We describe here a simple example for the one dimensional heat equation, over the domain . We set , where the final time is fixed. The control is either (i) over a part of the domain, with Dirichlet conditions, or (ii) at the boundary by the Neumann condition. So the state equation is in case (i)
where , and is the characteristic function of , and in case (ii)
The cost function is, for and :
We discretize in space by standard finite difference approximations.
Numerical simulations: problem heat
As an example, we take 50 space variables, with , , and a final time T=20.
The discretization method is implicit Euler with 200 steps.
We set here , which gives a singular arc for the control.
We display on Fig. 1 the results in the case of the Dirichlet boundary condition .
Fig. 2 shows the Neumann case, this time with .
We can clearly see the differences between the boundary conditions and .
Figure 1: Heat equation, Dirichlet condition, and .
Figure 2: Heat equation, Neumann condition, and .