Controlling Heterogeneous Stochastic Growth Processes on Lattices with Limited Resources

Abstract

We consider controlling a heterogeneous stochastic growth process defined on a lattice with a control resource constraint. We address heterogeneous effects in three respects: (i) the process grows at different rates for different directions on the lattice, (ii) the nodes of the lattice may have different dynamics, and (iii) nodes may have different priorities for control. We use a forest wildfire driven by a west-to-east wind near an urban region to illustrate our approach, where preserving the urban region is prioritized over the forest. We leverage the Galton-Watson branching process as an approximation to predict the process growth rate and stopping time and to construct effective control policies. Our approach is also applicable to processes with an underlying graph structure, such as robot swarms, disease epidemics, computer viruses, and social networks. In contrast to prior work, we directly address heterogeneous models and our framework allows for a broader class of control policy descriptions. Lastly, we characterize the conditions under which a control policy will stabilize a supercritical heterogeneous growth process.