necessary and sufficient small gain conditions for integral input to state stable systems a lyapunov perspective this paper is concerned with conditions for the stability of interconnected nonlinear systems consisting of integral input to state stable iiss systems with external inputs the treatment of iiss and input to state stable iss systems is unified both necessary conditions and sufficient conditions are investigated using a lyapunov formulation in the presence of model uncertainty this paper proves that for the stability of the interconnected system at least one subsystem is necessarily iss which is a stronger stability property in the set of iiss the necessity of a small gain type property is also demonstrated this paper proposes a common form of smooth lyapunov functions which can establish the iiss and the iss of the interconnection comprising iiss and iss subsystems whenever the small gain type condition is satisfied the result covers situations more general than the earlier study and removes technical conditions assumed in the previous literature global asymptotic stability is discussed as a special case

the effect of two stage sampling on ordinary least squares methods abstract we look at the effect of intracluster correlation on standard procedures in linear regression the ordinary least squares estimator of the coefficient vector performs well in most cases but the usual estimator of cov and procedures based on this such as confidence intervals and hypothesis tests can be seriously misleading the size of the effect however tends to be smaller than the corresponding effect on the variance of an estimated mean in two stage sampling provided that the cluster sample sizes are approximately equal

lyapunov based boundary feedback design for parabolic pdes this paper presents a methodology for the construction of simple control lyapunov functionals clfs for boundary controlled parabolic partial differential equations pdes the proposed methodolog

convenient estimators for the panel probit model the paper shows that several estimators for the panel probit model suggested in the literature belong to a common class of gmm estimators they are relatively easy to compute because they are based on conditional moment restrictions involving univariate moments of the binary dependent variable only applying nonparametric methods we discuss an estimator that is optimal in this class a monte carlo study shows that a particular variant of this estimator has good small sample properties and that the efficiency loss compared to maximum likelihood is small an application to the product innovation decisions of german firms reveals the expected efficiency gains

characterizations of input to state stability for infinite dimensional systems we prove characterizations of input to state stability iss for a large class of infinite dimensional control systems including some classes of evolution equations over banach spaces time delay systems ordinary differential equations ode and switched systems these characterizations generalize well known criteria of iss proved by sontag and wang for ode systems for the special case of differential equations in banach spaces we prove even broader criteria for iss and apply these results to show that under some mild restrictions the existence of a noncoercive iss lyapunov functions implies iss we introduce the new notion of strong iss siss that is equivalent to iss in the ode case but is strictly weaker than iss in the infinite dimensional setting and prove several criteria for the siss property at the same time we show by means of counterexamples that many characterizations which are valid in the ode case are not true for general infinite dimensional systems

a computationally efficient quadrature procedure for the one factor multinomial probit model a problem of estimation that has long confronted many economists is the difficulty of estimating the parameters of equations with limited dependent variables on cross section time series i e panel data while there are widely available packaged computer programs for estimating either a cross section probit and tobit models or b simple permanent transitory random effects panel models with continuous dependent variables there are no available computationally feasible methods of combining these two models this is because the likelihood function that arises in such a combined model contains multivariate normal integrals whose evaluation is quite difficult if not impossible with conventional approximation methods there is a widespread feeling among those working in the area that one possible method of evaluation the use of quadrature techniques is in principle possible but is in practice computationally too burdensome to consider e g albright et al 2 p 13 hausman and wise 6 p 12 in this note we point out that this is true only of standard quadrature techniques such as trapezoidal integration or its improved variants gaussian quadrature on the other hand is extremely efficient and is well within the bounds of computational feasibility on modern computers in what follows we state the nature of the integrals that need to be evaluated provide a brief exposition of gaussian quadrature and provide a numerical illustration of its use in

further facts about input to state stabilization previously published results about input to state stabilizability are shown to hold even for systems which are not linear in controls provided that a more general type of feedback is allowed applications to certain stabilization problems and coprime factorizations as well as comparisons to other results on input to state stability are briefly discussed

the gauss green theorem for fractal boundaries v is the 1 vectorfield dual to if 1 fi dx1 dxi dxn then v f1 fn there has been considerable effort in the literature e g jk m p to extend this formula to permit integrands of less regularity by generalizing the lebesgue integral on the other hand invariably the situations in which 1 holds require fairly strong hypotheses on the boundary e g that it should have sigmafinite n 1 measure or that the gradient of the characteristic function of be a vector valued measure with finite total variation f p however there is a natural way to expand the validity of 1 to much more general boundaries while still using the ordinary lebesgue integral this is the topic of the present paper for the case of lipschitz forms the results of this paper follow readily from whitney s theory of flat chains w2 however his approach to the gauss green theorem is not widely appreciated because he focused on chains and cochains where effectively 1 is used to define the exterior derivative in hn we extend whitney s method to treat the more general holder case only the case n 2 is discussed

exponential input to state stabilization of a class of diagonal boundary control systems with delay boundary control abstract this paper deals with the exponential input to state stabilization with respect to boundary disturbances of a class of diagonal infinite dimensional systems via delay boundary control the considered input delays are uncertain and time varying the proposed control strategy consists of a constant delay predictor feedback controller designed on a truncated finite dimensional model capturing the unstable modes of the original infinite dimensional system we show that the resulting closed loop system is exponentially input to state stable with fading memory of both additive boundary input perturbations and disturbances in the computation of the predictor feedback

on stabilizability and sampling for infinite dimensional systems some questions either motivated by or related to time sampling regarding the stabilizability of infinite dimensional or distributed parameter systems are considered in particular these include the characteristics of those systems which are finite rank stabilizable and the preservation of stabilizability under sampling a pathological class of systems which when sampled may be rendered unstabilizable by arbitrarily small perturbations in the sampling rate is identified the approach taken in establishing these results is based upon a spectral decomposition of the infinitesimal generator of the underlying open loop semigroup the results obtained apply to control systems with dynamics described by broad classes of partial and functional differential equations that arise frequently in applications

sampled data observers for semilinear damped wave equations under spatially sampled state measurements abstract sampled data observers controllers under the sampled in space and time measurements were suggested in the past for parabolic systems in the present paper for the first time a sampled data observer is constructed for a hyperbolic system governed by 1d semilinear wave equation with either viscous or boundary damping the measurements are sampled in space and time sufficient conditions for the exponential stability of the estimation error are derived by using the time delay approach to sampled data control and appropriate lyapunov krasovskii functionals the dual sampled data controller problems are formulated numerical examples including observer design for unstable damped sine gordon equation illustrate the efficiency of the method

cauchy transform on nonrectifiable surfaces in clifford analysis abstract the problem of reconstructing a monogenic clifford algebra valued function on the boundary of a general open set in r n 1 from a prescribed jump data u over the boundary is deeply connected with the study of the clifford cauchy transform c u x y x n y x n 1 n y u y d y x necessary and sufficient condition on non rectifiable is established guaranteeing the existence of continuous boundary values of this transform for all functions satisfying a holder type condition

a tight small gain theorem for not necessarily iss systems a new version of the small gain theorem is presented for nonlinear finite dimensional systems the result provides conditions for global asymptotic stability under relaxed assumptions in particular the two interconnected subsystems need not be input to state stable in open loop

sampled data boundary feedback control of 1 d parabolic pdes abstract the paper provides results for the application of boundary feedback control with zero order hold zoh to 1 d linear parabolic systems on bounded domains it is shown that the continuous time boundary feedback applied in a sample and hold fashion guarantees closed loop exponential stability provided that the sampling period is sufficiently small two different continuous time feedback designs are considered the reduced model design and the backstepping design the obtained results provide stability estimates for weighted 2 norms of the state and robustness with respect to perturbations of the sampling schedule is guaranteed

iss like estimates for nonlinear parabolic pdes with variable coefficients on higher dimensional domains abstract this paper presents a maximum principle based approach in the establishment of iss like estimates for a class of nonlinear parabolic partial differential equations pdes with variable coefficients and different types of nonlinear boundary conditions on higher dimensional domains comparing with the existing literature the iss like estimates established in this paper are independent of the nonlinear terms of the pdes technical development on iss analysis of the considered systems is detailed and an example of establishing iss like estimates for a nonlinear parabolic equation with respectively a nonlinear robin boundary condition and a nonlinear dirichlet boundary condition is provided to illustrate the application of the developed method

on the asymptotic properties of estimators of models containing limited dependent variables for the tobit model with independent observations amemiya 1 has established the strong consistency and asymptotic normality of a stationary point 9 of the log likelihood the likelihood for dependent observations may be computationally intractable so the behavior of 9 in the presence of serially correlated observations is of interest under a relaxation of amemiya s assumption of independence we prove that 9 is strongly consistent and asymptotically normal and give an expression for the limiting covariance matrix

small sample properties of probit model estimators abstract when maximum likelihood estimates of the coefficients in a nonlinear model such as the probit model are obtained there are a number of asymptotically equivalent covariance matrix estimators that can be used these covariance matrix estimators are typically associated with different computer algorithms for example with the newton raphson algorithm the inverse of the negative of the hessian matrix from the log likelihood function is used with the method of scoring the inverse of the information matrix is used and with a procedure proposed by berndt hall hall and hausman 1974 the inverse of the outer product of the first derivatives of the log likelihood function is used although these three estimators are asymptotically equivalent their performance can vary in finite samples the main objective of this article is to use a monte carlo experiment to investigate the finite sample properties of the three covariance matrix estimators in the context of maximum likelihood estimation of the pr

input output to state stability this work explores lyapunov characterizations of the input output to state stability ioss property for nonlinear systems the notion of ioss is a natural generalization of the standard zero detectability property used in the linear case the main contribution of this work is to establish a complete equivalence between the ioss property and the existence of a certain type of smooth lyapunov function as corollaries one shows the existence of norm estimators and obtains characterizations of nonlinear detectability in terms of relative stability and of finite energy estimates

stability of higher order discrete time lur e systems abstract we consider discrete time lur e systems obtained by applying nonlinear feedback to a system of higher order difference equations arma models the arma model relates the inputs and outputs of the linear system and does not involve any internal or state variables a stability theory subsuming results of circle criterion type is developed including criteria for input to output stability a concept which is very much reminiscent of input to state stability

the cauchy integral calderon commutators and conjugations of singular integrals in ⁿ we consider the cauchy integral and hilbert transform for lipschitz domains in the clifford algebra based on r 1 the hilbert transform is shown to be the generating function for the calderon commutators in r 1 we make use of an intrinsic characterization of these commutators to obtain l2 estimates these estimates are used to show the analyticity of the hilbert transform and of the conjugation of singular integral operators by bi lipschitz changes of variable in r 1

lyapunov characterization of input to state stability for semilinear control systems over banach spaces abstract we prove that input to state stability iss of nonlinear systems over banach spaces is equivalent to existence of a coercive lipschitz continuous iss lyapunov function for this system for linear infinite dimensional systems we show that iss is equivalent to existence of a non coercive iss lyapunov function and provide two simpler constructions of coercive and non coercive iss lyapunov functions for input to state stable linear systems

input to state stability of nonlinear impulsive systems we prove that impulsive systems which possess an input to state stable iss lyapunov function are iss for time sequences satisfying the fixed dwell time condition if an iss lyapunov function is the exponential one we provide a stronger result which guarantees uniform iss of the whole system over sequences satisfying the generalized average dwell time condition then we prove two small gain theorems that provide a construction of an iss lyapunov function for an interconnection of impulsive systems if the iss lyapunov functions for subsystems are known the construction of local iss lyapunov functions via the linearization method is provided relations between small gain and dwell time conditions as well as between different types of dwell time conditions are also investigated although our results are novel already in the context of finite dimensional systems we prove them for systems based on differential equations in banach spaces that makes obtained results considerably more general

brief paper capability and limitation of max and sum type construction of lyapunov functions for networks of iiss systems this paper addresses the problem of verifying stability of networks whose subsystems admit dissipation inequalities of integral input to state stability iiss we focus on two ways of constructing a lyapunov function satisfying a dissipation inequality of a given network their difference from one another is elucidated from the viewpoint of formulation relation fundamental limitation and capability one is referred to as the max type construction resulting in a lipschitz continuous lyapunov function the other is the sum type construction resulting in a continuously differentiable lyapunov function this paper presents geometrical conditions under which the lyapunov construction is possible for a network comprising n 2 subsystems although the sum type construction for general n 2 has not yet been reduced to a readily computable condition we obtain a simple condition of iiss small gain in the case of n 2 it is demonstrated that the max type construction fails to offer a lyapunov function if the network contains subsystems which are not input to state stable iss

misspecified heteroskedasticity in the panel probit model a small sample comparison of gmm and sml estimators abstract this paper compares generalized method of moments gmm and simulated maximum likelihood sml approaches to the estimation of the panel probit model both techniques circumvent multiple integration of joint density functions without the need to restrict the error term variance covariance matrix of the latent normal regression model particular attention is paid to a three stage gmm estimator based on nonparametric estimation of the optimal instruments for given conditional moment functions monte carlo experiments are carried out which focus on the small sample consequences of misspecification of the error term variance covariance matrix the correctly specified experiment reveals the asymptotic efficiency advantages of sml the gmm estimators outperform sml in the presence of misspecification in terms of multiplicative heteroskedasticity this holds in particular for the three stage gmm estimator allowing for heteroskedasticity over time increases the robustness with respect to misspecification in terms of multiplicative heteroskedasticity an application to the product innovation activities of german manufacturing firms is presented

a characterization of integral input to state stability the notion of input to state stability iss is now recognized as a central concept in nonlinear systems analysis it provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite l sup 2 gains it plays a central role in recursive design coprime factorizations controllers for nonminimum phase systems and many other areas in this paper a newer notion that of integral input to state stability iiss is studied the notion of iiss generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states in that sense generalizing the linear h sup 2 theory it allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance we obtain several necessary and sufficient characterizations of the iiss property expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations

small gain theorems for large scale systems and construction of iss lyapunov functions we consider interconnections of n nonlinear subsystems in the input to state stability iss framework for each subsystem an iss lyapunov function is given that treats the other subsystems as independent inputs a gain matrix is used to encode the mutual dependencies of the systems in the network under a small gain assumption on the monotone operator induced by the gain matrix a locally lipschitz continuous iss lyapunov function is obtained constructively for the entire network by appropriately scaling the individual lyapunov functions for the subsystems the results are obtained in a general formulation of iss the cases of summation maximization and separation with respect to external gains are obtained as corollaries

controllability of the semilinear parabolic equation governed by a multiplicative control in the reaction term a qualitative approach in this paper we are concerned with the global nonnegative approximate controllability property of a rather general semilinear heat equation with superlinear term governed in a bounded domain omega subset r n by a multiplicative bilinear control in the reaction term like v u x t where v is the control we show that any nonnegative target state in l 2 omega can approximately be reached from any nonnegative nonzero initial state by applying at most three static bilinear l infty omega controls subsequently in time this result is further applied to discuss the controllability properties of the nonhomogeneous version of this problem with bilinear term like v u x t theta x where theta is given our approach is based on an asymptotic technique allowing us to distinguish and make use of the pure diffusion and or pure reaction parts of the dynamics of the system at hand while suppressing the effect of a general nonlinear term

integral control of linear systems with actuator nonlinearities lower bounds for the maximal regulating gain closing the loop around an exponentially stable single input single output regular linear system subject to a globally lipschitz and nondecreasing actuator nonlinearity and compensated by an integral controller is known to ensure asymptotic tracking of constant reference signals provided that 1 the steady state gain of the linear part of the plant is positive 2 the positive integrator gain is sufficiently small and 3 the reference value is feasible in a very natural sense here lower bounds are derived for the maximal regulating gain for various special cases including systems with nonovershooting step response and second order systems with a time delay in the input or output the lower bounds are given in terms of open loop frequency step response data and the lipschitz constant of the nonlinearity and are hence readily obtainable

sampled data relay control of diffusion pdes we consider a vector reaction advection diffusion equation on a hypercube the measurements are weighted averages of the state over different subdomains these measurements are asynchronously sampled in time subject to matched disturbances the discrete control signals are applied through shape functions and zero order holds the feature of this work is that we consider generalized relay control the control signals take their values in a finite set this allows for networked control through low capacity communication channels first we derive linear matrix inequalities lmis whose feasibility guarantees the ultimate boundedness with a limit bound proportional to the sampling period then we construct a switching procedure for the controller parameters that ensures semi global practical stability for an arbitrarily large domain of initial conditions the trajectories converge to a set whose size does not depend on the domain size for the disturbance free system this procedure guarantees exponential convergence to the origin the results are demonstrated by two examples 2d catalytic slab and a chemical reactor

estimation and testing in the random effects probit model abstract this paper examines the finite sample properties of the random effects probit estimator in comparison to the standard probit estimator and the standard probit estimator with a corrected asymptotic covariance matrix the monte carlo experiment considers data generating processes consistent with longitudinal data and also data from sample surveys the probit estimator with corrected asymptotic covariance matrix works surprisingly well over a wide range of parametric configurations and is recommended as long as an estimate of the error correlation is not of high importance

stability of nonnegative lur e systems a stability instability trichotomy for a class of nonnegative continuous time lur e systems is derived asymptotic exponential and input to state stability concepts are considered the presented trichotomy rests on perron frobenius theory absolute stability theory and recent input to state stability results for lur e systems applications of the results derived arise in various fields including density dependent population dynamics and two examples are discussed in detail

state dependent scaling problems and stability of interconnected iiss and iss systems this paper addresses the problem of establishing stability of nonlinear interconnected systems this paper introduces a mathematical formulation of the state dependent scaling problems whose solutions directly provide lyapunov functions proving stability properties of interconnected dissipative systems in a unified manner stability criteria are interpreted as sufficient conditions for the existence of solutions to the state dependent scaling problems computing solutions to the problems is straightforward for systems covered by classical stability criteria it however could be too difficult for systems with strong nonlinearity the main purpose of this paper is to demonstrate the effectiveness beyond formal applicability by focusing on interconnected integral input to state stable iiss systems and input to state stable iss systems this paper derives small gain type theorems for interconnected systems involving iiss systems from the state dependent scaling formulation this paper provides solutions and lyapunov functions explicitly the new framework seamlessly generalizes the iss small gain theorem and classical stability criteria such as the lp small gain theorem the passivity theorems the circle and popov criteria state dependence of the scaling is crucial for effective treatment of essential nonlinearities while constants are sufficient for classical nonlinearities

construction of lyapunov functions for interconnected parabolic systems an iiss approach this paper is devoted to two issues the first one is to provide lyapunov based tools to establish integral input to state stability iiss and input to state stability iss for some classes of nonlinear parabolic equations the other one is to provide a stability criterion for interconnections of iiss parabolic systems the results addressing the former problem allow us to overcome obstacles arising in tackling the latter one the results for the latter problem are a small gain condition and a formula of lyapunov functions which can be constructed for interconnections whenever the small gain condition holds it is demonstrated that for interconnections of partial differential equations the choice of a right state and input spaces is crucial in particular for iiss subsystems which are not iss as illustrative examples stability of two highly nonlinear reaction diffusion systems is established by the proposed small gain criterion

a de giorgi iteration based approach for the establishment of iss properties for burgers equation with boundary and in domain disturbances this note addresses input to state stability iss properties with respect to w r t boundary and in domain disturbances for burgers equation the developed approach is a combination of the method of de giorgi iteration and the technique of lyapunov functionals by adequately splitting the original problem into two subsystems the iss properties in l 2 norm for burgers equation have been established using this method moreover as an application of de giorgi iteration iss in l infty norm w r t in domain disturbances and actuation errors in boundary feedback control for a one dimensional linear unstable reaction diffusion equation have also been established it is the first time that the method of de giorgi iteration is introduced in the iss theory for infinite dimensional systems and the developed method can be generalized for tackling some problems on multidimensional spatial domains and to a wider class of nonlinear partial differential equations

output to state stability and detectability of nonlinear systems the notion of input to state stability iss has proved to be useful in nonlinear systems analysis this paper discusses a dual notion output to state stability oss a characterization is provided in terms of a dissipation inequality involving storage lyapunov functions combining iss and oss there results the notion of input output to state stability ioss which is also studied and related to the notion of detectability the existence of observers and output injection

iss in different norms for 1 d parabolic pdes with boundary disturbances for one dimensional parabolic partial differential equations with disturbances at both boundaries and distributed disturbances we provide input to state stability iss estimates in various norms due to the lack of an iss lyapunov functional for boundary disturbances the proof methodology uses i an eigenfunction expansion of the solution and ii a finite difference scheme the iss estimate for the sup norm leads to a refinement of the well known maximum principle for the heat equation finally the obtained results are applied to quasi static thermoelasticity models that involve nonlocal boundary conditions small gain conditions that guarantee the global exponential stability of the zero solution for such models are derived

a method of simulated moments for estimation of discrete response models without numerical integration this paper proposes a simple modification of a conventional method of moments estimator for a discrete response model replacing response probabilities that require numerical integration with estimators obtained by monte carlo simulation this method of simulated moments msm does not require precise estimates of these probabilities for consistency and asymptotic normality relying instead on the law of large numbers operating across observations to control simulation error and hence can use simulations of practical size the method is useful for models such as high dimensional multinomial probit mnp where computation has restricted applications

iss property with respect to boundary disturbances for a class of riesz spectral boundary control systems abstract this paper deals with the establishment of input to state stability iss estimates for infinite dimensional systems with respect to both boundary and distributed disturbances first a new approach is developed for the establishment of iss estimates for a class of riesz spectral boundary control systems satisfying certain eigenvalue constraints second a concept of weak solutions is introduced in order to relax the disturbances regularity assumptions required to ensure the existence of classical solutions the proposed concept of weak solutions that applies to a large class of boundary control systems which is not limited to the riesz spectral ones provides a natural extension of the concept of both classical and mild solutions assuming that an iss estimate holds true for classical solutions we show the existence the uniqueness and the iss property of the weak solutions

on continuity of solutions for parabolic control systems and input to state stability abstract we study minimal conditions under which mild solutions of linear evolutionary control systems are continuous for arbitrary bounded input functions this question naturally appears when working with boundary controlled linear partial differential equations here we focus on parabolic equations which allow for operator theoretic methods such as the holomorphic functional calculus moreover we investigate stronger conditions than continuity leading to input to state stability with respect to orlicz spaces this also implies that the notions of input to state stability and integral input to state stability coincide if additionally the uncontrolled equation is dissipative and the input space is finite dimensional

input to state stability of differential inclusions with applications to hysteretic and quantized feedback systems input to state stability iss of a class of differential inclusions is proved every system in the class is of lur e type a feedback interconnection of a linear system and a set valued nonlinearity applications of the iss results in the context of feedback interconnections with a hysteresis operator or a quantization operator in the feedback path are developed

robust stability of networks of iiss systems construction of sum type lyapunov functions this paper gives a solution to the problem of verifying stability of networks consisting of integral input to state stable iiss subsystems the iiss small gain theorem developed recently has been restricted to interconnections of two subsystems for large scale systems stability criteria relying only on gain type information that were previously developed address only input to state stable iss subsystems to address the stability problem involving iiss subsystems interconnected in general structure this paper shows how to construct lyapunov functions of the network by means of a sum of nonlinearly rescaled individual lyapunov functions of subsystems under an appropriate small gain condition

infinite dimensional input to state stability and orlicz spaces in this work the relation between input to state stability and integral input to state stability is studied for linear infinite dimensional systems with an unbounded control operator although a special focus is laid on the case l infty general function spaces are considered for the inputs we show that integral input to state stability can be characterized in terms of input to state stability with respect to orlicz spaces since we consider linear systems the results can also be formulated in terms of admissibility for parabolic diagonal systems with scalar inputs both stability notions with respect to l infty are equivalent

stability and convergence properties of forced infinite dimensional discrete time lur e systems incremental stability and convergence properties for forced infinite dimensional discrete time lur e systems are addressed lur e systems have a linear and nonlinear component and arise as the fe

iss with respect to boundary disturbances for 1 d parabolic pdes due to unbounded input operators in partial differential equations pdes with boundary inputs there has been a long held intuition that input to state stability iss properties and finite gains cannot be established with respect to disturbances at the boundary this intuition has been reinforced by many unsuccessful attempts as well as by the success in establishing iss only with respect to the derivative of the disturbance contrary to this intuition we establish such a result for parabolic pdes our methodology does not rely on the transformation of the boundary disturbance to a distributed input and the stability analysis is performed in time varying subsets of the state space the obtained results are used for the comparison of the gain coefficients of transport pdes with respect to inlet disturbances and for the establishment of the iss property with respect to control actuator errors for parabolic systems under boundary feedback control

stability of infinite dimensional sampled data systems suppose that a static state feedback stabilizes a continuous time linear infinite dimensional control system we consider the following question if we construct a sampled data controller by applying an idealized sample and hold process to a continuous time stabilizing feedback will this sampled data controller stabilize the system for all sufficiently small sampling times here the state space x and the control space u are hilbert spaces the system is of the form x t ax t bu t where a is the generator of a strongly continuous semigroup on x and the continuous time feedback is u t fx t the answer to the above question is known to be yes if x and u are finite dimensional spaces in the infinite dimensional case if f is not compact then it is easy to find counterexamples therefore we restrict attention to compact feedback we show that the answer to the above question is yes if b is a bounded operator from u into x moreover if b is unbounded we show that the answer yes remains correct provided that the semigroup generated by a is analytic we use the theory developed for static state feedback to obtain analogous results for dynamic output feedback control

generalized sampled data stabilization of well posed linear infinite dimensional systems we consider well posed linear infinite dimensional systems the outputs of which are sampled in a generalized sense using a suitable weighting function under certain natural assump tions on the system the weighting function and the sampling period we show that there exists a generalized hold function such that unity sampled data feedback renders the closed loop system exponentially stable in the state space sense as well as l 2 stable in the input output sense to illustrate our main result we describe an application to a structurally damped euler bernoulli beam

stabilization of well posed infinite dimensional systems by dynamic sampled data feedback it is shown that a set of six natural conditions is necessary and sufficient for the existence of a finite dimensional stabilizing sampled data controller for a well posed infinite dimensional system the underlying stability concept for the sampled data system is reminiscent of the notion of input to state stability from nonlinear control theory

stability analysis of a text 2 times text 2 linear hyperbolic system with a sampled data controller via backstepping method and looped functionals this paper is concerned with the global exponential stability of a 2 times 2 linear hyperbolic system with a sampled data boundary feedback control designed by means of the backstepping method for a nominal continuous input we show that there exists a sufficiently small intersampling time that encompasses both periodic and aperiodic sampling for which the global exponential stability of the closed loop system is guaranteed in addition we provide easily tractable sufficient stability conditions that can be used to find an upper bound of the maximum intersampling time the results rely on the combination of the lyapunov method and looped functionals the effectiveness of the proposed results is illustrated with a numerical example

a strict control lyapunov function for a diffusion equation with time varying distributed coefficients in this paper a strict lyapunov function is developed in order to show the exponential stability and input to state stability iss properties of a diffusion equation for nonhomogeneous media such media can involve rapidly time varying distributed diffusivity coefficients based on this lyapunov function a control law is derived to preserve the iss properties of the system and improve its performance a robustness analysis with respect to disturbances and estimation errors in the distributed parameters is performed on the system precisely showing the impact of the controller on the rate of convergence and iss gains this is important in light of a possible implementation of the control since in most cases diffusion coefficient estimates involve a high degree of uncertainty an application to the safety factor profile control for the tore supra tokamak illustrates and motivates the theoretical results a constrained control law incorporating nonlinear shape constraints in the actuation profiles is designed to behave as close as possible to the unconstrained version albeit with the equivalent of a variable gain finally the proposed control laws are tested under simulation first in the nominal case and then using a model of tore supra dynamics where they show adequate performance and robustness with respect to disturbances

local input to state stability characterizations and counterexamples abstract we show that a nonlinear locally uniformly asymptotically stable infinite dimensional system is automatically locally input to state stable liss provided the nonlinearity possesses some sort of uniform continuity with respect to external inputs also we prove that liss is equivalent to existence of a liss lyapunov function we show by means of a counterexample that if this uniformity is not present then the equivalence of local asymptotic stability and local iss does not hold anymore using a modification of this counterexample we show that in infinite dimensions a uniformly globally asymptotically stable at zero globally stable and locally iss system possessing an asymptotic gain property does not have to be iss in contrast to finite dimensional case

input to state stability of a clamped free damped string in the presence of distributed and boundary disturbances this note establishes the exponential input to state stability eiss property for a clamped free damped string with respect to distributed and boundary disturbances while efficient methods for establishing iss properties for distributed parameter systems with respect to distributed disturbances have been developed during the last decades establishing iss properties with respect to boundary disturbances remains challenging one of the well known methods for well posedness analysis of systems with boundary inputs is the use of a lifting operator for transferring the boundary disturbance to a distributed one however the resulting distributed disturbance involves time derivatives of the boundary perturbation thus the subsequent iss estimate depends on its amplitude and may not be expressed in the strict form of iss properties to solve this problem we show for a clamped free damped string equation that the projection of the original system trajectories in an adequate riesz basis can be used to establish the desired eiss property

small gain stability analysis of certain hyperbolic parabolic pde loops abstract this work provides stability results in the spatial sup norm for hyperbolic parabolic loops in one spatial dimension the results are obtained by an application of the small gain stability analysis two particular cases are selected for the study because they contain challenges typical of more general systems to which the results are easily generalizable but at the expense of less pedagogical clarity and more notational clutter i the feedback interconnection of a parabolic pde with a first order zero speed hyperbolic pde with boundary disturbances and ii the feedback interconnection by means of a combination of boundary and in domain terms of a parabolic pde with a first order hyperbolic pde the first case arises in the study of the movement of chemicals underground and includes the wave equation with kelvin voigt damping as a subcase the second case arises when we apply backstepping to a pair of hyperbolic pdes that is obtained by ignoring diffusion phenomena moreover the second case arises in the study of parabolic pdes with distributed delays in the first case we provide sufficient conditions for iss in the spatial sup norm with respect to boundary disturbances in the second case we provide delay independent sufficient conditions for exponential stability in the spatial sup norm

input to state stability with respect to boundary disturbances for a class of semi linear parabolic equations abstract this paper studies the input to state stability iss properties for a class of semi linear parabolic partial differential equations pdes with respect to boundary disturbances in order to avoid the appearance of time derivatives of the disturbances in iss estimates some technical inequalities are first developed which allow directly dealing with the boundary conditions and establishing the iss based on the method of lyapunov functionals the well posedness analysis of the considered problem is carried out and the conditions for iss are derived two examples are used to illustrate the application of the developed method

brief input to state stability for a class of lurie systems we analyze input to state stability iss for the feedback interconnection of a linear block and a nonlinear element this study is of importance for establishing robustness against actuator nonlinearities and disturbances in the absolute stability framework we prove iss from a positive real property of the linear block by restricting the sector nonlinearity to grow unbounded as its argument tends to infinity when this growth condition is violated examples show that the iss property is lost the result is used to give a simple proof of boundedness for negative resistance oscillators such as the van der pol oscillator in a separate application we relax the minimum phase assumption of an earlier boundedness result for systems with nonlinearities that grow faster than linear