Dynamic Programming
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Given an nxr integrable matrix function Y(t), we extend the Lyapunov-Lindenstrauss theorem describing extreme points of the set {@!"0^TY(t)u(t)dt|u@?I} from the Cartesian product I of r Lipschitz classes to the Cartesian product I=H^@w[0,T]:=H^@w^"^1[0,T]x...xH^@w^"^r[0,T] of classes H^@w[0,T] of functions with the modulus of continuity majorized by the given concave modulus of continuity @w. 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(.)@?H^@w[0,T],x(0),u(0)=0,x(T),u(T)=(@L@^,@C@^),for locally integrable nxn- and nxr-matrix valued functions A(t) and B(t), the collection @w=(@w"1,...,@w"r) of concave moduli of continuity, and @L@^@?R^n, @C@^@?R^r. 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^)@?H^@w(R"+), connecting two given points in R^r^+^1. Then, we describe the extreme points of the setS"@w","r","@t","a:={(x(@t),x^'(@t),...,x^(^r^)(@t))|x^(^r^)@?H^@w[0,T]:x^(^i^)(0)=a"i,i=0,...,r}for a=(a"0,...,a"r)@?R^r^+^1, @t0. This problem is related to the Kolmogorov problem for intermediate derivatives where the triples (x(@t),x^(^m^)(@t),x^(^r^)(@t)) are considered for 0