Vertices of reachable sets in nonlinear control problems

For a system dx/dt=f(x,u ), x(0)=x⁰, with x∈IR ², u∈U⊆IR², and f sufficiently smooth and `generic´, it is shown that the number of `corners´ (or vertices) of the time T reachable set Ω_T does not exceed the number of vertices of U. Moreover, for any trajectory x(t), leading to a vertex xˆ=x(T) of Ω_T , the control u(t) satisfies u(t ) ≡uˆ, t∈ [0,T], with uˆ a vertex of U. A particular case of f(x,u)=Ax+(Bx)u is considered, where the above genericity conditions take especially simple form