Steering Opinion Dynamics in Signed Time-Varying Networks via External Control Input

This paper studies targeted opinion formation in multi-agent systems evolving over signed, time-varying directed graphs. The dynamics of each agent's state follow a Laplacian-based update rule driven by both cooperative and antagonistic interactions in the presence of exogenous factors. We formulate these exogenous factors as external control inputs and establish a suitable controller design methodology enabling collective opinion to converge to any desired steady-state configuration, superseding the natural emergent clustering or polarization behavior imposed by persistently structurally balanced influential root nodes. Our approach leverages upper Dini derivative analysis and Gr\"onwall-type inequalities to establish exponential convergence for opinion magnitude towards the desired steady state configuration on networks with uniform quasi-strong $\delta$-connectivity. Finally, the theoretical results are validated through extensive numerical simulations.