Boundary feedback controller over a bluff body for prescribed drag and lift coefficients

This paper presents an improved boundary feedback controller for the two and three-dimensional Navier-Stokes equations, in a bounded domain Ω, for prescribed drag and lift coefficients. In order to determine the feedback control law, we consider an extended system coupling the equations governing the Navier-Stokes problem with an equation satisfied by the control on the bluff body, which is a part of the domain boundary. By using the Faedo-Galerkin method and a priori estimation techniques, a stabilizing boundary control is built. This control law ensures the stability of the controlled discrete system. A compactness result then allows us to pass to the limit in the non linear system satisfied by the approximated solutions

Remarks on the internal exponential stabilization to a nonstationary solution for 1D Burgers equations

International audienceThe feedback stabilization of the Burgers system to a nonstationary solution using finite-dimensional internal controls is considered. Estimates for the dimension of the controller are derived. In the particular case of no constraint on the support of the controla better estimate is derived and the possibility of getting an analogous estimate for the general case is discussed; some numerical examplesare presented illustrating the stabilizing effect of the feedback control, and suggesting that the existence of an estimatein the general case analogous to that in the particular one is plausible

Stability of two-dimensional forced Navier-Stokes flow on a bounded circular domain

This research is concerned with the stability of a two-dimensional, electromagnetically forced, zonal flow on a circular domain. Flows like these are found in nature (e.g. shear flow in the atmosphere, Jovian disk) and experiment (e.g. plasma flow in a Fusion reactor) and a requirement for experiments is often that these types of flows remain stable and axi-symmetric. A numerical method is developed based on a spectral expansion into an infinite system of ordinary differential equations for velocity functions resulting from a Stokes eigenvalue problem. The system is truncated to gain a finite-dimensional system which is useful for computations of both equilibrium flows and strongly disturbed flows. Numerical results are compared to both finite difference method results and analytical results for the equilibrium basic flow. Both linear and nonlinear stability are explored for the Navier-Stokes equations on the circular domain and for the system of ordinary differential equations. Differences in stability and the evolution of perturbations are explained on the basis of discrepancies between infinite-dimensional partial differential equations like the Navier-Stokes equations and a finite-dimensional system of ordinary differential equations resulting from a Galerkin truncation. On the basis of both stability analyses a control system is developed which stabilizes the system of ordinary differential equations to stay in a desired equilibrium. It is argued that this control system is also usable for the control of the Navier-Stokes equations. This research is concerned with the stability of a two-dimensional, electromagnetically forced, zonal flow on a circular domain. Flows like these are found in nature (e.g. shear flow in the atmosphere, Jovian disk) and experiment (e.g. plasma flow in a Fusion reactor) and a requirement for experiments is often that these types of flows remain stable and axi-symmetric. A numerical method is developed based on a spectral expansion into an infinite system of ordinary differential equations for velocity functions resulting from a Stokes eigenvalue problem. The system is truncated to gain a finite-dimensional system which is useful for computations of both equilibrium flows and strongly disturbed flows. Numerical results are compared to both finite difference method results and analytical results for the equilibrium basic flow. Both linear and nonlinear stability are explored for the Navier-Stokes equations on the circular domain and for the system of ordinary differential equations. Differences in stability and the evolution of perturbations are explained on the basis of discrepancies between infinite-dimensional partial differential equations like the Navier-Stokes equations and a finite-dimensional system of ordinary differential equations resulting from a Galerkin truncation. On the basis of both stability analyses a control system is developed which stabilizes the system of ordinary differential equations to stay in a desired equilibrium. It is argued that this control system is also usable for the control of the Navier-Stokes equations