From Sontag s to Cardano-Lyapunov Formula for Systems Not Affine in the Control: Convection-Enabled PDE Stabilization
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866909152179126272 |
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| author | Belhadjoudja, Mohamed Camil Krstic, Miroslav Maghenem, Mohamed Witrant, Emmanuel |
| author_facet | Belhadjoudja, Mohamed Camil Krstic, Miroslav Maghenem, Mohamed Witrant, Emmanuel |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2403_18071 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | From Sontag s to Cardano-Lyapunov Formula for Systems Not Affine in the Control: Convection-Enabled PDE Stabilization Belhadjoudja, Mohamed Camil Krstic, Miroslav Maghenem, Mohamed Witrant, Emmanuel Systems and Control Analysis of PDEs 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. |
| title | From Sontag s to Cardano-Lyapunov Formula for Systems Not Affine in the Control: Convection-Enabled PDE Stabilization |
| topic | Systems and Control Analysis of PDEs |
| url | https://arxiv.org/abs/2403.18071 |