TY - JOUR

T1 - Directional convexity and finite optimality conditions

AU - Bressan, Alberto

N1 - Funding Information:
* Supported by the U.S. Army under Contract DAAG29-80-C-0041. + Present address: Department of Mathematics, University of Colorado, Boulder, Co.

PY - 1987/7

Y1 - 1987/7

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

AN - SCOPUS:38249035768

SN - 0022-247X

VL - 125

SP - 234

EP - 246

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -