### 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) 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 max_{tϵT} 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 language | English |
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Pages (from-to) | 421-453 |

Number of pages | 33 |

Journal | Transactions of the American Mathematical Society |

Volume | 346 |

Issue number | 2 |

DOIs | |

Publication status | Published - 1994 |

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### Keywords

- 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