Appl Math Optim 36:323–360 (1997)
1997 Springer-Verlag New York Inc.
Minimax Control of Parabolic Systems with
Dirichlet Boundary Conditions and
B. S. Mordukhovich and K. Zhang
Department of Mathematics, Wayne State University,
Detroit, MI 48202, USA
Communicated by I. Lasiecka
Abstract. In this paper we formulate and study a minimax control problem for a
class of parabolic systems with controlled Dirichlet boundary conditions and un-
certain distributed perturbations under pointwise control and state constraints. We
prove an existence theorem for minimax solutions and develop effective penalized
procedures to approximate state constraints. Based on a careful variational analy-
sis, we establish convergence results and optimality conditions for approximating
problems that allow us to characterize suboptimal solutions to the original minimax
problem with hard constraints. Then passing to the limit in approximations, we
prove necessary optimality conditions for the minimax problem considered under
proper constraint qualiﬁcation conditions.
Key Words. Approximations, Constraint qualiﬁcation, Dirichlet boundary con-
trols, Minimax criterion, Parabolic equations, State constraints, Uncertain distur-
bances, Variational inequalities.
AMS Classiﬁcation. Primary 49K20, 49K35, Secondary 49J20, 35K50.
This research was partly supported by the National Science Foundation under Grants DMS-9206989
and DMS-9404128 and by the NATO Contract CRG-950360.