = 0.4, ξ 3 = 0.5; σ = 0, and initial values Γ 0 = 6 sin(πξ),Γ 0 = 4 cos(πξ). Figure 1 displays the solution of open-loop system (2). We can see that the solution of open-loop system (2) is growing fast and is unstable. Figure 2a,b display the solutions of "Γpart" and "Γ-part" of closed-loop system (90), respectively. Trajectory of the controller (89) is displayed in Figure 3. So it is clearly seen that the solution of closed-loop system (90) decays to zero and is stable under controller (89).

Concluding Remarks
The paper is presented the boundary output feedback stabilization of a heat equation with multi-point heat source mainly by backstepping approach. We first design the state feedback controller based on backstepping transformation. The exponential stability of closed-loop system is guaranteed. Secondly, the observer-based output feedback controller is constructed for infinite-dimensional systems. Furthermore, we prove that closed-loop system is exponentially stable. In the future, we will consider the result without assumption (11) and the case that τ i is a variable function or a more general continuous function hold.

Conflicts of Interest:
The authors declare no conflict of interest.