Invariant sublevel set

If a Lyapunov function is nonincreasing on a sublevel set, trajectories cannot cross its boundary outward.

A smooth gradient flow confined to a compact invariant sublevel set has finite total integral of $\Vert \nabla V{\Vert }^2$ when $V$ is bounded below. Compactness supplies uniform continuity of that squared norm along the trajectory, so it tends to zero. Every accumulation point is a critical point. If the trapped component contains exactly one critical point, the trajectory converges to it. This supplies a direct convergence proof in place of an inference solely from an energy-contour picture.

Suppose a compact connected component $C$ of $\{x:V(x)\le c\}$ contains an equilibrium $x_{\ast }$ and $\dot{V}<0$ throughout $C\setminus \{x_{\ast }\}$. Then $C$ is an invariant sublevel set, and every trajectory in $C$ converges to $x_{\ast }$ by the LaSalle invariance principle. Thus the interior of $C$ lies in the basin of attraction of $x_{\ast }$. The largest such component supplies the strongest basin estimate available from that Lyapunov function alone.

For $\dot{z}=Az+\Vert z{\Vert }^{2}z$, suppose a positive-definite matrix $P$ solves $A^{T}P+PA=-2I$. The quadratic function $V=z^{T}Pz$ then obeys $\dot{V}=2\Vert z{\Vert }^2(V-1)$. Its unit ellipsoid separates decay toward the origin from escape. In a planar system with nonzero angular speed it is an invariant periodic orbit.

A boundary point with $\dot{V}=0$ can still enter the sublevel set; LaSalle analysis decides whether it belongs to an invariant zero-derivative trajectory.