  Autonomous vehicles (AVs) enable more efficient and sustainable
transportation systems. Ample studies have shown that controlling a small
fraction of AVs can smooth traffic flow and mitigate traffic congestion.
However, deploying AVs in real-world systems poses challenges due to safety and
cost concerns. A viable alternative approach that can be implemented in the
near future is $\textit{coarse-grained guidance}$, where human drivers are
guided by real-time instructions, updated every $\Delta$ seconds, to stabilize
the traffic. While previous theoretical studies consider stability analysis for
continuous AV control, this article presents the first integrated theoretical
analysis that directly relates the guidance provided to the human drivers to
the traffic flow stability outcome. Casting the problem into the Lyapunov
stability framework, this study derives sufficient conditions for
coarse-grained guidance with hold length $\Delta$ to stabilize the system, and
provides extensions of the analysis to incorporate additional human driving
behaviors such as magnitude error and reaction delay. Numerical simulations
reveal that the theoretical analysis closely matches simulated results. The
analysis further offers insights into the relationship between system parameter
and stability criteria, and can be leveraged to design improved controllers
with greater maximum hold length.
