Internal Optimal Controller Synthesis for Navier–Stokes Equations

Barbu and Triggiani (Indiana Univ. Math. J. 2004; 53:1443–1494) have proposed a solution of the internal feedback stabilization problem of Navier–Stokes equations with no-slip boundary conditions. They have shown that any unstable steady-state solution can be exponentially stabilized by a finite-dimensional feedback controller with support in an arbitrary open subset of positive measure. The finite dimension of the feedback controller is minimal and is related to the largest algebraic multiplicity of the unstable eigenvalues of the linearized equation. The feedback law is obtained as a solution of a linear-quadratic control problem. In this paper, we formulate a practical algorithm implementation of the proposed stabilization approach, based on the finite element method, and demonstrate its applicability and effectiveness using an example involving the stabilization of two-dimensional Navier–Stokes equations.     

Boundary output feedback stabilization for a class of coupled parabolic systems

In this paper, the boundary output feedback stabilization problem is addressed for a class of coupled nonlinear parabolic systems. An output feedback controller is presented by introducing a Luenberger‐type observer based on the measured outputs. To determine observer gains, a backstepping transform is introduced by choosing a suitable target system with nonlinearity. Furthermore, based on the state observer, a backstepping boundary control scheme is presented. With rigorous analysis, it is proved that the states of nonlinear closed‐loop system including state estimation and estimation error of plant system are locally exponentially stable in the L²norm. Finally, a numerical example is proposed to illustrate the effectiveness of the presented scheme.     

A new algebraic approach to stabilization for boundary control systems of parabolic type

We study the stabilization problem of linear parabolic boundary control systems. While the control system is described by a pair of standard linear differential operators (L,τ), the corresponding semigroup generator generally admits no Riesz basis of eigenvectors. In the sense that very little information on the fractional powers of this generator is needed, our approach has enough generality as a prototype to be used for other types of parabolic systems. We propose in this paper a new algebraic approach to the stabilization, which gives—to the best of the author's knowledge—the simplest framework of the problem. The control system with the scheme of boundary observation/boundary feedback is turned into the differential equations with no boundary input in usual and standard L²-spaces in a readable manner.     

Eigenvalue assignment of an augmented hyperbolic system by linear feedback

A class of augmented linear hyperbolic systems subject to a boundary control is studied. It is shown that the eigenvalues of the closed loop system subject to a feedback control, can be assigned arbitrarily, apart from an asymptotic condition on the distribution of the eigenvalues. The feedback vector is constructed in an appropriate Hilbert space. The results are applied to an important example of the synthesis of a distortionless transmission line system by linear feedback.     

Stability and asymptotic behaviour of solutions of the heat equation

In this paper we study the boundary stabilization of the heatequation. The stabilization is achieved by applying either Dirichletor Neumann feedback boundary control. Furthermore, we considerthe asymptotic behaviour of the heat equation with general lineardelay or nonlinear power time delay. We prove that the energydoes not grow faster than a polynomial.     

Feedback Stabilization for a Scalar Conservation Law with PID Boundary Control

This paper deals with a scalar conservation law in 1-D space dimension, and in particular, the focus is on the stability analysis for such an equation. The problem of feedback stabilization under proportional-integral-derivative (PID for short) boundary control is addressed. In the proportional-integral (PI for short) controller case, by spectral analysis, the authors provide a complete characterization of the set of stabilizing feedback parameters, and determine the corresponding time delay stability interval. Moreover, the stability of the equilibrium is discussed by Lyapunov function techniques, and by this approach the exponential stability when a damping term is added to the classical PI controller scheme is proved. Also, based on Pontryagin results on stability for quasipolynomials, it is shown that the closed-loop system subject to PID control is always unstable.     

Dynamic feedback control and exponential stabilization of a compound system

We studied the exponential stabilization problem of a compounded system composed of a flow equation and an Euler–Bernoulli beam, which is equivalent to a cantilever Euler–Bernoulli beam with a delay controller. We designed a dynamic feedback controller that stabilizes exponentially the system provided that the eigenvalues of the free system are not the zeros of controller. In this paper we described the design detail of the dynamic feedback controller and proved its stabilization property.     

Stability analysis of linear parabolic systems and removement of singularities in substructure: Static feedback

We analyze stability property of a class of linear parabolic systems via static feedback. Stabilization via static feedback scheme is most difficult and challenging when both actuators and observation weights admit spillovers. This arises typically in the boundary observation-boundary feedback scheme. We propose a simple static feedback law containing a parameter γ, and enhance the stability property or achieve (slightly) stabilization. In some situations, the evolution of the substructure of finite dimension contains singularities regarding γ. We show that these singularities are removed as long as the dimension is not large.     

The Riesz basis property of discrete operators and application to a Euler-Bernoulli beam equation with boundary linear feedback control

In this paper, we give an abstract condition of Riesz basisgeneration for discrete operators in Hilbert spaces, from whichwe show that the generalized eigenfunctions of a Euler–Bernoullibeam equation with boundary linear feedback control form a Rieszbasis for the state Hilbert space. As an consequence, the asymptoticexpression of eigenvalues together with exponential stabilityare readily presented.     

Stabilization of nonlinear systems in compound critical cases

In this paper, we study the stabilization of nonlinear systems in critical cases by using the center manifold reduction technique. Three degenerate cases are considered, wherein the linearized model of the system has two zero eigenvalues, one zero eigenvalue and a pair of nonzero pure imaginary eigenvalues, or two distinct pairs of nonzero pure imaginary eigenvalues; while the remaining eigenvalues are stable. Using a local nonlinear mapping (normal form reduction) and Liapunov stability criteria, one can obtain the stability conditions for the degenerate reduced models in terms of the original system dynamics. The stabilizing control laws, in linear and/or nonlinear feedback forms, are then designed for both linearly controllable and linearly uncontrollable cases. The normal form transformations obtained in this paper have been verified by using code MACSYMA.     

Boundary stabilization of vibration of nonlocal micropolar elastic media

In this paper, we study the stabilization problem of vibration of linearized three-dimensional nonlocal micropolar elasticity. For this purpose, we need to demonstrate the well-posedness of the system of equations governing the vibration of three-dimensional nonlocal micropolar media for both forced (i.e. with boundary feedback) and unforced cases. We assume the non-homogeneous system of equations for the unforced (uncontrolled) case to establish the well-posedness. It should be pointed out that the well-posedness of the evolution equations in micropolar case has been studied by many authors; but, the well-posedness in the nonlocal micropolar is an open problem. Our tools in well-posedness analysis are the semigroup techniques. Afterwards, we pursue the stabilization problem and show that the vibration of the nonlocal micropolar elastic media will be eventually dissipated under boundary feedback actions consisting of stress and couple stress feedback laws. These control laws are simple, linear and can be easily implemented in practical applications. The stabilization proof is accomplished using Lyapunov stability and LaSalle’s invariant set theorems.     

Adaptive stabilization of the sine‐Gordon equation by boundary control

This paper is concerned with adaptive global stabilization of the sine‐Gordon equation without damping by boundary control. An adaptive stabilizer is constructed by the concept of high‐gain output feedback. The closed‐loop system is shown to be locally well‐posed by the Banach fixed point theorem and then to be globally well‐posed by the Lyapunov method. Moreover, using a multiplier method global exponential stabilization of the system is proved. Copyright © 2004 John Wiley & Sons, Ltd.     

Exponential stabilization of thermoelastic system of type II with non-uniform bounded disturbance

The boundary stabilization problem of a thermoelastic system of type II with a tip-mass is considered with the assumption that there is an external non-uniform bounded disturbance at the control end. In order to estimate this kind of disturbance, a time-varying high-gain estimator is designed, where the idea of active disturbance rejection control is adopted. Based on the estimate of disturbance, we propose a boundary feedback controller so as to exponentially stabilize this system. Finally, some numerical simulations on the dynamical behavior of the closed-loop system are given.     

Boundary stabilization of a microbeam model

In this paper, we study the boundary stabilization of the deflection of a clamped-free microbeam, which is modeled by a sixth-order hyperbolic equation. We design a boundary feedback control, simpler than the one designed in Vatankhah et al,² that forces the energy associated to the deflection to decay exponentially to zero as the time goes to infinity. The rate in which the energy exponentially decays is explicitly given.     

Asymptotic stabilization of entropy solutions to scalar conservation laws through a stationary feedback law

In this paper, we study the problem of asymptotic stabilization by closed loop feedback for a scalar conservation law with a convex flux and in the context of entropy solutions. Besides the boundary data, we use an additional control which is a source term acting uniformly in space.     

Riesz basis generation and boundary stabilization of two strings connected by a point mass with variable coefficients

In this paper, we study the Riesz basis property and the problem of stabilization of two vibrating strings connected by a point mass with variable physical coefficients under a boundary feedback control acts at one extreme point and Dirichlet boundary condition on the other end. It is shown that the system has a sequence of generalized eigenfunctions which forms a Riesz basis for the state Hilbert space. By a detailed spectral analysis, it is proved that this hybrid system is asymptotically stable but not exponentially stable.     

Control of large flexible systems via eigenvalue relocation

For the vibration control of large flexible systems, a control scheme by which the eigenvalues of the closed-loop systems are assigned to predetermined locations within the feasible region through velocity-only feedback is presented. Owing to the properties of second-order λ-matrices and an efficient model decoupling technique, the control scheme makes it possible that selected modes are damped with the rest of the modes unchanged.     

具有分布反馈和边界反馈的非均质Timoshenko梁的指数镇定性

讨论具有分布反馈控制和边界反馈控制的非均质Timoshenko梁的指数镇定问题.首先利用已有的关于线性分布参数系统的渐进稳定性判据,证明所论梁系统的能量可仅由一个分布反馈控制指数镇定.进而利用频域分片乘子方法,在所论梁系统同时具有分布反馈控制和边界反馈控制的条件下,证明其相应的闭环系统能量指数稳定.     

SPECTRAL PROPERTIES OF DISCRETE STURM-LIOUVILLE PROBLEMS WITH TWO SQUARED EIGENPARAMETER-DEPENDENT BOUNDARY CONDITIONS

In this article, we consider a discrete right-definite Sturm-Liouville problems with two squared eigenparameter-dependent boundary conditions. By constructing some new Lagrange-type identities and two fundamental functions, we obtain not only the existence,the simplicity, and the interlacing properties of the real eigenvalues, but also the oscillation properties, orthogonality of the eigenfunctions, and the expansion theorem. Finally, we also give a computation scheme for computing eigenvalues and eigenfunctions of specific eigenvalue problems.     

Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations

We study the local stabilization of the three-dimensional Navier–Stokes equations around an unstable stationary solution w, by means of a feedback boundary control. We first determine a feedback law for the linearized system around w. Next, we show that this feedback provides a local stabilization of the Navier–Stokes equations. To deal with the nonlinear term, the solutions to the closed loop system must be in H^{3/2+ε,3/4+ε/2}(Q), with 0<ε. In [V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc. 852 (2006); V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal. 64 (2006) 2704–2746], such a regularity is achieved with a feedback obtained by minimizing a functional involving a norm of the state variable strong enough. In that case, the feedback controller cannot be determined by a well posed Riccati equation. Here, we choose a functional involving a very weak norm of the state variable. The compatibility condition between the initial state and the feedback controller at t=0, is achieved by choosing a time varying control operator in a neighbourhood of t=0.