simultaneous exact controllability and some applications we study the exact controllability of two systems by means of a common finite dimensional input function a property called simultaneous exact controllability most of the time we consider one system to be infinite dimensional and the other finite dimensional in this case we show that if both systems are exactly controllable in time t 0 and the generators have no common eigenvalues then they are simultaneously exactly controllable in any time t t0 moreover we show that similar results hold for approximate controllability for exactly controllable systems we characterize the reachable subspaces corresponding to input functions of class h1 and h2 we apply our results to prove the exact controllability of a coupled system composed of a string with a mass at one end finally we consider an example of two infinite dimensional systems we characterize the simultaneously reachable subspace for two strings controlled from a common end the result is obtained using a recent generalization of a classical inequality of ingham

asymptotic behavior of the growth fragmentation equation with bounded fragmentation rate we are interested in the large time behavior of the solutions to the growth fragmentation equation we work in the space of integrable functions weighted with the principal dual eigenfunction of the growth fragmentation operator this space is the largest one in which we can expect convergence to the steady size distribution although this convergence is known to occur under fairly general conditions on the coefficients of the equation we prove that it does not happen uniformly with respect to the initial data when the fragmentation rate in bounded first we get the result for fragmentation kernels which do not form arbitrarily small fragments by taking advantage of the dyson phillips series then we extend it to general kernels by using the notion of quasi compactness and the fact that it is a topological invariant

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

regular linear systems governed by neutral fdes the aim of this paper is to prove that a class of distributed parameter systems governed by neutral fdes provides regular linear systems employing the well established theory of representation transfer function and feedback of these later we then give new representations of the state and the output function of the neutral systems

non coercive lyapunov functions for infinite dimensional systems abstract we show that the existence of a non coercive lyapunov function is sufficient for uniform global asymptotic stability ugas of infinite dimensional systems with external disturbances provided the speed of decay is measured in terms of the norm of the state and an additional mild assumption is satisfied for evolution equations in banach spaces with lipschitz continuous nonlinearities these additional assumptions become especially simple the results encompass some recent results on linear switched systems on banach spaces finally we derive new non coercive converse lyapunov theorems and give some examples showing the necessity of our assumptions

new riccati equations for well posed linear systems we consider the classic problem of minimizing a quadratic cost functional for well posed linear systems over the class of inputs that are square integrable and that produce a square integrable output as is well known the minimum cost can be expressed in terms of a bounded nonnegative self adjoint operator x that in the finite dimensional case satisfies a riccati equation unfortunately the infinite dimensional generalization of this riccati equation is not always well defined we show that x always satisfies alternative riccati equations which are more suitable for algebraic and numerical computations

on invariance of p admissibility of control and observation operators to q type of perturbations of generator of c0 semigroup abstract in this paper it is proved in general setting that p admissibilities of control operators and observation operators are invariant to any q type of perturbations of generator of c 0 semigroups on banach space moreover some relations between the extensions of observation operators with respect to the original generator and the perturbed generator are also characterized so that the output can be expressed in the mild sense as an application the admissibility as well as the mild expressibility of output of a class of observation systems with time delay in state is deduced

uniform controllability of semidiscrete approximations of parabolic control systems controlling an approximation model of a controllable infinite dimensional linear control system does not necessarily yield a good approximation of the control needed for the continuous model in the present paper under the main assumptions that the discretized semigroup is uniformly analytic and that the control operator is mildly unbounded we prove that the semidiscrete approximation models are uniformly controllable moreover we provide a computationally efficient way to compute the approximation controls an example of application is implemented for the one and two dimensional heat equation with neumann boundary control

lyapunov based boundary control for a class of hyperbolic lotka volterra systems this paper considers a boundary feedback control problem for two first order nonlinearly coupled hyperbolic partial differential equations with lotka volterra type coupling boundary control action is used on one equation to drive the state at the end of the spatial domain to a desired constant reference value static and dynamic boundary controllers are designed based on a special lyapunov functional that is related to an entropy function the time derivative of the entropy function is made strictly negative by an appropriate choice of boundary conditions a unique classical solution is shown to exist globally in time and asymptotic exponential convergence to the desired steady state solution is shown in the norm the boundary control design is illustrated with simulations

regularity of a schrodinger equation with dirichlet control and colocated observation this paper studies the basic properties of the schrodinger equation defined on a bounded domain of r n n 2 with partial dirichlet control and colocated observation it is shown that the system is not only wellposed in the sense of d salamon but also regular in the sense of g weiss it is also shown that the corresponding feedthrough operator is zero 2005 elsevier b v all rights reserved

on moment sequences and infinitely divisible sequences we consider infinitely divisible sequences in the context of moment sequences in the same vein as r a horn j math anal appl 31 1970 130 135 considered renewal sequences and we compare the results

riesz bases and exact controllability of c0 groups with one dimensional input operators abstract this paper considers linear infinite dimensional systems with c 0 group generators and one dimensional admissible input operators the exact controllability and riesz basis generation property are discussed the corresponding results of jacob and zwart advances in mathematical systems theory birkhauser boston ma 2000 under the assumption of algebraic simplicity for eigenvalues of the generator are generalized to the case in which the eigenvalues are allowed to be algebraically multiple but with a uniform bound on the multiplicity

a class of infinitely divisible distributions connected to branching processes and random walks a class of infinitely divisible distributions on 0 1 2 is defined by requiring the discrete levy function to be equal to the probability function except for a very simple factor these distributions turn out to be special cases of the total offspring distributions in sub critical branching processes and can also be interpreted as first passage times in certain random walks there are connections with lambert s w function and generalized negative binomial convolutions

new blow up rates for fast controls of schrodinger and heat equations abstract we consider the null controllability problem for the schrodinger and heat equations with boundary control we concentrate on short time or fast controls we improve recent estimates see l miller geometric bounds on the growth rate of null controllability cost for the heat equation in small time j differential equations 204 2004 202 226 l miller how violent are fast controls for schrodinger and plate vibrations arch ration mech anal 172 2004 429 456 l miller controllability cost of conservative systems resolvent condition and transmutation j funct anal 218 2005 425 444 l miller the control transmutation method and the cost of fast controls siam j control optim 45 2006 762 772 on the norm of the operator associating to any initial state the minimal norm control driving the system to zero our main results concern the schrodinger and heat equations in one space dimension they yield new estimates concerning window problems for series of exponentials as described in t i seidman the coefficient map for certain exponential sums nederl akad wetensch indag math 48 1986 463 478 and in t i seidman s a avdonin s a ivanov the window problem for series of complex exponentials j fourier anal appl 6 2000 233 254 these results are used following l miller the control transmutation method and the cost of fast controls siam j control optim 45 2006 762 772 to deal with the case of several space dimensions

model reduction by balanced truncation for systems with nuclear hankel operators we prove the h infinity error bounds for lyapunov balanced truncation and for optimal hankel norm approximation under the assumption that the hankel operator is nuclear this is an improvement of the result from glover curtain and partington siam j control optim 26 1998 pp 863 898 where additional assumptions were made the proof is based on convergence of the schmidt pairs of the hankel operator in a sobolev space we also give an application of this convergence theory to a numerical algorithm for model reduction by balanced truncation

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

admissibility in input elements for diagonal semigroups on l 2 abstract we characterize the admissible unbounded input elements for diagonal semigroups on l 2 the result verifies the necessity of a condition whose sufficience was previously established by l f ho and d l russell

spectral conditions for admissibility and observability of wave systems applications to finite element schemes in this article we derive uniform admissibility and observability properties for the finite element space semi discretizations of ddot u a 0 u 0 where a 0 is an unbounded self adjoint positive definite operator with compact resolvent to address this problem we present a new spectral approach based on several spectral criteria for admissibility and observability of such systems our approach provides very general admissibility and observability results for finite element approximation schemes of ddot u a 0 u 0 which stand in any dimension and for any regular mesh in the sense of finite elements our results can be combined with previous works to derive admissibility and observability properties for full discretizations of ddot u a 0 u 0 we also present applications of our results to controllability and stabilization problems

minimum time and minimum energy for linear control systems abstract in this paper we give estimates concerning regularity properties of the minimum time function and the minimum norm controls involving the controllability function given by the observability inequalities we further study the continuity of the minimum time function in two variables the initial point and the radius of the ball that limits the control we prove a maximum principle for both time and norm optimal control problems

on application of the implemented semigroup to a problem arising in optimal control in this paper we show that the concept of an implemented semigroup provides a natural mathematical framework for analysis of the infinite dimensional differential lyapunov equation lyapunov equations of this form arise in various system theoretic and control problems with a finite time horizon infinite dimensional state space and unbounded operators in the mathematical model of the system the implemented semigroup approach allows us to derive a necessary and sufficient condition for the differential lyapunov equation with an unbounded forcing term to admit a bounded solution in a suitable space whilst our focus is on the differential lyapunov equation we show that the same framework is also appropriate for the algebraic version of this equation as an application we show that the approach can be used to solve a simple decoupling problem arising in optimal control the problem of infinite time admissibility of the control operator and an infinite dimensional version of the lyapunov theorem serve as ad

time optimal boundary controls for the heat equation the fact that the time optimal controls for parabolic equations have the bang bang property has been recently proved for controls distributed inside the considered domain interior control the main result in this article asserts that the boundary controls for the heat equation have the same property at least in rectangular domains this result is proved by combining methods from traditionally distinct fields the lebeau robbiano strategy for null controllability and estimates of the controllability cost in small time for parabolic systems on one side and a remez type inequality for muntz spaces and a generalization of turan s inequality on the other side

bounds on functions biorthogonal to sets of complex exponentials control of damped elastic systems abstract in this article we obtain explicit bounds on the norms of biorthogonal functions to sets of complex exponentials exp k t where the k belong to a sector of the positive real axis the bounds obtained are uniform within a class of sequences which exhibit similar growth and separation properties the dependence of the bounds upon the growth and separation parameters is carefully examined the main theorem then leads to some results regarding the solvability of moment problems of the form 0 t f t exp k t dt c k which arise in control problems associated with damped or dissipative systems as an application we establish an exact null controllability result for boundary control of a rectangular structurally damped plate

weighted admissibility and wellposedness of linear systems in banach spaces we study linear control systems in infinite dimensional banach spaces governed by analytic semigroups for p in 1 infty and alpha in mathbb r we introduce the notion of l p admissibility of type alpha for unbounded observation and control operators generalizing earlier work by le merdy j london math soc 2 67 2003 pp 715 738 and haak and le merdy houston j math 31 2005 pp 1153 1167 we give conditions under which l p admissibility of type alpha is characterized by boundedness conditions which are similar to those in the well known weiss conjecture we also study l p wellposedness of type alpha for the full system here we use recent ideas due to pruss and simonett arch math basel 82 2004 pp 415 431 our results are illustrated by a controlled heat equation with boundary control and boundary observation where we take lebesgue and besov spaces as state space this extends the considerations in c i byrnes e j dynam control systems 8 2002 pp 341 370 to non hilbertian settings and to p neq2

an evolution equation approach to nonautonomous linear systems with state input and output delays we propose a new approach which brings nonautonomous linear systems with state input and output delays in the line with the standard theory of nonautonomous linear systems to this purpose we establish using the concept of lebesgue extensions a new variation of constants formula for nonhomogenous delay equations from this we deduce another new one for nonautonomous linear systems with state and input delays inspiriting from this formula we show that a given delay system determines a nonautonomous absolutely regular linear system with the same input and output spaces as of the delay system our abstract results will be applied to investigate the existence and the uniqueness of the solutions for a class of nonautonomous neutral equations in banach spaces

performance output tracking for multidimensional heat equation subject to unmatched disturbance and noncollocated control this paper investigates performance output tracking for a boundary controlled multidimensional heat equation it is assumed that the so called total disturbance which is composed of internal possibly nonlinear uncertainty and external disturbance the equation is subject to is on one part of the boundary and that the control is applied on the rest of the boundary using only partial boundary measurement we first propose an extended state observer to estimate both system state and the total disturbance this allows us to design a servomechanism and then an output feedback controller under the condition that both the reference signal and the disturbance vanish or belong to spaces h 1 0 infty l 1 gamma 1 and l 2 0 infty l 2 gamma 0 respectively we show that the over all control strategy achieves three objectives on the system performance first exponentially output tracking for arbitrary given reference signal second uniformly boundedness of all internal signals and third the internal asymptotic stability of the closed loop system in addition the control strategy turns out to be robust to the measurement noise we provide numerical experiments to illustrate the effectiveness of the proposed control strategy

exponential decay for the fragmentation or cell division equation abstract we consider a classical integro differential equation that arises in various applications as a model for cell division or fragmentation in biology it describes the evolution of the density of cells that grow and divide we prove the existence of a stable steady distribution first positive eigenvector under general assumptions in the variable coefficients case we also prove the exponential convergence for large times of solutions toward such a steady state

low gain integral control of well posed linear infinite dimensional systems with input and output nonlinearities abstract time varying low gain integral control strategies are presented for asymptotic tracking of constant reference signals in the context of exponentially stable well posed linear infinite dimensional single input single output systems subject to globally lipschitz nondecreasing input and output nonlinearities it is shown that applying error feedback using an integral controller ensures that the tracking error is small in a certain sense 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 positive gain function t k t is ultimately sufficiently small and not of class l 1 under a weak restriction on the initial data it is shown that a b and c ensure asymptotic tracking if additionally the impulse response of the linear part of the system is a finite signed borel measure the global lipschitz assumption on the output nonlinearity may be considerably relaxed

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

feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations general second order parabolic and hyperbolic equations on a bounded domain are considered the input is applied in the neumann or mixed boundary condition and is expressed as a finite dimensional feedback in the parabolic case the feedback acts in particular on the dirichlet trace of the solution here it is shown that the resulting closed loop system defines a feedback c0 semigroup on l2 in fact on h32 2 0 that is both analytic and compact for positive times and whose generator has compact resolvent in the hyperbolic case the feedback acts on the position vector only or on its dirichlet trace in a special case here a similar result is established regarding the existence of a feedback c0 cosine operator moreover an example is given which hints that the class of prescribed feedbacks acting on the dirichlet trace cannot be substantially enlarged functional analytic techniques are employed in particular perturbation theory however perturbation theory for the original variable fails on l2 the space in which the final result is sought therefore our approach employs perturbation theory after a suitable continuous extension on the larger space h12 2

performance output tracking for a wave equation subject to unmatched general boundary harmonic disturbance in this paper we consider performance output reference tracking for a wave equation with general harmonic disturbance at one end and the control at the other end we first design a state reference system determined completely by the measured output and the reference signal where the parameter update law is presented an adaptive servomechanism output feedback control is then designed it is shown that the closed loop system is well posed three control objectives are achieved a the performance output is tracking the reference signal b all the internal loops are bounded c when the disturbance and reference are disconnected the closed loop is exponentially stable finally the state of system is shown to be tracking the reference state and the updated parameters are convergent to their true values

necessary conditions for exponential stabilizability of distributed parameter systems with infinite dimensional unbounded feedback abstract in this paper we consider distributed parameter control systems of the form x t ax t bu t where x t is in the state space x u t is in an infinite dimensional control space u and b is an unbounded input operator which is admissible in a standard sense we find necessary conditions for there to exist a compact operator k x u such that a bk generates a c 0 semigroup which is of exponential growth type in particular for every a must have at most finitely many eigenvalues of finite multiplicity in re z furthermore the part of a corresponding to the remaining spectrum must generate a semigroup with type at most we apply this to a model for a clamped euler bernoulli plate with dirichlet control applied to the entire boundary we show that a compact state feedback cannot exponentially stabilize the plate in the state space of maximum regularity

an eigenvalue problem related to cell growth abstract in this paper the eigenvalues of the operator corresponding to the partial differential equation which describes the evolution of a population reproducing by simple fission are investigated this is done by transforming the eigenvalue problem to an integral equation the theory concerning positive operators on a banach space appears to be very useful

necessary conditions for exact controllability with a finite dimensional input space we derive necessary conditions for exact controllability of infinite dimensional systems described by x ax bu where the generator a has a riesz basis of eigenvectors and the input space is finite dimensional these conditions are in terms of the eigenvalues of a and the degree of unboundedness of b but no further information about b is needed our results easily imply lack of exact controllability for a wide range of distributed parameter systems it is known from partial differential equation pde examples that the addition of damping into an exactly controllable second order system can destroy exact controllability we discuss the effect of damping on such systems using our general conditions we also give simple new proofs for results about the lack of exact controllability of hyperbolic systems in more than one space dimension

exponential stabilization of a rayleigh beam using collocated control we consider a hinged elastic beam described by the rayleigh beam equation on the interval 0 pi we assume the presence of two sensors one measures the angular velocity of the beam at a point xi epsiv 0 pi and the other measures the bending curvature of the beam at the same point if xi 0 or xi pi then the second output is not needed the corresponding operator semigroup is unitary on a suitable hilbert state space these two measurements are advantageous because they make the open loop system exactly observable regardless of the point xi we design the actuators and the feedback law in order to exponentially stabilize this system using the theory of collocated static output feedback developed in our recent paper we design the actuators such that they are collocated meaning that b c where b is the control operator and c is the observation operator it turns out that if xi epsiv 0 pi then the actuators cause a discontinuity of the bending exactly at this is the price in this example of having collocated actuators and sensors this obliges us to use an extension of to define the output signal in terms of the left and right limit of the bending at xi we prove that for all static output feedback gains in a suitable finite range the closed loop system is well posed and exponentially stable this follows from the general theory in our paper whose main points are recalled here

p admissible control elements for diagonal semigroups on l r spaces for linear systems described by dot x t a x t bu t where a is a diagonal operator on the state space l r for some 1 leq r infty we develop necessary and sufficient conditions for b to be p admissible this extends results by ho russell and weiss to the case r neq 2

boundary control of a linear thermoelastic beam abstract we show in this paper that the motions of a linear thermoelastic beam may be controlled exactly to zero in a finite time by a single boundary control that acts on one end of the beam the optimal time of controllability depends upon the moment of inertia parameter of the beam and becomes arbitrarily small if this parameter is omitted as in the euler bernoulli beam theory

coprime factorization for regular linear systems mild sufficient conditions are given for the existence of a doubly coprime factorization of the transfer function of a regular linear system as well as formulae for such a factorization the results are illustrated by two examples of delay systems one of which has infinitely many unstable poles

optimal control of systems with a unitary semigroup and with colocated control and observation abstract we solve the quadratic optimal control problem on an infinite time interval for a class of linear systems whose state space is a hilbert space and whose operator semigroup is unitary the difficulty is that the systems in this class having unbounded control and observation operators may be ill posed we show that there is a surprisingly simple solution to the problem the optimal feedback turns out to be output feedback our approach is to use a change of variables which transforms the system into a one which according to recent research is known to be conservative we show that under a mild assumption the transfer function of this conservative system is inner and then it follows that the optimal control of this conservative system is trivial we give an example with the wave equation on an n dimensional domain with neumann control and dirichlet observation of the velocity

the active disturbance rejection control approach to stabilisation of coupled heat and ode system subject to boundary control matched disturbance we consider stabilisation for a linear ordinary differential equation system with input dynamics governed by a heat equation subject to boundary control matched disturbance the active disturbance rejection control approach is applied to estimate in real time the disturbance with both constant high gain and time varying high gain the disturbance is cancelled in the feedback loop the closed loop systems with constant high gain and time varying high gain are shown respectively to be practically stable and asymptotically stable

stabilization of distributed parameter systems by piecewise polynomial control in this paper we consider the stabilization of continuous time distributed parameter systems by piecewise polynomial controls we obtain necessary and sufficient conditions for stabilization by piecewise polynomial controls

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

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

pointwise exponential stabilization of a linear parabolic pde system using non collocated pointwise observation this paper deals with the problem of exponential stabilization for a linear distributed parameter system dps using pointwise control and non collocated pointwise observation where the system is modeled by a parabolic partial differential equation pde the main objective of this paper is to construct an output feedback controller for pointwise exponential stabilization of the linear parabolic pde system using the non collocated pointwise observation the observer based output feedback control technique is utilized to overcome the design difficulty caused by the non collocation pointwise observation we construct a luenberger type pde state observer to exponentially track the state of the pde system a collocated pointwise feedback controller is proposed based on the estimated state a variation of poincare wirtinger inequality is derived and the spatial domain is decomposed into multiple sub domains according to the number of the actuators and sensors by using lyapunov s direct method integration by parts and the variation of poincare wirtinger inequality at each sub domain sufficient conditions for the existence of the observer based out feedback controller are developed such that the resulting closed loop system is exponentially stable and presented in terms of standard linear matrix inequalities lmis furthermore the closed loop well posedness analysis is also provided by the semigroup approach extensive numerical simulation results are presented to show the performance of the proposed output feedback controller

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

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

second order systems with acceleration measurements accelerometers are often used to measure the output of second order systems such as structural vibrations conditions under which these systems are well posed are obtained we also establish conditions under which these systems have minimum phase transfer functions

well posedness and exact controllability of fourth order schrodinger equation with boundary control and collocated observation in this paper we study the well posedness and exact controllability of a system described by a fourth order schrodinger equation on a bounded domain of mathbb r n n geqslant 2 with boundary control and collocated observation the neumann boundary control problem is first discussed it is shown that the system is well posed in the sense of d salamon this implies the exponential stability of the closed loop system under proportional output feedback control the well posedness result is then generalized to the dirichlet boundary control problem in particular in order to conclude feedback stabilization from well posedness we discuss the exact controllability with the dirichlet boundary control which is similar to the neumann boundary control case in addition we show that both systems are regular in the sense of g weiss and their feedthrough operators are zero

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

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

time varying regular bilinear systems the theory of infinite dimensional systems introduced and developed by salamon trans amer math soc 300 1987 pp 383 431 and weiss siam j control optim 27 1989 pp 527 545 has been recently applied for various systems see e g the jacob partington survey current trends in operator theory and its applications birkhauser boston 2004 in the same spirit schnaubelt in his recent work siam j control optim 41 2002 pp 1141 1165 has extended their approach to the time varying linear systems in this paper we follow this spirit and prove that one can extend the results recently obtained for infinite dimensional time invariant bilinear systems to the time varying setting in particular we show that the absolute regularity and detectability assumptions ensure the existence of an observer for this kind of systems

backstepping based adaptive error feedback regulator design for one dimensional reaction diffusion equation abstract in this paper we consider the error feedback regulator problem efrp for one dimensional reaction diffusion equation with unknown harmonic boundary disturbance where the regulated output is anti collocated with the control some auxiliary systems are constructed in order to make the control and the disturbance become collocated or make the measured tracking error become the output then the error feedback adaptive servomechanism is presented in which the parameters of the disturbance and the reference signals are estimated our design is based on the motion planning and the backstepping approaches in addition we give a brief description that our approach is applicable to the heat equation with distributed disturbance it is shown that the proposed adaptive control law regulates the tracking error to zero and keeps the states of all the internal loops bounded

discrete time and sampled data low gain control of infinite dimensional linear systems in the presence of input hysteresis we introduce a general class of causal dynamic discrete time nonlinearities which have certain monotonicity and lipschitz continuity properties in particular the discretizations of a large class of continuous time hysteresis operators obtained by applying the standard sampling and hold operations belong to this class it is shown that closing the loop around a power stable linear infinite dimensional discrete time single input single output system subject to an input nonlinearity from the class under consideration and compensated by a discrete time integral controller guarantees asymptotic tracking of constant reference signals provided that a the positive integrator gain is sufficiently small and b the reference value is feasible in a very natural sense we apply this result in the development of sampled data low gain integral control for exponentially stable regular linear infinite dimensional continuous time systems subject to input hysteresis

robustness with respect to delays for exponential stability of distributed parameter systems in this paper we address the question of whether the open loop exponential growth rate of a linear system can be improved by a feedback in such a way that this improvement is robust with respect to small delays in the feedback loop when the input operator is admissible and the class of possible feedbacks consists of compact operators we find that if a feedback can improve the exponential growth rate then it can do so robustly furthermore we find that if the control space is finite dimensional and a bounded feedback cannot be found to improve exponential stability then a large class of unbounded feedbacks cannot improve the exponential growth rate robustly even if such feedbacks can improve the exponential growth rate in the absence of delays

invertibility of riccati operators and controllability of a related systems abstract injectivity and existence of a bounded inverse of riccati operators is investigated for various classes of systems a unified approach to this problem based on few general properties verified by several control systems is proposed the relation between the invertibility of riccati operators and the controllability of certain related systems is pointed out

internal model based tracking and disturbance rejection for stable well posed systems in this paper we solve the tracking and disturbance rejection problem for infinite dimensional linear systems with reference and disturbance signals that are finite superpositions of sinusoids we explore two approaches both based on the internal model principle in the first approach we use a low gain controller and here our results are a partial extension of results by hamalainen and pohjolainen in their papers the plant is required to have an exponentially stable transfer function in the callier desoer algebra while in this paper we only require the plant to be well posed and exponentially stable these conditions are sufficiently unrestrictive to be verifiable for many partial differential equations in more than one space variable our second approach concerns the case when the second component of the plant transfer function from control input to tracking error is positive in this case we identify a very simple stabilizing controller which is again an internal model but which does not require low gain we apply our results to two problems involving systems modeled by partial differential equations the problem of rejecting external noise in a model for structure acoustics interactions and a similar problem for two coupled beams

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

null controllability with vanishing energy linear null controllable systems for which an arbitrary initial state can be transferred to the origin with arbitrarily small energy are characterized theorems are stated in terms of an associated algebraic riccati equation and in terms of the spectrum of the linear part of the system the results so obtained allow us to determine ornstein uhlenbeck operators for which the liouville theorem about bounded harmonic functions holds

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

on a feynman kac approach to growth fragmentation semigroups and their asymptotic behaviors abstract this work develops further a probabilist approach to the asymptotic behavior of growth fragmentation semigroups via the feynman kac formula which was introduced in a joint article with a r watson 6 here it is first shown that the sufficient condition for a malthusian behavior which was established in 6 is also necessary we then provide a simple criterion to ensure exponential speed of convergence which enables us to treat cases that were not covered previously in the literature