Local feedback stabilization of time-periodic evolution equations by finite dimensional controls

2020 ◽  
Vol 26 ◽  
pp. 101
Author(s):  
Mehdi Badra ◽  
Debanjana Mitra ◽  
Mythily Ramaswamy ◽  
Jean-Pierre Raymond
Keyword(s):  
Periodic Solutions ◽  
Evolution Equations ◽  
Stokes Equations ◽  
Finite Dimension ◽  
Parabolic Systems ◽  
Feedback Stabilization ◽  
Navier Stokes ◽  
Dimensional System ◽  
Unbounded Control ◽  
Finite Dimensional

We study the feedback stabilization around periodic solutions of parabolic control systems with unbounded control operators, by controls of finite dimension. We prove that the stabilization of the infinite dimensional system relies on the stabilization of a finite dimensional control system obtained by projection and next transformed via its Floquet-Lyapunov representation. We emphasize that this approach allows us to define feedback control laws by solving Riccati equations of finite dimension. This approach, which has been developed in the recent years for the boundary stabilization of autonomous parabolic systems, seems to be totally new for the stabilization of periodic systems of infinite dimension. We apply results obtained for the linearized system to prove a local stabilization result, around periodic solutions, of the Navier-Stokes equations, by finite dimensional Dirichlet boundary controls.

scholarly journals Feedback stabilization of one-dimensional parabolic systems related to formations

2015 ◽  
Vol 63 (1) ◽  
pp. 295-303
Author(s):  
H. Sano
Keyword(s):  
Finite Dimension ◽  
Time Derivative ◽  
Parabolic Systems ◽  
Feedback Stabilization ◽  
Boundary Values ◽  
One Dimensional ◽  
Finite Dimensional ◽  
Neumann Type ◽  
Sturm Liouville ◽  
Output Operator

Abstract This paper is concerned with the problem of stabilizing one-dimensional parabolic systems related to formations by using finitedimensional controllers of a modal type. The parabolic system is described by a Sturm-Liouville operator, and the boundary condition is different from any of Dirichlet type, Neumann type, and Robin type, since it contains the time derivative of boundary values. In this paper, it is shown that the system is formulated as an evolution equation with unbounded output operator in a Hilbert space, and further that it is stabilized by using an RMF (residual mode filter)-based controller which is of finite-dimension. A numerical simulation result is also given to demonstrate the validity of the finite-dimensional controller

Periodic Solutions of Non-Linear Evolution Equations in Banach Spaces

1971 ◽  
Vol 23 (1) ◽  
pp. 189-196 ◽  
Author(s):  
Bui An Ton
Keyword(s):  
Periodic Solutions ◽  
Banach Spaces ◽  
Evolution Equations ◽  
Stokes Equations ◽  
Inner Product ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Linear Evolution ◽  
Non Linear ◽  
Linear Evolution Equations

In this paper the theory of Browder [2] and of Lions [3] on periodic solutions of non-linear evolution equations in Banach spaces is put in a more general framework so as to include the Navier-Stokes equations and their variants.An abstract existence theorem is proved in § 1. Applications are given in § 2. The existence of periodic solutions of the Navier-Stokes equations without any restriction on the dimension of the space domain is established. Application of the abstract theorem to the following problem is given:1. Let H be a Hilbert space and (., .)H the inner product in H. Let V and W be two reflexive separable Banach spaces with W ⊂ V ⊂ H. W is dense in V and V is dense in H.

scholarly journals Effects of surfactant on propagation and rupture of a liquid plug in a tube

2019 ◽  
Vol 872 ◽  
pp. 407-437 ◽  
Author(s):  
M. Muradoglu ◽  
F. Romanò ◽  
H. Fujioka ◽  
J. B. Grotberg
Keyword(s):  
Shear Stress ◽  
Evolution Equations ◽  
Stokes Equations ◽  
Thin Liquid Film ◽  
Front Tracking ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Liquid Plug ◽  
Front Tracking Method ◽  
Airway Walls

Surfactant-laden liquid plug propagation and rupture occurring in lower lung airways are studied computationally using a front-tracking method. The plug is driven by an applied constant pressure in a rigid axisymmetric tube whose inner surface is coated by a thin liquid film. The evolution equations of the interfacial and bulk surfactant concentrations coupled with the incompressible Navier–Stokes equations are solved in the front-tracking framework. The numerical method is first validated for a surfactant-free case and the results are found to be in good agreement with the earlier simulations of Fujioka et al. (Phys. Fluids, vol. 20, 2008, 062104) and Hassan et al. (Intl J. Numer. Meth. Fluids, vol. 67, 2011, pp. 1373–1392). Then extensive simulations are performed to investigate the effects of surfactant on the mechanical stresses that could be injurious to epithelial cells, such as pressure and shear stress. It is found that the liquid plug ruptures violently to induce large pressure and shear stress on airway walls and even a tiny amount of surfactant significantly reduces the pressure and shear stress and thus improves cell survivability. However, addition of surfactant also delays the plug rupture and thus airway reopening.

scholarly journals On periodic solutions for one-phase and two-phase problems of the Navier–Stokes equations

2020 ◽  
Author(s):  
Thomas Eiter ◽  
Mads Kyed ◽  
Yoshihiro Shibata
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Periodic Solutions ◽  
Parabolic Equations ◽  
Stokes Equations ◽  
Transmission Conditions ◽  
Navier Stokes ◽  
Two Phase ◽  
Navier Stokes Equations ◽  
Linearized Equations

Abstract This paper is devoted to proving the existence of time-periodic solutions of one-phase or two-phase problems for the Navier–Stokes equations with small periodic external forces when the reference domain is close to a ball. Since our problems are formulated in time-dependent unknown domains, the problems are reduced to quasilinear systems of parabolic equations with non-homogeneous boundary conditions or transmission conditions in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value 0, which is avoided by changing the equations with the help of the necessary conditions for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal $$L_p$$ L p –$$L_q$$ L q regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous boundary conditions or transmission conditions, which is obtained by the systematic use of $${\mathcal R}$$ R -solvers developed in Shibata (Diff Int Eqns 27(3–4):313–368, 2014; On the $${{\mathcal {R}}}$$ R -bounded solution operators in the study of free boundary problem for the Navier–Stokes equations. In: Shibata Y, Suzuki Y (eds) Springer proceedings in mathematics & statistics, vol. 183, Mathematical Fluid Dynamics, Present and Future, Tokyo, Japan, November 2014, pp 203–285, 2016; Comm Pure Appl Anal 17(4): 1681–1721. 10.3934/cpaa.2018081, 2018; $${{\mathcal {R}}}$$ R boundedness, maximal regularity and free boundary problems for the Navier Stokes equations, Preprint 1905.12900v1 [math.AP] 30 May 2019) to the resolvent problem for the linearized equations and the transference theorem obtained in Eiter et al. ($${{\mathcal {R}}}$$ R -solvers and their application to periodic $$L_p$$ L p estimates, Preprint in 2019) for the $$L_p$$ L p boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper.

Unique Strong Solutions and V -Attractors of a Three Dimensional System of Globally Modified Navier-Stokes Equations

2006 ◽  
Vol 6 (3) ◽  
Author(s):  
Tomás Caraballo ◽  
José Real ◽  
Peter E. Kloeden
Keyword(s):  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Dimensional System ◽  
Navier Stokes Equations ◽  
Three Dimensional System

AbstractWe prove the existence and uniqueness of strong solutions of a three dimensional system of globally modified Navier-Stokes equations. The flattening property is used to establish the existence of global V -attractors and a limiting argument is then used to obtain the existence of bounded entire weak solutions of the three dimensional Navier-Stokes equations with time independent forcing.

Sign in / Sign up

Export Citation Format

Share Document