Optimal Control for Suppression of Singularity in Chemotaxis via Flow Advection

• Published in: Applied mathematics & optimization Vol. 89; no. 3; p. 57
• Main Authors: Hu, Weiwei, Lai, Ming-Jun, Lee, Jinsil
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.06.2024; Springer Nature B.V
• Subjects: Advection; Applied mathematics; Boundary conditions; Calculus of Variations and Optimal Control; Optimization; Collocation methods; Control; Design; Mathematical and Computational Physics; Mathematical Methods in Physics; Mathematics; Mathematics and Statistics; Nonlinear control; Numerical and Computational Physics; Optimal control; Optimization; Simulation; Singularities; Splines; Systems Theory; Theoretical; Two dimensional flow; Patlak–Keller–Segel system; Cellular flows; Spline collocation methods; Optimal control; Flow advection; Chemotaxis; Suppression of singularity
• ISSN: 0095-4616; 1432-0606
• DOI: 10.1007/s00245-024-10122-9

This work focuses on the optimal control design for suppressing the singularity formation in chemotaxis governed by the parabolic-elliptic Patlak–Keller–Segel (PKS) system via flow advection. The main idea of this work lies in utilizing flow advection for enhancing diffusion as to control the nonlinear behavior of the system. The objective is to determine an optimal strategy for adjusting flow strength so that the possible finite time blow-up of the solution can be suppressed. Rigorous proof of the existence of an optimal solution and derivation of first-order optimality conditions for solving such a solution are presented. Spline collocation methods are employed for solving the optimality conditions. Numerical experiments based on 2D cellular flows in a rectangular domain are conducted to demonstrate our ideas and designs.