Appl Math Optim 47:121–142 (2003)
2003 Springer-Verlag New York Inc.
Existence of Optimal Controls for Semilinear Parabolic Equations
without Cesari-Type Conditions
Mathematical Department, Fudan University,
Shanghai 200433, People’s Republic of China
Communicated by I. Lasiecka
Abstract. Optimal control problems governed by semilinear parabolic partial dif-
ferential equations are considered. No Cesari-type conditions are assumed. By prov-
ing the existence theorem and the Pontryagin maximum principle of optimal “state-
control” pairs for the corresponding relaxed problems, an existence theorem of
optimal pairs for the original problem is established.
Key Words. Existence, Semilinear parabolic equation, Optimal control, Relaxed
AMS Classiﬁcation. 49J20, 49J45, 35K20.
It is well known to researchers working in optimal control theory that to guarantee the
existence of (classical) optimal pairs we need a Cesari-type condition, which is a natural
generalization of optimal control problems for linear state equations with convex cost
functionals. Along this line, many results are available. We refer the readers to the books
by Berkovitz , Cesari , and Li and Yong  for detailed presentation.
When Cesari-type conditions are no longer satisﬁed, measure-valued controls (i.e.,
randomizing controls), called “relaxed controls,” are introduced. Relaxed controls have
other names such as “sliding regimes” , “generalized controls” , “relaxed curves”
This work was supported in part by the Science Foundation of the Education Ministry of China.