comparison of systems with complex behavior abstract we present a formalism for comparing the asymptotic dynamics of dynamical systems with physical systems that they model based on the spectral properties of the koopman operator we first compare invariant measures and discuss this in terms of a statistical takens theorem proved here we also identify the need to go beyond comparing only invariant ergodic measures of systems and introduce an ergodic theoretic treatment of a class of spectral functionals that allow for this the formalism is extended for a class of stochastic systems discrete random dynamical systems the ideas introduced in this paper can be used for parameter identification and model validation of driven nonlinear models with complicated behavior as an illustration we provide an example in which we compare the asymptotic behavior of a combustion system measured experimentally with the asymptotic behavior of a class of models that have the form of a random dynamical system

dynamic feedback stabilization of an unstable wave equation abstract in this paper we consider the stabilization of an unstable wave equation through a dynamic boundary compensator the measurement which is one internal point of the state of the wave equation is fluxed into the compensator while the output of the compensator is forced into the boundary of the wave equation the well posedness of the closed loop system is proved by using the semigroup method by choosing appropriate controller parameters and adopting nyquist criterion for distributed parameter systems the eigenvalues of the closed loop systems are shown to be inside the left half complex plane then the closed loop system is proved to be exponentially stable by using the riesz basis approach numerical simulations are presented to verify the effectiveness of the proposed compensator

strong stability of an unstable wave equation by boundary feedback with only displacement observation the stabilization of a 1 d wave equation that contains instability at its free end and control at the other end is considered the controller is designed through the estimated state that is designed in the case that only displacement is available the method of backstepping is adopted in investigation the c0 semigroup theory and lyapunov method are used to show that the closed loop system is asymptotically stable

backstepping boundary control of a wave pde with spatially distributed time invariant unknown disturbances an observer based boundary control scheme is suggested for an infinite dimensional system modeled as a wave partial differential equation pde with in domain and boundary disturbances we consider a collocated case where the sensors and the actuators are located at the same boundary the aim of the controller is to regulate the rate of change of the opposite boundary around a given reference the unknown in domain disturbance is spatially distributed but it is constant in time same as the unknown boundary disturbance the problem is reformulated as a stabilization of an linear time invariant lti system with simultaneous delays in input and output channels the exponential stability of the equilibrium of the closed loop system is proven the performance of the controller is shown using numerical simulations

closed form boundary state feedbacks for a class of 1 d partial integro differential equations in this paper a problem of boundary stabilization of a class of linear parabolic partial integro differential equations p i des in one dimension is considered using the method of backstepping avoiding spatial discretization required in previous efforts the problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic p i de the kernel p i de is then converted into an equivalent integral equation and by applying the method of successive approximations the equation s well posedness and the kernel s smoothness are established it is shown how to extend this approach to design optimally stabilizing controllers an adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller and the performance bounds are derived for a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed loop solutions are found in closed form a numerical scheme for the kernel p i de is proposed its numerical effort compares favorably with that associated with operator riccati equations

boundary stabilization of a 1 d wave equation with in domain antidamping we consider the problem of boundary stabilization of a 1 d one dimensional wave equation with an internal spatially varying antidamping term this term puts all the eigenvalues of the open loop system in the right half of the complex plane we design a feedback law based on the backstepping method and prove exponential stability of the closed loop system with a desired decay rate for plants with constant parameters the control gains are found in closed form our design also produces a new lyapunov function for the classical wave equation with passive boundary damping

models for stochastic climate prediction there has been a recent burst of activity in the atmosphere ocean sciences community in utilizing stable linear langevin stochastic models for the unresolved degree of freedom in stochastic climate prediction here several idealized models for stochastic climate modeling are introduced and analyzed through unambiguous mathematical theory this analysis demonstrates the potential need for more sophisticated models beyond stable linear langevin equations the new phenomena include the emergence of both unstable linear langevin stochastic models for the climate mean and the need to incorporate both suitable nonlinear effects and multiplicative noise in stochastic models under appropriate circumstances the strategy for stochastic climate modeling that emerges from this analysis is illustrated on an idealized example involving truncated barotropic flow on a beta plane with topography and a mean flow in this example the effect of the original 57 degrees of freedom is well represented by a theoretically predicted stochastic model with only 3 degrees of freedom

group actions on central simple algebras a geometric approach abstract we study actions of linear algebraic groups on central simple algebras using algebro geometric techniques suppose an algebraic group g acts on a central simple algebra a of degree n we are interested in questions of the following type a do the g fixed elements form a central simple subalgebra of a of degree n b does a have a g invariant maximal subfield c does a have a splitting field with a g action extending the g action on the center of a somewhat surprisingly we find that under mild assumptions on a and the actions one can answer these questions by using techniques from birational invariant theory i e the study of group actions on algebraic varieties up to equivariant birational isomorphisms in fact group actions on central simple algebras turn out to be related to some of the central problems in birational invariant theory such as the existence of sections stabilizers in general position affine models etc in this paper we explain these connections and explore them to give partial answers to questions a c

output feedback stabilization of an unstable wave equation we consider the problem of stabilization of a one dimensional wave equation that contains instability at its free end and control on the opposite end in contrast to classical collocated boundary damper feedbacks for the neutrally stable wave equations with one end satisfying a homogeneous boundary condition the controllers and the associated observers designed in the paper are more complex due to the open loop instability of the plant the controller and observer gains are designed using the method of backstepping which results in explicit formulae for the gain functions we prove exponential stability and the existence and uniqueness of classical solutions for the closed loop system we also derive the explicit compensators in frequency domain the results are illustrated with simulations

observer design and exponential stabilization for wave equation in energy space by boundary displacement measurement only in this technical note we consider finite time and exponential stabilization for one dimensional wave equations by boundary displacement measurement only we limit ourselves in the energy state space where the usual observability inequality is not valid anymore however in the optimal state space the boundary displacement is indeed exactly observable this motivates us to design observer via displacement output only for these systems we first discuss a simple case as a motivation an observer is designed and an output feedback control is then synthesized to make system finite time stable in energy space in this same spirit we consider the same problem for an unstable wave equation and the finite time stability is also achieved by displacement output feedback finally we consider an anti stable wave equation a direct delayed output feedback control can achieve exponential stability with arbitrary decay rate simulation results are presented to validate the theoretical conclusions

compensation of wave actuator dynamics for nonlinear systems the problem of stabilization of pde ode cascades has been solved in the linear case for several pde classes whereas in the nonlinear case the problem has been solved only for the transport delay pde namely for compensation of an arbitrary delay at the input of a nonlinear plant motivated by a specific engineering application in off shore drilling we solve the problem of stabilization of the cascade of a wave pde with a general nonlinear ode due to the presence of nonlinearities of arbitrary growth and the time reversibility of the wave pde and due to the possibility of using arguments based on lyapunov functionals or explicit solutions several stability analysis approaches are possible we present stability results in the h 2 times h 1 and c 1 times c 0 norms for general nonlinear odes as well as in the h 1 times l 2 norm for linear odes we specialize our general design for wave pde ode cascades to the case of a wave pde whose uncontrolled end does not drive an ode but is instead governed by a nonlinear robin boundary condition a nonlinear spring as in the friction law in drilling this is the first global stabilization result for wave equations that incorporate non collocated destabilizing nonlinearities of superlinear growth we present two numerical examples one with a nonlinear ode and one with a nonlinear spring at the uncontrolled boundary of the wave pde

boundary state feedback exponential stabilization for a one dimensional wave equation with velocity recirculation abstract in this paper we consider boundary state feedback stabilization of a one dimensional wave equation with in domain feedback recirculation of an intermediate point velocity we firstly construct an auxiliary control system which has a nonlocal term of the displacement at the same intermediate point then by choosing a well known exponentially stable wave equation as its target system we find one backstepping transformation from which a state feedback law for this auxiliary system is proposed finally taking the resulting closed loop of the auxiliary system as a new target system we obtain another backstepping transformation from which a boundary state feedback controller for the original system is designed by the equivalence of three systems the closed loop of original system is proved to be well posed and exponentially stable some numerical simulations are presented to validate the theoretical results

stabilization of one dimensional wave equation with nonlinear boundary condition subject to boundary control matched disturbance in this paper we consider the stabilization of a one dimensional wave equation with nonlinear van der pol type boundary condition that covers the antistable boundary and subject to boundary control matched disturbance on the other side due to the nonlinear boundary condition and disturbance the uncontrolled system may present spatiotemporal chaotic period doubling bifurcation and some other dynamical behaviors we will deal with this disturbance which is supposed to be bounded only by the integral sliding mode control the well posedness of the system for the closed loop system is proved and the reaching condition is obtained finally we provide some numerical simulations to illustrate the theoretical outcomes

active disturbance rejection control based on weighed moving average state observer abstract in this work we are concerned with the boundary stabilization of a one dimensional anti stable wave equation corrupted by a boundary disturbance we firstly propose by weighed moving average technique a state observer to make an estimation of the disturbance secondly we design a control by the active disturbance rejection strategy to stabilize the system

backstepping based estimation of thermoacoustic oscillations in a rijke tube with experimental validation we investigate the state observer design problem for thermoacoustic instabilities in a rijke tube using an infinite dimensional perspective the observer whose design is based on the backstepping methodology with volterra and full integral terms consists of a copy of the linearized plant model plus output injection terms and relies only on a single boundary acoustic pressure sensor the exponential convergence in the l 2 sense of the observed error dynamics is proved and the analytic expression of the observer gains are derived explicitly the results are tested in an experimental rijke tube prototype in order to illustrate the effectiveness of the method

rapid stabilization for a korteweg de vries equation from the left dirichlet boundary condition this paper deals with the stabilization problem for the korteweg de vries equation posed on a bounded interval the control acts on the left dirichlet boundary condition at the right end point dirichlet and neumann homogeneous boundary conditions are considered the proposed feedback law forces the exponential decay of the system under a smallness condition on the initial data moreover the decay rate can be tuned to be as large as desired the feedback control law is designed by using the backstepping method

stability results for the wave equation with indefinite damping abstract the one dimensional wave equation with damping of indefinite sign in a bounded interval with dirichlet boundary conditions is considered it is proved that solutions decay uniformly exponentially to zero provided the damping potential is in thebv class has positive average is small enough and satisfies a finite number of further constraints guaranteeing that the derivative of the real part of the eigenvalues is negative when the damping vanishes this sharp result completes a previous one by the first author showing that an indefinite sign damping always produces unstable solutions if it is large enough and it answers by the afirmative to a conjecture concerning small damping terms the method of proof relies on the developments by s cox and the second author on the high frequency asymptotic expansion of the spectrum for damped wave equations and on the characterization of the decay rate in terms of the spectral abscissa

boundary control of coupled reaction diffusion processes with constant parameters the problem of boundary stabilization is considered for some classes of coupled parabolic linear pdes of the reaction diffusion type with reference to n coupled equations each one equipped with a scalar boundary control input a state feedback law is designed with actuation at only one end of the domain and exponential stability of the closed loop system is proven the treatment is addressed separately for the case in which all processes have the same diffusivity and for the more challenging scenario where each process has its own diffusivity and a different solution approach has to be taken the backstepping method is used for controller design and particularly the kernel matrix of the transformation is derived in explicit form of series of bessel like matrix functions by using the method of successive approximations to solve the corresponding pde thus the proposed control laws become available in explicit form additionally the stabilization of an underactuated system of two coupled reaction diffusion processes is tackled under the restriction that only a scalar boundary input is available capabilities of the proposed synthesis and its effectiveness are supported by numerical studies made for three coupled systems with distinct diffusivity parameters and for underactuated linearized dimensionless temperature concentration dynamics of a tubular chemical reactor controlled through a boundary at low fluid superficial velocities when convection terms become negligible

null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach using the backstepping approach we recover the null controllability for the heat equations with variable coefficients in space in one dimension and prove that these equations can be stabilized in finite time by means of periodic time varying feedback laws to this end on the one hand we provide a new proof of the well posedness and the optimal bound with respect to damping constants for the solutions of the kernel equations this allows one to deal with variable coefficients even with a weak regularity of these coefficients on the other hand we establish the well posedness and estimates for the heat equations with a nonlocal boundary condition at one side

anti collocated observer based output feedback control of wave equation with cubic velocity nonlinear boundary and dirichlet control this study focuses on the observer and observer based output feedback control design for a one dimensional wave equation with van der pol type nonlinear boundary condition and dirichlet boundary co

parameter estimation and non collocated adaptive stabilization for a wave equation subject to general boundary harmonic disturbance this paper is concerned with the parameter estimation and asymptotic stabilization of a 1 d wave equation that is subject to general harmonic disturbances at the controlled end and suffers from instability at the other end first we design an adaptive observer in terms of measured position and velocity we then adopt the backstepping method for infinite dimensional systems to design an observer based output feedback law the resulting closed loop system is shown to be asymptotically stable and the estimates of the parameters converge to the unknown parameters

boundary stabilization of wave equation with velocity recirculation nonlocal terms have been the mainstay of the applications of partial differential equation pde backstepping methods to parabolic pdes the problem of similar nonlocal terms for wave equations is still open for wave equations similar nonlocal terms have not been studied in this paper we open the topic of exploration of control of wave pdes with nonlocal terms this paper is concerned with the wave equation with in domain feedback recirculation of a boundary velocity with a spatially constant coefficient due to this nonlocal term the passivity of the wave equation is destroyed we first design an explicit state feedback controller to achieve exponential stability for the closed loop system then we design the output feedback by using infinite dimensional observer the backstepping approach is adopted in investigation it is shown that by using two measurements only the output feedback makes the closed loop system exponentially stable

output feedback adaptive control of a wave pde with boundary anti damping abstract we develop an adaptive output feedback controller for a wave pde in one dimension with actuation on one boundary and with an unknown anti damping term on the opposite boundary this model is representative of a torsional stick slip instability in drillstrings in deep oil drilling as well as of various acoustic instabilities the key feature of the proposed controller is that it requires only the measurements of boundary values and not of the entire distributed state of the system our approach is based on employing riemann variables to convert the wave pde into a cascade of two delay elements with the first of the two delay elements being fed by control and the same element in turn feeding into a scalar ode this enables us to employ a prediction based design for systems with input delays suitably converted to the adaptive output feedback setting the result s relevance is illustrated with simulation example

analysis and stabilization for networked linear hyperbolic systems of rationally dependent conservation laws in this paper for a networked linear hyperbolic partial differential equations pdes system of conservation laws the propagation periods of which are rationally dependent with coupled boundary conditions we propose a novel approach to analyze its controllability and observability in addition we propose a design method of a stabilizing controller where a boundary input with boundary valued feedback is considered first we characterize the control properties such as controllability of such a pde system in terms of the corresponding ones of a finite dimensional discrete time system defined on the boundaries of the pde system which is derived by fully exploiting the method of characteristics since the obtained discrete time system is low dimensional its analysis is relatively easier next we propose a design method of a stabilizing controller based on this discrete time system finally numerical simulations are presented to show that the proposed method is effective

ude based robust boundary control for an unstable parabolic pde with unknown input disturbance a novel control strategy named uncertainty and disturbance estimator ude based robust control is applied to the stabilization of an unstable parabolic partial differential equation pde with a dirichlet type boundary actuator and an unknown time varying input disturbance the unstable pde is stabilized by the backstepping approach and the unknown input disturbance is compensated by the ude which constructs an estimation of the disturbance by filtering the system input and measurable state signals the proposed ude based robust boundary control only requires the spectrum information of the disturbance signal and one boundary output measurement stability analysis of the closed loop systems is carried out and a numerical example is provided to validate the proposed method

boundary control of partial differential equations using frequency domain optimization techniques abstract we present a frequency domain based h control strategy to solve boundary control problems for systems governed by parabolic or hyperbolic partial differential equation where controllers are constrained to be physically implementable and of simple structure suited for practical applications the efficiency of our technique is demonstrated by controlling a reaction diffusion equation with input delay and a wave equation with boundary anti damping

backstepping approach to the arbitrary decay rate for euler bernoulli beam under boundary feedback in this article we are concerned with the boundary stabilisation of the euler bernoulli beam equation for which all eigenvalues of the control free system are located on the imaginary axis of the complex plane the fourth order system in spacial variable is transformed into a coupled heat like system this enables us to make a natural backstepping transformation in vector form to transform the system into a target system which has arbitrary decay rate the state feedback is thus designed it is shown that the original closed loop system is exponentially stable with the given arbitrary decay rate

controllability for a class of area preserving twist maps abstract in this paper we study controllability of two dimensional integrable twist maps with bounded area preserving time dependent control perturbations in contrast to the time independent perturbation case of the kolmogorov arnold moser theorem there are no invariant sets other than the whole phase space if the perturbation is made a function of time we give necessary and sufficient conditions for global controllability of these maps

boundary control of an anti stable wave equation with anti damping on the uncontrolled boundary abstract much of the boundary control of wave equations in one dimension is based on a single principle passivity under the assumption that control is applied through neumann actuation on one boundary and the other boundary satisfies a homogeneous dirichlet boundary condition we have recently expanded the scope of tractable problems by allowing destabilizing anti stiffness a robin type condition on the uncontrolled boundary where the uncontrolled system has a finite number of positive real eigenvalues in this paper we go further and develop a methodology for the case where the uncontrolled boundary condition has anti damping which makes the real parts of all the eigenvalues of the uncontrolled system positive and arbitrarily high i e the system is anti stable exponentially stable in negative time using a conceptually novel integral transformation we obtain extremely simple explicit formulae for the gain functions for the case with only boundary sensing available at the same end with actuation we design backstepping observers which are dual to the backstepping controllers and have explicit output injection gains we then combine the control and observer designs into an output feedback compensator and prove the exponential stability of the closed loop system

backstepping observers for a class of parabolic pdes in this paper we design exponentially convergent observers for a class of parabolic partial integro differential equations p i des with only boundary sensing available the problem is posed as a problem of designing an invertible coordinate transformation of the observer error system into an exponentially stable target system observer gain output injection function is shown to satisfy a well posed hyperbolic pde that is closely related to the hyperbolic pde governing backstepping control gain for the state feedback problem for several physically relevant problems the observer gains are obtained in closed form the observer gains are then used for an output feedback design in both collocated and anti collocated setting of sensor and actuator the order of the resulting compensator can be substantially lowered without affecting stability explicit solutions of a closed loop system are found in particular cases

brief paper arbitrary decay rate for two connected strings with joint anti damping by boundary output feedback in this paper we are concerned with the boundary stabilization of two connected strings with middle joint anti damping for which all eigenvalues of the control free system are located on the right complex plane we first design an explicit state feedback controller to achieve exponential stability for the closed loop system consequently we design the output feedback by using infinite dimensional observer the backstepping approach is adopted in investigation it is shown that by using one boundary stabilizer only the output feedback can make the closed loop system exponentially stable with arbitrary decay rate

output feedback stabilization of an unstable wave equation with general corrupted boundary observation we consider boundary output feedback stabilization for an unstable wave equation with boundary observation subject to a general disturbance we adopt for the first time the active disturbance rejection control approach to stabilization for a system described by the partial differential equation with corrupted output feedback by the approach the disturbance is first estimated by a relatively independent estimator it is then canceled in the feedback loop as a result the control law can be designed almost as that for the system without disturbance we show that with a time varying gain properly designed the observer driven by the disturbance estimator is convergent and that all subsystems in the closed loop are asymptotically stable in the energy state space we also provide numerical simulations which demonstrate the convergence results and underline the effect of the time varying gain estimator on peaking value reduction