Lyapunov energy of the accelerated flow (strongly convex)

For \mu-strongly convex f the right flow has constant damping \ddot X + 2\sqrt{\mu}\,\dot X + \nabla f(X)=0 and converges linearly. The Lyapunov energy E(t)=\bigl(f(X(t))-f^*\bigr)+\tfrac12\bigl\|\dot X(t)+\sqrt{\mu}\,(X(t)-x^*)\bigr\|^2 satisfies \dot E\le-\sqrt{\mu}\,E, hence E(t)\le E(0)e^{-\sqrt{\mu}\,t} and f(X(t))-f^*\le E(0)e^{-\sqrt{\mu}\,t} — a straight line on the semi-log plot.

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