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```{eval-rst}
.. include:: ../../../global.rst
```

(demo_sparse_control)=
# Sparse Control

## Problem Formulation

In this demo, we investigate a possibility for obtaining sparse optimal controls.
To do so, we use a sparsity promoting $L^1$ regularization. Hence, our model problem
for this demo is given by

$$
\begin{align}
    &\min\; J(y,u) = \frac{1}{2} \int_{\Omega} \left( y - y_d \right)^2 \text{ d}x
    + \frac{\alpha}{2} \int_{\Omega} \lvert u \rvert \text{ d}x \\
    &\text{ subject to } \qquad
    \begin{alignedat}[t]{2}
        -\Delta y &= u \quad &&\text{ in } \Omega,\\
        y &= 0 \quad &&\text{ on } \Gamma.
    \end{alignedat}
\end{align}
$$

This is basically the same problem as in {ref}`demo_poisson`, but the regularization
is now not the $L^2$ norm squared, but just the $L^1$ norm.

## Implementation

The complete python code can be found in the file {download}`demo_sparse_control.py
</../../demos/documented/optimal_control/sparse_control/demo_sparse_control.py>`
and the corresponding config can be found in {download}`config.ini
</../../demos/documented/optimal_control/sparse_control/config.ini>`.

### Initialization

The implementation of this problem is completely analogous to the one of
{ref}`demo_poisson`, the only difference is the definition of the cost functional.

```python
from fenics import *

import cashocs

config = cashocs.load_config("config.ini")
mesh, subdomains, boundaries, dx, ds, dS = cashocs.regular_mesh(50)
V = FunctionSpace(mesh, "CG", 1)

y = Function(V)
p = Function(V)
u = Function(V)

e = inner(grad(y), grad(p)) * dx - u * p * dx
bcs = cashocs.create_dirichlet_bcs(V, Constant(0), boundaries, [1, 2, 3, 4])

y_d = Expression("sin(2*pi*x[0])*sin(2*pi*x[1])", degree=1)
alpha = 1e-4
```

Next, we define the cost function, now using the mentioned $L^1$ norm for the
regularization

```python
J = cashocs.IntegralFunctional(
    Constant(0.5) * (y - y_d) * (y - y_d) * dx + Constant(0.5 * alpha) * abs(u) * dx
)
```

::::{note}
Note that for the regularization term we now do not use
{python}`Constant(0.5*alpha)*u*u*dx`,  which corresponds to the $L^2(\Omega)$ norm
squared, but rather
:::{code-block} python
Constant(0.5 * alpha) * abs(u) * dx
:::

which corresponds to the $L^1(\Omega)$ norm. Other than that, the code is identical.
::::

We solve the problem analogously to the previous demos

```python
ocp = cashocs.OptimalControlProblem(e, bcs, J, y, u, p, config=config)
ocp.solve()
```

and we visualize the results with the code

```python
import matplotlib.pyplot as plt

plt.figure(figsize=(15, 5))

plt.subplot(1, 3, 1)
fig = plot(u)
plt.colorbar(fig, fraction=0.046, pad=0.04)
plt.title("Control variable u")

plt.subplot(1, 3, 2)
fig = plot(y)
plt.colorbar(fig, fraction=0.046, pad=0.04)
plt.title("State variable y")

plt.subplot(1, 3, 3)
fig = plot(y_d, mesh=mesh)
plt.colorbar(fig, fraction=0.046, pad=0.04)
plt.title("Desired state y_d")

plt.tight_layout()
# plt.savefig('./img_sparse_control.png', dpi=150, bbox_inches='tight')
```

which yields the following output
![](/../../demos/documented/optimal_control/sparse_control/img_sparse_control.png)

:::{note}
The oscillations in between the peaks for the control variable {python}`u` are just
numerical noise, which comes from the discretization error.
:::