The control of stochastic growth processes has attracted a large amount of interest recently, motivated, for instance, by the desire to control and contain the spread of epidemic diseases. In this paper, we consider the control of stochastic growth processes on lattices. Throughout the paper, we present our results using the forest wildfire as an example, in which the control is exerted by a limited number of autonomous vehicles that can be assigned to extinguish fires on individual trees planted in a grid formation. However, our results are broadly applicable to problems that involves nodes connected into lattice-like structures, such as control of transportation networks on a Manhattan grid. In this context, we define a notion of stability for stochastic growth processes on lattices, and we derive analytical bounds for control effort that guarantees stability. The lattice structure allows us to analytically characterize stabilizing control policies with the help of certain recent results from the statistical mechanics literature. This analysis leads to randomized policies that stabilize originally unstable stochastic growth processes on lattices almost surely, and guarantee asymptotic optimality both in terms of the allocation and the utilization of control effort.