Lyapunov function for the bounded-confidence Fokker–Planck model with liar control

Prove that V(t) = (m_T(t) − x_d)^2 is a Lyapunov function for the bounded-confidence Fokker–Planck equation governing the truth-teller density f_T(x,t) under liar control. Specifically, for solutions of ∂_t f_T(x,t) + (1/c_T) ∂_x(∫ P(x′−x)(x′−x) f_T(x′,t) dx′ · f_T(x,t)) + (ρ/c_L) ∂_x(P(y(x)−x)(y(x)−x) f_T(x,t)) = ((1/c_T) + (ρ/c_L)) (ζ/2) ∂_{xx}(D(x)^2 f_T(x,t)), with no-flux boundary conditions on the opinion interval and where the control satisfies y(x) = x_d − (1/κ)(x − x_d) ∂_y{P(y−x)(y−x)} projected onto the opinion interval, determine that d/dt V(t) ≤ 0 for all t ≥ 0 and appropriate smooth solutions, thereby establishing V(t) as a Lyapunov function.

Background

In the bounded-confidence kinetic formulation, the macroscopic dynamics of the truth-teller opinion density f_T(x,t) are governed by a Fokker–Planck equation derived via a Boltzmann-type binary interaction model and a quasi-invariant limit. The drift terms encode interaction among truth-tellers and influence from a liar whose apparent opinion y(x) is determined by an instantaneous optimal control subject to regularization.

The authors propose V(t) = (m_T(t) − x_d)^2, with m_T(t) the mean opinion of truth-tellers and x_d the liar’s goal, as a candidate Lyapunov function to capture convergence towards the liar’s target in the bounded-confidence regime. Numerical experiments indicate monotone decay of V(t), but an analytic proof is nontrivial due to the structure of the control term and the derivative of the interaction kernel.

Establishing V(t) as a Lyapunov function would provide rigorous global stability and convergence guarantees for the controlled kinetic model with bounded confidence and regularized liar control, clarifying when and how the population’s mean opinion approaches the liar’s target.