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

Z. Páles, Vera Zeidan

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39 Citations (Scopus)


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) ϵ Г, where Г is a closed convex set of a Banach space with nonempty interior. The inequality constraints g(s, x, u) ≤ 0 depend on a parameter 5 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 maxtϵT 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 a 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.

Original languageEnglish
Pages (from-to)421-453
Number of pages33
JournalTransactions of the American Mathematical Society
Issue number2
Publication statusPublished - 1994



  • Abstract control equation
  • Mixed state and/or control equality constraints
  • Nonsmooth functions
  • Optimal controls
  • Second-order necessary conditions
  • State and/or control inequality constraints with parameter

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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