Optimal strategies for controlled growth in metastable Kawasaki dynamics

In this paper, we develop a Markov decision process (MDP) formulation for the low–temperature metastable Ising model evolving according to Kawasaki dynamics in a finite box of the two–dimensional square lattice. We analyze how an external controller can guide the system to the all–occupied state by appropriately adding and moving particles at specified moments in time. To this end, we construct a reduced MDP on a constrained family of configurations having a single cluster, a regime where particle attachment is more likely than detachment. We investigate two reward structures: one that depends solely on the time to reach the target configuration, and another that incorporates action–dependent energy costs. Within this MDP framework, we characterize the exact optimal policies under both reward structures, which turn out to have a different behavior: while a purely efficiency–based criterion promotes the growth from the boundary centers of the cluster, an energy–based reward function favours the growth at the corners of the cluster.