Directional convexity and finite optimality conditions

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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
Issue number1
StatePublished - Jul 1987

All Science Journal Classification (ASJC) codes

  • Analysis
  • Applied Mathematics


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