papers/hanzo-hmm/sections/99-lyapunov-stability-proof.tex

\section{Lyapunov Stability Proof}

\label{app:lyapunov}

\begin{theorem}[Exponential Stability]
With fee dissipation rate $\alpha > 0$, the equilibrium $(\bar{\bm{q}}, \bar{\bm{p}})$ is exponentially stable with rate $\alpha$.
\end{theorem}

\begin{proof}
Define the Lyapunov function $V = |\mathcal{H} - \kappa|^2 + \gamma\sum_i(q_i - \bar{q}_i)^2$ for suitable $\gamma > 0$. Fee dissipation ensures $d\mathcal{H}/dt = -\alpha(\mathcal{H} - \kappa) + O(|\bm{q} - \bar{\bm{q}}|^2)$. Therefore:
\begin{align}
\frac{dV}{dt} &= 2(\mathcal{H} - \kappa)\frac{d\mathcal{H}}{dt} + 2\gamma\sum_i(q_i - \bar{q}_i)\dot{q}_i \\
&\le -2\alpha(\mathcal{H} - \kappa)^2 + 2\gamma\sum_i(q_i - \bar{q}_i)w_iq_i \\
&\le -\min(\alpha, \gamma\lambda) \cdot V,
\end{align}
for $\gamma$ chosen such that the cross-terms are dominated. This gives $V(t) \le V(0)e^{-\min(\alpha,\gamma\lambda)t}$.
\end{proof}

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\vspace{1em}
\noindent\textit{Disclaimer.} This document describes a proposed protocol. Security properties require formal verification and independent audit before production deployment.