2025/12/29 by Subrata Majumdar, Majumdar, Subrata, Debanjit Mondal +1
Computer Science · Engineering · Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Neural Networks Stability and Synchronization #Nonlinear Differential Equations Analysis #Optimization and Control (math.OC) #Stability and Controllability of Differential Equations
paper · doi:10.48550/arxiv.2512.23339
openalex publication_date 2025/12/29 · openalex created_date 2025/12/31 · openalex updated_date 2026/07/28
In this work, we investigate the small-time global controllability properties of a class of fourth-order nonlinear parabolic equations driven by a bilinear control posed on the one-dimensional torus. The controls depend only on time and act through a prescribed family of spatial profiles. Our first result establishes the small-time global approximate controllability of the system using three scalar controls, between states that share the same sign. This property is obtained by adapting the geometric control approach to the fourth-order setting, using a finite family of frequency-localized controls. We then study the small-time global exact controllability to non-zero constant states for the concerned system. This second result is achieved by analyzing the null controllability of an appropriate linearized fourth-order system and by deducing the controllability of the nonlinear bilinear model through a fixed-point argument together with the small-time global approximate control property.