Boundary stabilization of 1D hyperbolic systems

Hyperbolic systems model the phenomena of propagations at finite speeds. They are present in many fields of science and, consequently, in many human applications. For these applications, the question of stability or stabilization of their stationary state is a major issue. In this paper we present state-of-the-art tools to stabilize 1-D nonlinear hyperbolic systems using boundary controls. We review the power and limits of energy-like Lyapunov functions; the particular case of density–velocity systems; a method to stabilize shock steady-states; an extraction method allowing to use the spectral information of the linearized system in order to stabilize the nonlinear system; and some results on proportional-integral boundary control. We also review open questions and perspectives for this field, which is still largely open.     

Dirichlet feedback control for the stabilization of the wave equation: a numerical approach

In (J. Differential Equations 66 (1987) 340) a uniform stabilization method of the wave equation by boundary control à la Dirichlet has been discussed. In this article, we investigate the numerical implementation of the above stabilization process by a numerical scheme which mimics the energy decay properties of its continuous counterpart. The practical implementation of that scheme leads to a biharmonic problem of a new type which is solved by a method directly inspired by some related work of Glowinski and Pironneau on the solution of the Dirichlet problem for the biharmonic operator (SIAM Rev. 21(2) (1979) 167). Numerical experiments show that the decay properties of the energy are well-preserved by our numerical methodology.     

Distributed feedback design for systems governed by the wave equation

This article deals with the geometric control of a one-dimensional non-autonomous linear wave equation. The idea consists in reducing the wave equation to a set of first-order linear hyperbolic equations. Then, based on geometric control concepts, a distributed control law that enforces the exponential stability and output tracking in the closed-loop system is designed. The presented control approach is applied to obtain a distributed control law that brings a stretched uniform string, modelled by a wave equation with Dirichlet boundary conditions, to rest in infinite time by considering the displacement of the middle point of the string as the controlled output. The controller performances have been evaluated in simulation by considering both tracking and disturbance rejection problems. The robustness of the controller has also been studied when the string tension is subjected to sudden variations.     

On boundary feedback stabilization of Timoshenko beam with rotor inertia at the tip

The feedback stabilization problem of a nonuniform Timoshenko beam system with rotor inertia at the tip of the beam is studied. First, as a special kind of linear boundary force feedback and moment control is applied to the beam‘ s tip, the strict mathematical treatment, a suitable state Hilbert space is chosen, and the well-poseness of the corresponding closed loop system is proved by using the semigroup theory of bounded linear operators. Then the energy corresponding to the closed loop system is shown to be exponentially stable. Finally, in the special case of umform beam, some sufficient and necessary conditions for the corresponding closed loop system to be asymptotically stable are derived.     

An example for the switching delay feedback stabilization of an infinite dimensional system: The boundary stabilization of a string

For vibrating systems, a delay in the application of a feedback control may destroy the stabilizing effect of the control. In this paper we consider a vibrating string that is fixed at one end and stabilized with a boundary feedback with delay at the other end.We show that certain delays in the boundary feedback preserve the exponential stability of the system. In particular, we show that the system is exponentially stable with delays freely switching between the values 4L/c and 8L/c, where L is the length of the string and c is the wave speed.     

The solution of a nonclassic problem for one-dimensional hyperbolic equation using the decomposition procedure

In this article, we propose a new approach for solving an initial–boundary value problem with a non-classic condition for the one-dimensional wave equation. Our approach depends mainly on Adomian's technique. We will deal here with new type of nonlocal boundary value problems that are the solution of hyperbolic partial differential equations with a non-standard boundary specification. The decomposition method of G. Adomian can be an effective scheme to obtain the analytical and approximate solutions. This new approach provides immediate and visible symbolic terms of analytic solution as well as numerical approximate solution to both linear and nonlinear problems without linearization. The Adomian's method establishes symbolic and approximate solutions by using the decomposition procedure. This technique is useful for obtaining both analytical and numerical approximations of linear and nonlinear differential equations and it is also quite straightforward to write computer code. In comparison to traditional procedures, the series-based technique of the Adomian decomposition technique is shown to evaluate solutions accurately and efficiently. The method is very reliable and effective that provides the solution in terms of rapid convergent series. Several examples are tested to support our study.     

Asymptotic feedback stabilization: A sufficient condition for the existence of control Lyapunov functions

In this paper we study the feedback stabilization problem for a wide class of nonlinear systems that are affine in the control. We offer sufficient conditions for the existence of ‘Control Lyapunov functions’ that according to [3,23] and [28–30] guarantee stabilization at a specified equilibrium by means of a feedback law, which is smooth except possibly at the equilibrium. We note that the results of the paper present a local nature.