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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.