# Averaged dynamics and control for heat equations with random diffusion

Author: - 23 November 2020

### Background and motivation

Let us consider the random heat equation described by the following system:

for $G$ a domain, $G_0\subset G$ a subdomain, $f$ a control, $y^0$ the initial configuration and $\alpha$ the diffusivity coefficient, which is a positive random variable with density function $\rho$. We have that the averaged solution of (1) is given by:

We want to determine if, given a positive random variable $\alpha$ and an initial configuration $y^0\in L^2(G)$, there is some $f\in L^2((0,T)\times G_0)$ such that $\tilde y(T,\cdot)=0$.

In order to illustrate the effect of averaging in the dynamics, let us study the dynamics of (1) when $G=\mathbb R^d$ and $f=0$. As averaging and the Fourier transform commute, we work on the Fourier transform of the fundamental solution of the heat equation, which is given by:

Consequently, the Fourier transform of the average of the fundamental solutions is given by:

i.e. the Laplace transform of $\rho$ evaluated in $|\xi|^2t$. In particular, for $r\in(0,1)$ if $\rho(\alpha)\sim_{0^+} e^{-C\alpha^{-\frac{r}{1-r}}}$ we have that:

when $|\xi|^2t\to+\infty$, which can be proved by the Laplace method. Thus, for those density functions the averaged dynamics in $\mathbb R^d$ has a fractional nature. As it is proved in [1], for $G$ bounded this is also true and we have the usual controllability and observability results of fractional dynamics; that is, (2) implies that the averaged unique continuation is preserved, but (2) preserves the null averaged observability if and only if $r>1/2$, being the threshold density functions those which satisfy:

### Some numerical simulations

Let us now illustrate the difference between several probability distributions for the diffusion through numerical simulations. For that, we recall that the optimal control is given by $\varphi(t,x;\phi)1_{G_0}$, for $\varphi$ the averaged solution of

and $\phi$ the state which minimizes the functional:

Due to the hardness of the numerical computations in higher dimensions and to get better illustrations we work in $d=1$, and in particular in $G=(0,\pi)$. We also consider $G_0=(1,2)$, $T=1$ and $y^0=\frac{1}{2}$. Moreover, to illustrate the difference between diffusivities inside and outside the null controllability regime, we consider $\rho=1_{(1,2)}$, which is inside, and $\rho=1_{(0,1)}$, which is outside.

In order to numerically implement this problem, we approximate it by minimizing $J$ in $V_M:=\langle e_i \rangle_{i=1}^M$ for ${e_i}$ the eigenfunctions of the Dirichlet Laplacian and for $M\in{40,50,60}$. Since $V_M$ is a finite dimensional space, computing the minimum of $J$ is equivalent to solving numerically a linear system, which can be easily done by using any numerical computing environment (in our case MATLAB). We have the following illustrations:

First, we illustrate in Figure 1 (resp. in Figure 2) the controls induced by the minimum of $J$ for $\rho=1_{(1,2)}$ (resp. for $\rho=1_{(0,1)}$). For $\rho=1_{(1,2)}$ the sequence of controls converges, which is something that can be seen in an even more clear way when $t\in[0,1/2]$. Of course, the closer the time is to $1$, the more slowly the punctual values of the control converges pointwise with $M$ (and in $t=1$ it diverges), which is a well-known behaviour when controlling a parabolic dynamics (see, for instance, [2], [3] and [4]). However, for $\rho=1_{(0,1)}$ the sequence of controls diverges, which is something that we can appreciate in a more detailed way when $t\in[0,1/2]$.

Figure 1: The optimal control for $\rho=1_{(1,2)}$ and $y^0=\frac{1}{2}$ induced by the minimum of the functional $J$ in $V_{40}$, $V_{50}$ and $V_{60}$. In the left column we illustrate the whole controls, whereas in the right column we illustrate the controls with the time variable zoomed in $[0,1/2]$.
Figure 2: The optimal control for $\rho=1_{(0,1)}$ and $y^0=\frac{1}{2}$ induced by the minimum of the functional $J$ in $V_{40}$, $V_{50}$ and $V_{60}$. In the left column we illustrate the whole controls, whereas in the right column we illustrate the controls with the time variable zoomed in $[0,1/2]$.

Next, we show in Figure 3 the canonical prolongation of the previously obtained controls to $t=0$. Again, for $\rho=1_{(1,2)}$ we have a clear convergence, whereas for $\rho=1_{(0,1)}$ it diverges.

Figure 3: The natural extensions to $t=0$ of the controls induced by the minimum of the functional $J$ in $V_{40}$, $V_{50}$ and $V_{60}$ with $y^0=\frac{1}{2}$. In the left figure we have considered $\rho =1_{(1,2)}$ and in the right one $\rho=1_{(0,1)}$.

Finally, we illustrate in Figure 4 the state at $t=1$ of the respective solutions of the averaged heat equation with the previously obtained controls. For $\rho=1_{(1,2)}$ the solution converges smoothly to $0$, whereas for $\rho=1_{(0,1)}$ the solution diverges.

Figure 4: The state in time $t=1$ of the averaged solutions of the heat equation after applying the control induced by the minimum of $J$ in $V_{40}$, $V_{50}$ and $V_{60}$ with $y^0=\frac{1}{2}$. In the left figure we have considered $\rho =1_{(1,2)}$ and in the right one $\rho=1_{(0,1)}$.

### Bibliography

[1] J. A. Bárcena-Petisco, E. Zuazua, Averaged dynamics and control for heat equations with random diffusion . Preprint https://hal.archives-ouvertes.fr/hal-02958671/

[2] E. Fernández-Cara and A. Münch. Strong convergent approximations of null controls for the 1D heat equation. SeMA journal, 61(1):49-78, 2013.

[3] R. Glowinski and J. L. Lions. Exact and approximate controllability for distributed parameter systems. Acta Numer., 1:269-378, 1994.

[4] A. Münch and E. Zuazua. Numerical approximation of null controls for the heat equation: illposedness and remedies. Inverse Probl., 26(8):085018, 2010.