- [[Zeno's paradoxes]] # Idea To help us understand a system will reach [[equilibrium]]. A function $f(x)$ is a Lyaponuv function if the following holds: 1. it has a maximum (or minimum) value 2. there is a $k > 0$ such that if $x_{x+1} \neq x_t, F(x_{t+1}) >F(x+t) + k$ If those two assumptions cold, at some point, $x_{x+1} = x$ ([[equilibrium]]). The idea is easy, but the difficult part is constructing the Lyapunov function. It's possible for a Lyapunov function to stop at somewhere less than optimal ([[local minimum]] and [[gradient descent]])^[https://www.coursera.org/learn/model-thinking/lecture/gYxGW/time-to-convergence-and-optimality]. ## Examples We can use Lyapunov functions to describe how [[cities self-organize because people try to avoid crowds]]. # References - https://www.coursera.org/learn/model-thinking/lecture/GlPDe/lyapunov-functions