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

(demo_nonlinear_pdes)=
# Optimal Control with Nonlinear PDE Constraints

## Problem Formulation

In this demo, we take a look at the case of nonlinear PDE constraints for optimization
problems. As a model problem, we consider

$$
\begin{align}
    &\min\; J(y,u) = \frac{1}{2} \int_{\Omega} \left( y - y_d \right)^2 \text{ d}x
    + \frac{\alpha}{2} \int_{\Omega} u^2 \text{ d}x \\
    &\text{ subject to } \qquad
    \begin{alignedat}[t]{2}
        -\Delta y + c y^3 &= u \quad &&\text{ in } \Omega,\\
        y &= 0 \quad &&\text{ on } \Gamma.
    \end{alignedat}
\end{align}
$$

As this problem has a nonlinear PDE as state constraint, we have to modify the config
file slightly. In particular, in the Section
{ref}`StateSystem <config_ocp_state_system>` we have to write
:::{code-block} ini
:caption: config.ini

is_linear = False
:::

Note, that {ini}`is_linear = False` works for any problem, as linear equations are
just a special case of nonlinear ones, and the corresponding nonlinear solver
converges in a single iteration for these. However, in the opposite case, FEniCS will
raise an error, so that an actually nonlinear equation cannot be solved using
{ini}`is_linear = True`. Also, we briefly recall from {ref}`config_optimal_control`,
that the default behavior is {ini}`is_linear = False`, so that this is not an issue.

## Implementation

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

### Initialization

Thanks to the high level interface for implementing weak formulations, this problem
is tackled almost as easily as the one in {ref}`demo_poisson`. In particular, the
entire initialization, up to the definition of the weak form of the PDE constraint,
is identical, and we have

```python
from fenics import *

import cashocs

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)
```

For the definition of the state constraints, we use essentially the same syntax as
we would use for the problem in FEniCS, i.e., we write

```python
c = Constant(1e2)
e = inner(grad(y), grad(p)) * dx + c * pow(y, 3) * p * dx - u * p * dx
```

:::{note}
In particular, the only difference between the cashocs implementation of this weak
form and the FEniCS one is that, as before, we use {py:class}`fenics.Function` objects
for both the state and adjoint variables, whereas we would use
{py:class}`fenics.Function` objects for the state, and {py:class}`fenics.TestFunction`
for the adjoint variable, which would actually play the role of the test function.
Other than that, the syntax is, again, identical to the one of FEniCS.
:::

Finally, the boundary conditions are defined as before

```python
bcs = cashocs.create_dirichlet_bcs(V, Constant(0), boundaries, [1, 2, 3, 4])
```

### Solution of the optimization problem

To define and solve the optimization problem, we now proceed exactly as in
{ref}`demo_poisson`, and use

```python
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
)

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

We visualize the results with the lines

```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_nonlinear_pdes.png', dpi=150, bbox_inches='tight')
```

and the result looks as follows
![](/../../demos/documented/optimal_control/nonlinear_pdes/img_nonlinear_pdes.png)