On state feedback control of a class of nonlinear PDE systems in finite dimension. (English) Zbl 1531.35316
Summary: In this note, we propose some new results on state feedback stabilization of a class of non linear PDE systems that are described by Vlasov-Poisson equations. For doing so, thanks to the Galerkin discontinuous discretization approach, we establish first an explicit state space model in finite dimension. A state feedback controller is then proposed. One of the main futures is that the asymptotic stabilization may be guaranteed under an easy tractable LMI condition. Extension to the observer based control case, in the presence of disturbances or not, was also established. We provide an LMI condition to assure convergence in the \(\mathcal{H}_\infty\) sense.
MSC:
| 35Q83 | Vlasov equations |
| 35Q93 | PDEs in connection with control and optimization |
| 65M60 | Finite element, Rayleigh-Ritz and Galerkin methods for initial value and initial-boundary value problems involving PDEs |
| 65L06 | Multistep, Runge-Kutta and extrapolation methods for ordinary differential equations |
| 65M12 | Stability and convergence of numerical methods for initial value and initial-boundary value problems involving PDEs |
| 65M15 | Error bounds for initial value and initial-boundary value problems involving PDEs |
| 93B52 | Feedback control |
| 93B07 | Observability |
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35C09 | Trigonometric solutions to PDEs |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
| 35A02 | Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness |
References:
| [1] | Klibanov, Michael V.; Yamamoto, Masahiro, Exact controllability for the time dependent transport equation. SIAM J Control Optim, 6, 2071-2195 (2007) · Zbl 1149.93009 |
| [2] | Glass, Olivier, On the controllability of the Vlasov-Poisson system. J Differential Equations, 2, 332-379 (2003) · Zbl 1109.93007 |
| [3] | Knopf, Patrik, Optimal control of a Vlasov-Poisson plasma by an external magnetic field. Calc Var Partial Differential Equations, 1-37 (2018) · Zbl 1401.49070 |
| [4] | Young, Brent, Optimal-control for the global Cauchy problem of the relativistic Vlasov-Poisson system. Transport Theory Statist Phys, 6-7, 331-359 (2011) · Zbl 1459.35354 |
| [5] | Hafayed, Mokhtar; Boukaf, Samira; Shi, Yan; Meherrem, Shahlar, A mckean-vlasov optimal mixed regular-singular control problem for nonlinear stochastic systems with poisson jump processes. Neurocomputing, 133-144 (2016) |
| [6] | Glass, Olivier, Asymptotic stabilizability by stationary feedback of the two-dimensional Euler equation: the multiconnected case. SIAM J Control Optim, 3, 1105-1147 (2005) · Zbl 1130.93403 |
| [7] | Glass, Olivier, Controllability and asymptotic stabilization of the Camassa-Holm equation. J Differential Equations, 6, 1584-1615 (2008) · Zbl 1186.35185 |
| [8] | Mohamed, Ghattassi; Mohamed, Boutayeb, Nonlinear controller design for class of parabolic-hyperbolic system. J Nonlinear Syst Appl, 15, 20 (2016) |
| [9] | Coron, Jean-Michel, On the null asymptotic stabilization of the two-dimensional incompressible Euler equations in a simply connected domain. SIAM J Control Optim, 6, 1874-1896 (1999) · Zbl 0954.76010 |
| [10] | Coron, Jean-Michel, Sur la stabilisation des fluides parfaits incompressibles bidimensionnels. Sémin Équations Dériv Part (Polytech) Aussi Sémin Goulaouic-Schwartz, 1-15 (1999) · Zbl 1086.93511 |
| [11] | De Dios, Blanca Ayuso; Carrillo, José A.; Shu, Chi-Wang, Discontinuous Galerkin methods for the multi-dimensional Vlasov-Poisson problem. Math Models Methods Appl Sci, 12 (2012) · Zbl 1258.65084 |
| [12] | Vlasov, A. A., On high-frequency properties of electron gas. J Exp Theor Phys, 3, 291-318 (1938) |
| [13] | Vlasov, A. A., The vibrational properties of an electron gas. Sov Phys Uspekhi, 6, 721 (1968) |
| [14] | Di Pietro, Daniele Antonio; Ern, Alexandre, Mathematical aspects of discontinuous Galerkin methods, Vol. 69 (2011), Springer Science & Business Media |
| [15] | Hesthaven, Jan S.; Warburton, Tim, Nodal discontinuous galerkin methods: algorithms, analysis, and applications (2007), Springer Science & Business Media · Zbl 1134.65068 |
| [16] | Pham, Thi Trang Nhung, Méthodes numériques pour l’équation de Vlasov réduite (2016), Strasbourg, (Ph.D. thesis) |
| [17] | Zemouche, Ali; Boutayeb, Mohamed; Bara, Gabriela Iuliana, Observers for a class of Lipschitz systems with extension to Hinfinity performance analysis. Syst Control Lett, 18-27 (2008) · Zbl 1129.93006 |
| [18] | Cisse, Amadou; Boutayeb, Mohamed, Observers of Vlasov-Poisson system. IFAC-PapersOnLine, 2, 5946-5951 (2020), 21st IFAC World Congress |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
