11.3 Basins of attraction and the Lyapunov equation
If \(V\) is a strict Lyapunov function on a region \(D\) and \(\Omega _c = \{x : V(x) < c\}\) is a bounded set whose closure lies in \(D\), then \(\Omega _c\) is positively invariant and every solution starting in it tends to \(0\): \(\Omega _c\) is contained in the basin of attraction. Choosing \(c\) as large as possible gives an inner estimate of the basin. For \(x' = -x + x^3\), \(y' = -y\), with \(V = x^2 + y^2\), \(\dot V = -2x^2(1 - x^2) - 2y^2 < 0\) for \(0 < x^2 + y^2\) and \(x^2 < 1\), so the disc \(x^2 + y^2 < 1\) lies in the basin; the true basin is the strip \(|x| < 1\).
For a linear system \(x' = Ax\), quadratic Lyapunov functions are found by linear algebra.
Theorem 11.3.1. All eigenvalues of \(A\) have negative real part if and only if for some (equivalently, every) symmetric positive definite \(Q\) the Lyapunov equation \[ A^{\mathsf T}P + PA = -Q \] has a symmetric positive definite solution \(P\). The solution is then unique, \(P = \int _0^\infty e^{A^{\mathsf T}t}Qe^{At}\,dt\), and \(V(x) = x^{\mathsf T}Px\) is a strict Lyapunov function, \(\dot V = -x^{\mathsf T}Qx\).
Proof. If \(P > 0\) solves the equation, \(V = x^{\mathsf T}Px\) has \(\dot V = x^{\mathsf T}(A^{\mathsf T}P + PA)x = -x^{\mathsf T}Qx < 0\), so the origin is asymptotically stable by Theorem 11.1.2, and every eigenvalue has negative real part (an eigenvalue with \(\Real \lambda \geq 0\) would give a solution not tending to \(0\)). Conversely, if the eigenvalues have negative real parts, \(\|e^{At}\| \leq Ke^{-\alpha t}\), the integral converges, and it is symmetric and positive definite since \(x^{\mathsf T}Px = \int \|Q^{1/2}e^{At}x\|^2dt > 0\) for \(x \neq 0\). Then \(A^{\mathsf T}P + PA = \int _0^\infty \frac {d}{dt}\bigl (e^{A^{\mathsf T}t}Qe^{At}\bigr )dt = -Q\). Uniqueness: the map \(P \mapsto A^{\mathsf T}P + PA\) has eigenvalues \(\lambda _i + \lambda _j\), none zero. □
Combined with linearisation, the theorem gives a Lyapunov function for any non-linear system at a hyperbolic sink: \(V = x^{\mathsf T}Px\) with \(A = Df(0)\) works in a neighbourhood, since the non-linear terms are dominated there, and its sublevel sets give basin estimates.
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