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This paper investigates an inverse problem of determining the spatially variable diffusion coefficient of a one-dimensional heat equation with unknown spatial varying source term and initial value. The big challenge of the problem comes from the multiple unknowns and very limited available data that are only boundary control and boundary observation at one end, in addition to the ill-posed nature of the inverse problem. We first design a switch on/off boundary control and show that the diffusion coefficient can be uniquely determined by the boundary observation. Next, a stable numerical algorithm for reconstruction of the diffusion coefficient is proposed by means of the matrix pencil method and optimal perturbation regularization technique. Finally, some numerical examples are presented to illustrate the effectiveness of the proposed identification algorithm.

This article addresses fuzzy observer design for a nonlinear parabolic equation over an unit square domain $\Omega$ in terms of the time delayed spatially averaged measurement, where the observer is composed of $m$ -chain of subobservers. Due to 2-D domain, special emphases are made to the computational complexity. A Lyapunov argument is utilized to give constructive conditions ensuring the exponential stability of the resulting error system. The method used for the continuous-time fuzzy observer is applicable to the sampled-data implementation. Consistent simulation results that support the proposed theoretical statements are presented.

In this paper, we consider robust output tracking for an Euler–Bernoulli beam equation under the guidance of the internal model principle, where the disturbances in all possible channels are considered. Three typical cases are investigated in terms of different regulated outputs. The first case is based on boundary displacement output, for which only asymptotic convergence can be achieved due to the compactness of the observation operator. The second case considers two outputs of both boundary displacement and velocity. Since the control is one-dimensional, we can only arbitrarily regulate the boundary displacement and at the same time, the velocity is regulated to track the derivative of the reference. This is not the standard form investigated in the literature for robust error feedback control of abstract infinite-dimensional systems. The last case represents an extreme case that the system is non-well posed. In all the above cases, this paper demonstrates the same technique of an observer-based approach to robust control design. In the latter two cases, we can achieve exponential convergence and the closed loop is also shown to be robust to system uncertainties. Numerical simulations are carried out in all cases to illustrate the effectiveness of the proposed controls.

The active disturbance rejection control (AD RC), first proposed by Jingqing Han in late 1980s, is a powerful control technology being able to deal with external disturbances and internal uncertainties in large scale for control systems in engineering applications. This survey paper will articulate, from a theoretical perspective, the origin, ideology and progress of ADRC for not only uncertain finite-dimensional systems but also uncertain infinite-dimensional ones. Some recent theoretical developments, general framework and unsolved problems of ADRC for finite-dimensional systems with mismatched disturbances and uncertainties by output feedback, uncertain finite-dimensional stochastic systems, uncertain infinite-dimensional systems described by both the wave equation and the fractional-order partial differential equation are successively addressed, from which we see the challenges and opportunities for this remarkable emerging control technology to various types of control systems.

A time delay present in the observation represents a mathematical challenge in an output feedback stabilization for linear infinite-dimensional systems. It is well known that for a linear hyperbolic system, a stabilizing output feedback control may become unstable when the observation has a time delay. For the fixed time delay in observation, the problem for one-dimensional partial differential equations (PDEs) has been solved by the observer-based feedback in the time interval where the observation is available and the predictor where the observation is not available. However, the generalization to multidimensional PDE systems has been a long-standing unsolved problem. In this paper, we investigate the problem from operator point of view for abstract first-order equation setting of infinite-dimensional systems. We formulate the problem in the framework of the well-posed and regular linear systems and solve it in the operator form. The result is then applied to the stabilization of a multidimensional Schrodinger equation.

This paper considers boundary control of time fractional order diffusion–wave equation. Both boundary stabilization and disturbance rejection are considered. This paper, for the first time, has confirmed, via hybrid symbolic and numerical simulation studies, that the existing two schemes for boundary stabilization and disturbance rejection for (integer order) wave/beam equations are still valid for time fractional order diffusion–wave equations. The problem definition, the hybrid symbolic and numerical simulation techniques, outlines of the methods for boundary stabilization and disturbance rejection are presented together with extensive simulation results. Different dynamic behaviors are revealed for different fractional orders which may stimulate new future research opportunities.

In this paper, we consider simultaneous reconstruction of diffusion coefficient and initial state for a one-dimensional heat equation through boundary control and measurement. The boundary measurement is proposed to make the system approximately observable, and both the coefficient and initial state are shown to be identifiable by this measurement under a boundary switch on/off control. By a Dirichlet series representation for the observation, we can transform the problem into an inverse process of reconstruction of the spectrum and coefficients for the Dirichlet series in terms of observation. This happens to be the reconstruction of spectral data for an exponential sequence with measurement error, and it enables us to develop an algorithm based on the matrix pencil method in signal analysis. A theoretical error analysis for the algorithm concerning the coefficient reconstruction is carried out for the proposed method. The numerical simulations are presented to verify the proposed algorithm.

In this paper, we propose a novel space semidiscretized finite difference scheme for approximation of the one-dimensional wave equation under boundary feedback. This scheme, referred to as the orde...

In this paper, we consider boundary stabilization for a one-dimensional wave equation subject to periodic disturbance. By regarding the periodic signal as a boundary output of a free wave equation, we transform the controlled plant into a coupled wave system. We first design a state observer for the coupled system to estimate the disturbance and the system state simultaneously. An output feedback control is then designed to stabilize the original system. As an application, the result is applied to the stabilization of a wave equation with periodic disturbance suffering in output. Finally, some simulations are presented to validate the theoretical results.

In this paper, we consider bang–bang property for a kind of time-varying time optimal control problem of null controllable heat equation. The study is a continuation of a recent work (Chen et al. in Syst Control Lett 112:18–23, 2018), where an approximate null controllable heat equation was considered. We first establish the equivalence between optimal norm control and optimal time control and then prove the existence of the optimal norm control and of the optimal time control. The time-varying bang–bang property for the optimal time control is finally established.

In this paper, we address a two output tracking problem under the guidance of the internal model principle for an Euler-Bernoulli beam equation, which is motivated from a recent paper [SIAM J. Control Optim., 57 (2019), 1890-1928] where an adaptive control approach was used to estimate in real time all coefficients of the external harmonic disturbance. Compared with the aforementioned paper, we improve the results significantly from several aspects: a) both position and velocity are regulated simultaneously instead of a single position; b) the convergence is uniformly exponentially instead of asymptotically; c) the control contains essentially the internal model and is therefore conditionally robust; d) a manmade condition has been removed with a necessary assumption only. Our approach is systematic and straightforward in dealing with other PDEs with the same kind.

In this paper, event-triggered reliable dissipative filtering is investigated for a class of Takagi–Sugeno (T–S) fuzzy systems. First, a reliable event-triggered communication scheme is introduced to release sampled measurement outputs only if the variation of the sampled vector exceeds a prescribed threshold condition. Second, an asynchronous premise reconstruct method for T–S fuzzy systems is presented, which relaxes the assumption of the prior work that the premises of the plant and the filter are synchronous. Third, the resulting filtering error system is modeled under consideration of event-triggered communication, sensor failure, and asynchronous premise in a unified framework. By adopting the Lyapunov functional method and integral inequality approach, a delay-dependent criterion is developed to guarantee asymptotic stability for the filtering error systems and achieve strict $(Q, S,R)-\alpha$ dissipativity. Consequently, suitable filters and the event parameters can be derived by solving a set of linear matrix inequalities. Finally, an example is given to show the effectiveness of the proposed method.

In this paper, we consider output tracking for a class of MIMO nonlinear systems which are composed of coupled subsystems with vast mismatched uncertainties. First, all uncertainties influencing the performance of controlled outputs, which include internal unmodelled dynamics, external disturbances, and uncertain nonlinear interactions between subsystems, are refined into the total disturbance in the control channels of subsystems. The total disturbance is shown to be sufficiently reflected in the measured output of each subsystem so that it can be estimated in real time by an extended state observer (ESO) in terms of the measured outputs. Second, we decouple approximately the MIMO systems by cancelling the total disturbance based on ESO estimation so that each subsystem becomes approximately independent linear time invariant one without uncertainty and interaction with other subsystems. Finally, we design an ESO based output feedback for each subsystem separately to ensure that the closed-loop state is bounded, and the closed-loop output of each subsystem tracks practically a given reference signal. This is completely in comply with the spirit of active disturbance rejection control (ADRC). Some numerical simulations are presented to demonstrate the effectiveness of the proposed output feedback control scheme.

This note investigates sampled-data control for chaotic systems. A memory sampled-data control scheme that involves a constant signal transmission delay is employed for the first time to tackle the stabilization problem for Takagi–Sugeno fuzzy systems. The advantage of the constructed Lyapunov functional lies in the fact that it is neither necessarily positive on sampling intervals nor necessarily continuous at sampling instants. By introducing a modified Lyapunov functional that involves the state of a constant signal transmission delay, a delay-dependent stability criterion is derived so that the closed-loop system is asymptotically stable. The desired sampled-data controller can be achieved by solving a set of linear matrix inequalities. Compared with the existing results, a larger sampling period is obtained by this new approach. A simulation example is presented to illustrate the effectiveness and conservatism reduction of the proposed scheme.

In this paper, we consider boundary feedback stabilization for unstable time fractional reaction diffusion equations. New state feedback controls with actuation on one end are designed by the backstepping method for both Dirichlet and Neumann boundary controls. By the Riesz basis approach and the fractional Lyapunov method, we prove the existence and uniqueness and the Mittag--Leffler stability for the closed-loop systems. For both cases, the observers and the observer-based output feedback are designed to stabilize the systems.

In this paper, a spacial semidiscretized finite difference scheme developed in our previous work (Liu and Guo, 2019) is used to approximate exact observability and controllability for Euler–Bernoulli beam control system. The uniform observability inequality is proved by discrete energy multiplier technique. The uniform controllability and uniform boundedness of the discrete controls are also developed. Compared with the existing literature, the proposed approach has achieved potentially the following advantages: (a) It removes the introduction of the numerical viscosity term to achieve uniformity; (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 proofs of the uniform observability and controllability are simplified significantly with the similar techniques in dealing with the continuous counterpart.

In this paper, we consider boundary output feedback stabilization for a multi-dimensional wave equation with boundary control matched unknown nonlinear internal uncertainty and external disturbance. A new unknown input type extended state observer is proposed to recover both state and total disturbance which consists of internal uncertainty and external disturbance. A key feature of the proposed observer in this paper is that we do not use the high-gain to estimate the disturbance. By the active disturbance rejection control (ADRC) strategy, the total disturbance is compensated (canceled) in the feedback loop, which together with a collocated stabilizing controller without uncertainty, leads to an output feedback stabilizing feedback control. It is shown that the resulting closed-loop system is well-posed and asymptotically stable under weak assumption on internal uncertainty and external disturbance. The numerical experiments are carried out to show the effectiveness of the proposed scheme.

In this paper, we study optimal actuator location of the minimum norm controls for a multi-dimensional heat equation with control defined in the space L 2 ( Ω × ( 0 , T ) ) . The actuator domain is time-varying in the sense that it is only required to have a prescribed Lebesgue measure for any moment. We select an optimal actuator location so that the optimal control takes its minimal norm over all possible actuator domains. We build a framework of finding the Nash equilibrium so that we can develop a sufficient and necessary condition to characterize the optimal relaxed solutions for both actuator location and corresponding optimal control of the open-loop system. The existence and uniqueness of the optimal classical solutions are therefore concluded. As a result, we synthesize both optimal actuator location and corresponding optimal control into a time-varying feedbacks.

In this paper,the problem of H∞ filtering for a class of nonlinear discrete-time delay systems is investigated. The time delay is assumed to be belonging to a given interval, and the designed filter includes additive gain variations which are supposed to be random and satisfy the Bernoulli distribution. By the augmented Lyapunov functional approach, a sufficient condition is developed to ensure that the filtering error system is asymptotically mean-square stable with a prescribed H∞ performance. In addition, an improved result of H∞ filtering for linear system is also derived. The filter parameters are obtained by solving a set of linear matrix inequalities. For nonlinear systems, the applicability of the developed filtering result is confirmed by a longitudinal flight system, and an additional example for linear system is presented to demonstrate the less conservativeness of the proposed design method.

In the last two decades, there is a big discovery about the numerical realization of feedback control of systems described by partial differential equations (PDEs) that for an exponentially stable hyperbolic PDEs, after semi-discretization on the spatial variable, the semi-discrete system is not uniformly decaying with respect to the step size. This prevents obviously the application of feedback control of PDEs in engineering, in particular, in computer digitalization. It is also found that with some vanishing viscosity term in the discrete scheme, the decay rate of semi-discrete scheme becomes uniform in step size. In this paper, we try to understand this problem from mathematical point of view. We first consider an Euler-Bernoulli beam equation without the viscosity term. It is found that the semi -discrete eigenvalues are not convergent to the continuous counterparts. However, when the continuous system has structural damping term, then the semi-discrete eigenvalues are uniformly convergent. The reason behind we believe is that the solution of the system without structural damping is not smooth enough, whereas for the system with structural damping, the solution is analytic. Many engineering control researchers are wondering about the smoothness of the solution of PDEs. This paper severs as an example for importance of smoothness of PDEs in control design.

In this paper, we study controllability for a parabolic system of chemotaxis. With one control only, the local exact controllability to positive trajectory of the system is obtained by applying Kakutani's fixed point theorem and the null controllability of associated linearized parabolic system. The positivity of the state is shown to be remained in the state space. The control function is shown to be in $L^\infty(Q)$, which is estimated by using the methods of maximal regularity and $L^p$-$L^q$ estimate for parabolic equations.

In this paper, we formulate state estimation problem for usually non-time invertible evolutionary systems into a nonstandard finite horizon linear quadratic output tracking problem. This problem directly leads to the classical filtering problem with disturbance constraint. For a given reference trajectory, we give the explicit expressions of optimal initial state and optimal control. In addition, based on some well-known results for classical finite horizon linear quadratic optimal control problems, we obtain, in terms of measured output, the pointwise linear state feedbacks to optimal initial state and optimal control, and the representation of optimal cost is also given. As an important consequence, the differential equation satisfied by the optimal state estimate for filtering problem is developed. Some examples are given as applications.

We consider optimal locations of heat diffusion and related optimal control to achieve null controllability for multi-dimensional heat equations. Both time optimal control and norm optimal control problems are considered. The reason behind combining these two problems together is that these two problems are actually equivalent: The energy to be used to drive the system to zero in minimal time interval is actually the minimal energy of driving the system to zero in this minimal time interval and visa versa. We formulate the optimal locations for time optimal control and norm optimal control into two types of shape optimization problems. One is seeking the optimal domain of heat diffusion with a fixed interior actuator domain. This can be considered as a domain perturbation problem in shape optimization. Another is to seek both the optimal locations of the optimal heat diffusion domain and the related optimal actuator domain. The existences of these two types of shape optimization problems over some class of open sets in general R N space have been proved separately.

We consider parameter estimation and stabilization for a one-dimensional Schrodinger equation with an unknown constant disturbance suffered from the boundary observation at one end and the non-collocated control at other end. An adaptive observer is designed in terms of measured position with unknown constant by the Lyapunov functional approach. By a backstepping transformation for infinite-dimensional systems, it is shown that the resulting closed-loop system is asymptotically stable. Meanwhile, the estimated parameter is shown to be convergent to the unknown parameter as time goes to infinity. The numerical experiments are carried out to illustrate the proposed approach.

We consider stabilization for a class of abstract second order infinite-dimensional systems with collocated control and observation. We show that the closed-loop system under a proportional direct output feedback control is asymptotically stable if and only if the closed-loop system under some dynamic output feedback control is asymptotically stable. A Hautus test is developed to ensure the asymptotic stability. Two types of dynamic output feedback controls are investigated. The results are applied to some coupled wave-heat equations where the heat system is considered as a controller of the wave system. This study provides a different view in the study of the coupled systems described by partial differential equations.

In this paper, the quasi-compactness and irreducibility of two queueing systems are investigated by abstract functional analytical methods. More precisely, the systems fit into Greiner’s framework and Greiner’s idea of boundary perturbation of the generator is utilized. Quasi-compactness or irreducibility of the semigroup generated by the system operator is obtained through special properties of the boundary perturbation operators. Moreover, the exponential stability of the queueing models is analyzed based on the quasi-compactness and irreducibility.

We consider minimal time and minimal norm of the optimal controls for a multi-dimensional internally controlled heat equation with control domain varying over a class of open sets. Two problems are formulated separately into different types of shape optimization problems over this open set class. The governing equation with any given initial value and admissible control are considered as constraints, and minimal time or minimal norm of the optimal controls is considered as cost for related shape optimization. The solution of the shape optimization leads to the optimal actuator location for optimal controls. The existence of such an optimal location domain for both minimal time and minimal norm controls is presented. A different problem that relates the balance between minimal time and minimal norm controls is also discussed. This study builds a link between optimal control and shape optimization for this multi-dimensional heat equation.

In this paper, new extended Grammian determinant solutions to a (3 + 1)-dimensional KP equation are presented by using Hirora's bilinear method, and a broad set of sufficient conditions of systems of linear partial differential equations is given. Moreover, some special solutions of the representative systems are obtained through a systematic analysis.

We consider the controllability of a chemotaxis system of parabolic–elliptic type. By linearizing the nonlinear system into two separate linear equations, we can bypass the obstacle caused by the nonlinear drift term and establish local null controllability of the original nonlinear system. This approach is different from the usual method for dealing with coupled parabolic systems.

In this paper, a novel space semi-discretized numerical scheme which is based on finite volume method is proposed for approximation of uniformly exponential decay of Euler–Bernoulli beam system, which turns out to be an alternative of finite-difference scheme from order reduction point of view. The new scheme is constructed on equidistant grid points without using any numerical viscosity terms. The uniformly exponential decay is proved by the Lyapunov function method and the energy multiplier technique. With construction of a new gradient recovery function, the numerical solution is proved to be convergent to the (weak) solution of the original continuous system. Compared with the existing literature, the proposed approach has potentially achieved the following objectives: a) It removes the introduction of the numerical viscosity term to achieve uniform convergence; 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.

This paper deals with the design of boundary control to stabilize one-dimensional Schrodinger equation with general external disturbance. The backstepping method is first applied to transform the anti-stability from the free end to the control end. A variable structure feedback stabilizing controller is then designed to achieve arbitrary assigned decay rate. The Galerkin approximation scheme is used to show the existence of the solution to the closed-loop system. The exponential stability of the closed-loop system is obtained by the Lyapunov functional method. A numerical example demonstrates the efficiency of the proposed control scheme.

In this paper, an algorithm is developed to reject time and spatially varying boundary disturbances from a multidimensional Kirchhoff plate via boundary control. The disturbance and control input are assumed to be matched. The active disturbance rejection control approach is adopted for developing the algorithm. A state feedback scheme is designed to estimate the disturbance based on an infinite number of ordinary differential equations obtained from the original multidimensional system using infinitely many time-dependent test functions. The proposed control law cancels the disturbance using its estimated value. All subsystems in the closed loop are shown to be asymptotically stable. Simulation results are presented to validate the theoretical conclusions and to exhibit the reduction in the peaking phenomenon due to the use of time varying gains instead of constant high gains.

In this paper, we are concerned with the stabilization of a coupled system of Euler–Bernoulli beam or plate with heat equation, where the heat equation (or vice versa the beam equation) is considered as the controller of the whole system. The dissipative damping is produced in the heat equation via the boundary connections only. The one-dimensional problem is thoroughly studied by Riesz basis approach: The closed-loop system is showed to be a Riesz spectral system and the spectrum-determined growth condition holds. As the consequences, the boundary connections with dissipation only in heat equation can stabilize exponentially the whole system, and the solution of the system has the Gevrey regularity. The exponential stability is proved for a two dimensional system with additional dissipation in the boundary of the plate part. The study gives rise to a different design in control of distributed parameter systems through weak connections with subsystems where the controls are imposed.

This is the second part of our recent work [SIAM J. Control Optim., 50 (2012), pp. 222--242] on new open sets class and related shape optimization. In this paper, we are concerned with a boundary shape optimization problem. It is shown that the convergence of the open sets class under the Hausdorff distance implies the convergence of the Hausdorff measure on the boundary. The existence of the boundary shape optimization is concluded.

In this paper, we are concerned with the output feedback stabilization of a one-dimensional wave equation with an unstable term at one end, and the observation suffered by a general harmonic disturbance with unknown magnitudes at the other end. An adaptive observer is designed in terms of the corrupted observation. The backstepping method for infinite-dimensional systems is adopted in the design of the feedback law. It is shown that the resulting closed-loop system is asymptotically stable. Meanwhile, the estimated parameters are shown to be convergent to the unknown parameters as time goes to infinity.

In this paper, the global and semiglobal convergence of the nonlinear active distur- bance rejection control (ADRC) for a class of multi-input multi-output nonlinear systems with large uncertainty that comes from both dynamical modeling and external disturbance are proved. As a result, a class of linear systems with external disturbance that can be dealt with by the ADRC is classified. The ADRC is then compared both analytically and numerically to the well-known in- ternal model principle. A number of illustrative examples are presented to show the efficiency and advantage of the ADRC in dealing with unknown dynamics and in achieving fast tracking with lower overstriking.

In this paper, we consider the well-posedness and exact controllability of a fourth-order multi-dimensional Schrodinger equation with hinged boundary by either moment or Dirichlet boundary control and collocated observation, respectively. It is shown that in both cases, the systems are well posed in the sense of D. Salamon, which implies that the systems are exactly controllable in some finite time interval if and only if its corresponding closed loop systems under the direct output proportional feedback are exponentially stable. This leads us to discuss further the exact controllability of the systems. In addition, the systems are consequently shown to be regular in the sense of G. Weiss as well, and the feedthrough operators are zero.

In this paper, a delayed HIV-1 infection model with nonlinear incidence of infection is reinvestigated. It is shown that if the reproduction number R > 1 , then the system is permanent, and the infective equilibrium of the system is globally asymptotically stable. Thus, the global dynamics of the system is completely determined by the reproduction number R . The results obtained enrich and improve the corresponding results given by Wang et al. [X. Wang, Y. Tao, X. Song, A delayed HIV-1 infection model with Beddington–DeAngelis functional response, Nonlinear Dynamics 62 (2010) 67–72]. The conclusions we established also verify the numerical simulation results on the global asymptotic stability of the infective equilibrium in the paper [D. Li, W. Ma, Asymptotic properties of an HIV-1 infection model with time delay, J. Math. Anal. Appl. 335 (2007) 683–691].

In this paper, we introduce three new classes of open sets in a general Euclidean space $\mathbb{R}^N$. It is shown that every class of open sets is compact under the Hausdorff distance. The result is then applied to a shape optimization problem of elliptic equation. The existence of the optimal solution is presented.

The open-loop systems of a Schrödinger equation and an Euler-Bernoulli equation with variable coefficients and boundary controls and collocated observations are considered. It is shown, with the help of a multiplier method on a Riemannian manifold, that both systems are well-posed in the sense of D. Salamon and regular in the sense of G. Weiss. The feed-through operators are found to be zero. The results imply particularly that the exact controlability of each open-loop system is equivalent to the exponential stability of the associated closed-loop system under the output proportional feedback.

In this paper, a shape optimization problem over a multi-dimensional starlike domain with boundary payoff is considered. The function, which characterizes the boundary of the domain with respect to some ball contained inside domain, is shown to be Lipschitz continuous. The existence of an optimal solution is proved.

The tracking differentiator was first proposed by Han in 1989 and the proof of convergence was presented the first time in Han and Wang (Han, J.Q., and Wang, W. (1994), ‘Nonlinear Tracking-differentiator’, Journal of Systems Science and Mathematical Science, 14, 177–183 (in Chinese)). Unfortunately, the proof there is incomplete. This problem has been open for over two decades. In this article, we give a rigorous proof under some additional conditions. An application for online estimation of the unknown frequencies for the finite sum of the sinusoidal signals is presented. The numerical simulations illustrate the effectiveness of the estimation for both linear and nonlinear tracking differentiators.

In this paper, we study a class of nonlinear reaction-diffusion equations with Neumann boundary control. Some sufficient conditions under which the system admits a unique global solution are given. A time-dependent gain nonlinear proportional boundary feedback control is designed to make the closed-loop system blow-up in a finite time. Finally, some concrete examples are presented to illustrate the applications.

The spectral property of an Euler–Bernoulli beam equation with clamped boundary conditions and internal Kelvin–Voigt damping is considered. The essential spectrum of the system operator is rigorously identified to be an interval on the left real axis. Under some assumptions on the coefficients, it is shown that the essential spectrum contains continuous spectrum only, and the point spectrum consists of isolated eigenvalues of finite algebraic multiplicity. The asymptotic behavior of eigenvalues is presented.