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First order necessary conditions

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

 (2.1)

holds for all satisfying the constraints (1.2-1.4), where belongs to a sufficiently small -neighborhood of . Suppose further that is regular. Then there exist Lagrange multipliers (the adjoint state) and such that the adjoint equation
 (2.2)

the variational inequality
 (2.3)

and the complementary slackness condition
 (2.4)

are fulfilled, see [3] or [11]. We mention that (2.3) is equivalent to the well-known projection property
 (2.5)

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.

Next: Second order sufficient optimality Up: First and second-order optimality Previous: First and second-order optimality
Hans D. Mittelmann
2003-01-25