In this talk, he will discuss some of our recent results on the analysis and control of the complex Ginzburg–Landau equation (CGLE) on a finite spatial domain. This equation models a wide range of physical phenomena, from instability waves to superconductivity and Bose–Einstein condensation.

Our first focus is the local and global well-posedness of the CGLE with mixed Dirichlet–Neumann boundary conditions. He will present a framework for establishing local well-posedness in fractional Sobolev spaces using sharp space-time estimates derived via the Fokas method, also known as the unified transform. These estimates allow us to rigorously treat inhomogeneous boundary data and nonlinear effects.

In the second part, he turn to the problem of chaos suppression through boundary control. Specifically, he introduce a nonlocal boundary feedback law that uses only a finite number of Fourier modes of the solution. This exploits the fact that long-time dynamics of the CGLE are governed by a finite-dimensional component, while high-frequency modes decay rapidly. He will explain how to design a controller that achieves exponential stabilization and identify the minimal number of modes required to ensure this behavior.

Finally, He will present numerical results that validate the theory and highlight how our control strategy effectively suppresses chaotic dynamics. The methods developed here pave the way for finite-dimensional stabilization techniques in broader classes of nonlinear dissipative systems.