Optimal trajectories for quadratic variational processes via invariant imbedding

Consider minimizing the integral $$I = \int_0^T {\dot w^2 + g(y)w^2 ] dy}$$ where $$w = w(y), \dot w = dw/dy, w(T) = 1, w(0) = free$$ ForT sufficiently small, it is shown that $$w_{opt} = x(t,T), 0 \leqslant t \leqslant T$$ where the functionx, viewed as a function ofT, is a solution of the Cauchy problem $$\begin{gathered} x_T (t,T) = r(T)x(t,T), T \geqslant t \hfill \\ x(t,t) = 1 \hfill \\\end{gathered}$$ and the auxiliary functionr satisfies the Riccati system $$\begin{gathered} r_T = ---g(T) + r^2 , T \geqslant 0 \hfill \\ r(0) = 0 \hfill \\\end{gathered}$$ In the derivation of the Cauchy problem, no use is made of Euler equations, dynamic programming, or Pontryagin's maximum principle. Only ordinary differential equations are employed. The Cauchy problem provides a one-sweep integration procedure; it is intimately connected with the theory of the second variation.