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

Mohamed Camil Belhadjoudja, Miroslav Krstic, Mohamed Maghenem, Emmanuel Witrant

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.

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.
Paper Structure (12 sections, 1 theorem, 51 equations, 1 figure)

This paper contains 12 sections, 1 theorem, 51 equations, 1 figure.

Table of Contents

  1. Introduction and Main Result
  2. Sontag's formula
  3. Going beyond control-affine systems: Framework
  4. Going beyond control-affine systems: Motivation
  5. Main result
  6. Construction of the Universal Controllers
  7. Upperbounding $\dot{V}$
  8. Flow Convection
  9. Counter-convection
  10. Buckmaster Convection
  11. Numerical Example
  12. Conclusion

Key Result

Theorem 1

Consider the control system eq1, eq2-1, eq2-2, along with $V$ defined in eqdefV. For each of the convection terms in flow, linear, and buckmaster, there exists a continuous feedback with the property $v(0)=0$ and guaranteeing that I3 holds for the closed-loop regular solutions. $\square$

Figures (1)

  • Figure 1: Open-loop response (top) of system \ref{['blow_up']}. Closed-loop response (bottom) of system \ref{['blow_up']}.

Theorems & Definitions (3)

  • Theorem 1
  • Remark 1
  • Remark 2