In this work we solve the optimal control problem
where is the control variable,
the state variable and
a target function. The minimization problem is subject to the elliptic partial differential equation
In order to use the conjugate gradient method, the state variable is separated in two terms as
where solves the state equation with zero Dirichlet boundary conditions,
and is the control-free solution to the state equation,
With the above separation of the state variable, the cost functional can be expressed as
We define a linear operator
and its adjoint
with solution to
The directional derivative of the cost function then reads as
After having identified and
we can use the conjugate gradient method to reach the optimal control faster.
The solver must be compiled in the terminal. It is advisable to first clean previous compilations with
wclean
and then use
wmake
OpenFOAM C++ library must be installed in order to compile the code.
The OpenFOAM distribution provided by the OpenFOAM Foundation was used.
In order to run the solver move to the case folder poissonCGAdjoinFoamCase and type in the command line
./Allprepare
poissonCGAdjointFoam
The poissonCGAdjointFoam solver has been tested in a square domain with zero Dirichlet boundary conditions and
. The target function is
.
It might be needed to use
sed -i -e 's/\r$//' filename
and
chmod +x filename
in order to be able to execute
./filename
- F. Tröltzsch. Optimal control of partial differential equations: theory, methods, and applications. American Mathematical Soc., 2010.