scholarly journals Exact boundary controllability of Galerkin approximations of Navier-Stokes system for soret convection

2015 ◽  
Vol 5 (2) ◽  
pp. 41-49
Author(s):  
Sivaguru S. Ravindran
Keyword(s):  
Soret Effect ◽  
Galerkin Approximation ◽  
Type System ◽  
Exact Controllability ◽  
Navier Stokes ◽  
Finite Dimensional ◽  
Galerkin Approximations ◽  
Diffusive Convection ◽  
Hilbert Uniqueness Method ◽  
Bounded Smooth

We study the exact controllability of finite dimensional Galerkin approximation of a Navier-Stokes type system describing doubly diffusive convection with Soret effect in a bounded smooth domain in Rd (d = 2, 3) with controls on the boundary. The doubly diffusive convection system with Soret effect involves a difficult coupling through second order terms. The Galerkin approximations are introduced undercertain assumptions on the Galerkin basis related to the linear independence of suitable traces of its elements over the boundary. By Using Hilbert uniqueness method in combination with a fixed point argument, we prove that the finite dimensional Galerkin approximations are exactly controllable.

scholarly journals Stabilization and Observability of a Rotating Timoshenko Beam Model

2007 ◽  
Vol 2007 ◽  
pp. 1-19 ◽  
Author(s):  
Alexander Zuyev ◽  
Oliver Sawodny
Keyword(s):  
Timoshenko Beam ◽  
Galerkin Approximation ◽  
Sufficient Conditions ◽  
Beam Equation ◽  
Distributed Parameter ◽  
Dimensional System ◽  
Beam Model ◽  
Finite Dimensional ◽  
Galerkin Approximations ◽  
Rotating Timoshenko Beam

A control system describing the dynamics of a rotating Timoshenko beam is considered. We assume that the beam is driven by a control torque at one of its ends, and the other end carries a rigid body as a load. The model considered takes into account the longitudinal, vertical, and shear motions of the beam. For this distributed parameter system, we construct a family of Galerkin approximations based on solutions of the homogeneous Timoshenko beam equation. We derive sufficient conditions for stabilizability of such finite dimensional system. In addition, the equilibrium of the Galerkin approximation considered is proved to be stabilizable by an observer-based feedback law, and an explicit control design is proposed.

scholarly journals Asymptotically stable attracting sets in the Navier-Stokes equations

1986 ◽  
Vol 34 (1) ◽  
pp. 37-52 ◽  
Author(s):  
P. E. Kloeden
Keyword(s):  
Stokes Equations ◽  
Galerkin Approximation ◽  
Navier Stokes ◽  
Asymptotically Stable ◽  
Compact Sets ◽  
Navier Stokes Equations ◽  
Attracting Set ◽  
Galerkin Approximations ◽  
Steady Solutions ◽  
The Stability

The planar Navier-Stokes equations with periodic boundary conditions are shown to have a nearby asymptotically stable attracting set whenever a Galerkin approximation involving the eigenfunctions of the Stokes operator has such an attracting set, provided the approximation has sufficiently many terms and its attracting set is sufficiently strongly stable. Lyapunov functions are used to characterize the stability of these attracting sets, which are compact sets of arbitrary geometric shape. This generalizes earlier results of Constantin, Foias and Temam and of the author for asymptotically stable steady solutions in the Navier-Stokes equations and such Galerkin approximations.

scholarly journals A note on the exact boundary controllability for an imperfect transmission problem

2021 ◽  
Author(s):  
S. Monsurrò ◽  
A. K. Nandakumaran ◽  
C. Perugia
Keyword(s):  
Exact Controllability ◽  
Dirichlet Condition ◽  
Adjoint Problem ◽  
Imperfect Interface ◽  
External Boundary ◽  
Exact Boundary Controllability ◽  
Multipliers Method ◽  
Exact Boundary ◽  
Hilbert Uniqueness Method ◽  
Uniqueness Method

AbstractIn this note, we consider a hyperbolic system of equations in a domain made up of two components. We prescribe a homogeneous Dirichlet condition on the exterior boundary and a jump of the displacement proportional to the conormal derivatives on the interface. This last condition is the mathematical interpretation of an imperfect interface. We apply a control on the external boundary and, by means of the Hilbert Uniqueness Method, introduced by J. L. Lions, we study the related boundary exact controllability problem. The key point is to derive an observability inequality by using the so called Lagrange multipliers method, and then to construct the exact control through the solution of an adjoint problem. Eventually, we prove a lower bound for the control time which depends on the geometry of the domain, on the coefficients matrix and on the proportionality between the jump of the solution and the conormal derivatives on the interface.

Long-time dynamics of a class of nonlocal extensible beams with time delay

2021 ◽  
pp. 108128652110194
Author(s):  
Fengjuan Meng ◽  
Cuncai Liu ◽  
Chang Zhang
Keyword(s):  
Time Delay ◽  
Stability Property ◽  
Beam Equation ◽  
Exponential Attractors ◽  
Time Dynamics ◽  
Bounded Smooth Domain ◽  
Finite Dimensional ◽  
Long Time ◽  
Bounded Smooth ◽  
Long Time Dynamics

This work is devoted to the following nonlocal extensible beam equation with time delay: [Formula: see text] on a bounded smooth domain [Formula: see text]. The main purpose of this paper is to consider the long-time dynamics of the system. Under suitable assumptions, the quasi-stability property of the system is established, based on which the existence and regularity of a finite-dimensional compact global attractor are obtained. Moreover, the existence of exponential attractors is proved.

A simple proof of existence of weak and statistical solutions of Navier–Stokes equations

1992 ◽  
Vol 436 (1896) ◽  
pp. 1-11 ◽  
Keyword(s):  
Weak Solution ◽  
Initial Data ◽  
Natural Number ◽  
Simple Proof ◽  
Stokes Equations ◽  
Galerkin Approximation ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Existence Proofs ◽  
Statistical Solutions

The Galerkin approximation to the Navier–Stokes equations in dimension N , where N is an infinite non-standard natural number, is shown to have standard part that is a weak solution. This construction is uniform with respect to non-standard representation of the initial data, and provides easy existence proofs for statistical solutions.

scholarly journals On Galerkin Approximations of the Surface Active Quasigeostrophic Equations

2016 ◽  
Vol 46 (1) ◽  
pp. 125-139 ◽  
Author(s):  
Cesar B. Rocha ◽  
William R. Young ◽  
Ian Grooms
Keyword(s):  
Galerkin Approximation ◽  
Three Dimensional ◽  
Active Surface ◽  
Inversion Problem ◽  
Conservation Of Energy ◽  
Galerkin Approximations ◽  
Representation Of Solutions ◽  
Surface Active ◽  
Surface Buoyancy ◽  
Quasigeostrophic Equations

AbstractThis study investigates the representation of solutions of the three-dimensional quasigeostrophic (QG) equations using Galerkin series with standard vertical modes, with particular attention to the incorporation of active surface buoyancy dynamics. This study extends two existing Galerkin approaches (A and B) and develops a new Galerkin approximation (C). Approximation A, due to Flierl, represents the streamfunction as a truncated Galerkin series and defines the potential vorticity (PV) that satisfies the inversion problem exactly. Approximation B, due to Tulloch and Smith, represents the PV as a truncated Galerkin series and calculates the streamfunction that satisfies the inversion problem exactly. Approximation C, the true Galerkin approximation for the QG equations, represents both streamfunction and PV as truncated Galerkin series but does not satisfy the inversion equation exactly. The three approximations are fundamentally different unless the boundaries are isopycnal surfaces. The authors discuss the advantages and limitations of approximations A, B, and C in terms of mathematical rigor and conservation laws and illustrate their relative efficiency by solving linear stability problems with nonzero surface buoyancy. With moderate number of modes, B and C have superior accuracy than A at high wavenumbers. Because B lacks the conservation of energy, this study recommends approximation C for constructing solutions to the surface active QG equations using the Galerkin series with standard vertical modes.

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