### Abstract

For Λ ε{lunate} R^{d}, we say that a set A⊆R^{d} is Λ-convex if the segment pq is contained in A whenever p, q ε{lunate} A and p - q ε{lunate} Λ. For the control system x ̇(t)=f(x(t), u(t)), x(0)=0εR^{d} with u(t)εΩ⊆R^{m} for tε[0, T], results are provided which establish the Λ-convexity of the reachable set R(T). They rely on a uniqueness assumption for solutions of Pontrjagin's equations and are proven by means of a generalized Mountain Pass Theorem. When the reachable set is Λ-convex, trajectories reaching the boundary of R(T) satisfy some additional necessary conditions. A stronger version of the Pontrjagin Maximum Principle can thus be proven.

Original language | English (US) |
---|---|

Pages (from-to) | 234-246 |

Number of pages | 13 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 125 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1987 |

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### All Science Journal Classification (ASJC) codes

- Analysis
- Applied Mathematics

### Cite this

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*Journal of Mathematical Analysis and Applications*, vol. 125, no. 1, pp. 234-246. https://doi.org/10.1016/0022-247X(87)90178-8

**Directional convexity and finite optimality conditions.** / Bressan, Alberto.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Directional convexity and finite optimality conditions

AU - Bressan, Alberto

PY - 1987/1/1

Y1 - 1987/1/1

N2 - For Λ ε{lunate} Rd, we say that a set A⊆Rd is Λ-convex if the segment pq is contained in A whenever p, q ε{lunate} A and p - q ε{lunate} Λ. For the control system x ̇(t)=f(x(t), u(t)), x(0)=0εRd with u(t)εΩ⊆Rm for tε[0, T], results are provided which establish the Λ-convexity of the reachable set R(T). They rely on a uniqueness assumption for solutions of Pontrjagin's equations and are proven by means of a generalized Mountain Pass Theorem. When the reachable set is Λ-convex, trajectories reaching the boundary of R(T) satisfy some additional necessary conditions. A stronger version of the Pontrjagin Maximum Principle can thus be proven.

AB - For Λ ε{lunate} Rd, we say that a set A⊆Rd is Λ-convex if the segment pq is contained in A whenever p, q ε{lunate} A and p - q ε{lunate} Λ. For the control system x ̇(t)=f(x(t), u(t)), x(0)=0εRd with u(t)εΩ⊆Rm for tε[0, T], results are provided which establish the Λ-convexity of the reachable set R(T). They rely on a uniqueness assumption for solutions of Pontrjagin's equations and are proven by means of a generalized Mountain Pass Theorem. When the reachable set is Λ-convex, trajectories reaching the boundary of R(T) satisfy some additional necessary conditions. A stronger version of the Pontrjagin Maximum Principle can thus be proven.

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U2 - 10.1016/0022-247X(87)90178-8

DO - 10.1016/0022-247X(87)90178-8

M3 - Article

VL - 125

SP - 234

EP - 246

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

SN - 0022-247X

IS - 1

ER -