Directional convexity and finite optimality conditions

Research output: Contribution to journalArticle

8 Citations (Scopus)

Abstract

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.

Original languageEnglish (US)
Pages (from-to)234-246
Number of pages13
JournalJournal of Mathematical Analysis and Applications
Volume125
Issue number1
DOIs
StatePublished - Jan 1 1987

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Reachable Set
Maximum principle
Optimality Conditions
Convexity
Trajectories
Control systems
Mountain Pass Theorem
Maximum Principle
Uniqueness
Control System
Trajectory
Necessary Conditions

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics

Cite this

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title = "Directional convexity and finite optimality conditions",
abstract = "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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Directional convexity and finite optimality conditions. / Bressan, Alberto.

In: Journal of Mathematical Analysis and Applications, Vol. 125, No. 1, 01.01.1987, p. 234-246.

Research output: Contribution to journalArticle

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