Appl Math Optim 49:145–157 (2004)
2004 Springer-Verlag New York Inc.
Dirichlet Boundary Control of Hyperbolic Equations in the
Presence of State Constraints
Boris S. Mordukhovich
and Jean-Pierre Raymond
Department of Mathematics, Wayne State University,
Detroit, MI 48202, USA
Laboratoire MIP, Universit´e Paul Sabatier,
31062 Toulouse Cedex 4, France
Communicated by I. Lasiecka
Abstract. We study optimal control problems for hyperbolic equations (focusing
on the multidimensional wave equation) with control functions in the Dirichlet
boundary conditions under hard/pointwise control and state constraints. Imposing
appropriate convexity assumptions on the cost integral functional, we establish the
existence of optimal control and derive new necessary optimality conditions in the
integral form of the Pontryagin Maximum Principle for hyperbolic state-constrained
Key Words. Optimal control, Hyperbolic equations, Dirichlet boundary controls,
State constraints, Integral maximum principle.
AMS Classiﬁcation. Primary 49K20, 49J20, Secondary 93C20, 35L20.
This paper is devoted to the study of optimal control problems for state-constrained
hyperbolic equations with controls in Dirichlet boundary conditions. We pay the main
attention to the following problem governed by the multidimensional wave equation.
The research of the ﬁrst author was partly supported by the National Science Foundation under Grants
DMS-0072179 and DMS-0304989.