  This work is concerned with optimal control of partial differential equations
where the control enters the state equation as a coefficient and should take on
values only from a given discrete set of values corresponding to available
materials. A "multi-bang" framework based on convex analysis is proposed where
the desired piecewise constant structure is incorporated using a convex penalty
term. Together with a suitable tracking term, this allows formulating the
problem of optimizing the topology of the distribution of material parameters
as minimizing a convex functional subject to a (nonlinear) equality constraint.
The applicability of this approach is validated for two model problems where
the control enters as a potential and a diffusion coefficient, respectively.
This is illustrated in both cases by numerical results based on a semi-smooth
Newton method.
