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Karush-Kuhn-Tucker theorem for operator constraints and the local Pontryagin maximum principle. (English) Zbl 0912.49022

For the following optimization problem:
\( \min F(x,u) \) under constraints \( H(x,u) = \Theta_1 \), \( G(x,u) \leq \Theta_2 \), \( h_i(x,u) = 0 \), \( i = 1,\dots,m \), \( g_j(x,u) \leq 0 \), \( j = 1, \dots,n \), \( (x,u) \in X \times U \), \( u \in {\mathcal U} \), where \( F \), \( h_i \), \( g_j \) are functionals, \( H \), \( G \) are operators with values in \( Y \) and \( Z \), respectively, \( X \), \( Y \), \( U \) are Banach spaces, \( Z \) is a normed space, ordered by some convex cone, \( \Theta_1\), \( \Theta_2 \) denote zeros in \( Y \) and \( Z \), respectively, and \( \mathcal U \) is a subset of \( U \),
first order necessary conditions (Karush-Kuhn-Tucker) for a local minimum are derived using the general approach of Dubovitskij-Milyutin (see, A. J. Dubovitskij and A. A. Milyutin [Zh. Vychisl. Mat. Mat. Fiz. 5, 395-453 (1965; Zbl 0158.33504)]and I. V. Girsanov [“Lectures on mathematical theory of extremum problems” (Russian 1970; Zbl 0214.14502; English translation 1972; Zbl 0234.49016)]). Further, this result is used to obtain directly the local Pontryagin Maximum Principle for some optimal control problem. In the paper a clear link between Karush-Kuhn-Tucker Theorem and the Pontryagin Maximum Principle, both being a necessary condition for optimal control, is shown.

MSC:

49K27 Optimality conditions for problems in abstract spaces
90C48 Programming in abstract spaces
93C25 Control/observation systems in abstract spaces
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