### Abstract

Second order necessary conditions are presented 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 Γ, where Γ is a closed convex set of a Banach space with nonempty interior. The inequality constraints g(s, x, u) ≤ 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_{tεT} f(t, x, u) where T is a compact metric space, f is upper semicontinuous in t and Lipshitz in (x, u). The results are in terms a function σ 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) | 3998-4003 |

Number of pages | 6 |

Journal | Proceedings of the IEEE Conference on Decision and Control |

Volume | 4 |

Publication status | Published - Dec 1 1994 |

Event | Proceedings of the 33rd IEEE Conference on Decision and Control. Part 1 (of 4) - Lake Buena Vista, FL, USA Duration: Dec 14 1994 → Dec 16 1994 |

### ASJC Scopus subject areas

- Control and Systems Engineering
- Modelling and Simulation
- Control and Optimization

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## Cite this

*Proceedings of the IEEE Conference on Decision and Control*,

*4*, 3998-4003.