Technical CommuniqueAn exponential stability result for the wave equation☆
Introduction
In recent years, boundary control of infinite-dimensional systems has become an important research area, see e.g. Luo, Guo, and Morgül (1999) for more information and references. In this note, we will consider a system described by the one-dimensional wave equation in a bounded domain. We assume that a dynamic boundary control is applied to the system for stabilization. We propose a (rational) controller transfer function, which contains a strictly positive real part and some simple poles on the imaginary axis. The residues associated with the imaginary axis poles are assumed to be positive. Such transfer functions have been proposed to stabilize the wave equation, see Morgül 1994, Morgül 1998, where it was shown that with these controllers, the resulting closed-loop system is asymptotically stable under some conditions. In many cases, exponential stability is desired, due to e.g. the robustness of the resulting closed-loop system, and in infinite-dimensional systems, asymptotic stability may not imply exponential stability.
Note that exponential stability for this system could be achieved by static output feedback, see Chen (1979). However, if we also want to achieve tracking and/or disturbance rejection for certain classes of output signals, then we need a dynamic controller containing an internal model, see Hämäläinen and Pohjolainen (2000), Morgül (1998). The dynamic controllers discussed in this paper would be suitable for reference and disturbance signals which are the superposition of a constant and a sinusoid at frequency ω1. We show that the resulting closed-loop system is exponentially stable under some conditions. We do not discuss the tracking error.
Section snippets
Problem statement
In this note, we consider the following system:where, without the loss of generality various coefficients, including the length of the spatial domain, are assumed to have unit values, x∈(0,1) denotes the spatial variable, t⩾0 denotes time, u(σ,τ) denotes the solution of the wave equation at x=σ,t=τ, a subscript as in ut denotes the partial derivative with respect to the corresponding variable, and is the boundary control applied at the end point x
Exponential stability
Our main result is the following: Theorem 3 Consider the system given by (11). Let the assumptions stated above hold, and letγ>0, (see (7)). Ifω1≠mπfor all natural numbersm, thenT(t) is exponentially stable. Proof The case k1=0 was proven in Morgül (1994), hence we consider the case k1>0. Note that the operator is compact for λ⩾0; hence, the spectrum of L consists entirely of isolated eigenvalues and moreover λ=0 is not an eigenvalue of L, see Morgül (1998). In the proof, we will use Theorem 2 stated
Conclusion
In this note, we considered the stabilization of the wave equation in a bounded domain by means of a dynamic boundary control law. The transfer function of the controller may contain simple poles on the imaginary axis. This type of controllers were proposed for the stabilization of the wave equation, however only asymptotic stability results were given. In this note, we proved that with the proposed controller the closed-loop system is actually exponentially stable under some conditions.
References (13)
A dynamic control law for the wave equation
Automatica
(1994)Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain
Journal de Mathematiques Pures et Appliquees
(1979)- et al.
An introduction to infinite-dimensional linear systems theory
(1995) - et al.
A finite-dimensional robust controller for systems in the CD-algebra
IEEE Transactions on Automatic Control
(2000) - et al.
Conditions for robustness and nonrobustness of the stability of feedback systems with respect to small delays in the feedback loop
SIAM Journal of Control and Optimization
(1996) - et al.
Stability and stabilization of infinite dimensional systems with applications
(1999)
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This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Rodolphe Sepulchre under the direction of Editor Paul Van den Hof.