---
jupytext:
  main_language: python
  text_representation:
    extension: .md
    format_name: myst
    format_version: 0.13
    jupytext_version: 1.16.2
---

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

(demo_control_boundary_conditions)=
# Boundary conditions for control variables

## Problem Formulation

In this demo we investigate cashocs for solving PDE constrained optimization problems
with additional boundary conditions for the control variables. Our problem is given by

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

Here, we consider the control variable $u$ in
$H^1_\Gamma(\Omega) = \{ v \in H^1(\Omega) \vert v = u_\Gamma \text{ on } \Gamma \}$.

In the following, we will describe how to solve this problem using cashocs.

## Implementation

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

### Initialization

The beginning of the program is nearly the same as for {ref}`demo_poisson`

```python
from fenics import *

import cashocs

cashocs.set_log_level(cashocs.LogLevel.INFO)
config = cashocs.load_config("config.ini")
mesh, subdomains, boundaries, dx, ds, dS = cashocs.regular_mesh(25)
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-6
J = cashocs.IntegralFunctional(
    Constant(0.5) * (y - y_d) * (y - y_d) * dx + Constant(0.5 * alpha) * u * u * dx
)
```

### Scalar product and boundary conditions for the control variable

Now, we can first define the scalar product for the control variable $u$, which we
choose as the standard $H^1_0(\Omega)$ scalar product. This can be implemented as
follows in cashocs

```python
scalar_product = dot(grad(TrialFunction(V)), grad(TestFunction(V))) * dx
```

Moreover, we define the list of boundary conditions for the control variable as usual
in FEniCS and cashocs with the help of {py:func}`cashocs.create_dirichlet_bcs`

```python
control_bcs = cashocs.create_dirichlet_bcs(V, Constant(100.0), boundaries, [1, 2, 3, 4])
```

Here, we have chosen a value of $u_\Gamma = 100$ for this particular demo, in
order to be able to visually see, whether the proposed method works.

Finally, we can set up and solve the optimization problem as usual

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

where the only additional parts in comparison to {ref}`demo_poisson` are the keyword
arguments {python}`riesz_scalar_products`, which was already covered in
{ref}`demo_neumann_control`, and {python}`control_bcs_list`, which we have defined
previously.

After solving this problem with cashocs, we visualize the solution 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_control_boundary_conditions.png", dpi=150, bbox_inches="tight")
```

and the result should look like this
![](/../../demos/documented/optimal_control/control_boundary_conditions/img_control_boundary_conditions.png)