From Sontag s to Cardano-Lyapunov Formula for Systems Not Affine in the Control: Convection-Enabled PDE Stabilization
<|reference_start|>From Sontag s to Cardano-Lyapunov Formula for Systems Not Affine in the Control: Convection-Enabled PDE Stabilization: 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.<|reference_end|>
@article{belhadjoudja2024from,
    title={From Sontag s to Cardano-Lyapunov Formula for Systems Not Affine in the
  Control: Convection-Enabled PDE Stabilization},
    author={Mohamed Camil Belhadjoudja, Miroslav Krstic, Mohamed Maghenem,
  Emmanuel Witrant},
    journal={arXiv preprint arXiv:2403.18071},
    year={2024},
    archivePrefix={arXiv},
    eprint={2403.18071},
    primaryClass={eess.SY cs.SY math.AP}
}