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First- and second-order necessary conditions for control problems with constraints

Authors: Zsolt Páles and Vera Zeidan
Journal: Trans. Amer. Math. Soc. 346 (1994), 421-453
MSC: Primary 49K15; Secondary 49J52
MathSciNet review: 1270667
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Abstract: Second-order necessary conditions are developed for an abstract nonsmooth control problem with mixed state-control equality and inequality constraints as well as a constraint of the form $ G(x,u) \in \Gamma $, where $ \Gamma $ is a closed convex set of a Banach space with nonempty interior. The inequality constraints $ g(s,x,u) \leqslant 0$ depend on a parameter $ s$ belonging to a compact metric space $ S$. The equality constraints are split into two sets of equations $ K(x,u) = 0$ and $ H(x,u) = 0$, where the first equation is an abstract control equation, and $ H$ is assumed to have a full rank property in $ u$. The objective function is $ {\max _{t \in T}}f(t,x,u)$ where $ T$ is a compact metric space, $ f$ is upper semicontinuous in $ t$ and Lipschitz in $ (x,u)$. The results are in terms of a function $ \sigma $ that disappears when the parameter spaces $ T$ and $ S$ are discrete. We apply these results to control problems governed by ordinary differential equations and having pure state inequality constraints and control state equality and inequality constraints. Thus we obtain a generalization and extension of the existing results on this problem.

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Keywords: Nonsmooth functions, second-order necessary conditions, mixed state and/or control equality constraints, state and/or control inequality constraints with parameter, abstract control equation, optimal controls
Article copyright: © Copyright 1994 American Mathematical Society