pointwise stabilisation of a string with time delay in the observation abstractpointwise stabilisation is proposed in this paper for a string equation where the observation signal is subject to a time delay different from the boundary control the feedback stabiliser is acting at the middle joint of the string well posedness of the open loop system and solvability of the observer are shown first an observer system is then designed to estimate the state at the time interval when the observation is available and a predictor system is designed to predict the state at the time interval when the observation is not available pointwise output feedback controller is introduced to make the closed loop system asymptotically stable for the non smooth initial values and exponentially stable for the smooth initial values respectively simulation results demonstrate that the output feedback based on the observer and predictor effectively stabilises the pointwise control system with time delay

brief paper global stabilization of semilinear systems using switching controls this paper considers feedback stabilization of distributed semilinear systems using switching controls the semilinear processes together with the switched controllers constitute a type of switched nonlinear system three kinds of stabilizabilities namely weak strong and exponential stabilizabilities are investigated one by one sets of sufficient conditions are obtained for each case a necessary condition for strong stabilizability is given the stabilizing control is characterized using a minimization problem applications to parabolic and hyperbolic like equations are considered

optimizability and estimatability for infinite dimensional linear systems an infinite dimensional linear system described by dot x t ax t bu t t geq 0 is said to be optimizable if for every initial state x 0 an input u in l 2 can be found such that x in l 2 here a is the generator of a strongly continuous semigroup on a hilbert space and b is an admissible control operator for this semigroup in this paper we investigate optimizability also known as the finite cost condition and its dual estimatability we explore the connections with stabilizability and detectability we give a very general theorem about the equivalence of input output stability and exponential stability of well posed linear systems the two are equivalent if the system is optimizable and estimatable we conclude that a well posed system is exponentially stable if and only if it is dynamically stabilizable and input output stable we illustrate the theory by two examples based on pdes in two or more space dimensions the wave equation and a structural acoustics model

basis property and stabilization of a translating tensioned beam through a pointwise control force abstract introducing a recently developed powerful method riesz basis method as it is called and applying it to the stabilization of a translating tensioned beam through a pointwise control force are the two purposes of the paper an asymptotic analysis which is different from the one in treating the same problem in literature is used to get a complete asymptotic expansion of the eigenvalues and eigenfunctions for the beam the controlled system is written as an evolution equation in a state hilbert space whose energy norm induced by inner product is the total energy of the beam the spectrum determined growth condition is obtained by showing that there is a set of generalized eigenfunctions of the system which forms a riesz basis for the state space as a consequence the stability particularly the exponential stability of the system is developed the analysis used here can be extended to other distributed systems and the results significantly improve that in 1

on the dynamics of nonautonomous periodic general dynamical systems and differential inclusions abstract we establish the existence and stability results for periodic nonautonomous uniform forward attractors of periodic general dynamical systems set valued dynamical systems we also investigate the dynamical behavior of nonautonomous periodic differential inclusion x t f t x t on r m with only upper semi continuous right hand side by applying the abstract results firstly we show that if the system has a compact uniformly attracting set then it has a periodic nonautonomous uniform forward attractor a secondly we prove that a is robust with respect to both internal and external perturbations finally we apply the robustness result to discuss the effects of small time delays to asymptotic stability of the system

convexity of stokes waves of extreme form existence is established of a piecewise convex periodic planar curve s below which is defined a harmonic function which simultaneously satisfies prescribed dirichlet and neumann boundary conditions on s in hydrodynamics this corresponds to the existence of a periodic stokes wave of extreme form which has a convex profile between consecutive stagnation points where there is a corner with a contained angle of 120

an example for the switching delay feedback stabilization of an infinite dimensional system the boundary stabilization of a string abstract for vibrating systems a delay in the application of a feedback control may destroy the stabilizing effect of the control in this paper we consider a vibrating string that is fixed at one end and stabilized with a boundary feedback with delay at the other end we show that certain delays in the boundary feedback preserve the exponential stability of the system in particular we show that the system is exponentially stable with delays freely switching between the values 4 l c and 8 l c where l is the length of the string and c is the wave speed

feedback boundary stabilization of wave equations with interior delay in this paper we consider a boundary stabilization problem for the wave equation with interior delay we prove an exponential stability result under some lions geometric condition the proof of the main result is based on an identity with multipliers that allows us to obtain a uniform decay estimate for a suitable lyapunov functional

robustness of the stability of feedback systems with respect to small time delays it is shown that a feedback system is robustly stable with respect to small time delays if and only if it is stable for zero time delay and a structured singular value is less than one

stabilization of the wave equation with variable coefficients and a delay in dissipative boundary feedback we consider the stabilization of the wave equation with variable coefficients and a delay in the dissipative boundary feedback by virtue of the riemannian geometry methods the energy perturbed approach and the multiplier skills we establish the uniform stability of the energy of the closed loop system

nondeterministic space is closed under complementation in this paper we show that nondeterministic space s n is closed under complementation for s n greater than or equal to log n it immediately follows that the context sensitive languages are closed under complementation thus settling a question raised by kuroda inform and control 7 1964 pp 207 233

stabilization of distributed control systems with delay abstract in this paper the asymptotic stabilization of linear distributed parameter control systems with delay is considered specifically we are concerned with the class of control systems described by the equation x t a x t l x t b u t where a is the infinitesimal generator of a strongly continuous semigroup on a banach space x assuming appropriate conditions we will show that the usual spectral controllability assumption implies the feedback stabilization of the system applications to systems described by partial differential equations with delay are given

stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks in this paper we consider in a bounded and smooth domain the wave equation with a delay term in the boundary condition we also consider the wave equation with a delayed velocity term and mixed dirichlet neumann boundary condition in both cases under suitable assumptions we prove exponential stability of the solution these results are obtained by introducing suitable energies and by using some observability inequalities if one of the above assumptions is not satisfied some instability results are also given by constructing some sequences of delays for which the energy of some solutions does not tend to zero

two examples of ill posedness with respect to time delays revisited we revisit two examples of elastic control systems which are stabilized by boundary feedback and then destabilized when time delays occur in the feedback in the first example a wave equation with a feedback gain greater than or equal to one arbitrary divergence rates are obtained for small time delays in the second example a euler bernoulli beam any time delay results in an arbitrary divergence rate

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

the state feedback regulator problem for regular linear systems this paper is about the state feedback regulator problem for infinite dimensional linear systems the plant assumed to be an exponentially stable regular linear system is driven by a linear possibly infinite dimensional exosystem via a disturbance signal the exosystem has its spectrum in the closed right half plane and also generates the reference signal for the plant output the regulator problem is to design a controller that while guaranteeing the stability of the closed loop system without the exosystem drives the tracking error to zero a particular version of this problem is the state feedback regulator problem in which the states of the exosystem and the plant are known to the controller under suitable assumptions we show that the latter problem is solvable if and only if a pair of algebraic equations called the regulator equations is solvable we derive conditions in terms of the transfer function of the plant and eigenvalues of the exosystem for the solvability of the regulator equations three examples illustrating the theory are presented

on the stabilization and stability robustness against small delays of some damped wave equations in this paper we consider a system which can be modeled by two different one dimensional damped wave equations in a bounded domain both parameterized by a nonnegative damping constant we assume that the system is axed at one end and is controlled by a boundary controller at the other end we consider two problems namely the stabilization and the stability robustness of the closed loop system against arbitrary small time delays in the feedback loop we propose a class of dynamic boundary controllers and show that these controllers solve the stabilization problem when the damping coefficient is nonnegative and the stability robustness problem when the damping coefficient is strictly positive

optimal neumann control for the 1d wave equation finite horizon infinite horizon boundary tracking terms and the turnpike property we consider a vibrating string that is fixed at one end with neumann control action at the other end we investigate the optimal control problem of steering this system from given initial data to rest in time t by minimizing an objective functional that is the convex sum of the l 2 norm of the control and of a boundary neumann tracking term we provide an explicit solution of this optimal control problem showing that if the weight of the tracking term is positive then the optimal control action is concentrated at the beginning and at the end of the time interval and in between it decays exponentially we show that the optimal control can actually be written in that case as the sum of an exponentially decaying term and of an exponentially increasing term this implies that if the time t is large the optimal trajectory approximately consists of three arcs where the first and the third short time arcs are transient arcs and in the middle arc the optimal control and the corresponding state are exponentially close to 0 this is an example for a turnpike phenomenon for a problem of optimal boundary control if t infinite horizon time problem then only the exponentially decaying component of the control remains and the norms of the optimal control action and of the optimal state decay exponentially in time in contrast to this situation if the weight of the tracking term is zero and only the control cost is minimized then the optimal control is distributed uniformly along the whole interval 0 t and coincides with the control given by the hilbert uniqueness method

on the unique satisfiability problem unique sat is the problem of deciding whether a given boolean formula has exactly one satisfying truth assignment this problem is a typical moreover complete representative of a natural class of problems about unique solutions all these problems belong to the class dife l1 l2 l1 lz np studied by papadimitriou and yannakakis we consider the relationship between these two classes particularly whether unique sat is dife complete it is if np co np we construct an oracle relative to which unique sat is not complete for dif and another oracle relative to which unique sat is complete for dif e whereas np v co np

new conformal mapping for adaptive resolving of the complex singularities of stokes wave a new highly efficient method is developed for computation of travelling periodic waves stokes waves on the free surface of deep water a convergence of numerical approximation is determined by the complex singularities above the free surface for the analytical continuation of the travelling wave into the complex plane an auxiliary conformal mapping is introduced which moves singularities away from the free surface thus dramatically speeding up numerical convergence by adapting the numerical grid for resolving singularities while being consistent with the fluid dynamics the efficiency of that conformal mapping is demonstrated for the stokes wave approaching the limiting stokes wave the wave of the greatest height which significantly expands the family of numerically accessible solutions it allows us to provide a detailed study of the oscillatory approach of these solutions to the limiting wave generalizations of the conformal mapping to resolve multiple singularities are also introduced

a hierarchy of propositional horn formulas a hierarchy of propositional horn formulas is introduced the levels hk and hk of the hierarchy are defined by way of the number of alternations between players in a certain game related to the satisfiability of horn formulas the satisfiability problems for formulas from a given level of the hierarchy are shown to be complete in nspace log n a certain relationship between the hierarchy and the bounded depth circuits is exhibited using it we show that for some hk and hk formulas the equivalent formulas in the lower levels of the hierarchy must be exponentially longer

wave equations fractional derivatives and a new instance of the lack of robustness of velocity feedbacks due to the attenuation of disturbances under integration it has been proposed that the time derivative of order in the stabilizing feedback of a hyperbolic system be replaced with caputo fractional derivatives of lower order it has already been noted that in this way exponential stability is not preserved we first complement this result we then prove that even dissipativity is not preserved if the caputo fractional derivative has to be numerically computed

boundary feedback stabilization of the telegraph equation decay rates for vanishing damping term abstract we study a semilinear mildly damped wave equation that contains the telegraph equation as a special case we consider neumann velocity boundary feedback and prove the exponential stability of the closed loop system we show that for vanishing damping term in the partial differential equation the decay rate of the system approaches the rate for the system governed by the wave equation without damping term in particular this implies that arbitrarily large decay rates can occur if the velocity damping in the partial differential equation is sufficiently small

optimal energy control in finite time by varying the length of the string we consider a finite string where at both end points a homogeneous dirichlet boundary condition holds one boundary point is fixed and the other is moving hence the length of the string is changing in time the string is controlled through the movement of this boundary point we consider movements of the boundary that are lipschitz continuous only movements for which at the given finite terminal time the string has the same length as at the beginning are admissible moreover we impose an upper bound for the lipschitz constant of the movement that is smaller than the speed of wave propagation we consider the optimal control problem to find an admissible movement for which at the given terminal time the energy of the string is minimal we give a sufficient condition for the existence and uniqueness of an optimal movement and construct an optimal control movement

robustly stabilizing controllers for dissipative infinite dimensional systems with collocated actuators and sensors we derive formulas for normalized doubly coprime factorizations over h for a class of nonexponentially stabilizable systems using these results we solve the problem of robust stabilization with respect to normalized coprime factor perturbations for a class of infinite dimensional systems s a b b d with finite rank inputs and outputs and dissipative operator a these systems are not exponentially stabilizable we give a parameterization of controllers that achieve a given robustness margin

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

two approaches for the stabilization of nonlinear kdv equation with boundary time delay feedback this paper concerns the nonlinear korteweg de vries equation with boundary time delay feedback under appropriate assumption on the coefficients of the feedbacks delayed or not we first prove that this nonlinear infinite dimensional system is well posed for small initial data the main results of our study are two theorems stating the exponential stability of the nonlinear time delay system two different methods are employed a lyapunov function approach allowing to have an estimation on the decay rate but with a restrictive assumption on the length of the spatial domain of the kdv equation and an observability inequality approach with a contradiction argument for any noncritical lengths but without estimation on the decay rate some numerical simulations are given to illustrate the results

the spectral approximation of linear operators with applications to the computation of eigenelements of differential and integral operators we are concerned with the numerical solution of the eigenvalue problem t varphi lambda varphi varphi ne 0 where t is a linear operator in a bana ch space t may represent a bounded integral operator or a closed differential operator with bounded inverse the linear operator t and its approximation t n are defined in the same space perturbation theory is then a suitable framework for our problem we present in the first part a systematic study of the various notions of convergence t n to t which imply the convergence of the eigenvalues with preservation of the multiplicities the results are then applied to practical methods for integral and differential operators in the second part we present convergence rates analytic perturbation theory is used to refine on the computed eigenelements of an integral operator and to produce localization results on the eigenelements finally superconvergence results are discussed both for integral and differential operators

strong stability of neutral equations with an arbitrary delay dependency structure the stability theory for linear neutral equations subjected to delay perturbations is addressed it is assumed that the delays cannot necessarily vary independently of each other but depend on a possibly smaller number of independent parameters as a main result necessary and sufficient conditions for strong stability are derived along with bounds on the spectrum which take into account the precise dependency structure of the delays in the derivation of the stability theory results from realization theory and determinantal representations of multivariable polynomials play an important role the observations and results obtained in the paper are first illustrated and validated with a numerical example next the effects of small feedback delays on the stability of a boundary controlled hyperbolic partial differential equation and of a control system involving state derivative feedback are analyzed

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

robustness with respect to small delays for exponential stability of non autonomous systems robustness of stability with respect to small delays e g motivated by feedback systems in control theory is of great theoretical and practical important but this property does not hold for many systems in this paper we introduce the conception of robustness with respect to small time varying delays for exponential stability of the non autonomous linear systems sufficient conditions are given for the non autonomous systems to be robust and examples are provided to illustrate that the conditions are satisfied for a large class of the non autonomous parabolic systems

well posedness of a structural acoustics control model with point observation of the pressure abstract we consider a controlled and observed partial differential equation pde which describes a structural acoustics interaction physically this pde describes an acoustic chamber with a flexible chamber wall the control is applied to this flexible wall and the class of controls under consideration includes those generated by piezoceramic patches the observation we consider is point measurements of acoustic pressure inside the cavity mathematically the model consists of a wave equation coupled through boundary trace terms to a structurally damped plate or beam equation and the point controls and observations for this system are modeled by highly unbounded operators we analyze the map from the control to the observation since the properties of this map are central to any control design which is based upon this observation we also show there exists an appropriate state space x so that if the initial state is in x and the control is in l2 then the state evolves continuously in x and the observation is in l2 the analysis of this system entails a microlocal analysis of the wave component of the system and the use of pseudodifferential machinery

exponential stabilization of the wave equation by dirichlet integral feedback we consider the problem of boundary feedback stabilization of a vibrating string that is fixed at one end and with control action at the other end in contrast to previous studies that have require

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

sensitivity to infinitesimal delays in neutral equations the stability of a steady state solution of a neutral functional differential equation can be sensitive to infinitesimal changes in the delays this phenomenon is caused by the behavior of the essential spectrum and is determined by the roots of an exponential polynomial avellar and hale j math anal appl 73 1980 pp 434 452 have considered the case of multiple fixed and nonzero delays in the first part of this paper their results are illustrated by means of spectral plots in the second part we extend the theory of avellar and hale to the limit case whereby some of the delays are brought to zero which may lead to characteristic roots with arbitrarily large real part necessary and sufficient conditions are provided using these results we show that the ratio of the delays plays a crucial role when several delays tend to zero simultaneously as an illustration of the theory we analyze the robustness of a boundary controlled pde in the presence of a small feedback delay

a paradigm of ill posedness with respect to time delays the authors present a simple general model of an infinite dimensional control system which is stabilized by a velocity feedback but then destabilized by small or large time delays in the feedback this model is in some sense generic for boundary stabilized conservative hyperbolic linear partial differential equations

technical communique an exponential stability result for the wave equation we consider a system described by the one dimensional linear wave equation in a bounded domain with appropriate boundary conditions to stabilize this system we propose a dynamic boundary controller applied at the free end of the system the transfer function of the proposed controller is a proper rational function which consists of a strictly positive real function and some poles on the imaginary axis we then show that under some conditions the closed loop system is exponentially stable

the computation of water waves modelled by nekrasov s equation nekrasov s integral equation describing water waves of almost extreme form is solved numerically the method consists of applying a simple quadrature rule to a rearranged version of the original equation strongly graded meshes are used to resolve an expected boundary layer in the solution for methods based on the trapezoidal rule global bifurcation theory is used to prove for fixed discretization parameter n the existence a continuous branch of positive numerical solutions these are parametrized by mu a natural parameter occurring in the original integral equation for fixed mu collective compactness arguments then prove subsequential convergence of these solutions as the mesh is refined i e as n to infty numerical experiments using higher order quadrature rules are reported these reveal that the method is capable of detecting a boundary layer and gibbs phenomenon type oscillations of maximum height about 0 37 circ in a region of width o 70 mathord left vphantom

brief paper dynamic stabilization of an euler bernoulli beam equation with time delay in boundary observation an euler bernoulli beam equation under boundary control and delayed observation is considered an observer predictor based scheme is developed to stabilize the equation on a time interval where the observation is available the state is tracked by the observer when the observation is not available due to the delay the state is estimated by the predictor the estimation is then used in proportional feedback it is shown that the state of closed loop system decays exponentially for smooth initial values

an example on the effect of time delays in boundary feedback stabilization of wave equations this note is concerned with the erect of time delays in boundary feedback stabilization schemes for wave equations the question to be addressed is whether such delays can destabilize a system which is uniformly asymptotically stable in the absence of delays

numerical computation of water and solitary waves by the double exponential transform we propose a numerical method for solving integral equations whose solutions possess singularities at the end points our method is based on the double exponential transform and effective use of the fast fourier transform its accuracy is tested by nekrasov s integral equation for water waves and yamada s equation for solitary waves