### Abstract

In this paper the notion of critical tangent cone CT(x|Q) is introduced. When Q is closed, convex with nonempty interior, x ∈ Q, then the nonemptiness of the Dubovitskii-Milyutin set of second-order admissible variations, V (x, d|Q), is characterized by the condition d ∈ CT(x|Q). More verifiable characterization is obtained for the cases where Q is the set of continuous or L^{∞} selections of a certain set-valued map. In the latter case, a strong normality condition in terms of CT(x(t)|Q(t)) is defined in order that the Lagrange multiplier corresponding to the L^{∞}-selections set be represented via integrable functions. Finally these results are applied to a general optimal control problem and second-order optimality conditions are derived in terms of the original data.

Original language | English |
---|---|

Pages (from-to) | 1149-1161 |

Number of pages | 13 |

Journal | Nonlinear Analysis |

Volume | 47 |

Issue number | 2 |

DOIs | |

Publication status | Published - Aug 2001 |

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### Keywords

- Critical cone
- Critical tangent cone
- First- and second-order optimality conditions
- Set-valued constraints

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics
- Mathematics(all)

### Cite this

*Nonlinear Analysis*,

*47*(2), 1149-1161. https://doi.org/10.1016/S0362-546X(01)00254-1

**The critical tangent cone in second-order conditions for optimal control.** / Páles, Z.; Zeidan, V.

Research output: Contribution to journal › Article

*Nonlinear Analysis*, vol. 47, no. 2, pp. 1149-1161. https://doi.org/10.1016/S0362-546X(01)00254-1

}

TY - JOUR

T1 - The critical tangent cone in second-order conditions for optimal control

AU - Páles, Z.

AU - Zeidan, V.

PY - 2001/8

Y1 - 2001/8

N2 - In this paper the notion of critical tangent cone CT(x|Q) is introduced. When Q is closed, convex with nonempty interior, x ∈ Q, then the nonemptiness of the Dubovitskii-Milyutin set of second-order admissible variations, V (x, d|Q), is characterized by the condition d ∈ CT(x|Q). More verifiable characterization is obtained for the cases where Q is the set of continuous or L∞ selections of a certain set-valued map. In the latter case, a strong normality condition in terms of CT(x(t)|Q(t)) is defined in order that the Lagrange multiplier corresponding to the L∞-selections set be represented via integrable functions. Finally these results are applied to a general optimal control problem and second-order optimality conditions are derived in terms of the original data.

AB - In this paper the notion of critical tangent cone CT(x|Q) is introduced. When Q is closed, convex with nonempty interior, x ∈ Q, then the nonemptiness of the Dubovitskii-Milyutin set of second-order admissible variations, V (x, d|Q), is characterized by the condition d ∈ CT(x|Q). More verifiable characterization is obtained for the cases where Q is the set of continuous or L∞ selections of a certain set-valued map. In the latter case, a strong normality condition in terms of CT(x(t)|Q(t)) is defined in order that the Lagrange multiplier corresponding to the L∞-selections set be represented via integrable functions. Finally these results are applied to a general optimal control problem and second-order optimality conditions are derived in terms of the original data.

KW - Critical cone

KW - Critical tangent cone

KW - First- and second-order optimality conditions

KW - Set-valued constraints

UR - http://www.scopus.com/inward/record.url?scp=0035425783&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0035425783&partnerID=8YFLogxK

U2 - 10.1016/S0362-546X(01)00254-1

DO - 10.1016/S0362-546X(01)00254-1

M3 - Article

AN - SCOPUS:0035425783

VL - 47

SP - 1149

EP - 1161

JO - Nonlinear Analysis, Theory, Methods and Applications

JF - Nonlinear Analysis, Theory, Methods and Applications

SN - 0362-546X

IS - 2

ER -