A novel semi-discrete scheme preserving uniformly exponential stability for an Euler–Bernoulli beam☆
Introduction
It has been known for a long time that for PDEs, the finite-difference method or other numerical discretization methods may have problems to approximate the original PDEs. One of the problems is that the exponential decay rate of an exponentially stable PDE cannot be kept uniformly in the process of the discretization. Let us look at the following one-dimensional Euler–Bernoulli beam with boundary control [1]: where is the initial value, is the control (input), is the (measured) output. The prime ‘’ is used to denote the partial derivative with respect to time. System (1.1) arises in vibrating control of a flexible beam where the left end of the structure is clamped and the right end is controlled. Consider system (1.1) in the energy state space where . The energy of system (1.1) is given by Finding the derivative of along the solution of (1.1) gives which means that system (1.1) is a passive system. The stabilizing output feedback control is naturally designed as and the closed-loop system is then described by Since the energy of the closed-loop satisfies the energy of the system is decreasing as time goes on. Actually, it has been known for a long time that for any given initial state , the solution of (1.2) decays exponentially in time and the decay rate is uniform for all initial states in the state space , i.e., for some independent of the initial values [1], [2].
The approximation of uniformly exponential stabilization for infinite dimensional systems has been studied extensively since from 1990s, for which a big concern is that whether or not the exponential decay rate of the discretized energy is preserved to be uniform with respect to the mesh size. Banks et al. in [3] first pointed out that the exponential decay of the discretized energy might not be uniform with respect to the mesh size for the classical finite difference and finite element schemes. This is largely due to numerical spurious oscillations for high frequencies presented in the numerical schemes (see, e.g., [4]). To remedy this problem, the authors suggested to use mixed finite element methods or polynomial based Galerkin methods to preserve the uniformly exponential decay [5], [6]. There are also some other remedies to damp out these spurious high frequencies like Tychonoff regularizations [7]; two-grid algorithms [8]; and filtering techniques [9], [10], [11], just to name a few.
Among all these numerical methods, the numerical viscosity method turns out to be an effective method to damp out the high frequencies and keep at the same time the engineering popular finite-difference nature of the numerical scheme. In [12], the uniform boundary controllability and convergence of controls of 1-d wave equation was studied by adding a numerical viscosity in a finite difference scheme. The uniform boundary stabilization of the 1-d wave equation was also considered in [13] by numerical viscosity method. It is worth mentioning that the uniform observability can also be achieved by finite difference space semi-discretization without numerical viscosity, if a nonuniform numerical mesh is adopted [14]. In addition, for some fully discretized finite difference schemes, one can also obtain uniform observability [15], [16] and uniformly exponential decay [17], [18] without numerical viscosity term, provided that the space step coincides with the time step.
As far as uniform approximation for beam system is concerned, the filtering technique was used in [19] to achieve the uniform boundary controllability for hinged beam system. The numerical viscosity method was also adopted to study uniform approximation for controlled hinged beam system in [20]. The success of numerical viscosity method to hinged beam systems is attributed highly on the analytical computation of eigenvalues and eigenvectors for discretized systems, which is difficult to be applied to deal with other type of boundary conditions. The filtering technique was applied to approximate the uniform observability of semi-discretized clamped beam system in [21]. In [22], the filtering technique was also used to study the uniform controllability of clamped beam equation, where a numerical asymptotic estimation was used to localize all the eigenvalues of the associated discrete operator. However, either filtration technique or numerical viscosity method depends on the extent of filtration for high frequencies and the viscosity coefficient.
In this paper we propose a new space semi-discretized numerical scheme on an equidistant mesh grids, based on finite volume method, which preserves the uniform exponential decay of system (1.2), without numerical viscosity. It turns out that this scheme is an alternative of finite-difference scheme from order reduction point of view. The advantages of the scheme include: (a) It removes the introduction of the numerical viscosity term to achieve uniform convergence yet keeps finite-difference scheme nature; (b) It can deal with any type of boundary conditions without help of the spectral analysis which is limited only for some special boundary conditions; (c) the convergence proof is simplified significantly with the similar techniques in dealing with the continuous counterpart.
The rest of this paper is organized as follows. Section 2 presents construction of the numerical scheme and shows that the discretized energy is non-increasing. The Lyapunov function method is adopted to prove the uniformly exponential decay of discretized energy in Section 3. The convergence of numerical solution to original continuous system is shown in Section 4, following up the conclusions presented in Section 5.
Section snippets
Preliminary
Let and let the spatial mesh size . The interval is discretized equidistantly as Here we call the grids. In order to discretize the boundary conditions of the system, we introduce two external points and outside the spatial domain . Let , , , where , are the midpoints of . Then, . These s are called control volumes in
Uniformly exponential decay
In this section, we consider the uniformly exponential stability of the semi-discretized system (2.7). The energy of the semi-discretized system (2.7) is defined as The following Lemma 3.1 shows that the energy is non-increasing in time.
Lemma 3.1 The derivative of energy of system (2.7) satisfies
Proof Multiplying both sides of the first equation of (2.7) by and summing up from 1 to , we obtain
Convergence to the continuous solution
To begin with, we introduce some additional notations. Denote a vector as . For every , we define the extension operators as follows For every , define by , i.e., It is a routine task to check that For
Conclusions
In this paper, we propose a new numerical scheme for an Euler–Bernoulli beam system. Constructed on equidistant grid points and without resorting to numerical viscosity, the new scheme is shown to preserve the uniformly exponential stability of the original continuous beam system. Though initially obtained by using a finite volume method, it turns out that the scheme can also be derived via the finite difference method based on order reduction. Consequently, the scheme is essentially a finite
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
The authors would like to thank the anonymous reviewers for their very careful reading and valuable comments and suggestions to improve the manuscript.
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This work was supported partially by National Natural Science Foundation of China under Project (No, 11901365, 61873260), and the Project of Department of Education of Guangdong Province (No. 2017KZDXM087).