Turing’s model has been widely used to explain how simple, uniform structures can give rise to complex, patterned structures during the development of organisms. However, it is very hard to establish rigorous theoretical results for the dynamic evolution behavior of Turing’s model since it is described by nonlinear partial differential equations. We focus on controllability of Turing’s model by linearization and spatial discretization. This linearized model is a networked system whose agents are second order linear systems and these agents interact with each other by Laplacian dynamics on a graph. A control signal can be added to agents of choice. Under mild conditions on the parameters of the linearized Turing’s model, we prove the equivalence between controllability of the linearized Turing’s model and controllability of a Laplace dynamic system with agents of first order dynamics. When the graph is a grid graph or a cylinder grid graph, we then give precisely the minimal number of control nodes and a corresponding control node set such that the Laplace dynamic systems on these graphs with agents of first order dynamics are controllable. 

Publication: 

Automatica, Volume 162, April 2024, 111507 

http://dx.doi.org/10.1016/j.automatica.2023.111507 

Author: 

Tianhao Li 

Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100190, PR China 

Ruichang Zhang 

Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100190, PR China 

Zhixin Liu 

Key Laboratory of Systems and Control, Academy of Mathematics and Systems Science, Chinese Academy of Sciences 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100190, PR China 

Email: lzx@amss.ac.cn 

Zhuo Zou 

Fudan University, Shanghai 200433, PR China 

Xiaoming Hu 

Optimization and Systems Theory, KTH Royal Institute of Technology, Stockholm 10044, Sweden