Uniform Turnpike Property

1 Introduction

In this post, we analyze a 1 − d heat equation with rapidly oscillating coefficients dependent on a parameter ε > 0, with a distributed control. We show that the uniform null controllability implies the uniform turnpike property, i.e., the turnpike property with constants independent of the ε-parameter.

The main conclusions of this post are the followings:

1.- The uniform controllability property implies the uniform turnpike property.

2.- Once we have a uniform turnpike property and if we know results from homogenization theory that guarantee that we can pass the limit in the \varepsilon equation, we can conclude the turnpike property for the limit system.

In the post, we illustrate how to prove this property for a particular equation. However, this procedure is entirely general for parabolic equations. The only difference lies in the assumption of the coefficient to guarantee the uniform controllability when the spatial domain is in \R^d with d>1.

The post is organized as follows: In the section 2 we introduce our equation and introduce two optimization problems, an evolutive problem and a stationary problem. Then, in the section 3, we show our main results, the integral turnpike property and the exponential turnpike property. Also, we show some necessary lemmas that allow us to conclude our results. After that, in section 4, we illustrate the turnpike property with some numerical simulation. Finally, we conclude the post with some commentaries and open problems.

 

2 Problem Settings

Let T>0 and \varepsilon>0. Consider the following heat equation with rapidly oscillating coefficients

\begin{cases} y^\varepsilon_t -(a(\frac{x}{\varepsilon})y^\varepsilon_x)_x=\chi_\omega f^\varepsilon\quad & (x,t)\in(0,1)\times(0,T),\\ y^\varepsilon(0,t)=y^\varepsilon(1,t)=0 &t\in(0,T),\\ y^\varepsilon(x,0)=y_0(x)&x\in (0,1). \end{cases}  (1)

where the initial condition y_0\in L^{2}(0,1), the control f^\varepsilon\in L^{2}(0,T;0,1) and \omega\subset(0,1) is a nonempty set. Additionally, we assume that the function a(\cdot)\in W^{2,\infty}(\R) is a periodic function satisfying

0 \lt a_{0} \leq a(x) \leq a_{1} \lt \infty,  (2)

for all x\in \R. Consider the space W(0,T) defined by

W(0,T)=\biggr\{ y\in L^2(0,T;H^1_0(0,1))\,:\, y_t \in L^2(0,T;H^{-1}(0,1))\biggr\}.

With the above setting, this equation admits a unique solution y^\varepsilon in the class W(0,T) for each \varepsilon>0 (see [3]).

Concerning the controllability properties of the equation (1), in [4], with boundary control, is proved the uniform controllability property. However, the following theorem shows that the uniform null controllability property holds to the system (1), with interior control.

Theorem 2.1.
Assume that a\in W^{2,\infty} is a periodic function satisfying (2). Let T>0. Then for any y_0\in L^2(0,1), and \varepsilon\in(0,1) the solution of the system (1) satisfy the null controllability property, that is, there exists a control f^\varepsilon\in L^2(0,1) such that

y^\varepsilon(x,T)=0, \quad \forall x\in(0,1).

Furthermore, the control satisfy

\|f^\varepsilon\|_{L^2(0,1)}^2\leq C\|y_0\|^2_{L^2(0,1)},

with C>0, the same constant given by the theorem 2.1, which is independent of \varepsilon.

As it is classical (see [1]), the above theorem assures us that the system (1) satisfies an observability inequality. In concrete, we have the following result.

Corollary 2.1.1.
Consider the following system

\begin{cases} -\phi^\varepsilon_t -(a(\frac{x}{\varepsilon})\phi^\varepsilon_x)_x=0\quad & (x,t)\in(0,1)\times(0,T),\\ \phi^\varepsilon(0,t)=\phi^\varepsilon(1,t)=0 &t\in(0,T),\\ \phi^\varepsilon(x,T)=\phi^\varepsilon_0&x\in (0,1). \end{cases}  (3)

Assume that a\in W^{2,\infty} is a periodic function satisfying (2). Let T>0. Then for any \phi^\varepsilon_0\in L^2(0,1), and \varepsilon\in(0,1) there exists a constant C>0 independent of \varepsilon such that

\|\phi^\varepsilon(\cdot,0)\|^{2}_{L^{2}(0,1)}\leq C\int_{0}^{T}\int_\omega|\phi^\varepsilon(x,t)|^{2}dxdt.

The proof of the theorem 2.1 is followed by using a control strategy in three steps in which: we first control the low frequencies of the system using Fourier decomposition. Then let the system evolve freely. Finally, we control to zero the whole solution and estimate the cost of controllability using Carleman inequalities. Indeed, in the last step is impossible to obtain estimations independent of ε. However is possible to see that if we control the correct low frequencies (all the frequencies with λk ≤ C/ε2), the resultant control from the tree steps strategy satisfies the uniform controllability property. As a sketch of the proof, we can see the following figure.

FIGURE 1. The final control is given by fε = χ[0,T/3]f1ε +χ[2T/3,T]f3ε. This control satisfy and drive to zero the state in time T.

When the spatial domain is in \R^d with d>1 is necessary to introduce a more restrictive condition over a. If we assume that a is Lipschits with Lipschits constant and derivate positives and uniformly bounded. Then, by the classical Carleman estimation developed in [7], we can ensure that the heat equation (4) is uniform null controllable.

 

2.1 Optimal Control Problems

Consider the following optimization problem (Optimal control problem)

\min_{f^\varepsilon\in L^{2}(0,T;0,1)}\biggr\{ J_\varepsilon^T(f^\varepsilon)=\frac{1}{2}\int_0^T \|f^\varepsilon(\cdot,t)\|_{L^2(0,1)}^2 dt +\|y^\varepsilon(\cdot,t)-y_d\|^2_{L^2(0,1)}dt\biggr\}  (4)

where y_d is a given target, and y^\varepsilon is the solution of (1). Note that we can ensure that there exist a unique optimal control via the direct methods of calculus of variation. Also, the optimal control is characterized as f^\varepsilon(x,t)=-\chi_\omega\psi^\varepsilon(x,t), where \psi^\varepsilon is the solution of the adjoint system

\begin{cases} -\psi^\varepsilon_t -( a(\frac{x}{\varepsilon})\psi^\varepsilon_x)_x=y^\varepsilon - y_d \quad & (x,t)\in(0,1)\times(0,T),\\ \psi^\varepsilon(0,t)=\psi^\varepsilon(1,t)=0 & t\in(0,T),\\ \psi^\varepsilon(x,T)=0 & x\in (0,1). \end{cases}  (5)

Since y_d\in L^2(0,1), there exist a unique solution \psi^\varepsilon of (5) in the class of W(0,T). In the follow, we will denote by (y^\varepsilon,f^\varepsilon,\psi^\varepsilon) the optimal variables associate to the problem (4)

By the other hand, consider the following stationary optimization problem

\min_{f\in L^2(0,1)}\biggr\{J_\varepsilon^s(f)= \frac{1}{2}\biggr(\|f\|_{L^2(0,1)}^2+\|y(\cdot)-y_d\|^{2}_{L^2(0,1)}\biggr)\biggr\},  (6)

where y_d\in L^2(0,1) is the same as above, and y satisfy the elliptic equation

\begin{cases} -( a(\frac{x}{\varepsilon})y_x)_x=f,\quad & x\in(0,1),\\ y(0)=y(1)=0. \end{cases}  (7)

We can see since (2), there exist a unique solution y\in H_0^1(0,1) to the problem (7).

As in (4), there exist a unique optimal control \overline{f}^\varepsilon\in L^2(0,1) and by the well-posed, a unique state \overline{y}^\varepsilon\in H^1(0,1) solution of (7). Furthermore, we can characterize the minimum of (6) as \overline{f}^\varepsilon(x) =-\chi_\omega\overline{\psi}^\varepsilon(x), where \overline{\psi}^\varepsilon is the solution of the adjoint stationary equation

\begin{cases} -( a(\frac{x}{\varepsilon})\overline{\psi}^\varepsilon_x)_x=\overline{y}^\varepsilon-y_d\quad & x\in(0,1),\\ \overline{\psi}^\varepsilon(0)=\overline{\psi}^\varepsilon(1)=0. \end{cases}  (8)

In the follow, we will denote by (\overline{y}^\varepsilon,\overline{f}^\varepsilon,\overline{\psi}^\varepsilon) the optimal variables associate to the problem (7).

We are in conditions to state our main results with the previous setting. As the first result, we have the called integral turnpike property.

Consider the optimal pairs (y^\varepsilon,f^\varepsilon) and (\overline{y}^\varepsilon,\overline{f}^\varepsilon) of the problems (4) and (6) respectively. Then

\frac{1}{T}\int_0^T y^\varepsilon(x,t)dt \to \overline{y}^\varepsilon, \quad \text{and } \quad \frac{1}{T}\int_0^T f^\varepsilon(x,t)dt \to \overline{f}^\varepsilon \quad \text{in } L^2(0,1),

as T\to\infty.

Our second result corresponds to the exponential turnpike property.

There exist constants K>0 and \mu>0 independent of T and \varepsilon such that

\|y^\varepsilon(\cdot,t)-\overline{y}^\varepsilon(\cdot)\|_{L^2(0,1)}+\|f^\varepsilon(\cdot,t)-\overline{f}^\varepsilon(\cdot)\|_{L^2(0,1)}\leq K(e^{-\mu t}+e^{-\mu(T-t)}),

for every t\in(0,T). Where (y^\varepsilon,f^\varepsilon) and (\overline{y}^\varepsilon,\overline{f}^\varepsilon) are the optimal pairs of the problems (4) and (6) respectively.

 

3 Main Results

In this section, we show the main results of this work.

3.1 Integral convergence

We start by stating the integral turnpike property

Theorem 3.1. Consider the optimal pairs (y^\varepsilon,f^\varepsilon) and (\overline{y}^\varepsilon,\overline{f}^\varepsilon) of the problems (4) and (6) respectively. Then

\frac{1}{T}\int_0^T y^\varepsilon(x,t)dt \to \overline{y}^\varepsilon, \quad \text{and } \quad \frac{1}{T}\int_0^T f^\varepsilon(x,t)dt \to \overline{f}^\varepsilon \quad in L^2(0,1),

as T\to\infty.

The proof of this theorem is based on the article [5], and it is necessary to prove a previous lemma that we show below. This lemma corresponds to a fundamental ingredient for the proof of our main results and is a direct consequence of the uniform null controllability property. (Theorem 2.1)

Lemma 3.1.
Let y^\varepsilon, \psi^\varepsilon, \overline{y}^\varepsilon the optimal variables of the problems (4) and (6) respectively. Then there exist a constant C_1>0 independent of the time horizon T>0 and ε >0 such that for the evolutive optimal variables

\|y^\varepsilon(\cdot,T)\|^{2}_{L^{2}(0,1)}\leq C_{1} \biggr(\int_{0}^T \|\chi_\omega f^\varepsilon(\cdot,t)\|^2_{L^{2}(0,1)} +\|y^\varepsilon(\cdot,t)\|^2_{L^2(0,1)}dt\biggr).

and

\|\psi^\varepsilon(\cdot,0)\|^{2}_{L^{2}(0,1)}\leq C_{2}\biggr(\int_{0}^T\|\chi_\omega\psi^\varepsilon(\cdot,t)\|^{2}_{L^2(0,1)}+\|y^\varepsilon(\cdot,t)-y_d\|^2_{L^2(0,1)}dt\biggr),

holds.
Furthermore, for the stationary optimal variables we have

\|\overline{y}^\varepsilon(\cdot)\|^{2}_{L^{2}(0,1)}+\|\overline{\psi}^\varepsilon(\cdot)\|^{2}_{L^{2}(0,1)}+\|\overline{f}^\varepsilon(\cdot)\|^2_{L^2(0,1)} \leq C_3\|y_d(\cdot)\|^{2}_{L^{2}(0,1)}.

3.2 Exponential convergence

We state our second main result, the uniform exponential turnpike property.

Theorem 3.2.
There exist constants K>0 and \mu>0 independent of T and \varepsilon such that

\|y^\varepsilon(\cdot,t)-\overline{y}^\varepsilon(\cdot)\|_{L^2(0,1)}+\|f^\varepsilon(\cdot,t)-\overline{f}^\varepsilon(\cdot)\|_{L^2(0,1)}\leq K(e^{-\mu t}+e^{-\mu(T-t)}),

for every t\in(0,T). Where (y^\varepsilon,f^\varepsilon) and (\overline{y}^\varepsilon,\overline{f}^\varepsilon) are the optimal pairs of the problems (5) and (7) respectively.

The proof of this theorem is based on a decoupling strategy, in which is necessary to introduce the Riccaty operator. Let’s consider the problem (4) with y_d\equiv0, that is

\min_{f^\varepsilon\in L^{2}(0,T;0,1)}\biggr\{ J_\varepsilon^{T,0}(f^\varepsilon)=\frac{1}{2}\int_0^T \|f^\varepsilon(\cdot,t)\|_{L^2(0,1)}^2 +\|y^\varepsilon(\cdot,t)\|^2_{L^2(0,1)}dt\biggr\},  (10)

where y^\varepsilon solve (1). Now define the operator \mathcal{E}(T): L^2(0,1) \to L^2(0,1) by setting

\mathcal{E}(T)y_0(x) :=\psi^\varepsilon(x,0)

We have the following lemma

Lemma 3.2.
The operator \mathcal{E}(T) is well-define, linear and continuous. Also, we have the following
1. Denoting by (\cdot,\cdot)_{L^2(0,1)} the inner product in L^2(0,1) we have

(\mathcal{E}(T)y_0(\cdot),y_0(\cdot))_{L^2(0,1)}= \min_{f^\varepsilon\in L^{2}(0,T;0,1)}J_\varepsilon^{T,0}(f^\varepsilon)

2. There exist an operator \hat{E}\in \mathcal{L}(L^2(0,1)) such that.

\mathcal{E}(t)y_0 \overset{t\to\infty}{\to}\hat{E}y_0 \text{ in }L^2(0,1).

Moreover, there exist \mu>0 and C>0 independent of \varepsilon>0 such that

\|\mathcal{E}-\hat{E}\|_{\mathcal{L}(L^2(0,1))}\leq Ce^{-\mu t},

for any t\geq0.

The last lemma allows us to decouple the optimal system. More careful development of these techniques can be seen in the classic book [3]. Related to this, we have a corollary below.

Corollary 3.2.1.
The optimal control f^\varepsilon of (4) is given by the affine law

f^\varepsilon(x,t)=\overline{f}^\varepsilon(x)-\chi_\omega\left[\mathcal{E}(T-t)\left(y^\varepsilon(x,t)-\overline{y}^\varepsilon(x)\right)+h(x,t)\right]

where h solves the evolution problem

\begin{cases} -h_{t}+\left(-(a(\frac{x}{\varepsilon})\,\cdot\,)_x+\chi_\omega\mathcal{E}(T-t)\right) h=0 \quad & (x,t)\in(0,1)\times(0,T) \\ h(0,t)=h(1,t)=0 &t\in(0,T),\\ h(T)=-\overline{\psi}^\varepsilon&x\in (0,1). \end{cases}  (11)

Finally, as a consequence of the results given by the lemma 3.2 and the corollary 3.2.1, we can conclude the proof of the Theorem 3.2.

Once we have the uniform turnpike property, we can try to answer the question of singular limits aboard. From [8], we know that we can pass the limit in the equation (1) and (7) to obtain two homogenized systems. Then we can also pass the limit in the optimal variables solution of (4) and (6) to get the optimal variables (y,f) and (\overline{y},\overline{f}) for the limit problems, (following [6]). Now we state our final result.

Corollary 3.2.2.
Let (y,f) and (\overline{y},\overline{f}) the optimal variable of the optimization problems (4) and (6) subject to they respectively homogenizated systems. Then, with the same constants C and \mu, of the theorem 3.2 we have

\|y(\cdot,t)-\overline{y}(\cdot)\|_{L^2(\Omega)}+\|f(\cdot,t)-\overline{f}(\cdot)\|_{L^2(\Omega)}\leq C(e^{-\mu t}+e^{-\mu(T-t)}), for every t\in(0,T).

 

4 Numerical simulations

In this section, we will show some numerical examples of how the turnpike property remains uniform in \varepsilon. To illustrate the property, let us consider the following optimal control problem

\min_{\substack{ f^{\varepsilon}\in L^{2}(0,T;(0,1))}}\biggr\{J(f^{\varepsilon})=\frac{1}{2}\int_{0}^{T}\biggr(\|f^{\varepsilon}\|^{2}+\|y^{\varepsilon}-1\|^{2}\biggr) dt\biggr\}.

where y^{\varepsilon} is the solution of the equation (1). On the other hand, we introduce the corresponding stationary optimization problem

\min_{\overline{f}^{\varepsilon}\in L^{2}((0,1))}\biggr\{ J^{s}(\overline{f}^{\varepsilon})=\frac{1}{2}\biggr(\|\overline{f}^{\varepsilon}\|^{2}_{L^{2}(0,1)}+\|\overline{y}^{\varepsilon}-1\|^{2}_{L^{2}((0,1))}\biggr)\biggr\}.

where in this case \overline{y}^{\varepsilon} is the solution of the respective stationary system (7). Here we will consider a(x)=\arctan(x+1), g(x)=\frac{x\left(46-206x+320x^2-160x^3\right)}{165} and T=100. This problem was solved numerically by solving the semi-discrete problem using the Gekko library in Python. This library is based on orthogonal collocation of finite elements, this is a form of implicit Runga–Kutta methods which provides stability and convergence of the algorithm. In the temporal discretization 101 steps were used, and in the spatial discretization 80 steps were used. Furthermore, for 150 equispaced values of \varepsilon, in the interval (0.001, 0.5), the respective solutions were obtained. This allowed us to obtain the following animations

Capture. Uniform Turnpike

On the left, we can observe the turnpike property for different values of ε vs. the limit system. We illustrate the uniform exponential turnpike property on the right, taking K = 0.15 and μ = 1.5 fixed for all ε.

In both animations, we can observe that the turnpike property is preserved when \varepsilon goes to 0. In the left plot, we can see that for fixed constants K,\mu>0, the turnpike property is preserved for all \varepsilon>0.

Moreover, in the plot on the right-hand side, we can observe how the evolutionary trajectory remains constant over the stationary trajectory (the turnpike) for all \varepsilon. In addition to the above, we observe that a(x)=\arctan(x+1) goes to \pi/2 when x\to\infty. Thus, we will call the limit system evolutive and stationary, the system corresponding to equation (1) and (7) with a(x)=\pi/2 respectively. From the figure on the right, we can observe that when \varepsilon\to 0, the epsilon-dependent gets closer and closer to the turnpike of the limit system.

 

5 Further Comments

In this section, we comment on the results presented in this post and discuss some open problems, which are natural extensions.
1. Summarizing, our main result, the uniform exponential turnpike property, i.e., the turnpike property with constants independent of the parameter ε, is a consequence of our heat equation with oscillating coefficients satisfies the uniform null controllability property. This is a crucial property for the development of our problem. Although in this article, we have developed the turnpike property for a particular equation. However, in general, if a system satisfies the turnpike property for each parameter ε, and the system is uniformly null controllable, we can follow the same proof to prove the uniformly turnpike property. The latter can be verified by rigorously tracing the dependence of the ε parameter on the constants and using the uniform null controllability.

2. It is possible to adopt other approach to prove the turnpike property. In [2] the property is proved via dissipativity assumptions over the main operators. If we consider an unbounded operator A_{\varepsilon}\in \mathcal{L}(X,V^{*}) with D(A)=V for each \varepsilon>0, and the operator B\in \mathcal{L}(U,V^{*}) is a bounded operator. Consider u^{\varepsilon}\in L^2(0,T;U) and define the space U, X and V such that the linear abstract problem

\begin{cases} y^{\varepsilon}_{t}+A_{\varepsilon}y^{\varepsilon}=Bu^{\varepsilon}\quad t\in (0,T),\\ y^{\varepsilon}(0)=y_{0}, \end{cases}  (12)

is well posed. Then considered the problem

\min_{\substack{ u^{\varepsilon}\in L^{2}(0,T;U)}}\biggr\{J(u^{\varepsilon})=\frac{1}{2}\int_{0}^{T}\biggr(\|u^{\varepsilon}\|^{2}_{U}+\|C(y^{\varepsilon}-y_{d})\|^{2}_X\biggr) dt\biggr\}.

with C\in \mathcal{L}(X). Consider the following assumption

Hypotheses 1.
For the pair (A,C) and the pair (A,B), we assume the following conditions. There exists a bounded feedback operator K_{C}, and a positive constant \alpha, independent of \varepsilon such that

\langle(A_{\varepsilon}+K_{C}C)v,v\rangle\geq \alpha\|v\|^{2}_{X}\quad \forall v\in X.

The above hypotheses allow us to ensure that the constants on the right-hand side of the turnpike property are uniform in ε > 0. With these hypotheses the theorems 3.1 and 3.2 can be proved by following [2]. This kind of hypothesis can be founded in the literature as exponentially detectable and exponentially stabilizable.

3. One of the main advantages of this work is using a controllability result to prove the turnpike property. The uniform null controllability of yε solution of (1), allow us ensure that z^{\varepsilon}(x,t)=h(x,t) y^{\varepsilon}(x,t) is also uniform null controllable, where is h(x,t) is a smooth enough non zero function. However, if we had built this result on a dissipativity assumption for the system (1), then the conclusion could not be valid for z^{\varepsilon}. In particular, let us take z^{\varepsilon}(x,t)=e^{e^{-Kt}} y^{\varepsilon}(x,t) for some constant K>0. Then the main operator of the system that satisfies z^{\varepsilon}, has eigenvalues on the right-side of the complex plane. That is, the system is not stable.

4. A example of interest to study the turnpike property may be the following equations

\begin{cases} \varepsilon y^{\varepsilon}_{tt}+ y^{\varepsilon}_t-\Delta y^{\varepsilon} + y^{\varepsilon}= f^\varepsilon &\text{ in }\Omega\times(0,T),\\ y^{\varepsilon}=0 &\text{ on }\partial\Omega\times(0,T),\\ y^{\varepsilon}(x,0)=y_0(x),\quad y^{\varepsilon}_t(x,0)=y_1(x)&\text{ in }\Omega. \end{cases}

Observe that for \varepsilon\gg 0, the hyperbolic behavior is predominant in the equation, and when \varepsilon\approx 0, the equation follows a parabolic behavior. This equation has been studied in the context of uniform controllability and homogenization in [9]. The results proved in [9] can be directly used in order to prove the uniform turnpike property, for this equation subject to (4).

 

References

[1] Caroline Fabre, Jean-Pierre Puel, and Enrike Zuazua. “Approximate controllability of the semilinear heat equation”. In: Proc. Roy. Soc. Edinburgh Sect. A (1995), Vol. 125, No. 1, pp. 31–61. ISSN: 0308-2105. doi: 10.1017/S0308210500030742. url: https://doi- org.usm.idm.oclc.org/10.1017/S0308210500030742.
[2] Lars Grüne, Manuel Schaller, and Anton Schiela. “Sensitivity analysis of optimal control for a class of parabolic PDEs motivated by model predictive control”. In: SIAM J. Control Optim. (2019), Vol. 57, No. 4, pp. 2753–2774. issn: 0363-0129. doi: 10.1137/18M1223083. url: https://doi-org.usm.idm.oclc.org/10.1137/18M1223083.
[3] J.-L. Lions. Optimal control of systems governed by partial differential equations. Die Grundlehren der mathematischen Wissenschaften, Band 170. Translated from the French by S.K. Mitter. Springer-Verlag, New York-Berlin (1971), pp. xi+396.
[4] A. López and E. Zuazua. “Uniform null-controllability for the one-dimensional heat equation with rapidly oscillating periodic density”. In: Ann. Inst. H. Poincaré Anal. Non Linéaire (2002), Vol. 19, No. 5, pp. 543–580. issn: 0294-1449. doi: 10.1016/S0294- 1449(01)00092- 0. url:https://doi-org.usm.idm.oclc.org/10.1016/S0294-1449(01)00092-0.
[5] Alessio Porretta and Enrique Zuazua. “Long time versus steady state optimal control”. In: SIAM J. Control Optim (2013),  Vol. 51, No. 6, pp. 4242–4273. issn: 0363-0129. doi: 10.1137/130907239. url: https://doi.org.usm.idm.oclc.org/10.1137/130907239.
[6] Enrique Zuazua. “Approximate controllability for linear parabolic equations with rapidly oscillating coefficients”. Modelling, identification, sensitivity analysis and control of structures (1994), vol. 23, no. 4, pp. 793–801.
[7] A. Fursiko, O. Imanuvilov. “Controllability of evolution equations”. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul (1996)
[8] D. Cioranescu and P. Donato. An introduction to homogenization. (The Clarendon Press, Oxford University Press, New York,1999)
[9] A. López, X. Zhang and E. Zuazua. Null controllability of the heat equation as singular limit of the exact controllability of dissipative wave equations. J. Math. Pures Appl. (2000) vol. 79, no. 9, pp. 741-808, https://doi-org.usm.idm.oclc.org/10.1016/S0021-7824(99)00144-0

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