Controllability of a Reaction-Diffusion System Describing Predator–Prey Model

Abstract

This article establishes the controllability to the trajectories of a reaction-diffusion-advection system describing predator–prey model by using distributed controls acting on a single equation with the no-flux boundary conditions. We first prove the exact null controllability of an associated linearized problem by establishing an observability estimate, derived from a global Carleman type inequality, for the adjoint system. The proof of the nonlinear problem relies on the suitable regularity of the control and Kakutani's fixed point theorem.

Keywords:

• Carleman estimates
• Controllability
• Observability
• Predator–prey model

AMS Subject Classification:

• 35K55
• 93B05
• 93C20
• 93C35

ACKNOWLEDGMENTS

The authors are thankful to the referees for their comments and suggestions that led to the final version of this article.

Notes

K.S.'s work was supported by the Brain Korea 21 Project at Yonsei University, 2008. J.-H.K.'s work was supported by the Korea Research Foundation Grant funded by the Korean Government (MOEHRD, Basic Research Promotion Fund; KRF-2008-314-C00045).