Lectures on the Calculus of Variations and Optimal Control Theory

admissible curves boundary bounded C₁ C₂ calculus of variations canonical Chapter clearly completes the proof conjugate consists constant continuous function continuously differentiable convergence convex convex function corresponding defined definition deformation denote derivative differential equation domain dual Euclidean Euclidean space Euler equation existence theory expression extremal arc finite number follows function x(t further geodesic given Hamiltonian Hence homogeneous hyperplane inequality infimum integral integrand intersection interval Lagrange Lagrange brackets Lagrange problem Lagrangian lemma limit line element linear function mathematics matrix maximum principle minimum neighborhood nonparametric nonsingular notion optimal control origin pair parametric polygonal flow positive positive cone rectifiable curve restriction Riesz measure satisfy scalar secondary Section segment sequence simply solution space subset suppose supremum suspected optimal t₁ tangent term theorem trajectory vanishes variables variational problem vector write x-space