Let the control be locally optimal for (P) with associated state ,
i.e.

the

and the

are fulfilled, see [3] or [11]. We mention that (2.3) is equivalent to the well-known projection property

where denotes projection onto . Moreover, we recall that these conditions can be derived by variational principles applied to the Lagrange function ,

Defining in this way, we tacitly assume that the homogeneous initial and boundary conditions of are formally included in the state space. The conditions (2.2-2.3) follow from for all admissible increments and . Let be given. We define

It holds on and on . These sets indicate strongly active control constraints.

2003-01-25