From Sontag s to Cardano-Lyapunov Formula for Systems Not Affine in the Control: Convection-Enabled PDE Stabilization

Abstract

We propose the first generalization of Sontag s universal controller to systems not affine in the control, particularly, to PDEs with boundary actuation. 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 prove our result in the context of convectionreaction-diffusion PDEs with Dirichlet actuation. We show that if the convection has a certain structure, then the L2 norm of the state is a CLF. In addition to generalizing Sontag s formula to some non-affine systems, we present the first general Lyapunov approach for boundary control of nonlinear PDEs. We illustrate our results via a numerical example.