Convection​-Diffusion​-Reaction Equation ( 2D) with a given Constrained Optimal Control Problem

I want solve: The Convection-Diffusion-Reaction Equation ( 2D) with a given Constrained Optimal Control Problem , I have used the programme https://www.mathworks.com/matlabcentral/fileexchange/60867-2d-unsteady-convection-diffusion-problem

But this programme give only the general solution of the Convection-Diffusion-Reaction Equation and not the Constrained Optimal Control Problem. any help to this work

Answer 1

[Torsten]

What do you want to control with which constraints and how ?

Optimal control problems together with ordinary or partial differential equations need a coupling of an integrator for the ordinary or partial differential equations and an optimizer for the constraint optimal control.

As you wrote, you found the integrator. Now you will have to couple it with an optimizer. The usual choice for this optimizer in MATLAB is "fmincon".

[Adams]

I'm interated to solve a similar problems as in the given work in :

Banholzer, S., Makarov, E. and Volkwein, S. POD-based multiobjective optimal control of time-variant heat phenomena, Numerical Mathematics and Advanced Applications ENUMATHA, (2017), 881–888.

https://www.google.com/url?sa=t&source=web&rct=j&opi=89978449&url=https://spp1962.wias-berlin.de/preprints/043.pdf&ved=2ahUKEwj70eiy8aiDAxVEU6QEHXIEBiwQFnoECBMQAQ&usg=AOvVaw0ThxySNkXQKsdtvPZRwf6M

The goal is to find a optimum bounded control u with somes constraints on the solution y

[Torsten]

As I said:

Either use "fmincon" or licence "PROPT":

https://en.wikipedia.org/wiki/PROPT

I think the approach with "fmincon" will give the greatest flexibility - PROPT only seems to support ODEs and DAEs, not PDEs.

If you have multiple conflicting objective functions, use "gamultiobj".

[Adams]

Thank you very much Mr Torsten, in our case we want used a differench approch as it mentioned in this article:( non smooth newton approachs ) https://www.inderscienceonline.com/doi/abs/10.1504/IJMMNO.2023.129922