Minimal Control Placement of Turing's Model Using Symmetries

In this paper, the minimal control placement prob-lem for Turing's reaction-diffusion model is studied. Turing's model describes the process of morphogens diffusing and reacting with each other and is considered as one of the most fundamental models to explain pattern formation in a devel-oping embryo. Controlling pattern formation artificially has gained increasing attention in the field of development biology, which motivates us to investigate this problem mathematically. In this work, the two-dimensional Turing's reaction-diffusion model is discretized into square grids. The minimal control placement problem for the diffusion system is investigated first. The symmetric control sets are defined based on the symmetry of the network structure. A necessary condition is provided to guarantee controllability. Under certain circumstances, we prove that this condition is also sufficient. Then we show that the necessary condition can also be applied to the reaction-diffusion system by means of suitable extension of the symmetric control sets. Under similar circumstances, a sufficient condition is given to place the minimal control for the reaction-diffusion system.