Generalizations of the time optimal problem and Lyapunov theorem on the range of vector measures

Given an n×rn×r integrable matrix function Y(t)Y(t), we extend the Lyapunov–Lindenstrauss theorem describing extreme points of the set {∫0TY(t)u(t)dt|u∈I} from the Cartesian product II of rr Lipschitz classes to the Cartesian product I=Hω[0,T]:=Hω1[0,T]×⋯×Hωr[0,T]I=Hω[0,T]:=Hω1[0,T]×⋯×Hωr[0,T] of classes Hω[0,T]Hω[0,T] of functions with the modulus of continuity majorized by the given concave modulus of continuity ωω.We also explain the intimate relationship between the aforementioned problem and the characterization of extremal functions in the classical time minimization problem of optimal controlT→inf;x˙(t)=A(t)x(t)+B(t)u(t),u(·)∈Hω[0,T],x(0),u(0)=0,x(T),u(T)=(Λ^,Γ^),for locally integrable n×nn×n- and n×rn×r-matrix valued functions A(t)A(t) and B(t)B(t), the collection ω=(ω1,…,ωr)ω=(ω1,…,ωr) of concave moduli of continuity, and Λ^∈Rn, Γ^∈Rr.Relying on these results, we solve the classical rendezvous   problem of finding the optimal trajectory in the phase space (x,x˙,x¨,…,x(r)), x(r)∈Hω(R+)x(r)∈Hω(R+), connecting two given points in Rr+1Rr+1. Then, we describe the extreme points of the setSω,r,τ,a:={(x(τ),x′(τ),…,x(r)(τ))|x(r)∈Hω[0,T]:x(i)(0)=ai,i=0,…,r}for a=(a0,…,ar)∈Rr+1a=(a0,…,ar)∈Rr+1, τ>0τ>0. This problem is related to the Kolmogorov problem for intermediate derivatives where the triples (x(τ),x(m)(τ),x(r)(τ))(x(τ),x(m)(τ),x(r)(τ)) are considered for 0