eigenvalues of the laplacian in two dimensions the eigenvalue problem for the laplace operator in two dimensions is classical in mathematics and physics nevertheless computational methods for estimating the eigenvalues are still of much current interest particularly in applications to acoustic and electromagnetic waveguides although our primary interest is with the computational methods there are a number of theoretical results on the behavior of the eigenvalues and eigenfunctions that are useful for understanding the methods and in addition are of interest in themselves these results are discussed first and then the various computational methods that have been used to estimate the eigenvalues are reviewed with particular emphasis on methods that give error bounds some of the more powerful techniques available are illustrated by applying them to a model problem

interval estimation for lpv systems applying high order sliding mode techniques in this paper the problem of design of interval observers for linear parameter varying lpv systems containing non detectable or non strongly observable parts is addressed firstly a high order sliding mode hosm method is applied to the strongly observable subsystem obtained by an appropriate change of coordinates to estimate the state and its derivative secondly this information is used to decrease the level of uncertainty in the rest of the system which leads to improvement of accuracy of the set membership estimates generated by an interval observer moreover it is shown that hosm techniques allows us to relax the applicability conditions of standard interval observer design methods the efficiency of the proposed approach is demonstrated through simulation examples

brief paper interval observer design for consistency checks of nonlinear continuous time systems this paper deals with fault detection for nonlinear continuous time systems a procedure based on interval analysis is proposed to build a guaranteed qlpv quasi linear parameter varying approximation of the nonlinear model the interval qlpv approximation makes it possible to derive two point observers which estimate respectively the lower and the upper bound of the state vector using cooperativity theory a set guaranteed to contain the actual value of the residual is then designed the modelling uncertainties and measurement errors are taken into account at the design stage the proposed methodology is illustrated through numerical simulations

high gain interval observer for partially linear systems with bounded disturbances in this paper a high gain interval observer is proposed for a class of partially linear systems affected by unknown but bounded additive disturbances term and measurements noise the proposed obse

nonlinear parabolic equations with measure data in this paper we give summability results for the gradients of solutions of nonlinear parabolic equations whose model isu div u p 2 u on 0 t p with homogeneous cauchy dirichlet boundary conditions wherep 1 and is a bounded measure on 0 t we also study how the summability of the gradient improves if the measure is a function inlm 0 t withm small moreover we give a new proof of the existence of a solution for problem p

ellipsoidal parameter or state estimation under model uncertainty ellipsoidal outer bounding of the set of all feasible state vectors under model uncertainty is a natural extension of state estimation for deterministic models with unknown but bounded state perturbations and measurement noise the technique described in this paper applies to linear discrete time dynamic systems it can also be applied to weakly non linear systems if non linearity is replaced by uncertainty many difficulties arise because of the non convexity of feasible sets combined quadratic constraints on model uncertainty and additive disturbances are considered in order to simplify the analysis analytical optimal or suboptimal solutions of the basic problems involved in parameter or state estimation are presented which are counterparts in this context of uncertain models to classical approximations of the sum and intersection of ellipsoids the results obtained for combined quadratic constraints are extended to other types of model uncertainty

summability of solutions of the dirichlet problem for nonlinear elliptic equations with right hand side in logarithmic classes close to l1 abstract in this paper we consider the dirichlet problem for a class of nonlinear elliptic second order equations with right hand side in l 1 we describe the summability properties of entropy and weak solutions of this problem in the case where the right hand side belongs to classes close to l 1 and defined by the logarithmic function or its compositions of arbitrary multiplicity

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

set membership state and parameter estimation for systems described by nonlinear differential equations this paper investigates the use of guaranteed methods to perform state and parameter estimation for nonlinear continuous time systems in a bounded error context a state estimator based on a prediction correction approach is given where the prediction step consists in a validated integration of an initial value problem for an ordinary differential equation ivp for ode using interval analysis and high order taylor models while the correction step uses a set inversion technique the state estimator is extended to solve the parameter estimation problem an illustrative example is presented for each part

interval estimation for continuous time switched linear systems this paper deals with the design of interval observers for switched linear systems sls a class of hybrid systems under the assumption that the disturbances and the measurement noise are bounded upper and lower bounds for the state are calculated new conditions of cooperativity in discrete time instants are firstly proposed then some techniques for interval estimation are developed in continuous time it is shown that it is possible to calculate the observer gains making the estimation error dynamics cooperative and stable via some change of coordinates under arbitrary switching sequences the performances of the developed techniques are illustrated through numerical examples

a note on interval observer design for unknown input estimation this paper addresses the problem of interval observer design for unknown input estimation in linear time invariant systems although the problem of unknown input estimation has been widely studied in the literature the design of joint state and unknown input observers has not been considered within a set membership context while conventional interval observers could be used to propagate with some additional conservatism unknown inputs by considering them as disturbances the proposed approach allows their estimation under the assumption that the measurement noise and the disturbances are bounded lower and upper bounds for the unmeasured state and unknown inputs are computed numerical simulations are presented to show the efficiency of the proposed approach

detection filter design for lpv systems a geometric approach this paper investigates fault detection and isolation of linear parameter varying lpv systems by using parameter varying c a invariant subspace and parameter varying unobservability subspaces the so called detection filter approach formulated as the fundamental problem of residual generation fprg for linear time invariant lti systems is extended for a class of lpv systems the question of stability is addressed in the terms of lyapunov quadratic stability by using linear matrix inequalities the results are applied to the model of a generic small commercial aircraft

some elliptic equations with w01 1 solutions abstract we consider some nonlinear dirichlet problems and we study how lower order terms can give a regularizing effect on the solutions the existence of distributional solutions with minimal properties solutions in w 0 1 1 functional space not so usual for finding solutions of elliptic problems or finite energy solutions even with nonregular data

leray lions operators with logarithmic growth abstract in this paper we prove existence uniqueness and regularity results for solutions of nonlinear elliptic equations in which the differential operator has logarithmic growth with respect to the gradient the solution will belong to the sobolev space w 0 1 1 to the orlicz sobolev space generated by the function t log 1 t but not to any sobolev space w 0 1 p with p 1

nonlocal equations with measure data we develop an existence regularity and potential theory for nonlinear integrodifferential equations involving measure data the nonlocal elliptic operators considered are possibly degenerate and cover the case of the fractional p laplacean operator with measurable coefficients we introduce a natural function class where we solve the dirichlet problem and prove basic and optimal nonlinear wolff potential estimates for solutions these are the exact analogs of the results valid in the case of local quasilinear degenerate equations established by boccardo and gallouet j funct anal 87 149 169 1989 partial differ equ 17 641 655 1992 and kilpelainen and mal ann scuola norm sup pisa cl sci iv 19 591 613 1992 acta math 172 137 161 1994 as a consequence we establish a number of results that can be considered as basic building blocks for a nonlocal nonlinear potential theory fine properties of solutions calderon zygmund estimates continuity and boundedness criteria are established via wolff potentials a main tool is the introduction of a global excess functional that allows us to prove a nonlocal analog of the classical theory due to campanato ann mat pura appl iv 69 321 381 1965 our results cover the case of linear nonlocal equations with measurable coefficients and the one of the fractional laplacean and are new already in such cases

the equation δ u u α 1 u f for 0 ga soit 0 et f l loc 1 r n n 1 on considere le probleme suivant u u 1 u f dans r n u l loc 1 r n u 1 u l loc 1 r n on demontre que si f 0 et s il existe une solution positive alors on a une solution positive minimale u on etudie l existence de solutions positives pour 0 1

brief paper near optimal interval observers bundle for uncertain bioreactors in this paper we design an interval observer for the estimation of unmeasured variables of uncertain bioreactors the observer is based on a bounded error observer as proposed in lemesle v gouze j l 2005 hybrid bounded error observers for uncertain bioreactor models bioprocess and biosystems engineering 27 311 318 that makes use of a loose approximation of the bacterial kinetics we first show how to generate guaranteed upper and lower bounds on the state provided that known intervals for the initial condition and the uncertainties are available these so called framers depend on a tuning gain they can be run in parallel and the envelope provides the best estimate an optimality criterion is introduced leading to the definition of an optimal observer we show that this criterion provides directly a gain set containing the best framers the method is applied to the estimation of the total biomass of an industrial wastewater treatment plant demonstrating its efficiency

dirichlet problems with singular convection terms and applications abstract in this paper we study dirichlet problems with singular convection terms then applications to the study of some elliptic systems of two equations are given

resolution of a semilinear equation in l1 let n or be a bounded regular open set of n and let x s be a continuous nondecreasing function in s measurable in x such that x 0 0 almost everywhere we solve for f l 1 the problem p u u f in u 0 on in fact for this result instead of assuming that is nondecreasing in s we need only that x s s 0 we deduce an almost necessary and sufficient condition on in order that p has a solution roughly speaking this condition is f v g with g l 1 and v l 1

interval slopes for rational functions and associated centered forms for an arithmetic expression f x involving n rational operations an o n algorithm is given which computes an interval enclosure for the set of slopes f x z where x ranges over an interval x applications to real and complex centered forms are given resulting in improvements over previous results by ratschek siam j numer anal 17 1980 pp 656 662 and petkovic freiburger intervall berichte 83 2 1983 pp 33 50

technical communique nonlinear bounded error state estimation of continuous time systems this paper presents a first study on the application of interval analysis and consistency techniques to state estimation of continuous time systems described by nonlinear ordinary differential equations the approach is presented in a bounded error context and the resulting methodology is illustrated by an example

w01 1 solutions of some unilateral problems abstract in this paper we prove the existence of w 0 1 1 solutions of some unilateral problems with non regular data

empirical convergence speed of inclusion functions for facility location problems one of the key points in interval global optimization is the selection of a suitable inclusion function which allows to solve the problem efficiently usually the tighter the inclusions provided by the inclusion function the better because this will make the accelerating devices used in the algorithm more effective at discarding boxes on the other hand whereas more sophisticated inclusion functions may give tighter inclusions they require more computational effort than others providing larger overestimations in an earlier paper the empirical convergence speed of inclusion functions was defined and studied and it was shown to be a good indicator of the inclusion precision if the empirical convergence speed is analyzed for a given type of functions then one can select the appropriate inclusion function to be used when dealing with those type of functions in this paper we present such a study dealing with functions used in competitive facility location problems

w01 1 solutions in some borderline cases of calderon zygmund theory abstract in this paper we study the existence of w 0 1 1 distributional solutions of dirichlet problems whose simplest example is div u p 2 u f x in u 0 on

nonlinear elliptic equations in rn without growth restrictions on the data abstract we show existence and regularity of solutions in rn to nonlinear elliptic equations of the form div a x du g x u when is just a locally integrable function under appropriate growth conditions on a and g but not on roughly speaking in the model case p u u s 1u with p 2 1 n existence of a nonnegative solution in rn is guaranteed for every nonnegative l1loc rn if and only if s p 1

brief guaranteed state estimation by zonotopes this paper presents a new approach to guaranteed state estimation for non linear discrete time systems with a bounded description of noise and parameters the main result is an algorithm to compute a set that contains the states consistent with the measured output and the given noise and parameters this set is represented by a zonotope the size of the zonotope is minimized each sample time by an analytic expression or by solving a convex optimization problem interval arithmetic is used to calculate a guaranteed trajectory of the process state two examples have been provided to clarify the algorithm

construction of interval observers for continuous time systems with discrete measurements we consider continuous time systems with input output and additive disturbances in the particular case where the measurements are only available at discrete instants and have disturbances to solve a state estimation problem we construct continuous discrete interval observers that are asymptotically stable in the absence of disturbances these interval observers are composed of two copies of the studied system and of a framer accompanied with appropriate outputs which give componentwise upper and lower bounds for the solutions of the studied system

design of interval observers and controls for pdes using finite element approximations synthesis of interval state estimators is investigated for the systems described by a class of parabolic partial differential equations pdes first a finite element approximation of a pde is constructed and the design of an interval observer for the derived ordinary differential equation is given second the interval inclusion of the state function of the pde is calculated using the error estimates of the finite element approximation finally the obtained interval estimates are used to design a dynamic output stabilizing control the results are illustrated by numerical experiments with an academic example and the black scholes model of financial market

bifurcation mechanism underlying echelon mode formation abstract this paper presents a theory on the underlying mathematical mechanism of the echelon mode a series of parallel short wrinkles that looks like a flight of stairs or wild geese arranged in formation which has been observed ubiquitously with uniform materials but which has long denied successful numerical simulations it is shown by means of the group theoretic bifurcation theory that the echelon mode formation can be explained as a recursive secondary tertiary symmetry breaking bifurcation if o 2 o 2 is chosen as the underlying symmetry to model the local uniformity of materials this implies for example that the use of periodic boundaries is essential to successfully realize the oblique stripe patterns and the subsequent echelon mode formation in numerical simulations in fact a recursive bifurcation analysis of a rectangular domain with periodic boundaries subject to uniform uniaxial compression yields various kinds of patterns such as diamond stripe and echelon modes which are often observed for materials under shear

design of reduced order interval observers for lti systems with bounded time varying disturbances a parametric approach this paper investigates the design problem of reduced order interval observers for linear time invariant systems with bounded time varying disturbances although there are many excellent design met

echelon modes in uniform materials abstract the mechanism of the echelon mode a series of parallel short wrinkles of soils that looks like a flight of stairs or wild geese arranged in formation observed in the triaxial compression test is revealed by means of the group theoretic bifurcation theory the study of the direct bifurcation of uniform materials in cylindrical domains with periodic boundaries has revealed the existence of various kinds of deformation patterns in particular a classical diamond checkerboard pattern solution and a pair of oblique stripe pattern solutions simultaneously branch at a bifurcation point of multiplicity four with the use of the periodic boundaries these oblique stripe solutions which cannot be directly obtained by the customary bifurcation analysis with non periodic boundaries have thus been made accessible the secondary bifurcation from these stripe solutions realizes the echelon mode the triaxial compression tests on soil specimens have demonstrated that the pattern of wrinkles shear bands and echelon modes of the specimens do indeed follow the present theory and can be understood as a natural consequence of pattern selection or in other words symmetry breaking bifurcation

interval observer design and control of uncertain non homogeneous heat equations the problems of state estimation and observer based control for heat non homogeneous equations under distributed in space point measurements are considered first an interval observer is designed in the form of partial differential equations pdes without galerkin projection second conditions of boundedness of the interval observer solutions with non zero boundary conditions and measurement noise are proposed third the obtained interval estimates are used to design a dynamic output feedback stabilizing controller the advantages of the pde based interval observer over the approximation based one are clearly shown in the numerical example

state estimation and decoupling of unknown inputs in uncertain lpv systems using interval observers abstractthis paper proposes a linear parameter varying lpv interval observer for state estimation and unknown inputs decoupling in uncertain continuous time lpv systems two different problems are considered and solved 1 the evaluation of the set of admissible values for the state at each instant of time and 2 the unknown input observation i e the design of the observer in such a way that some information about the nature of the unknown inputs affecting the system can be obtained in both cases analysis and design conditions which rely on solving linear matrix inequalities lmis are provided the effectiveness and appeal of the proposed method is demonstrated using an illustrative application to a two joint planar robotic manipulator

interval state observer for nonlinear time varying systems this paper is devoted to the design of interval observers for linear time varying ltv systems and a class of nonlinear time varying systems in the output canonical form an interval observer design is feasible if it is possible to calculate the observer gains making the estimation error dynamics cooperative and stable it is shown that under some mild conditions the cooperativity of an ltv system can be ensured by a static linear transformation of coordinates the efficiency of the proposed approach is demonstrated through numerical simulations

interval observers for continuous time lpv systems with l 1 l 2 performance an approach to interval observer design for linear parameter varying lpv systems is proposed it is assumed that the vector of scheduling parameters in lpv models is not available for measurement two different interval observers are constructed for nonnegative systems and for a generic case stability conditions are expressed in terms of matrix inequalities which can be solved with respect to the observer gains using standard numerical solvers applying l 1 l 2 framework the robustness and estimation accuracy with respect to model uncertainty are analyzed the efficiency of interval estimation for lpv models is demonstrated through numerical experiments for a microfluidic system and an academic example

expansion and estimation of the range of nonlinear functions many verification algorithms use an expansion f x f x s x x f r n r n for x x where the set of matrices s is usually computed as a gradient or by means of slopes in the following an expansion scheme is described which frequently yields sharper inclusions for s this allows also to compute sharper inclusions for the range of f over a domain roughly speaking f has to be given by means of a computer program the process of expanding f can then be fully automatized the function f need not be differentiable for locally convex or concave functions special improvements are described moreover in contrast to other methods x x may be empty without implying large overestimations for s this may be advantageous in practical applications

some cases of weak continuity in nonlinear dirichlet problems abstract we show how to pass to the limit in some nonlinear elliptic problems despite the weak convergence of the right hand sides and the weak convergence of the solutions

an effective method to interval observer design for time varying systems an interval observer for linear time varying ltv systems is proposed in this paper usually the design of such observers is based on monotone systems theory monotone properties are hard to satisfy in many situations to overcome this issue in a recent work it has been shown that under some restrictive conditions the cooperativity of an ltv system can be ensured by a static linear transformation of coordinates however a constructive method for the construction of the transformation matrix and the observer gain making the observation error dynamics positive and stable is still missing and remains an open problem in this paper a constructive approach to obtain a time varying change of coordinates ensuring the cooperativity of the observer error in the new coordinates is provided the efficiency of the proposed approach is shown through computer simulations

brief causal fault detection and isolation based on a set membership approach this paper presents a diagnostic methodology relying on a set membership approach for fault detection and on a causal model for fault isolation set membership methods are a promising approach to fault detection because they take into account a priori knowledge of model uncertainties and measurement errors every uncertain model parameter and or measurement is represented by a bounded variable in this paper detection consists of verifying the membership of measurements to an interval first order discrete time models are used and their output is explicitly computed with interval arithmetic fault isolation relies on a causal analysis and the exoneration principle which allows focusing the consistency tests on simple local models the isolation strategy consists of two steps performing minimal tests found with the causal graph and determining on line additional relevant tests that reduce the final diagnosis an application for a nuclear process is used in order to illustrate the method s efficiency