Summary: OPTIMAL CONTROL PROBLEMS GOVERNED BY
SEMILINEAR PARABOLIC EQUATIONS
WITH LOW REGULARITY DATA
H. AMANN AND P. QUITTNER
Abstract. We study the existence of optimal controls for problems governed
by semilinear parabolic equations. The nonlinearities in the state equation
need not be monotone and the data need not be regular. In particular, the
control may be any bounded Radon measure. Our examples include problems
with nonlinear boundary conditions and parabolic systems.
In  we developed a general existence and uniqueness theory for semilinear par-
abolic problems involving measures and low regularity data. The proofs were based
on a generalized variation-of-constants formula in suitable extrapolated spaces and
the Banach fixed point theorem. Other papers on this topic mostly use approxima-
tion of singular data by regular ones and, consequently, require a priori estimates
(usually based on maximum principles) for the approximating solutions in order to
solve the original problem. The approach in  is much simpler and more flexible.
In particular, it can be easily used for problems with non-monotone nonlinearities
and for systems. In  we also established stability estimates and compactness
properties which play an important role in control theory.