Past exam of the mathematics course of the University of Cambridge / 2011 / ii / Paper 4 / 14C (Dynamical Systems)

The First Lyapunov theorem states that for a locally Lipschitz autonomous system $\dot{x}=f(x)$ with equilibrium zero, if a $C^1$ function $V$ is positive definite near zero and $\dot{V}=\nabla V\cdot f\le 0$, then the equilibrium is Lyapunov stable. To prove it, choose a small sphere $|x|=\epsilon$ inside that neighborhood. Compactness and positivity give $m={\min }_{|x|=\epsilon }V(x)>0$. Continuity at zero gives a smaller ball $|x|<\delta$ where $V<m$. Since $V$ cannot increase along a trajectory, a trajectory starting there cannot first cross the sphere where $V\ge m$. It stays within the sphere for all future time, with continuation justified by confinement in a compact subset of the vector field's domain. This proves stability for every sufficiently small $\epsilon$.

The LaSalle invariance principle states that on a compact positively invariant set on which $\dot{V}\le 0$, every trajectory approaches the largest invariant subset of $\{\dot{V}=0\}$. If that subset is just the equilibrium, the trapping and stability argument proves asymptotic stability.

For example take $\dot{x}=y$, $\dot{y}=-x-y$ and $V=(x^2+y^2)/2$. Then $\dot{V}=-y^2$, which is not strictly negative on the punctured plane. Nevertheless its sublevel discs are compact and invariant, and a trajectory confined to $y=0$ must have $\dot{y}=-x=0$, hence $x=0$. LaSalle gives asymptotic stability despite the non-strict orbital derivative of this particular energy function.

The printed phrase about nonexistence of a strict Lyapunov function must be understood as a limitation of the chosen candidate. In this example a strict function actually exists:More generally the converse Lyapunov theorem supplies local strict functions for asymptotically stable sufficiently smooth systems. LaSalle avoids having to find such a function; it does not show that none exists.

Choose the ellipsoidal Lyapunov function $V=x^2+2y^2$. Direct substitution of the printed vector field givesThe weighted Cauchy-Schwarz inequality gives $(x-y{)}^2\le \frac{3}{2}(x^2+2y^2)=3V/2$, soFor any initial point with $V(0)<2/3$, choose $c$ with $V(0)<c<2/3$. The compact ellipse $V\le c$ is positively invariant, since $\dot{V}<0$ on its boundary. Inside it,This establishes global forward existence for those trajectories and convergence to the origin, while the First Lyapunov theorem establishes stability. The origin is asymptotically stable and its basin contains $\boxed{x^2+2y^2<2/3}$. The PDF coefficient in the second equation is $x^{2}y/2$, with the entire linear term $-x$; placing the factor $1/2$ on the linear term would destroy this factorization.