conditions for the equivalence of internal and external stability for distributed parameter systems conditions are given on a controlled observed distributed parameter system under which external stability also known as input output stability implies internal stability which is the exponential stability of the underlying semigroup generator it is shown that when the system satisfies a general definition of stabilizability and detectability external stability is equivalent to internal stability unlike the case previous definitions of stabilizability and detectability a system which satisfies these does not necessarily have a spectrum decomposition therefore in order to check internal stability it is often sufficient to check the boundedness of the transfer function in the right half complex plane this is illustrated with a simple example

an adaptive tracking problem with a control input constraint abstract in this paper we consider the problem of controlling an uncertain linear time invariant lti continuous time plant when there is a control input constraint a switching controller is presented which will carry out asymptotic error regulation for constant reference and constant disturbance signals which are compatible with the control input constraint assuming that i the plant is asymptotically stable ii the plant has no transmission zeros at zero and iii the plant has as many inputs as outputs this controller is quite easy to implement especially in the single input single output siso case and is noise tolerant some examples including a model of a three input three output distillation column are included to illustrate the controller behaviour

the dominguez lorenzo condition and multivalued nonexpansive mappings abstract let e be a nonempty bounded closed convex separable subset of a reflexive banach space x which satisfies the dominguez lorenzo condition i e an inequality concerning the asymptotic radius of a sequence and the chebyshev radius of its asymptotic center we prove that a multivalued nonexpansive mapping t e 2 x which is compact convex valued and such that t e is bounded and satisfies an inwardness condition has a fixed point as a consequence we obtain a fixed point theorem for multivalued nonexpansive mappings in uniformly nonsquare banach spaces which satisfy the property worth extending a known result for the case of nonexpansive single valued mappings we also prove a common fixed point theorem for two nonexpansive commuting mappings t e e and t e kc e where kc e denotes the class of all compact convex subsets of e when x is a uniformly convex banach space

transfer functions of regular linear systems part ii the system operator and the lax phillips semigroup this paper is a sequel to a paper by the second author on regular linear systems 1994 referred to here as part i we introduce the system operator of a well posed linear system which for a finite dimensional system described by x ax bu y cx du would be the s dependent matrix s s a si c b d in the general case s s is an unbounded operator and we show that it can be split into four blocks as in the finite dimensional case but the splitting is not unique the upper row consists of the uniquely determined blocks a si and b as in the finite dimensional case but the lower row is more problematic for weakly regular systems which are introduced and studied here there exists a special splitting of s s where the right lower block is the feedthrough operator of the system using s 0 we give representation theorems which generalize those from part i to well posed linear systems and also to the situation when the initial time is we also introduce the lax phillips semigroup t induced by a well posed linear system which is in fact an alternative representation of a system used in scattering theory our concept of a lax phillips semigroup differs in several respects from the classical one for example by allowing an index r which determines an exponential weight in the input and output spaces this index allows us to characterize the spectrum of a and also the points where s s is not invertible in terms of the spectrum of the generator of for various values of the system is dissipative if and only if with index zero is a contraction semigroup

fixed point theorems for multivalued nonexpansive mappings satisfying inwardness conditions abstract let x be a banach space whose characteristic of noncompact convexity is less than 1 and satisfies the nonstrict opial condition let c be a bounded closed convex subset of x kc x the family of all compact convex subsets of x and t a nonexpansive mapping from c into kc x with bounded range we prove that t has a fixed point the nonstrict opial condition can be removed if in addition t is an 1 contractive mapping

admissibility of unbounded control operators for linear systems described by dot x t ax t bu t where a generates a semigroup on the state space x and b is an unbounded operator some necessary as well as some sufficient conditions are given for b to be admissible i e for any t the state x t should be in x and should depend continuously on the input u in l p this approach begins with an axiomatic description of such a system in terms of a functional equation the results are applied to the wave equation on a bounded interval

integral control of infinite dimensional linear systems subject to input saturation 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 shown to ensure asymptotic tracking of constant reference signals provided that a the steady state gain of the linear part of the plant is positive b the positive integrator gain is sufficiently small and c the reference value is feasible in a very natural sense the class of actuator nonlinearities under consideration contains standard nonlinearities important in control engineering such as saturation and deadzone

on multivariate baskakov operator abstract in this paper we study multivariate baskakov operator b n d f x we first show that the operator can retain some properties of the original function f such as monotony semi additivity and lipschitz condition etc secondly we discuss the monotony on the sequence of multivariate baskakov operator b n d f x for n when the function f is convex then we propose for estimating the rate of approximation a new modulus of smoothness and prove the modulus to be equivalent to certain k functional finally with the modulus of smoothness as metric we establish a strong direct theorem by using a decomposition technique for the operator

conditions for robustness and nonrobustness of thestability of feedback systems with respect to small delays inthe feedback loop it has been observed that for many stable feedback control systems the introduction of arbitrarily small time delays into the loop causes instability in this paper we present a systematic frequency domain treatment of this phenomenon for distributed parameter systems we consider the class of all matrix valued transfer functions which are bounded on some right half plane and which have a limit at spinfty along the real axis such transfer functions are called regular under the assumption that a regular transfer function is stabilized by unity output feedback we give sufficient conditions for the robustness and for the nonrobustness of the stability with respect to small time delays in the loop these conditions are given in terms of the high frequency behavior of the open loop system moreover we discuss robustness of stability with respect to small delays for feedback systems with dynamic compensators in particular we show that if a plant with infinitely many poles in the closed right half plane is stabilized by a controller then the stability is not robust with respect to delays we show that the instability created by small delays is itself robust to small delays three examples are given to illustrate these results

infinite dimensional continuous time linear systems stabilityand structure analysis the question of exponential and asymptotic stability of infinite dimensional continuous time state space systems is investigated it is shown that every par balanced realization is asymptotically stable conditions are given for par balanced input normal or output normal ealizations to be asymptotically and or exponentially stable the boundedness of the system operators is also studied examples of delay systems are given to illustrate the theory

the regularity of the wave equation with partial dirichlet control and colocated observation in this paper we analyze a multidimensional controlled wave equation on a bounded domain subject to partial dirichlet control and colocated observation by means of a partial fourier transform it is shown that the system is well posed and regular in the sense of d salamon and g weiss the corresponding feedthrough operator is found to be the identity operator on the input space

dynamic stabilization of regular linear systems we consider a general class of infinite dimensional linear systems called regular linear systems for which convenient representations are known to exist both in time and in frequency domain for this class of systems we investigate the concepts of stabilizability and detectability in particular their invariance under feedback and their relationship to exponential stability we introduce two concepts of dynamic stabilization the first formulated as usual with the plant and the controller connected in feedback and the second with two feedback loops even for finite dimensional systems the second concept stabilization with an internal loop in the controller is more general we argue that the more general concept is the natural one and we derive sufficient conditions under which an observer based stabilizing controller with an internal loop can be constructed

on the concept of point value in the infinite dimensional realization theory in this article we study the effect of the chosen representation of a point value and point evaluation on the class of periodic signals realizable using a certain type of infinite dimensional linear system by suitably representing the point evaluation at the origin in a hilbert space we are able to give a complete characterization of its extensions these extensions involve a new concept called sequence the use of which as an observation operator of an infinite dimensional linear system is studied in this article in particular we consider their use in the realization of periodic signals we also investigate how the use of sequences affects the convergence properties of such realizations we consider the rate and character of convergence and the removal of the gibbs phenomenon as still a further demonstration of the significance of the chosen concept of a point value we discuss the use of distributional point values in the realization of periodic distributions the possible applications of this work lie in regulator problems of infinite dimensional control theory as is indicated by the well known internal model principle

robust stability of systems with integral control a number of necessary and sufficient conditions are derived which must be satisfied by the plant d c gain matrix of a linear time invariant system in order for an integral controller to exist for which the closed loop system is stable based on these results the robustness of integral control systems is analyzed i e the family of plants is defined which are stable when controlled with the same integral controller conditions for actuator sensor failure tolerance of systems with integral control are also given finally parallels are drawn between the results of this paper and the bifurcation theory of nonlinear systems

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

bilinear transformation of infinite dimensional state space systems and balanced realizations of nonrational transfer functions the bilinear transform maps the open right half plane to the open unit disk and is therefore a suitable tool for carrying over results for continuous time systems to discrete time systems and vice versa corresponding state space formulae are widely used and well understood for the case of finite dimensional systems in this paper infinite dimensional generalizations of these formulae are studied for a general class of infinite dimensional state space systems in particular it is shown that reachability and observability are carried over and that the reachability and observability gramians are preserved under this transformation young showed that a wide class of nonrational discrete time transfer functions admit a balanced state space representation it is shown that this result carries over to the continuous time situation via the bilinear transformation

on a chlodovsky variant of a multivariate beta operator in this work we state a chlodovsky variant of a multivariate beta operator to be called hereafter the multivariate beta chlodovsky operator we show that the multivariate beta chlodovsky operator can preserve properties of a general function of modulus of continuity and also the lipschitz constant of a lipschitz continuous function furthermore we set an h w d class by the function of modulus of continuity w and give some results for functions belonging to h w d

robust multivariable pi controller for infinite dimensional systems a robust multivariable controller is introduced for a class of distributed parameter systems the system to be controlled is given as dot x ax bu y cx in a banach space the purpose of the control which is based on the measurement y is to stabilize and regulate the system so that y t rightarrow y r as t rightarrow infty where y r is a constant reference vector under the assumptions that operator a generates a holomorphic stable semigroup b is linear and bounded c is linear and a bounded and the input and output spaces are of the same dimension a necessary and sufficient condition is found for the existence of a robust multivariable controller this controller appears to be a multivariable pi controller also a simple necessary criterion for the existence of a decentralized controller is derived the tuning of the controller is discussed and it is shown that the i part of the controller can be tuned on the basis of step responses without exact knowledge of the system s parameters the presented theory is then used as an example to control the temperature profile of a bar with the dirichlet boundary conditions

time varying and adaptive integral control of infinite dimensional regular linear systems with input nonlinearities closing the loop around an exponentially stable single input single output regular linear system subject to a globally lipschitz nondecreasing actuator nonlinearity and compensated by an integral controller with time dependent gain k t is shown to ensure asymptotic tracking of a constant reference signal r provided that a the steady state gain of the linear part of the system is positive b the reference value r is feasible in an entirely natural sense and c the function t mapsto k t monotonically decreases to zero at a sufficiently slow rate this result forms the basis of a simple adaptive control strategy that ensures asymptotic tracking under conditions a and b

exponential stability of coupled beams with dissipative joints a frequency domain approach two examples of coupled euler bernoulli beams with a dissipative joint are considered the joint placements that lead to exponential stability for these systems are characterized the technique used shows input output stability of a related controlled observed system and then shows that in these examples input output stability implies exponential stability in the first example the energy dissipation arises from a discontinuity in the shear at the joint in the second example the energy dissipation arises from a discontinuity in the bending moment at the joint the analysis of this system involves a complete spectrum analysis of the zero dynamics of the associated controlled observed system

transfer functions of regular linear systems part i characterizations of regularity we recall the main facts about the representation of regular linear systems essentially that they can be described by equations of the form x t ax t bu t y t cx t du t like finite dimensional systems but now a b and c are in general unbounded operators regular linear systems are a subclass of abstract linear systems we define transfer functions of abstract linear systems via a generalization of a theorem of foures and segal we prove a formula for the transfer function of a regular linear system which is similar to the formula in finite dimensions the main result is a simple to state but hard to prove necessary and sufficient condition for an abstract linear system to be regular in terms of its transfer function other conditions equivalent to regularity are also obtained the main result is a consequence of a new tauberian theorem which is of independent interest

justification of input output methods for systems with unbounded control and observation the authors present definitions of feedback stabilizability and detectability that are weaker than those previously appearing in the literature stabilizability and detectability in this sense are sufficient to ensure an equivalence between internal and external stability for well posed systems furthermore stabilizability and detectability are shown to be preserved under feedback and other connections these results are linked to present a justification of input output controller design methods for systems with unbounded control and observation

low gain integral control for multi input multioutput linear systems with input nonlinearities we consider the inclusion of a static antiwindup component in a continuous time low gain integral controller in feedback with a multi input multi output stable linear system subject to an input nonlinearity from a class of functions that includes componentwise diagonal saturation we demonstrate that the output of the closed loop system asymptotically tracks every constant reference vector which is feasible in a natural sense provided that the integrator gain is sufficiently small robustness properties of the proposed control scheme are investigated and three examples are discussed in detail

set valued dynamic systems and random iterations of set valued weaker contractions in uniform spaces set valued weaker contractions in uniform locally convex and metric spaces are defined and dynamic systems of such weaker contractions are studied conditions guaranteeing the convergence of generalized sequences of random iterations and iterations and the existence and uniqueness of endpoints of set valued weaker contractions are established our definitions and results are new for set valued maps in uniform locally convex and metric spaces and even for single valued maps

a general multi valued hybrid fixed point theorem and perturbed differential inclusions abstract combining three basic multi valued versions of banach schauder and tarski fixed point theorems a general hybrid fixed point theorem for multi valued mappings in banach spaces is proved via measure of noncompactness and it is further applied to perturbed differential inclusions for proving the existence results under mixed lipschitz compactness and monotone conditions

lp approximation by multivariate baskakov kantorovich operators abstract the concern of this paper is to introduce and study multivariate baskakov kantorovich operators which are non tensor product generalization of the one variable we establish a strong direct theorem of l p approximation by using a decomposition technique

lack of time delay robustness for stabilization of a structural acoustics model in this paper we consider a natural robustness question for a model for structural acoustics this model which has been of great interest in recent years is represented by a wave equation in r 2 coupled to a kelvin voigt beam the coupling is natural physically and is represented mathematically by highly unbounded operators we assume that the observation consists of point evaluation of the beam position the beam velocity and the wave velocity we are interested in the effect of arbitrarily small delays in the feedback loop on a controller that uses these observations we show that it is not possible to construct a dynamic stabilizer of a very general form including static feedback such that the stabilization is robust with respect to delays in the feedback loop in order to do this we need to carefully analyze the input to output map finally we relate these results to already existing numerical results obtained for a galerkin approximation of the system

coprime factorizations and well posed linear systems we study the basic notions related to the stabilization of an infinite dimensional well posed liner system in the sense of salamon and weiss we first introduce an appropriate stabilizability and detectability notion and show that if a system is jointly stabilizable and detectable then its transfer function has a doubly coprime factorization in h infty the converse is also true every function with a doubly coprime factorization in h infty is the transfer function of a jointly stabilizable and detectable well posed linear system we show further that a stabilizable and detectable system is stable if and only if its input output map is stable finally we construct a dynamic possibly non well posed stabilizing compensator the notion of stability that we use is the natural one for the quadratic cost minimization problem and it does not imply exponential stability

transfer functions for infinite dimensional systems in this paper we study three definitions of the transfer function for an infinite dimensional system the first one defines the transfer function as the expression c si a 1 b d in the second definition the transfer function is defined as the quotient of the laplace transform of the output and input with initial condition zero in the third definition we introduce the transfer function as the quotient of the input and output when the input and output are exponentials we show that these definitions always agree on the right half plane bounded to the left by the growth bound of the underlying semigroup but that they may differ elsewhere

adaptive sampled data integral control of stable infinite dimensional linear systems abstract in this paper adaptive discrete time low gain integral control strategies are presented for tracking constant reference signals for infinite dimensional discrete time power stable linear systems the discrete time results are applied in the development of adaptive sampled data low gain integral control of well posed infinite dimensional exponentially stable linear systems our results considerably extend improve and simplify previous work by two of the authors h logemann s townley discrete time low gain control of uncertain infinite dimensional systems ieee trans automat control 42 1997 22 37

robustness and continuity of the spectrum for uncertain distribution parameter systems we analyze the spectrum of a class of abstract partial differential equations with boundary feedback control if we interpret the feedback as a perturbation then we obtain a robustness radius for spectrum placement we obtain continuity of the spectrum using this radius and rouche s theorem we exploit extensively recent developments in the representation theory for abstract linear systems especially regular systems we also analyze ill posed systems by interchanging the roles of input and output we apply these results to a wave equation and an ill posed beam equation

fixed point theorems for nonexpansive mappings and suzuki generalized nonexpansive mappings on a banach lattice abstract we prove the existence of fixed points for multivalued nonexpansive nonself mappings on a weakly orthogonal reflexive banach lattice with uniformly monotone norm moreover for single valued mappings we extend betiuk pilarska and prus s result a betiuk pilarska s prus banach lattices which are order uniformly noncreasy j math anal appl 342 2008 1271 1279 on the weak fixed point property to continuous mappings satisfying condition c on a w weakly orthogonal ounc banach lattice

on the iterates of some bernstein type operators abstract in this paper we establish two basic functional type identities between the iterates of the bleimann butzer hahn operator and those of the bernstein operator on the one hand and the iterates of the modified meyer konig and zeller operator and those of the baskakov operator on the other these identities allow us to transfer the properties of these operators from one to another attention is focused on the limit behavior of the iterates and the linear combinations of iterates of fejer korovkin type

a fixed point theorem for pointwise eventually nonexpansive mappings in nearly uniformly convex banach spaces abstract the purpose of this paper is to prove the existence of a fixed point for a pointwise eventually nonexpansive mapping in a nearly uniformly convex banach space this provides an affirmative answer to a question given by kirk and xu w a kirk hong kun xu asymptotic pointwise contraction nonlinear anal 69 2008 4706 4712

quadratic optimal control of stable well posed linear systems we consider the infinite horizon quadratic cost minimization problem for a stable time invariant well posed linear system in the sense of salamon and weiss and show that it can be reduced to a spectral factorization problem in the control space more precisely we show that the optimal solution of the quadratic cost minimization problem is of static state feedback type if and only if a certain spectral factorization problem has a solution if both the system and the spectral factor are regular then the feedback operator can be expressed in terms of the riccati operator and the riccati operator is a positive self adjoint solution of an algebraic riccati equation this riccati equation is similar to the usual algebraic riccati equation but one of its coefficients varies depending on the subspace in which the equation is posed similar results are true for unstable systems as we have proved elsewhere

uniform approximation by exponential type operators abstract the so called exponential type operators include the most important approximation processes of bernstein szasz mirakjan baskakov etc their global behaviour in the weighted case was settled by k sato now we describe their approximation theoretical properties from the point of view of global uniform approximation

quadratic optimal control of well posed linear systems we study the infinite horizon quadratic cost minimization problem for a well posed linear system in the sense of salamon and weiss the quadratic cost function that we seek to minimize need not be positive but it is convex and bounded from below we assume the system to be jointly stabilizable and detectable and give a feedback solution to the cost minimization problem moreover we connect this solution to the computation of either a j s inner or an s normalized coprime factorization of the transfer function depending on how the problem is formulated we apply the general theory to get factorization versions of the bounded and positive real lemmas in the case where the system is regular it is possible to show that the feedback operator can be expressed in terms of the riccati operator and that the riccati operator is a stabilizing self adjoint solution of an algebraic riccati equation this riccati equation is nonstandard in the sense that the weighting operator in the quadratic term differs from the expected one and the computation of the correct weighting operator is a nontrivial task

bivariate bernstein type operators in this paper we introduce bivariate extension of bernstein type operators defined in 11 we show that these operators preserve some properties of the original function f such as lipschitz constant and monotonicity furthermore we present the monotonicity of the sequence of bivariate bernstein type operators for n when f is convex

on the characterization of admissible control and observation operators abstract in this note we give a short proof of a recent result in 2 integral equations operator theory 25 1996 182 198 which characterizes the admissibility of unbounded observation operators in linear infinite dimensional control theory moreover we present the analogous result for the admissibility of unbounded control operators

infinite dimensional linear systems with unbounded control and observation a functional analytic approach abstract the object of these notes is to develop a unifying framework for the functional analytic representation of infinite dimensional linear systems with unbounded input and output operators on the basis of the general approach new results are derived on the wellposedness of feedback systems and on the linear quadratic control problem the implications of the theory for large classes of functional and partial differential equations are discussed in detail keywords representation of infinite dimensional systems semigroups boundary control feedback linear quadratic control

exponential stabilization of well posed systems by colocated feedback we consider well posed linear systems whose state trajectories satisfy dot x ax bu where u is the input and a is an essentially skew adjoint and dissipative operator on the hilbert space x this means that the domains of a and a are equal and a a q where q geq 0 is bounded on x the control operator b is possibly unbounded but admissible and the observation operator of the system is b such a description fits many wave and beam equations with colocated sensors and actuators and it has been shown for many particular cases that the feedback u ka y v with ka 0 stabilizes the system strongly or even exponentially here y is the output of the system and v is the new input we show by means of a counterexample that if b is sufficiently unbounded then such a feedback may be unsuitable the closed loop semigroup may even grow exponentially our counterexample is a simple regular system with feedthrough operator zero however we prove that if the original system is exactly controllable and observable and if ka is sufficiently small then the closed loop system is exponentially stable

banach ultrapowers and multivalued nonexpansive mappings the banach ultrapower construction is applied in fixed point theory for multivalued mappings we introduce the notion of ultra asymptotic centers and use it to remove the separability assumption from the results of dominguez benavides lorenzo ramirez 2004 and dhompongsa kaewcharoen kaewkhao 2006

discrete time low gain control of uncertain infinite dimensional systems using a frequency domain analysis it is shown that the application of a feedback controller of the form k z 1 or kz z 1 where k spl isin r to a power stable infinite dimensional discrete time system with square transfer function matrix g z will result in a power stable closed loop system which achieves asymptotic tracking of arbitrary constant reference signals provided that i all the eigenvalues of g 1 have positive real parts and ii the gain parameter k is positive and sufficiently small moreover if g 1 is positive definite we show how the gain parameter gain x can be tuned adaptively the resulting adaptive tracking controllers are universal in the sense that they apply to any power stable system with g 1 0 in particular they are not based on system identification or plant parameter estimation algorithms nor is the injection of probing signals required finally we apply these discrete time results to obtain adaptive sampled data low gain controllers for the class of regular systems a rather general class of infinite dimensional continuous time systems for which convenient representations are known to exist both in state space and in frequency domain we emphasize that our results guarantee not only asymptotic tracking at the sampling instants but also in the sampling interval

robust stability of linear evolution operators on banach spaces this paper introduces a stability radius for a wide class of linear infinite dimensional time varying systems under structured time varying perturbations a framework is presented that allows the same degree of unboundedness in the perturbations as in the generator of the nominal model

adaptive low gain integral control of multivariable well posed linear systems the principle of low gain integral control for finite dimensional systems is well known more recently low gain integral control results have been obtained for classes of infinite dimensional systems in this paper we show that integral control with a simple and natural adaptation of the integrator gain achieves tracking of constant reference signals for every exponentially stable multivariable well posed infinite dimensional linear system whose steady state gain matrix has its spectrum in the open right half plane our results considerably extend improve and simplify previous work by the authors siam j control optim 35 1997 pp 78 116

low gain control of uncertain regular linear systems it is well known that closing the loop around an exponentially stable finite dimensional linear time invariant plant with square transfer function matrix bg s compensated by a controller of the form k s gamma 0 where k in bbb r and gamma 0 in bbb r mm will result in an exponentially stable closed loop system which achieves tracking of arbitrary constant reference signals provided that i all the eigenvalues of bg 0 gamma 0 have positive real parts and ii the gain parameter k is positive and sufficiently small in this paper we consider a rather general class of infinite dimensional linear systems called regular systems for which convenient representations are known to exist both in time and in frequency domain the purpose of the paper is twofold i we extend the above result to the class of exponentially stable regular systems and ii we show how the parameters k and gamma 0 can be tuned adaptively the resulting adaptive tracking controllers are not based on system identification or parameter estimation algorithms nor is the injection of probing signals required