 
Summary: OPTIMAL CONTROL PROBLEMS WITH FINAL OBSERVATION
GOVERNED BY EXPLOSIVE PARABOLIC EQUATIONS
H. AMANN AND P. QUITTNER
Abstract. 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.
1. Introduction
In his book [21], J.L. Lions studied several optimal control problems governed
by nonlinear parabolic equations of the form
# t y #y = y # + u, x
## , t # [0, T ], (1.1)
where# is a bounded domain in R n , # # {2, 3}, u = u(x, t) is the control and y =
y(x, t) is the state variable. Equation (1.1) is complemented by suitable boundary
and initial conditions, for example
y = 0 on
## × (0, T ), y(·, 0) = y 0 , (1.2)
