Solve a semilinear elliptic PDE in Matlab as part of an distributed optimal control problem

Hey everyone,

I need to put in the following PDE into Matlab and solve it via FEM. The PDE is part of an complete optimal control problem, but if I know, how to put in the PDE, I will probably also know the rest.

Here, y is the solution to the PDE in H^1_0(D) (the state operator) and u in L^2(D) is the control operator (fixed function). D is the unitsquare in R^2 and x \in R^2.

a and r are just random functions (also fixed).

My problem is, that I don't know, how to put in nonconstant values or functions into the standard specifyCoefficients enviroment.

PDE:

- \nabla * ( a(x,\omega) \nabla y(x,\omega)) + r(x,\omega) (y(x,\omega))^3 = u(x)

y(x,\omega)=0

I know how to put in the dirichlet boundary condition, but struggle with the nonboundary part. In general I would like to have something like similar to −∇⋅(c∇u)+au=f, but with nonfixed c,a,f.

model=createpde();

geometryFromEdges(model,@squareg);

applyBoundaryCondition(model,'dirichlet','Edge',1:model.Geometry.NumEdges,'u',0);

rightHandSide= ' (put in the fixed funtion here) '

coefficient= 'my function a'

distributedControl= 'u.^2' ; % so that overall I have u^3 in matlab

specifyCoefficients(model,'m',0,...

'd',0,...

'c',coefficient,...

'a',distributedControl,...

'f',righHandSide);

generateMesh(model,'Hmax',hmax);

pdemesh(model);

Unfortunatly this approach doesn't work and I'm not sure how I can make it work. I would appreciate any help.

Best,

Tobias