Symmetries, dynamics, and control for the 2D Kolmogorov flow. (English) Zbl 1390.93403
Summary: The symmetries, dynamics, and control problem of the two-dimensional (2D) Kolmogorov flow are addressed. The 2D Kolmogorov flow is known as the 2D Navier-Stokes (N-S) equations with periodic boundary conditions and with a sinusoidal external force along the \(x\)-direction. First, using the Fourier Galerkin method on the original 2D Navier-Stokes equations, we obtain a seventh-order system of nonlinear ordinary differential equations (ODEs) which approximates the behavior of the Kolmogorov flow. The dynamics and symmetries of the reduced seventh-order ODE system are analyzed through computer simulations for the Reynolds number range \(0 < R_e < 26.41\). Extensive numerical simulations show that the obtained system is able to display the different behaviors of the Kolmogorov flow. Then, we design Lyapunov based controllers to control the dynamics of the system of ODEs to different attractors (e.g., a fixed-point, a periodic orbit, or a chaotic attractor). Finally, numerical simulations are
undertaken to validate the theoretical developments.
MSC:
| 93C20 | Control/observation systems governed by partial differential equations |
| 35Q30 | Navier-Stokes equations |
| 93B17 | Transformations |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |
Keywords:
2D Kolmogorov flow; symmetries; dynamics; control; 2D Navier-Stokes (N-S) equations; periodic boundary conditions; sinusoidal external force; Fourier Galerkin methodSoftware:
DSToolReferences:
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