9 Lyapunov’s indirect method
Near an equilibrium a nonlinear system behaves like its linearization: if the Jacobian is Hurwitz the equilibrium is locally exponentially stable, and if the Jacobian has an eigenvalue with positive real part it is unstable. Both branches run the same way. A quadratic certificate for the linear system (theorem 389, theorem 405) is a certificate for the nonlinear one as well, once the linearization error is absorbed near the equilibrium.
9.1 The linearization error
Let \(A\) be the Fréchet derivative of \(f\) at \(x_0\). For every \(\varepsilon {\gt} 0\) there exists \(\delta {\gt} 0\) such that
This is the \(\varepsilon \)–\(\delta \) form of \(f(x) - f(x_0) - A(x-x_0) = o(\| x-x_0\| )\).
Let \(f\) be \(C^{1}\) with \(Df(x_{\mathrm{eq}}) = A\), let \(M\) be a matrix, and let \(c {\gt} 0\). Then there is \(r {\gt} 0\) such that for every \(x\) with \(\| x - x_{\mathrm{eq}}\| {\lt} r\),
where \(q_{M}\) is the centred quadratic form \(q_{M}(x) = \langle M(x - x_{\mathrm{eq}}),\, x - x_{\mathrm{eq}}\rangle \).
The bracket is the first-order remainder of \(f\) at \(x_{\mathrm{eq}}\). The point is that \(c\) is arbitrary: near enough to the equilibrium the remainder perturbs \(\dot{q}_{M}\) by less than any prescribed multiple of \(\| x - x_{\mathrm{eq}}\| ^{2}\), which is the order of \(\dot{q}_{M}\) itself. This is why the linearization decides stability.
Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 4.7 (the quadratic-Lyapunov proof of Lyapunov’s indirect method).
Write \(y = x - x_{\mathrm{eq}}\) and \(e\) for the remainder. Differentiability of \(f\) at \(x_{\mathrm{eq}}\) gives, for any \(\eta {\gt} 0\), a radius within which \(\| e\| \le \eta \| y\| \). The derivative of a centred quadratic form is bounded by \(|Dq_{M}(x)[e]| \le 2\| M\| \, \| y\| \, \| e\| \), so the product is at most \(2\| M\| \eta \, \| y\| ^{2}\). Choosing \(\eta = c / (4(\| M\| + 1))\) makes \(2\| M\| \eta \le c\).
9.2 The stable branch
The stable-branch theorem gives uniform local exponential decay on every finite forward solution segment, without silently excluding solutions that are not globally defined.
Let \(f : \mathbb {R}^n \to \mathbb {R}^n\) be \(C^1\), and let \(x_{\rm eq}\) be an equilibrium. If its Jacobian \(A\) at the equilibrium is Hurwitz, then \(x_{\rm eq}\) is locally exponentially stable (definition 284) for any system whose state equation is \(\dot x = f(x)\).
Reference: Khalil, Nonlinear Systems.
Solve the identity-forced Lyapunov equation for a positive-definite \(P\); after absorbing the \(o(\| x-x_{\rm eq}\| )\) linearization remainder on a small ball, \(V(x)=(x-x_{\rm eq})^{\mathsf T}P(x-x_{\rm eq})\) is an exponential Lyapunov function with exponent \(2\) there, so theorem 318 applies.
9.3 The unstable branch
The quadratic certificate is fed to Chetaev’s theorem (theorem 327).
Let \(f:\mathbb R^n\to \mathbb R^n\) be \(C^1\) with \(f(x_{\rm eq})=0\). If the Jacobian at \(x_{\rm eq}\) has a complex eigenvalue with positive real part, then \(x_{\rm eq}\) is unstable (definition 276) for any system whose state equation is \(\dot x = f(x)\).
Reference: Khalil, Nonlinear Systems (Lyapunov’s indirect method).
Build the real quadratic certificate supplied by theorem 405, absorb the first-order nonlinear remainder on a small ball, and apply the exponential Chetaev criterion.
If \(A \in \mathbb {R}^{n \times n}\) has a complex eigenvalue \(\mu \) with \(\operatorname {Re}\mu {\gt} 0\), then the origin is unstable (definition 276) for any system on \(\mathbb {R}^n\) whose state equation is \(\dot x = A x\).
Reference: Khalil, Nonlinear Systems (3rd ed.), Theorem 4.7.
The field \(x \mapsto Ax\) is \(C^1\), vanishes at \(0\) and is its own Jacobian, so this is theorem 435 at \(x_{\mathrm{eq}} = 0\).