Appl Math Optim 39:143–177 (1999)
1999 Springer-Verlag New York Inc.
Hamiltonian Pontryagin’s Principles for Control Problems
Governed by Semilinear Parabolic Equations
J. P. Raymond and H. Zidani
Universit´e Paul Sabatier, UMR CNRS MIP,
31062 Toulouse cedex 04, France
Abstract. In this paper we study optimal control problems governed by semilinear
parabolic equations. We obtain necessary optimality conditions in the form of an
exact Pontryagin’s minimum principle for distributed and boundary controls (which
can be unbounded) and bounded initial controls. These optimality conditions are
obtainedthanks to newregularityresultsfor linear andnonlinearparabolic equations.
Key Words. Optimal control, Nonlinear boundary controls, Semilinear parabolic
equations, Pontryagin’s minimum principle, Unbounded controls.
In this paper we consider optimal control problems governed by semilinear parabolic
equations with nonlinear boundary conditions. We obtain optimality conditions in the
form of three decoupled Pontryagin minimum principles, one for distributed controls,
the second one for boundary controls, and the last one for initial controls.
The proof of these optimality conditions requires some new regularity results for
linear and nonlinear parabolic equations, that we obtain in Section 3. We stress that
in the nonlinear equations considered in Section 3, the nonlinear term in the boundary
conditions is neither monotone nor Lipschitz continuous. Moreover, the distributed and
the boundary controls are not necessarily bounded. To deal with such equations we ﬁrst
study linear equations in which some coefﬁcients are not bounded, and others are only
bounded from below (see Propositions 3.3 and 3.4). Since we plan to consider control
problems with pointwise state constraints in future papers, we look for C