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

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 admissibility of observation for perturbed c0 semigroups on banach spaces abstract this paper concerns linear infinite dimensional systems with unbounded observation operators we study the invariance of admissibility for observation operators under some unbounded perturbations of the generator we give also some relations between the extensions of such observation operators with respect to the original generator and the perturbed generator our results are applied to functional differential linear systems

a uniform approach to define complexity classes abstract complexity classes are usually defined by referring to computation models and by putting suitable restrictions on them following this approach many proofs of results are tightly bound to the characteristics of the computation model and of its restrictions and therefore they sometimes hide the essential properties which insure the obtained results in order to obtain more general results a uniform family of computation models which encompasses most of the complexity classes of interest is introduced as a first initial set of results derivable from the proposed approach we will give a sufficient and necessary condition for proving separations of relativized complexity classes a characterization of complexity classes with complete languages and a sufficient condition for proving strong separations of relativized complexity classes examples of applications of these results to some specific complexity classes are then given additional results related to separations by sparse oracles can be found in bovet 1991

well posedness of systems of linear elasticity with dirichlet boundary control and observation an open loop system of linear elasticity with dirichlet boundary control and collocated observation is considered the main result we obtained states that this system is well posed in the sense of salamon the result deduces the exponential stability of the closed loop system under proportional output feedback hence it answers positively an open question proposed by liu and krstic ima j appl math 65 2000 pp 109 121 moreover the well posedness together with the regularity of the open loop system in the sense of weiss that was obtained in our separate paper s g chai and b z guo feedthrough operator for linear elasticity system with boundary control and observation preprint school of computational and applied mathematics university of the witwatersrand south africa 2008 makes this infinite dimensional system parallel in many ways to a linear finite dimensional system in the framework of well posed and regular linear infinite dimensional systems

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

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

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

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

complexity classes without machines on complete languages for up this paper develops techniques for studying complexity classes that are not covered by known recursive enumerations of machines often counting classes probabilistic classes and intersection classes lack such enumerations concentrating on the counting class up we show that there are relativizations for which upa has no complete languages and other relativizations for which pb upb npb and upb has complete languages among other results we show that p up if and only if there exists a set s in p of boolean formulas with at most one satisfying assignment such that s sat is not in p p up coup if and only if there exists a set s in p of uniquely satisfiable boolean formulas such that no polynomial time machine can compute the solutions for the formulas in s if up has complete languages then there exists a set r in p of boolean formulas with at most one satisfying assignment so that sat r is complete for up finally we indicate the wide applicability of our techniques to counting and probabilistic classes by using them to examine the probabilistic class bpp there is a relativized world where bppa has no complete languages if bpp has complete languages then it has a complete language of the form b majority where b p and majority f f is true for at least half of all assignments is the canonical pp complete set

regularity theory of hyperbolic equations with non homogeneous neumann boundary conditions ii general boundary data abstract this paper studies the regularity of solutions of general mixed second order time dependent hyperbolic problems of neumann type in a previous paper i lasiecka and r triggiani ann mat pura appl iv clvii 1990 285 367 using pseudo differential calculus we have provided sharp regularity results of the solutions and their traces when the non homogeneous data are in l 2 now we complement this study by providing a regularity when the non homogeneous data are more regular than as well as less regular than l 2 in contrast with our previous paper a functional analytic approach based on the l 2 results of our previous paper is used throughout

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

unambiguous computations and locally definable acceptance types abstract hertrampf s locally definable acceptance types show that many complexity classes can be defined in terms of polynomial time bounded ntms with simple local conditions on the nodes of its computation tree rather than global concepts like number of accepting paths etc we introduce a modification of hertrampf s locally definable acceptance types which allows to get a larger number of characterizable complexity classes among others the newly characterizable classes are up and mod z k p it is shown how different types of oracle access e g guarded access can be characterized by this model this sheds new light on the discussion on how to access unambiguous computation we present simple functions that describe precisely objects of current research as the unambiguous oracle alternation and promise hierarchies we exhibit the new class uap which seems to be an unambiguous analogue of wagner s p uap and thus p contains few and is currently the smallest class known with this property

complexity measures for public key cryptosystems a general theory of public key cryptography is developed that is based on the mathematical framework of complexity theory two related approaches are taken to the development of this theory and th

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

pointwise pressure observations of a canonical structural acoustics model in this paper we study regularity properties of a physically motivated observation associated with a controlled partial differential equation pde model on a rectangular domain this particular pde describes the active control of acoustic pressure in a chamber by means of piezoelectric activation and involves a wave coupled to a structurally damped beam equation with the coupling being accomplished by boundary trace terms the observation here consists of pointwise in space measurements of the velocity of the wave component with this velocity representing the acoustic pressure of the system as defined then the observation of the system like the control imput is modeled by an unbounded operator the main result of this paper is the discovery of an energy space such that for initial data taken from this space the desired observation of the acoustic pressure is well defined

well posedness and regularity of naghdi s shell equation under boundary control and observation abstract a system of naghdi s shell equation with dirichlet boundary control and collocated observation is considered results on the associated nonhomogeneous boundary value problem are presented based on these results it is shown that the system is well posed in the sense of d salamon and regular in the sense of g weiss the expression of the corresponding feedthrough operator is explicitly found by means of riemannian geometric method and partial fourier transform these properties make this partial differential control system parallel in many ways finite dimensional ones in the general framework of well posed and regular infinite dimensional systems

using inductive counting to simulate nondeterministic computation abstract immerman and szelepcsenyi s inductive counting technique demonstrated that for space classes the nondeterministic acceptance mechanism can simulate with no space penalty any reasonable acceptance mechanism based on censuses of configurations however the efficiency with which other acceptance mechanisms can simulate nondeterminism remains an open question this paper uses inductive counting to study the cost of simulating nondeterminism with valiant s paradigm of unique computation nondeterministic computation in which each input generates at most one accepting computation we show that unique computation can simulate nondeterministic computation with a space penalty logarithmic in the ambiguity of the nondeterministic computation tree relatedly we show that unique auxpdas and restricted sac1 circuits can efficiently simulate ambiguity bounded nondeterministic computation in particular all nondeterministic logspace languages of polynonmial ambiguity are in weakunambrac1 and thus are accepted by uniform weak unambiguous shallow circuits

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

reduced order model feedback control design numerical implementation in a thin shell model reduced order models employing the lagrange and popular proper orthogonal decomposition pod reduced basis methods in numerical approximation and feedback control of systems are presented and numerically tested the system under consideration is a thin cylindrical shell with surface mounted piezoceramic actuators donnell mushtari equations modified to include kelvin voigt damping are used to model the system dynamics basis functions constructed from fourier polynomials tensored with cubic splines are employed in the galerkin expansion of the full order model reduced basis elements are then formed from full order approximations of the exogenously excited shell taken at different time instances numerical examples illustrating the features of the reduced basis methods are presented as a first step toward investigating the behavior of the methods when implemented in physical systems the use of reduced order model feedback control gains in the full order model is considered and numerical examples are presented

well posedness of a model for structural acoustic coupling in a cavity enclosed by a thin cylindrical shell abstract a fully coupled mathematical model describing the interactions between a vibrating thin cylindrical shell and an enclosed acoustic field is presented because the model will ultimately be used in control applications involving piezoceramic actuators the loads and material contributions resulting from piezoceramic patches bonded to the shell are included in the discussion theoretical and computational issues lead to the consideration of a weak form of the modeling set of partial differential equations pdes and through the use of a semigroup formulation well posedness results for the system model are obtained

fuzzy systems and controllers lyapunov tools for a regionwise approach this paper reviews a class of tools which can help to check the stability of fuzzy linguistic controllers which is a difficult task due to their regionwise structure the application of conventional methods when considering a fuzzy controller as a global nonlinear controller often leads to conservative stability results the use of appropriate continuous or piecewise continuous lyapunov functions allows to address the stability of nonlinear systems under fuzzy control while taking into account the local structure of the fuzzy controllers some methods are presented which enforce the global lyapunov function to decrease when leaving one subspace for another closer to the equilibrium using concentric domains and providing less conservative conditions

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

some remarks on the stability of mamdani fuzzy control systems this paper presents a passivity approach to the stability analysis of mamdani fuzzy control systems the main idea of the paper is that the fuzzy controller can be considered from an input output viewpoint as a nonlinear dissipative operator an input output approach is provided by considering a dynamical dependency of inputs of the fuzzy controller then the passivity theory is applied in order to obtain conditions for absolute stability of the closed loop control system

inverse problem for structural acoustic interaction abstract we consider an inverse problem of determining a source term for a structural acoustic partial differential equation pde model that is comprised of a two or a three dimensional interior acoustic wave equation coupled to an elastic plate equation the coupling takes place across a boundary interface for this pde system we obtain uniqueness and stability estimates for the source term from a single measurement of boundary values of the structure acceleration of the elastic plate the proof of uniqueness is based on a carleman estimate first version of the wave problem within the chamber the proof of stability relies on three main points i a more refined carleman estimate second version and its resulting implication a continuous observability type estimate ii a compactness uniqueness argument iii an operator theoretic approach for obtaining the needed regularity in terms of the initial conditions

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

uniform stability in structural acoustic models with flexible curved walls abstract the aim of this paper is twofold first we develop an explicit extension of the kirchhoff model for thin shells based on the model developed by michel delfour and jean paul zolesio this model relies heavily on the oriented distance function which describes the geometry once this model is established we investigate the uniform stability of a structural acoustic model with structural damping the result no longer requires that the active wall be a plate it can be virtually any shell provided that the shell is thin enough to accommodate the curvatures

comments on practical design of nonlinear fuzzy controllers with stability analysis for regulating processes with unknown mathematical models it is shown in this note that the fuzzy controller with locally linear control rules used in ying 1994 cannot be linearized around the origin and therefore local stability of closed loop systems including this type of fuzzy controller cannot be analyzed by means of the lyapunov s first method besides it is proved that with another choice of inference law the fuzzy controller with locally linear control rules can be linearized in a neighbourhood of the origin copyright c 1996 elsevier science ltd

solving the linear quadratic optimal control problem for infinite dimensional systems abstract calculation of the solutions to linear quadratic optimal control problems for infinite dimensional systems is considered the sequence of solutions to a sequence of approximating finite dimensional problems converges to the optimal control for the infinite dimensional system if certain assumptions such as uniform stabilizability are satisfied we use this result to calculate controllers for the infinite dimensional system that are arbitrarily close to optimal a one dimensional heat equation and a problem of acoustic noise control are used to illustrate the algorithm the numerical results are discussed

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

the complexity of promise problems with applications to public key cryptography a promise problem is a formulation of partial decision problem complexity issues about promise problems arise from considerations about cracking problems for public key cryptosystems using a notion of turing reducibility between promise problems this paper disproves a conjecture made by even and yacobi 1980 that would imply nonexistence of public key cryptosystems with np hard cracking problems in its place a new conjecture is raised having the same consequence in addition the new conjecture implies that np complete sets cannot be accepted by turing machines that have at most one accepting computation for each input word

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

robust algorithms a different approach to oracles abstract a new notion of an oracle machine being helped by an oracle set is introduced it is required that the oracle machine is robust i e it always computes the same set independent of the oracle the main result states that the class of sets that can be computed by deterministic polynomial time algorithms being helped by some oracle set is exactly np co np some connections to probabilistic classes are also investigated

logspace and logtime leaf languages the computation tree of a nondeterministic machinemwith inputxgives rise to aleaf stringformed by concatenating the outcomes of all the computations in the tree in lexicographical order we may characterize problems by considering for a particular leaf language y the set of allxfor which the leaf string ofmis contained iny in this way in the context of polynomial time computation leaf languages were shown to capture many complexity classes in this paper we study the expressibility of the leaf language mechanism in the contexts of logarithmic space and of logarithmic time computation we show that logspace leaf languages yield a much finer classification scheme for complexity classes than polynomial time leaf languages capturing also many classes withinp in contrast logtime leaf languages basically behave like logtime reducibilities both cases are more subtle to handle than the polynomial time case we also raise the issue of balanced versus nonbalanced computation trees underlying the leaf language we indicate that it is a nontrivial problem to obtain information about the leaf string of a nonbalanced computation tree and present conditions under which it does not matter whether the computation tree is balanced or not

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