abel lemma based finite sum inequality and its application to stability analysis for linear discrete time delay systems this paper is concerned with stability of linear discrete time delay systems note that a tighter estimation on a finite sum term appearing in the forward difference of some lyapunov functional leads to a less conservative delay dependent stability criterion by using abel lemma a novel finite sum inequality is established which can provide a tighter estimation than the ones in the literature for the finite sum term applying this abel lemma based finite sum inequality a stability criterion for linear discrete time delay systems is derived it is shown through numerical examples that the stability criterion can provide a larger admissible maximum upper bound than stability criteria using a jensen type inequality approach and a free weighting matrix approach

auxiliary function based summation inequalities and their applications to discrete time systems auxiliary function based summation inequalities are addressed in this technical note by constructing appropriate auxiliary functions several new summation inequalities are obtained a novel sufficient criterion for asymptotic stability of discrete time systems with time varying delay is obtained in terms of linear matrix inequalities the advantage of the proposed method is demonstrated by two classical examples from the literature

pattern search algorithms for bound constrained minimization we present a convergence theory for pattern search methods for solving bound constrained nonlinear programs the analysis relies on the abstract structure of pattern search methods and an understanding of how the pattern interacts with the bound constraints this analysis makes it possible to develop pattern search methods for bound constrained problems while only slightly restricting the flexibility present in pattern search methods for unconstrained problems we prove global convergence despite the fact that pattern search methods do not have explicit information concerning the gradient and its projection onto the feasible region and consequently are unable to enforce explicitly a notion of sufficient feasible decrease

on the global convergence of derivative free methods for unconstrained optimization in this paper starting from the study of the common elements that some globally convergent direct search methods share a general convergence theory is established for unconstrained minimization methods employing only function values the introduced convergence conditions are useful for developing and analyzing new derivative free algorithms with guaranteed global convergence as examples we describe three new algorithms which combine pattern and line search approaches

technical communique improved stability criterion and its applications in delayed controller design for discrete time systems this paper is concerned with the problem of delay dependent stability analysis for discrete time systems with interval like time varying delays and the problem of stabilization for discrete time linear systems via time delayed controllers the first problem is solved by applying a novel lyapunov functional and an improved delay dependent stability criterion is obtained in terms of a linear matrix inequality based on this a sufficient condition for the solvability of the second problem is presented the reduced conservatism of the proposed stability result is shown through a numerical example while the applicability of the time delayed controller design method is demonstrated by an inverted pendulum system

understanding propagation failure as a slow capture near a limit point this paper investigates the phenomenon of propagation failure in spatially discrete reaction diffusion systems with excitable kinetics excitable kinetics are modeled by the fitzhugh nagumo equations in the limit of weak coupling strength a traveling pulse propagating in a chain of cells is constructed the asymptotic solution for the first three cells is studied in detail a single riccati equation describes propagation through each cell from this equation a first approximation for the critical coupling strength below which the pulse is blocked is obtained near this critical coupling strength propagation or failure results from a slow passage near a limit point leading either to a successful jump propagation or to capture by a slowly varying steady state failure the analysis of the behavior of the first three cells suggests an algorithmic method to analyze propagation through an arbitrary number of coupled cells failure may appear at any cell in the chain but is described by the same riccati e

pulse propagation in discrete systems of coupled excitable cells propagation of pulses in myelinated fibers may be described by appropriate solutions of spatially discrete fitzhugh nagumo systems in these systems propagation failure may occur if either the coupling between nodes is not strong enough or the recovery is too fast we give an asymptotic construction of pulses for spatially discrete fitzhugh nagumo systems which agrees well with numerical simulations and discuss the evolution of initial data into pulses and pulse generation at a boundary formulas for the speed and length of pulses are also obtained

pattern search methods for linearly constrained minimization we extend pattern search methods to linearly constrained minimization we develop a general class of feasible point pattern search algorithms and prove global convergence to a karush kuhn tucker point as in the case of unconstrained minimization pattern search methods for linearly constrained problems accomplish this without explicit recourse to the gradient or the directional derivative of the objective key to the analysis of the algorithms is the way in which the local search patterns conform to the geometry of the boundary of the feasible region

stationarity results for generating set search for linearly constrained optimization we present a new generating set search gss approach for minimizing functions subject to linear constraints gss is a class of direct search optimization methods that includes generalized pattern search one of our main contributions in this paper is a new condition to define the set of conforming search directions that admits several computational advantages for continuously differentiable functions we also derive a bound relating a measure of stationarity which is equivalent to the norm of the gradient of the objective in the unconstrained case and a parameter used by gss algorithms to control the lengths of the steps with the additional assumption that the derivative is lipschitz we obtain a big o bound as a consequence of this relationship we obtain subsequence convergence to a kkt point even though gss algorithms lack explicit gradient information numerical results indicate that the bound provides a reasonable estimate of stationarity

stability analysis of piecewise discrete time linear systems presents a stability analysis method for piecewise discrete time linear systems based on a piecewise smooth lyapunov function it is shown that the stability of the system can be established if a piecewise lyapunov function can be constructed and moreover the function can be obtained by solving a set of linear matrix inequalities lmis that is numerically feasible with commercially available software

stability of front solutions in inhomogeneous media abstract we use the fitzhugh nagumo f n model to study front solutions in inhomogeneous media inhomogeneity is modeled by adding a space dependent term z x in the equations changes in z shift the cubic nullcline up and down in the phase plane of the f n model this is used to mimic shifts in the cubic shaped nullcline of a bidomain model of calcium waves in egg cells physica d in press caused by changes in the total calcium concentration conditions for the existence and stability of stationary front solutions for a general class of z functions are obtained analytically when the dynamics are piecewise linear we show that spatial inhomogeneities cause pinning and oscillations of front solutions this is best demonstrated when z x is a linear ramp when the slope is large a linear ramp breaks the translational invariance of the stationary front and stabilizes the front the front becomes less stable as the slope decreases at a critical slope the front becomes unstable through a hopf bifurcation beyond which oscillations in the front occur this also occurs for nonlinear spatial inhomogeneities including a step increase with varying magnitude and steepness this bifurcation has been found previously physica d 106 1997 270 phys rev e 55 1997 366 in a local analysis of the f n model near a co dimension two point sometimes called a non equilibrium ising bloch nib bifurcation and with inhomogeneities of small magnitude the analysis presented here applies globally in parameter space and is valid for inhomogeneities of arbitrary magnitude we show that this bifurcation need not occur in conjunction with the nib point it is caused exclusively by spatial inhomogeneity a rich variety of wave phenomena including front oscillation front pinning and front reflection are shown to occur in inhomogeneous media

objective derivative free methods for constrained optimization we propose feasible descent methods for constrained minimization that do not make explicit use of the derivative of the objective function the methods iteratively sample the objective function value along a finite set of feasible search arcs and decrease the sampling stepsize if an improved objective function value is not sampled the search arcs are obtained by projecting search direction rays onto the feasible set and the search directions are chosen such that a subset approximately generates the cone of first order feasible variations at the current iterate we show that these methods have desirable convergence properties under certain regularity assumptions on the constraints in the case of linear constraints the projections are redundant and the regularity assumptions hold automatically numerical experience with the methods in the linearly constrained case is reported

generalized pattern search methods for a class of nonsmooth optimization problems with structure we propose a generalized pattern search gps method to solve a class of nonsmooth minimization problems where the set of nondifferentiability is included in the union of known hyperplanes and therefore is highly structured both unconstrained and linearly constrained problems are considered at each iteration the set of poll directions is enforced to conform to the geometry of both the nondifferentiability set and the boundary of the feasible region near the current iterate this is the key issue to guarantee the convergence of certain subsequences of iterates to points which satisfy first order optimality conditions numerical experiments on some classical problems validate the method

convergence properties of an augmented lagrangian algorithm for optimization with a combination of general equality and linear constraints we consider the global and local convergence properties of a class of augmented lagrangian methods for solving nonlinear programming problems in these methods linear and more general constraints are handled in different ways the general constraints are combined with the objective function in an augmented lagrangian the iteration consists of solving a sequence of subproblems in each subproblem the augmented lagrangian is approximately minimized in the region defined by the linear constraints a subproblem is terminated as soon as a stopping condition is satisfied the stopping rules that we consider here encompass practical tests used in several existing packages for linearly constrained optimization our algorithm also allows different penalty parameters to be associated with disjoint subsets of the general constraints in this paper we analyze the convergence of the sequence of iterates generated by such an algorithm and prove global and fast linear convergence as well as show that potentially troublesome penalty parameters remain bounded away from zero

wirtinger based integral inequality application to time delay systems in the last decade the jensen inequality has been intensively used in the context of time delay or sampled data systems since it is an appropriate tool to derive tractable stability conditions expressed in terms of linear matrix inequalities lmis however it is also well known that this inequality introduces an undesirable conservatism in the stability conditions and looking at the literature reducing this gap is a relevant issue and always an open problem in this paper we propose an alternative inequality based on the fourier theory more precisely on the wirtinger inequalities it is shown that this resulting inequality encompasses the jensen one and also leads to tractable lmi conditions in order to illustrate the potential gain of employing this new inequality with respect to the jensen one two applications on time delay and sampled data stability analysis are provided

optimization by direct search new perspectives on some classical and modern methods direct search methods are best known as unconstrained optimization techniques that do not explicitly use derivatives direct search methods were formally proposed and widely applied in the 1960s but fell out of favor with the mathematical optimization community by the early 1970s because they lacked coherent mathematical analysis nonetheless users remained loyal to these methods most of which were easy to program some of which were reliable in the past fifteen years these methods have seen a revival due in part to the appearance of mathematical analysis as well as to interest in parallel and distributed com puting this review begins by briefly summarizing the history of direct search methods and considering the special properties of problems for which they are well suited our focus then turns to a broad class of methods for which we provide a unifying framework that lends itself to a variety of convergence results the underlying principles allow general ization to handle bound constraints and linear constraints we also discuss extensions to problems with nonlinear constraints

wave block in excitable media due to regions of depressed excitability we study propagation failure using the one dimensional scalar bistable equation with a passive gap region by applying comparison principles for this type of equation the problem of finding conditions for block is reduced to finding conditions for the existence of steady state solutions we present a geometrical method that allows one to easily compute the critical gap length above which a steady state solution and thus block first occurs the method also helps to uncover the general bifurcation structure of the problem including the stability of the steady state solutions in obtaining these results we characterize the relationship between the properties of the system and propagation failure the method can easily be extended to other gap dynamics we use it to show that block associated with any local inhomogeneity must be associated with a limit point bifurcation

generating set search methods for piecewise smooth problems we consider a direct search approach for solving nonsmooth minimization problems where the objective function is locally lipschitz continuous and piecewise continuously differentiable on a finite family of polyhedra a generating set search method is proposed which is named structured because the structure of the set of nondifferentiability near the current iterate is exploited to define the search directions at each iteration some numerical results are presented to validate the approach

implementing generating set search methods for linearly constrained minimization we discuss an implementation of a derivative free generating set search method for linearly constrained minimization with no assumption of nondegeneracy placed on the constraints the convergence guarantees for generating set search methods require that the set of search directions possesses certain geometrical properties that allow it to approximate the feasible region near the current iterate in the hard case the calculation of the search directions corresponds to finding the extreme rays of a cone with a degenerate vertex at the origin a difficult problem we discuss here how state of the art computational geometry methods make it tractable to solve this problem in connection with generating set search we also discuss a number of other practical issues of implementation such as the careful treatment of equality constraints and the desirability of augmenting the set of search directions beyond the theoretically minimal set we illustrate the behavior of the implementation on several problems from the cuter test suite we have found it to be successful on problems with several hundred variables and linear constraints

wave blocking phenomena in bistable reaction diffusion systems the following bistable reaction variable diffusion systems including a parameter sigma is considered u 1 d x u x x frac 1 sigma f u v u 1 d x u x x sigma g u v traveling wave blocking problems are the motivation for studying the dependency of the heterogeneity of d x on the existence of stationary solutions which play the role of variers of traveling front or back waves

onset of unidirectional pulse propagation in an excitable medium with asymmetric heterogeneity heterogeneity is one of the most important and ubiquitous types of external perturbations in dissipative systems to know the behaviors of pulse waves in such media is closely related to studying the collision process between the pulse and the heterogeneity induced ordered pattern in particular we focus on unidirectional propagation of pulses in a medium with an asymmetric bump heterogeneity this topic has attracted much interest recently with respect to potential computational aspects of chemical pulse propagation as well as with respect to pulse propagation in biological signal processing we employ a three component reaction diffusion system with one activator and two inhibitor species to illustrate these issues the reduced dynamics near a drift bifurcation describes the phenomena in the full partial differential equations by ordinary differential equations such a reduced dynamics is able to capture unidirectional propagation properties of pulses near an asymmetric heterogeneity in a qualitatively correct way a remarkable feature is that such unidirectional behavior is linked to the imperfection of global bifurcation structure and the resulting asymmetric locations of critical points

traveling waves in a spring block chain sliding down a slope travelling waves in a spring block chain sliding down an inclined plane are studied for a piecewise linear spinodal friction force we construct analytically front waves pulse waves are obtained as the matching of two travelling fronts with identical speeds explicit formulas are obtained for the wavespeed and the wave form in the anti continuum limit the link with propagating phenomena in the burridge knopoff model is briefly discussed

output feedback stabilization for a discrete time system with a time varying delay this study employs the free weighting matrix approach to investigate the output feedback control of a linear discrete time system with an interval time varying delay first the delay dependent stability is analyzed using a new method of estimating the upper bound on the difference of a lyapunov function without ignoring any terms and based on the results a design criterion for a static output feedback sof controller is derived since the conditions thus obtained for the existence of admissible controllers are not expressed strictly in terms of linear matrix inequalities a modified cone complementarity linearization algorithm is employed to solve the nonconvex feasibility sof control problem furthermore the problem of designing a dynamic output feedback controller is formulated as one of designing an sof controller numerical examples demonstrate the effectiveness of the method and its advantage over existing methods

oscillatory wave fronts in chains of coupled nonlinear oscillators wave front pinning and propagation in damped chains of coupled oscillators are studied there are two important thresholds for an applied constant stress f for f f cs static peierls stress there are only stable moving wave fronts for piecewise linear models extending an exact method of atkinson and cabrera s to chains with damped dynamics corroborates this description for smooth nonlinearities an approximate analytical description is found by means of the active point theory generically for small or zero damping stable wave front profiles are nonmonotone and become wavy oscillatory in one of their tails

stability and passivity analysis of discrete time linear systems with time varying delay abstract this paper focuses on the stability and passivity analysis for a class of discrete time linear systems a novel general free matrix based summation inequality is proposed and two improved stability criteria in terms of linear matrix inequalities lmis are derived by employing the new summation inequality furthermore when an external disturbance appears a sufficient condition is established to ensure the considered linear system to be passive three numerical examples are provided to demonstrate the improvement of the proposed method

event based h filtering for sampled data systems this paper is concerned with event based h filtering for sampled data systems first an event based data packet processor is introduced to release sampled measurement outputs only if an event condition is violated as a result communication resources can be saved significantly while preserving the desired h performance second the resulting filtering error system is modeled as a system with an interval time varying delay by employing the lyapunov krasovskii functional approach a new bounded real lemma brl is established such that the filtering error system is asymptotically stable with the prescribed h performance third by performing an invertible linear transformation on the filtering error system a linear matrix inequality lmi based sufficient condition which is equivalent to the condition in the brl is obtained on the feasibility of the event based h filtering problem consequently suitable h filters and the event parameters in the event condition can be co designed provided that a set of lmis are satisfied finally a mechanical system with two masses and two springs is given to show the effectiveness of the proposed method

asymptotic construction of pulses in the discrete hodgkin huxley model for myelinated nerves a quantitative description of pulses and wave trains in the spatially discrete hodgkin huxley model for myelinated nerves is given predictions of the shape and speed of the waves and the thresholds for propagation failure are obtained our asymptotic predictions agree quite well with numerical solutions of the model and describe wave patterns generated by repeated firing at a boundary

nerve impulse propagation in a branching nerve system a simple model abstract local spatial changes of nerve axon geometry such as diameter increase and branching may cause that action potential waves approaching a region of geometric change fail to propagate beyond it in this paper this effect will be examined for a special kind of nonuniformity within the framework of a simple model an initial value problem for a single nonlinear diffusion equation on an unbounded domain

models of unidirectional propagation in heterogeneous excitable media two differential equation models of excitable media threshold and recovery kinetics with solutions that exhibit unidirectional propagation are presented it is shown that unidirectional propagation in heterogeneous excitable media with non oscillatory kinetics can be initiated from homogeneous initial data simulations on a reaction diffusion model with fitzhugh nagumo kinetics and spatially heterogeneous parameters yields a rotating wave on a one dimensional circular spatial domain an ordinary differential equation model with four semi coupled excitable cells and heterogeneous parameters is analyzed to determine a critical parameter region over which unidirectional propagation may occur

active set identification for linearly constrained minimization without explicit derivatives we consider active set identification for linearly constrained optimization problems in the absence of explicit information about the derivative of the objective function we begin by presenting some general results on active set identification that are not tied to any particular algorithm these general results are sufficiently strong that given a sequence of iterates converging to a karush kuhn tucker point it is possible to identify binding constraints for which there are nonzero multipliers we then focus on generating set search methods a class of derivative free direct search methods we discuss why these general results which are posed in terms of the direction of steepest descent apply to generating set search even though these methods do not have explicit recourse to derivatives nevertheless there is a clearly identifiable subsequence of iterations at which we can reliably estimate the set of constraints that are binding at a solution we discuss how active set estimation can be used to accelerate generating set search methods and illustrate the appreciable improvement that can result using several examples from the cuter test suite we also introduce two algorithmic refinements for generating set search methods the first expands the subsequence of iterations at which we can make inferences about stationarity the second is a more flexible step acceptance criterion

transition fronts in time heterogeneous and random media of ignition type abstract the current paper is devoted to the investigation of wave propagation phenomenon in reaction diffusion equations with ignition type nonlinearity in time heterogeneous and random media it is proven that such equations in time heterogeneous media admit transition fronts with time dependent profiles and that such equations in time random media admit transition fronts with random profiles important properties of transition fronts including the boundedness of propagation speeds and the uniform decaying estimates of the propagation fronts are also obtained