The following elliptic control problem with control and state constraints
generalizes the elliptic problems considered in
Bergounioux, Kunisch , Ito, Kunisch .
When admitting Dirichlet boundary conditions,
the functions in the cost functional (2.1),
in the boundary conditions (2.2),
and in the constraints (2.3) do not depend on the
state variable . This leads to the problem of
finding a control
that minimizes the functional
When treating Dirichlet boundary conditions, an intrinsic difficulty arises from the fact that the solution operator for (3.2) is not sufficiently smooth as to give an appropriate form of first order necessary conditions: cf., Lions , Lions, Magenes , Chapter 2. A weak formulation of first order necessary conditions for linear elliptic equations may be found in Bergounioux, Kunisch . Instead of trying to prove first order necessary conditions under strong assumptions we content ourselves with deriving first order necessary conditions in a purely formal way. This form of the necessary conditions will be justified by its analogy in the first order necessary conditions for the discretized version of the elliptic problem; cf. Section 4.2.
Denote an optimal solution of problem (3.1)-(3.4)
by and . The active sets
of inequality constraints (3.3) and (3.4) are
Let and be multipliers associated with the elliptic equation and the Dirichlet boundary condition in (3.2), and let resp. be the multiplier resp. the Borel measure associated with the control and state inequality constraints (3.3) and (3.4). Then the Lagrangian for problem (3.1)-(3.4) becomes
Note that the minimum condition (3.8) agrees with the condition (2.9) if we replace formally by . Hence, assuming functionals of tracking type (2.12), box constraints (2.13) for control and state and the property , we obtain the counterparts of the control laws (2.16), (2.17).
Case : the control is determined by
Case : the optimal control is of bang-bang
or singular type