an observer for infinite dimensional dissipative bilinear systems we consider bilinear systems of the form t a t u t b t y t t c on an infinite dimensional hilbert space h where a is the generator of a semigroup of contraction b is a bounded dissipative operator and c h the input signal u l r such that u t 0 for almost every t r we present a simple observer for this class of systems with the estimation error converging weakly to zero in h for every sufficiently rich input inputs that we call regularly persistent our result is a generalization of the previous results in 1 2

a study of fragmentation processes using a discrete element method abstract we present a model of solids made from polygonal cells connected via beams we calculate the macroscopic elastic moduli from the beam and cell parameters this modellisation is particularly suited for the simulation of fragmentation processes we study the effects of an explosion inside a circular disk and the impact of a projectile and obtain the fragment size distribution we find that if breaking only happens under tensile forces a layer on the free wall opposed to impact is first ejected in that case the distribution follows a power law with an exponent that in most cases is around two

luenberger boundary observer synthesis for sturm liouville systems a static luenberger observer of a system with sturm liouville operator is synthesised with the aid of a boundary control formulation to this aim approximate observability detectability and stability of the system is studied and design results are worked out for a typical biochemical case study

dynamical analysis of distributed parameter tubular reactors this paper is dedicated to the dynamical analysis of tubular reactor models namely plug flow and axial dispersion reactors involving sequential reactions for which the kinetics only depends on the concentrations of the reactants involved in the reaction this analysis is performed by describing these models as infinite dimensional state space systems with bounded observation and control operators it is shown that these systems are exponentially stable and that a the plug flow reactor model is observable when the component concentrations are measured at the reactor output and observable with respect to its physical domain when only the product concentration is measured and it is reachable with respect to its physical domain when it is controlled at the reactor input and b any finite set of dominant modes of the axial dispersion reactor model is observable when either all the component concentrations or only the product concentration are measured and reachable by using appropriate sensors and actuators the dynamical properties of related finite dimensional models are also discussed

trajectory analysis of nonisothermal tubular reactor nonlinear models the existence and uniqueness of the state trajectories temperature and reactant concentration are analyzed for nonisothermal plug flow and axial dispersion tubular reactor models it is mainly shown that these trajectories exist on the whole nonnegative real time axis and the set of all physically feasible state values is invariant under the dynamics equations the main nonlinearity in the model originates from the arrhenius type kinetics term in the model equations the analysis essentially uses lipschitz and dissipativity properties of the nonlinear operator involved in the dynamics and the concept of state trajectory positivity

fragmentation of random trees we study fragmentation of a random recursive tree into a forest by repeated removal of nodes the initial tree consists of n nodes and it is generated by sequential addition of nodes with each new node attaching to a randomly selected existing node as nodes are removed from the tree one at a time the tree dissolves into an ensemble of separate trees namely a forest we study statistical properties of trees and nodes in this heterogeneous forest and find that the fraction of remaining nodes m characterizes the system in the limit we obtain analytically the size density of trees of size s the size density has power law tail with exponent therefore the tail becomes steeper as further nodes are removed and the fragmentation process is unusual in that exponent increases continuously with time we also extend our analysis to the case where nodes are added as well as removed and obtain the asymptotic size density for growing trees

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

a series analysis of forced tranverse oscillations in a spring mass system when a mass is attached by two springs to two opposing parallel walls its motion parallel to the walls is generally governed by a nonlinear ordinary differential equation periodic solutions to this equation are sought with period equal to that of an applied sinusoidal forcing term the motion of the mass is described in terms of a fourier series with coefficients that are themselves expressed as perturbation series in the amplitude of the forcing term accurate nonlinear solutions and phase plane orbits are presented limiting values of the forcing amplitude are identified and are associated either with the existence of folds in the solution surface or with the phenomenon of nonlinear resonance

parameterization of positively stabilizing feedbacks for single input positive systems abstract this paper provides parameterizations of all positively stabilizing feedbacks for a particular class of finite dimensional single input positive systems a method to yield these parameterizations by means of tools of linear programming and a convergence analysis which allows to extend the results to a particular infinite dimensional system described by a parabolic partial differential equation it also provides an academic standard example the pure diffusion equation to support the theoretical results

lq optimal control of infinite dimensional systems by spectral factorization abstract we consider the standard lq optimal control of an exponentially stabilizable and detectable infinite dimensional semigroup state space system with bounded sensing and control it is reported that the reachable restriction of the lq optimal state feedback operator can be identified 1 by solving a spectral factorization problem delivering a bistable spectral factor with entries in the distributed proper stable transfer function algebra and 2 by obtaining any constant solution of an operator diophantine equation over the properties of the restricted solution feedback operator are discussed it is shown in particular that the lq optimal state feedback operator and its reachable restriction coincide whenever the unreachable state components are unobservable these theoretical results are applied to a simple model of heat diffusion with collocated sensor and actuator leading to an approximation procedure converging exponentially fast to the lq optimal state feedback operator this procedure is based on approximate spectral factorization by symmetric extraction and on simple residue calculus

scaling laws for impact fragmentation of spherical solids we investigate the impact fragmentation of spherical solid bodies made of heterogeneous brittle materials by means of a discrete element model computer simulations are carried out for four different system sizes varying the impact velocity in a broad range we perform a finite size scaling analysis to determine the critical exponents of the damage fragmentation phase transition and deduce scaling relations in terms of radius r and impact velocity v 0 the scaling analysis demonstrates that the exponent of the power law distributed fragment mass does not depend on the impact velocity the apparent change of the exponent predicted by recent simulations can be attributed to the shifting cutoff and to the existence of unbreakable discrete units our calculations reveal that the characteristic time scale of the breakup process has a power law dependence on the impact speed and on the distance from the critical speed in the damaged and fragmented states respectively the total amount of damage is found to have a similar behavior which is substantially different from the logarithmic dependence on the impact velocity observed in two dimensions

power series solution to a simple pendulum with oscillating support the problem of determining some of the effects of a small forcing term on a regular perturbation solution to a nonlinear oscillation problem is studied via a simple example in particular we investigate the periodic solution of a simple pendulum with an oscillating support a power series solution is constructed in terms of epsilon frac omega omega o 2 frac a l where omega o and omega are the natural and driving frequencies respectively a is the amplitude of the support oscillation and l is the length of the pendulum these solutions are analyzed for three cases above resonance omega omega o below resonance omega omega o and at resonance omega omega o in each case the approximate location of the nearest singularities which limit the convergence of the power series are obtained by using pade approximants using this information a new expansion parameter delta is introduced where the radius of convergence of the transformed series

generation of localization in a discrete chain with periodic boundary conditions numerical and analytical results we study the generation of localization in a discrete chain composed of n subsystems and periodic boundary conditions strongly localized motions are studied that is time periodic motions where nearly all of the energy is spatially confined to a single subsystem for varying n numerical results indicate that the strongly localized solutions are generated through a bifurcation from an in phase spatially extended solution however in the limit as n rightarrow infty the bifurcation point tends to infinity and a smooth transition from localization to nonlocalization occurs we then present an analytic technique to complement the numerical results it is based on the matching local asymptotic expansions of a solution branch using pade approximants this leads to global analytic representations of the considered solutions valid over the entire range of the control parameter

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

brief paper lq control design of a class of hyperbolic pde systems application to fixed bed reactor a general linear controller design method for a class of hyperbolic linear partial differential equation pdes systems is presented this is achieved by using an infinite dimensional hilbert state space description with infinite dimensional distributed input and output a state lq feedback operator is computed via the solution of a matrix riccati differential equation in the space variable the proposed method is applied to a fixed bed chemical reactor control problem where one elementary reaction takes place an optimal controller is designed for linearized fixed bed reactor model it is applied to the original nonlinear model and the resulting closed loop stability is analyzed numerical simulations are performed to show the performance of the designed controller

resonances and power series solutions of the forced van der pol oscillator to the first approximation the nonresonant steady state motion of the forced van der pol oscillator is a combination of a free and a forced response when the forcing amplitude increases beyond a certain critical value which depends on the forcing frequency the free response will decay and the motion is represented by just a particular solution that has the same frequency as the forcing term in the first part of this paper a general expression is determined for all of the higher order resonant frequencies associated with a formal multiple timescale perturbation expansion of the solution these frequencies are shown to be dense on the real axis then a formal power series expansion is developed for a particular solution in terms of the small damping parameter varepsilon which is shown to be valid when the forcing frequency is not close to a certain subset of the resonant frequencies using pade approximants information is obtained about the location and nature of the singularities in the complex

state and observer based feedback control of normal flow equations abstract we investigate wellposedness and stability of normal flow equations with two different control schemes the first one considers the velocity field as a control action and relies on a proportional regulator that is proved to be stable in the second one control acts on the source term and is based on an observer providing an estimate of the gradient of the normal flow solution convergence properties are demonstrated in this case numerical results are presented to show the effectiveness of the proposed approaches

finite dimensional compensator design for parabolic distributed systems with point sensors and boundary input a compensator design is proposed for a large class of parabolic distributed systems basically it is a modal approach utilizing known finite dimensional algorithms for observer and stabilizer design the observer is constructed from observations taken from point or averaging sensors and the compensator is a feedback of this observer and can be a distributed or boundary implementation explicit sufficient conditions are given for the convergence of this scheme

lq boundary control of a diffusion convection reaction system in this work the boundary control of a distributed parameter system modelled by linear parabolic partial differential equations pdes with spatially varying coefficients is studied an infinite dimensional state space setting is considered and an exact transformation of the boundary actuation is realised to obtain an evolutionary model the evolutionary model which incorporates the spatially varying coefficients of the underlying set of the pdes is used for subsequent linear quadratic regulator synthesis the formulated linear quadratic state feedback controller is applied to a nonlinear model of the reactor and its performance is studied

lq optimal control of hyperbolic pdaes the linear quadratic control synthesis for a set of coupled first order hyperbolic partial differential and algebraic equations is presented by using the infinite dimensional hilbert state space representation of the system and the well known operator riccati equation ore method solving the algebraic equations and substituting them into the partial differential equations pdes results in a model consisting of a set of pure hyperbolic pdes the resulting pde system involves a hyperbolic operator in which the velocity matrix is spatially varying non symmetric and its eigenvalues are not necessarily negative through of the domain the c0 semigroup generation property of such an operator is proven and it is shown that the generated c0 semigroup is exponentially stable and consequently the ore has a unique and non negative solution conversion of the ore into a matrix riccati differential equation allows the use of a numerical scheme to solve the control problem

backstepping observer design for parabolic pdes with measurement of weighted spatial averages this paper is concerned with the observer design for one dimensional linear parabolic partial differential equations whose output is a weighted spatial average of the state over the entire spatial domain we focus on the backstepping approach which provides a systematic procedure to design an observer gain for systems with boundary measurement if the output is not a boundary value of the state the backstepping approach is not directly applicable to obtaining an observer gain that stabilizes the error dynamics therefore we attempt to convert the error system into another system to which backstepping is applicable the conversion is successfully achieved for a class of weighting functions and the resultant observer realizes exponential convergence of the estimation error with an arbitrary decay rate in terms of the l 2 norm in addition an explicit expression of the observer gain is available in a special case the effectiveness of the proposed observer is also confirmed by numerical simulations

stabilization of a coupled transport diffusion system with boundary input we consider the problem of stabilizing a coupled transport diffusion system with boundary input the system is described by two linear transport diffusion equations and is not asymptotically stable in order to stabilize the system with boundary input sensor influence functions are assumed to be located at interior of the domain first we formulate the system as an evolution equation with unbounded output operators in a hilbert space using variable transformation next we derive a reduced order model with a finite dimensional state variable for the infinite dimensional system then a stabilizing controller is constructed for the reduced order model under an additional assumption it is shown that the finite dimensional controller together with a residual mode filter plays a role of a finite dimensional stabilizing controller for the original infinite dimensional system if the order of the residual mode filter is chosen sufficiently large finally the validity of the design method is demonstrated through a numerical simulation