Core Concepts
Generalizing Sontag's formula to non-affine systems with polynomial derivatives for boundary-actuated PDEs.
Abstract
The content discusses the generalization of Sontag's formula to systems not affine in control, focusing on PDEs with boundary actuation. It introduces a continuous universal controller for systems with depressed cubic, quadratic, or depressed quartic dependence on the control. The article proves the results in the context of convection-reaction-diffusion PDEs with Dirichlet actuation. It extends Sontag's formula to non-affine systems and presents a new Lyapunov approach for boundary control of nonlinear PDEs. The content is structured into sections discussing the introduction and main result, going beyond control-affine systems, the motivation, the main result, construction of universal controllers, a numerical example, and a conclusion. The article also references related works and provides insights into the stabilization of challenging PDE systems.
Stats
"We assume that the system admits a control Lyapunov function (CLF) whose derivative, rather than being affine in the control, has either a depressed cubic, quadratic, or depressed quartic dependence on the control."
"For each case, a continuous universal controller that vanishes at the origin and achieves global exponential stability is derived."
"We show that if the convection has a certain structure, then the L2 norm of the state is a CLF."
Quotes
"We propose the first generalization of Sontag’s universal controller to systems not affine in the control."
"We illustrate our results via a numerical example."
"Our result not only generalizes Sontag formula to non-affine cases but also introduces a new method for boundary control of PDEs."