We study optimal controls problems with ﬁnal observation. The
governing parabolic equations or systems involve superlinear nonlinearities and
their solutions may blow up in ﬁnite time. Our proof of the existence, regularity
and optimality conditions for an optimal pair is based on uniform a priori
estimates for the approximating solutions. Our conditions on the growth of the
nonlinearity are essentially optimal. In particular, we also solve a longstanding
open problem of J.L. Lions concerning singular systems.