10. [HM50] Consider the dynamics arising from the day–night boundary conditions (6.5.162).

What is the dependence on the initial values wi? Under which conditions after many days of life the agent behavior becomes independent of wi?

(Hint) Consider that the day–light boundary conditions do not affect the values of wi,j. The explicit presence on the Lagrangian of a penalizing term on wi,j might change the dynamics significantly.