Finite-preparation residual budget and its remote-boundary limit

Fix arbitrary finite real constants $\mathrm{\Gamma }>0$ and $\tau \ge 0$. Fix positive integers $m,n$, a field $\mathbb{F}\in \{\mathbb{R},\mathbb{C}\}$ with its usual absolute value, and the matrix space $E=M_{m\times n}(\mathbb{F})$ with any fixed norm $\mathrm{\parallel }\cdot \mathrm{\parallel }:E\to {\mathbb{R}}_{\ge 0}$. Let $B(t):E\to E$ be a fixed family of maps, defined for all real $t$; linearity is not required. Let $(t_{\nu }{)}_{\nu \ge 1}$ be a real sequence with $t_{\nu }\to -\mathrm{\infty }$, and let ${\rho }_{\nu }:(t_{\nu },\mathrm{\infty })\to E$ be differentiable curves. Derivatives are taken with respect to real time in real coordinates, using the real and imaginary entries when $\mathbb{F}=\mathbb{C}$, and are reassembled as matrices in $E$. Suppose $\mathrm{\parallel }{\dot{\rho }}_{\nu }(t)-B(t)[{\rho }_{\nu }(t)]\mathrm{\parallel }\le {\mathrm{\Gamma }}^2\tau +\mathrm{\Gamma }\tau \mathrm{/}(t-t_{\nu })$ for every $t>t_{\nu }$.

For every fixed finite observation interval $[u,v]$ with $u\le v$, choose ${\nu }_0$ such that $t_{\nu }<u$ for all $\nu \ge {\nu }_0$. For these indices define $R_{\nu }={\sup }_{t\in [u,v]}\mathrm{\parallel }{\dot{\rho }}_{\nu }(t)-B(t)[{\rho }_{\nu }(t)]\mathrm{\parallel }$ and ${\delta }_{\nu }=\mathrm{\Gamma }\tau \mathrm{/}(u-t_{\nu })$. Then $0\le R_{\nu }\le {\mathrm{\Gamma }}^2\tau +{\delta }_{\nu }$, ${\delta }_{\nu }\to 0$, and ${\mathrm{lim\hspace{0.17em}sup}}_{\nu \ge {\nu }_0}{R}_{\nu }\le {\mathrm{\Gamma }}^2\tau$. The limit superior is taken on this tail, reindexed as a sequence. This is a limit of error budgets; it asserts no convergence, existence or uniqueness of the matrix curves themselves.