improved estimation of eigenvalues and eigenvectors of covariance matrices using their sample estimates the problem of estimating the eigenvalues and eigenvectors of the covariance matrix associated with a multivariate stochastic process is considered the focus is on finite sample size situations whereby the number of observations is limited and comparable in magnitude to the observation dimension using tools from random matrix theory and assuming a certain eigenvalue splitting condition new estimators of the eigenvalues and eigenvectors of the covariance matrix are derived that are shown to be consistent in a more general asymptotic setting than the traditional one indeed these estimators are proven to be consistent not only when the sample size increases without bound for a fixed observation dimension but also when the observation dimension increases to infinity at the same rate as the sample size numerical evaluations indicate that the estimators have an excellent performance in small sample size scenarios where the observation dimension and the sample size are comparable in magnitude

mach reflection for the two dimensional burgers equation abstract we study shock reflection for the two 2d burgers equation this model equation is an asymptotic limit of the euler equations and retains many of the features of the full equations a von neumann type analysis shows that the 2d burgers equation has detachment sonic and crocco points in complete analogy with gas dynamics numerical solutions support the detachment sonic criterion for transition from regular to mach reflection there is also strong numerical evidence that the reflected shock in the 2d burgers mach reflection forms a smooth wave near the mach stem as proposed by colella and henderson in their study of the euler equations

a linear approximation for the regular reflection of a weak shock at a wedge abstract the problem of shock reflection by a wedge in the flow dominated by the unsteady potential flow equation is an important problem in weak regular reflection the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream the reflected shock is transonic its position is a free boundary for the unsteady potential equation which is degenerate at the sonic line in self similar coordinates applying the special partial hodograph transformation used in zhouping xin huicheng yin transonic shock in a nozzle i 2 d case comm pure appl math lvii 2004 1 51 zhouping xin huicheng yin transonic shock in a nozzle ii 3 d case ims preprint 2003 we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain when the angle between incident shock and wedge is small we can see the weak regular reflection as the disturbance of normal reflection as in chen shuxing linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 by linearizing the resulted nonlinear equation and boundary conditions with the above viewpoint in chen shuxing linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 we obtain a linear degenerate elliptic equation with mixed boundary conditions in a curved quadrilateral domain by means of elliptic regularization techniques a delicate a priori estimate and compact arguments we show that the solution of the linearized problem is smooth in the interior and lipschitz continuous up to the degenerate boundary

stability of a mach configuration we prove the stability of a mach configuration which occurs in shock reflection off an obstacle or shock interaction in compressible flow the compressible flow is described by a full steady euler system of gas dynamics the unperturbed mach configuration is composed of three straight shock lines and a slip line carrying contact discontinuity among four regions divided by these four lines in the neighborhood of the intersection two are supersonic regions and other two are subsonic regions we prove that if the constant states in the supersonic regions are slightly perturbed then the structure of the whole configuration holds while the other two shock fronts and the slip line as well as the flow field in the subsonic regions are also slightly perturbed such a conclusion asserts the existence and stability of the general mach configuration in shock theory in order to prove the result we reduce the problem to a free boundary value problem where two unknown shock fronts are free boundaries while the slip line is transformed to a fixed line by a lagrange transformation in the region where the solution is to be determined we have to deal with an elliptic hyperbolic composed system by decoupling this system and combining the technique for both hyperbolic equations and elliptic equations we establish the required estimates which are crucial in the proof of the existence of a solution to the free boundary value problem 2005 wiley periodicals inc

existence and regularity of solution of mixed boundary value problem of a busemann equation with nonlinear absorb term abstract the busemann equation is a classical equation coming from fluid dynamics the well posed problem and regularity of solution of busemann equation with nonlinear term are interesting and important the busemann equation is elliptic in y 0 and is degenerate at the line y 0 in r 2 with a special nonlinear absorb term we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain by means of elliptic regularization technique a delicate prior estimate and compact argument we show that the solution of mixed boundary value problem of busemann equation is smooth in the interior and lipschitz continuous up to the degenerate boundary on some conditions the result is better than the result of classical boundary degenerate elliptic equation

existence and regularity of the solution of a mixed boundary value problem for the keldysh equation with a nonlinear absorption term abstract the keldysh equation is a more general form of the classic tricomi equation from fluid dynamics its well posedness and the regularity of its solution are interesting and important the keldysh equation is elliptic in y 0 and is degenerate at the line y 0 in r 2 adding a special nonlinear absorption term we study a nonlinear degenerate elliptic equation with mixed boundary conditions in a piecewise smooth domain similar to the potential fluid shock reflection problem by means of an elliptic regularization technique a delicate a priori estimate and compact argument we show that the solution of a mixed boundary value problem of the keldysh equation is smooth in the interior and lipschitz continuous up to the degenerate boundary under some conditions we believe that this kind of regularity result for the solution will be rather useful

compressible flow and transonic shock in a diverging nozzle the paper is devoted to the study of compressible flows and transonic shocks in diverging nozzles in the framework of the full compressible euler system consider a nozzle having a shape as a diverging truncated sector with generic opening angle if the upstream flow at the entrance is supersonic and is near to an axial symmetric flow and if all parameters of the upstream flow and the receiver pressure at the exit are suitably assigned then a transonic shock appears in the nozzle to determine the transonic shock and the flow in the nozzle leads to a free boundary value problem for a nonlinear partial differential equation we prove that the receiver pressure can uniquely determine the location of the transonic shock as well as the flow behind the shock such a conclusion was conjectured by courant and friedrichs and is confirmed theoretically in this paper for the divergent nozzles

a class of singular gierer meinhardt systems of elliptic boundary value problems abstract we establish existence results for singular gierer meinhardt elliptic systems with zero dirichlet boundary conditions with the different range of growth rates of the source term we use the schauder fixed point theorem and upper lower solution methods to obtain the existence results we also study boundary estimates for certain singularities

optimal holder regularity for mixed boundary value problems abstract we continue our study of mixed boundary value problems lu 1 in mu 2 on u 3 on for second order elliptic operators l and first order oblique operators m a best possible estimate of the regularity of u is proved under weak hypotheses on the coefficients of l and m

existence and uniqueness analysis of a detached shock problem for the potential flow abstract we study a problem for two dimensional steady potential and isentropic euler equations in a bounded domain where an artificial detached shock interacts with a wedge using the stream function we obtain a free boundary problem for the subsonic state and the detached artificial shock curve and we prove that such configuration admits a unique solution in certain weighted holder spaces the proof is based on various holder and schauder estimates for second order elliptic equations and fixed point theorems moreover we pose an energy principle and remark that the physical attached shock is the minimizer of the energy functional

sampled data observers for 1 d parabolic pdes with non local outputs abstract the present work provides a systematic approach for the design of sampled data observers to a wide class of 1 d parabolic pdes with non local outputs the studied class of parabolic pdes allows the presence of globally lipschitz nonlinear and non local terms in the pde two different sampled data observers are presented one with an inter sample predictor for the unavailable continuous measurement signal and one without an inter sample predictor explicit conditions on the upper diameter of the uncertain sampling schedule for both designs are derived for exponential convergence of the observer error to zero in the absence of measurement noise and modeling errors moreover explicit estimates of the convergence rate can be deduced based on the knowledge of the upper diameter of the sampling schedule when measurement noise and or modeling errors are present input to output stability ios estimates of the observer error hold for both designs with respect to noise and modeling errors the main results are illustrated by two examples which show how the proposed methodology can be extended to other cases e g boundary point measurements

a class of asymptotic tests for principal component vectors in this paper the hypothesis that a set of vectors lie in the subspace spanned by a prescribed subset of the principal component vectors for a normal population is considered a class of invariant asymptotic tests based on the sample covariance matrix is derived tests in this class are shown to be consistent and their local power functions are given the arguments used in deriving the class of tests are not heavily dependent on the assumption of normality nor on the use of the sample covariance matrix the results are shown to generalize when the procedures are based on any affine invariant m estimate of scatter and when the population is elliptical

potential theory for regular and mach reflection of a shock at a wedge if a plane shock hits a wedge a self similar pattern of reflected shocks travels outward as the shock moves forward in time the nature of the pattern is explored for weak incident shocks strength b and small wedge angles 2 w through potential theory a number of different scalings some study of mixed equations and matching asymptotics for the different scalings the self similar equations are of mixed type a linearization gives a linear mixed flow valid away from a sonic curve near the sonic curve a shock solution is constructed in another scaling except near the zone of interaction between the incident shock and the wall where a special scaling is used the parameter c1 2w 1 b ranges from 0 to here is the polytropic constant and c1 is the sound speed behind the incident shock for 2 regular reflection weak or strong can occur and the whole pattern is reconstructed to lowest order in shock strength for 1 2 mach reflection occurs and the flow behind the reflection is subsonic and can be constructed in principle with an open elliptic problem and matched the case 0 can be solved for 1 2 2 or even larger the flow behind a mach reflection may be transonic and further investigation must be made to determine what happens the basic pattern of reflection is an almost semi circular shock issuing for regular reflection from the reflection point on the wedge and for mach reflection matched with a local interaction flow assuming their nature choosing the least entropy generation the weak regular reflection will occur for sufficiently large von neumann paradox an accumulation point of vorticity occurs on the wedge above the leading point 1994 john wiley sons inc

asymptotics for kernel estimate of sliced inverse regression to explore nonlinear structures hidden in high dimensional data and to estimate the effective dimension reduction directions in multivariate nonparametric regression li and duan proposed the sliced inverse regression sir method which is simple to use in this paper the asymptotic properties of the kernel estimate of sliced inverse regression are investigated it turns out that regardless of the kernel function the asymptotic distribution remains the same for a wide range of smoothing parameters

e h type mach configuration and its stability this paper is devoted to the study of the classification and stability of mach configurations in the mach reflection of a shock wave by a rigid wall the flow behind the reflected shock front can be supersonic or subsonic correspondingly the mach configuration can be classified as e e type and e h type in this paper we proved the stability of e h type mach configurations provided the reflected shock is weak the result is the complement of the stability of e e type mach configurations in our earlier work

regularity of solutions to regular shock reflection for potential flow the shock reflection problem is one of the most important problems in mathematical fluid dynamics since this problem not only arises in many important physical situations but also is fundamental for the mathematical theory of multidimensional conservation laws that is still largely incomplete however most of the fundamental issues for shock reflection have not been understood including the regularity and transition of different patterns of shock reflection configurations therefore it is important to establish the regularity of solutions to shock reflection in order to understand fully the phenomena of shock reflection on the other hand for a regular reflection configuration the potential flow governs the exact behavior of the solution in c 1 1 across the pseudo sonic circle even starting from the full euler flow that is both of the nonlinear systems are actually the same in a physically significant region near the pseudo sonic circle thus it becomes essential to understand the optimal regularity of solutions for the potential flow across the pseudo sonic circle the transonic boundary from the elliptic to hyperbolic region and at the point where the pseudo sonic circle the degenerate elliptic curve meets the reflected shock a free boundary connecting the elliptic to hyperbolic region in this paper we study the regularity of solutions to regular shock reflection for potential flow in particular we prove that the c 1 1 regularity is optimal for the solution across the pseudo sonic circle and at the point where the pseudo sonic circle meets the reflected shock we also obtain the c 2 regularity of the solution up to the pseudo sonic circle in the pseudo subsonic region the problem involves two types of transonic flow one is a continuous transition through the pseudo sonic circle from the pseudo supersonic region to the pseudo subsonic region the other a jump transition through the transonic shock as a free boundary from another pseudo supersonic region to the pseudo subsonic region the techniques and ideas developed in this paper will be useful to other regularity problems for nonlinear degenerate equations involving similar difficulties

marginal tests with sliced average variance estimation summary we present a new computationally feasible test for the dimension of the central subspace in a regression problem based on sliced average variance estimation we also provide a marginal coordinate test under the null hypothesis both the test of dimension and the marginal coordinate test involve test statistics that asymptotically have chi squared distributions given normally distributed predictors and have a distribution that is a linear combination of chi squared distributions in general

a distribution free method of interval estimation for unsigned directional data summary a method of moments is suggested for large samples of unsigned directional data the point estimators obtained coincide with the natural estimates obtained from maximum likelihood estimation if the data have a bingham distribution however the calculations necessary for interval and regional estimation are considerably simpler the statistical analysis is based upon the asymptotic sampling properties of principal components in nonnormal populations davis 1977 tyler 1981 a numerical example first treated by bingham 1974 is examined and found to yield very similar regional estimates the hypothesis of rotational symmetry is discounted by using a test statistic suggested by watson 1965

on global transonic shocks for the steady supersonic euler flows past sharp 2 d wedges abstract in this paper under certain downstream pressure condition at infinity we study the globally stable transonic shock problem for the perturbed steady supersonic euler flow past an infinitely long 2 d wedge with a sharp angle as described in the book of courant and friedrichs r courant k o friedrichs supersonic flow and shock waves interscience new york 1948 pages 317 318 when a supersonic flow hits a sharp wedge it follows from the rankine hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the edge of the sharp wedge in terms of the different pressure states in the downstream region which correspond to the supersonic shock and the transonic shock respectively it has frequently been stated that the strong shock is unstable and that therefore only the weak shock could occur however a convincing proof of this instability has apparently never been given the aim of this paper is to understand this open problem more concretely we will establish the global existence and stability of a transonic shock solution for 2 d full euler system when the downstream pressure at infinity is suitably given meanwhile the asymptotic state of the downstream subsonic solution is determined

input to error stable observer for nonlinear sampled data systems with application to one sided lipschitz systems in this paper the design of sampled data state observers for nonlinear plants is investigated under the effect of system and measurement disturbance signals we establish general design principles using the standard approaches of i direct discrete time design via approximation and ii discretization of a continuous time observer emulation by interpreting the disturbances as exogenous inputs affecting the error dynamics sufficient conditions are derived which ensure the input to state stability iss of the observer error system in a semiglobal practical sense for the unknown exact discrete time model next we focus on systems whose vector fields are one sided lipschitz to develop constructive design techniques via linear matrix inequalities lmis numerical simulations of an academic example and a chaotic attractor corroborate the effectiveness of the proposed sampled data estimators

a class of quasilinear degenerate elliptic problems abstract we establish the existence of solutions for a class of quasilinear degenerate elliptic equations the equations in this class satisfy a structure condition which provides ellipticity in the interior of the domain and degeneracy only on the boundary equations of transonic gas dynamics for example satisfy this property in the region of subsonic flow and are degenerate across the sonic surface we prove that the solution is smooth in the interior of the domain but may exhibit singular behavior at the degenerate boundary the maximal rate of blow up at the degenerate boundary is bounded by the degree of degeneracy in the principal coefficients of the quasilinear elliptic operator our methods and results apply to the problems recently studied by several authors which include the unsteady transonic small disturbance equation the pressure gradient equations of the compressible euler equations and the singular quasilinear anisotropic elliptic problems and extend to the class of equations which satisfy the structure condition such as the shallow water equation compressible isentropic two dimensional euler equations and general two dimensional nonlinear wave equations our study provides a general framework to analyze degenerate elliptic problems arising in the self similar reduction of a broad class of two dimensional cauchy problems

compensating actuator and sensor dynamics governed by diffusion pdes for possibly unstable ode systems with actuator delay predictor based infinite dimensional feedback can compensate for actuator delay of arbitrary length and achieve stabilization we extend this concept to another class of pde ode cascades where the infinite dimensional part of the plant is of diffusive rather than convective type we derive predictor like feedback laws and observers with explicit gain kernels the gain kernels involve second order matrix exponentials of the system matrix of the ode plant which is the result of the second order in space character of the actuator sensor dynamics the construction of the kernel functions is performed using the continuum version of the backstepping method robustness to small perturbations in the diffusion coefficient is proved

sliced inverse regression for dimension reduction abstract modern advances in computing power have greatly widened scientists scope in gathering and investigating information from many variables information which might have been ignored in the past yet to effectively scan a large pool of variables is not an easy task although our ability to interact with data has been much enhanced by recent innovations in dynamic graphics in this article we propose a novel data analytic tool sliced inverse regression sir for reducing the dimension of the input variable x without going through any parametric or nonparametric model fitting process this method explores the simplicity of the inverse view of regression that is instead of regressing the univariate output variable y against the multivariate x we regress x against y forward regression and inverse regression are connected by a theorem that motivates this method the theoretical properties of sir are investigated under a model of the form y f 1 x k x e where the k s are the unknown

a framework for nonlinear sampled data observer design via approximate discrete time models and emulation we study observer design for sampled data nonlinear systems using two approaches i the observer is designed via an approximate discrete time model of the plant ii the observer is designed based on the continuous time plant model and then discretized for sampled data implementation emulation we investigate under what conditions and in what sense these designs achieve convergence for the unknown exact discrete time model we present examples which show that designs that violate our conditions may indeed lead to instability when implemented on the exact model

asymptotics of eigenprojections of correlation matrices with some applications in principal components analysis summary the asymptotic distribution of an eigenprojection for a sample correlation matrix is obtained in particular it is shown that the rank of the asymptotic covariance matrix depends on distributional parameters in a somewhat complicated manner the results obtained in this paper can be used to determine this rank some applications of the asymptotic distribution of these eigenprojections to inferential problems involving principal components subspaces are given many statistical applications involve the analysis of a sample covariance matrix or sample correlation matrix as a result of the complexity of the exact joint distribution of the elements of these matrices most of these applications utilise the corresponding asymptotic distributions since the asymptotic distribution for the sample covariance matrix is somewhat simpler than that of the sample correlation matrix covariance matrix based analyses have received more attention in the literature unfortunately in practice analyses based on correlation matrices are more prevalent than those based on covariance matrices as a result in those situations in which correlation matrix based procedures have not yet been developed practitioners will often use procedures that have been developed for a corresponding analysis based on a sample covariance matrix this results in a procedure which at best can only be described as a crude approximation in this paper we review some of the asymptotic properties of the sample correlation matrix in particular we give the asymptotic distribution of eigenprojections computed from the sample correlation matrix the associated asymptotic covariance matrix of this distribution is obtained and its rank is determined throughout this paper we focus on the sample correlation matrix computed from the usual sample covariance matrix however it should be noted that the asymptotic results obtained can be easily generalised to the correlation matrix computed from any asymptotically normal estimate of the covariance matrix we also consider some applications which utilise eigenprojections of correlation matrices these are some inference problems in a principal components analysis of a correlation matrix in some of these situations straightforward generalisations of the covariance matrix based test statistics are not possible because of some complications associated with the rank of the asymptotic covariance matrix of the eigenprojections in these cases

dimension reduction in binary response regression abstract the idea of dimension reduction without loss of information can be quite helpful for guiding the construction of summary plots in regression without requiring a prespecified model focusing on the central subspace we investigate such sufficient dimension reduction in regressions with a binary response three existing methods sliced inverse regression principal hessian direction and sliced average variance estimation and one new method difference of covariances are studied for their ability to estimate the central subspace and produce sufficient summary plots combining these numerical methods with the graphical methods proposed earlier by cook leads to a novel paradigm for the analysis of binary response regressions

free boundary and mixed boundary value problems for quasilinear elliptic equations tangential oblique derivative and degenerate dirichlet boundary problems abstract we present holder estimates and holder gradient estimates for a class of free boundary problems with tangential oblique derivative boundary conditions provided the oblique vector does not vanish at any point on the boundary we also establish the existence result for a general class of quasilinear degenerate problems of this type including nonlinear wave systems and the unsteady transonic small disturbance equation

shock diffraction by convex cornered wedges for the nonlinear wave system we are concerned with rigorous mathematical analysis of shock diffraction by two dimensional convex cornered wedges in compressible fluid flow through the nonlinear wave system this shock diffraction problem can be formulated as a boundary value problem for second order nonlinear partial differential equations of mixed elliptic hyperbolic type in an unbounded domain it can be further reformulated as a free boundary problem for nonlinear degenerate elliptic equations of second order with a degenerate oblique derivative boundary condition we establish a global theory of existence and optimal regularity for this shock diffraction problem to achieve this we develop several mathematical ideas and techniques which are also useful for other related problems involving similar analytical difficulties

transonic nozzle flows and free boundary problems for the full euler equations abstract we establish the existence and uniqueness of transonic flows with a transonic shock through a two dimensional nozzle of slowly varying cross sections the transonic flow is governed by the steady full euler equations given an incoming smooth flow that is close to a constant supersonic state i e smooth cauchy data at the entrance and the subsonic condition with nearly horizontal velocity at the exit of the nozzle we prove that there exists a transonic flow whose downstream smooth subsonic region is separated by a smooth transonic shock from the upstream supersonic flow this problem is approached by a one phase free boundary problem in which the transonic shock is formulated as a free boundary the full euler equations are decomposed into an elliptic equation and a system of transport equations for the free boundary problem an iteration scheme is developed and its fixed point is shown to exist which is a solution of the free boundary problem by combining some delicate estimates for the elliptic equation and the system of transport equations with the schauder fixed point argument the uniqueness of transonic nozzle flows is also established by employing the coordinate transformation of euler lagrange type and detailed estimates of the solutions

a proof of existence of perturbed steady transonic shocks via a free boundary problem we prove the existence of a solution of a free boundary problem for the transonic small disturbance equation the free boundary is the position of a transonic shock dividing two regions of smooth flow assuming inviscid irrotational flow as modeled by the transonic small disturbance equation the equation is hyperbolic upstream where the flow is supersonic and elliptic in the downstream subsonic region to study the stability of a uniform planar transonic shock we consider perturbation by a steady c1 upstream disturbance if the upstream disturbance is small in a c1 sense then there is a steady solution in which the downstream flow and the transonic shock are holder continuous perturbations of the uniform configuration this result provides a new use of inviscid perturbation techniques to demonstrate in two dimensions the stability of transonic shock waves of the type that appear for example over the wing of an airplane along an airfoil or as bow shocks in a flow with a supersonic free stream velocity 2000 john wiley sons inc

supersonic flow onto a solid wedge we consider the problem of two dimensional supersonicflow onto a solid wedge or equivalently in a concave corner formed by two solid walls for mild corners there are two possible steady state solutions one with a strong and one with a weak shock emanating from the corner the weak shock is observed in supersonic flights a longstanding natural conjecture is that the strong shock is unstable in some sense we resolve this issue by showing that a sharp wedge will eventually produce weak shocks at the tip when accelerated to a supersonic speed more precisely we prove that for upstream state as initial data in the entire domain the timedependent solution is self similar with a weak shock at the tip of the wedge we construct analytic solutions for self similar potential flow both isothermal and isentropic with arbitrary 1 in the process of constructing the self similar solution we develop a large number of theoretical tools for these elliptic regions these tools allow us to establish large data results rather than a small perturbation we show that the wave pattern persists as long as the weak shock is supersonic supersonic when this is no longer true numerics show a physical change of behavior in addition we obtain rather detailed information about the elliptic region including analyticity as well as bounds for velocity components and shock tangents c 2007 wiley periodicals inc

a singular quasilinear anisotropic elliptic boundary value problem ii we consider the problem uauxx ubuyy p x 0 with a 0 b 0 on a smooth convex bounded region in r2 with dirichlet boundary conditions we show that if the positive function p is uniformly bounded away from zero then the problem has a classical solution

riemann problems for the two dimensional unsteady transonic small disturbance equation we study a two parameter family of riemann problems for the unsteady transonic small disturbance utsd equation also called the two dimensional burgers equation which is used to model the transition from regular to mach reflection for weak shock waves the related initial value problem consists of oblique shock data in the upper half plane with two parameters a and b corresponding to the slopes of the initial shock waves the study of quasi steady solutions leads to a problem that changes type when written in self similar coordinates the problem is hyperbolic in the region where the flow is supersonic and elliptic where the flow is subsonic in this paper we give a complete description of the flow in the hyperbolic region by resolving the hyperbolic wave interactions in the form of quasi one dimensional riemann problems in the region of physical space where the flow is subsonic we pose the related free boundary problems and discuss the behavior of the subsonic solution using results from our previo

sampled boundary observer for strict feedback nonlinear ode systems with parabolic pde sensor abstract we design an observer for ode pde cascades where the ode is nonlinear of strict feedback structure and the pde is a linear and of parabolic type the observer provides online estimates of the finite dimensional ode state vector and the infinite dimensional state of the pde based only on sampled boundary measurements a design that simultaneously addresses nonlinear odes and boundary measurement sampling is the paper s key contribution our observer design combines the backstepping design approach and the high gain observer methodology and our analysis employs a lyapunov krasovskii type functional to establish exponential convergence our sufficient conditions for convergence involve the maximum sampling interval and the pde domain length

global exponential sampled data observers for nonlinear systems with delayed measurements abstract this paper presents new results concerning the observer design for certain classes of nonlinear systems with both sampled and delayed measurements by using a small gain approach we provide sufficient conditions which involve both the delay and the sampling period ensuring exponential convergence of the observer system error the proposed observer is robust with respect to measurement errors and perturbations of the sampling schedule moreover new results on the robust global exponential state predictor design problem are provided for wide classes of nonlinear systems

a free boundary problem for a quasi linear degenerate elliptic equation regular reflection of weak shocks we prove the existence of a solution to the weak regular reflection problem for the unsteady transonic small disturbance utsd model for shock reflection by a wedge in weak regular reflection the state immediately behind the reflected shock is supersonic and constant the flow becomes subsonic further downstream the equation in self similar coordinates is degenerate at the sonic line the reflected shock becomes transonic and begins to curve there its position is the solution to a free boundary problem for the degenerate equation using the rankine hugoniot conditions along the reflected shock we derive an evolution equation for the transonic shock and an oblique derivative boundary condition at the unknown shock position by regularizing the degenerate problem we construct uniform bounds we apply local compactness arguments to extract a limit that solves the problem the solution is smooth in the interior and continuous up to the degenerate boundary this work completes a stage in our program to construct self similar solutions of two dimensional riemann problems in a series of papers we developed techniques for solving the degenerate elliptic equations that arise in self similar reductions of hyperbolic conservation laws in other papers especially in joint work with gary lieberman we developed techniques for solving free boundary problems of the type that arise from rankine hugoniot relations for the first time in this paper we combine these approaches and show that they are compatible although our construction is limited to a finite part of the unbounded subsonic region it suggests that this approach has the potential to solve a variety of problems in weak shock reflection including mach and von neumann reflection in the utsd equation and the analogous problems for the unsteady full potential equation 2002 john wiley sons inc

on the existence of positive solutions of quasilinear elliptic boundary value problems abstract we establish the existence of positive solutions to a class of quasilinear anisotropic problems which have either sublinear or superlinear nonlinearity with a b nonnegative constants and positive constants one example is if b a 0 1 has a solution on the other hand if b a 1 superlinear case then there exists a 0 such that for 0 1 has at least one solution and for no solution exists

inverse regression based uncertainty quantification algorithms for high dimensional models many uncertainty quantification uq approaches suffer from the curse of dimensionality that is their computational costs become intractable for problems involving a large number of uncertainty parameters in these situations the classic monte carlo often remains the preferred method of choice because its convergence rate o n 1 2 where n is the required number of model simulations does not depend on the dimension of the problem however many high dimensional uq problems are intrinsically low dimensional because the variation of the quantity of interest qoi is often caused by only a few latent parameters varying within a low dimensional subspace known as the sufficient dimension reduction sdr subspace in the statistics literature motivated by this observation we propose two inverse regression based uq algorithms iruq for high dimensional problems both algorithms use inverse regression to convert the original high dimensional problem to a low dimensional one which is then efficiently solved by building a response surface for the reduced model for example via the polynomial chaos expansion the first algorithm which is for the situations where an exact sdr subspace exists is proved to converge at rate o n 1 hence much faster than mc the second algorithm which doesn t require an exact sdr employs the reduced model as a control variate to reduce the error of the mc estimate the accuracy gain could still be significant depending on how well the reduced model approximates the original high dimensional one iruq also provides several additional practical advantages it is non intrusive it does not require computing the high dimensional gradient of the qoi and it reports an error bar so the user knows how reliable the result is

singular gierer meinhardt systems of elliptic boundary value problems abstract we establish existence results for singular gierer meinhardt elliptic systems with zero dirichlet boundary conditions gierer meinhardt systems are model problems for pattern formations of spatial tissue structures of morphogenesis the mathematical difficulties are that the system becomes singular near the boundary and it is non quasimonotone we show the existence of positive solutions for the activator inhibitor model with common sources

on multilinear principal component analysis of order two tensors summary principal component analysis is commonly used for dimension reduction in analysing highdimensional data multilinear principal component analysis aims to serve a similar function for analysing tensor structure data and has empirically been shown effective in reducing dimensionality in this paper we investigate its statistical properties and demonstrate its advantages conventional principal component analysis which vectorizes the tensor data may lead to inefficient and unstable prediction due to the often extremely large dimensionality involved multilinear principal component analysis in trying to preserve the data structure searches for low dimensional projections and thereby decreases dimensionality more efficiently the asymptotic theory of order two multilinear principal component analysis including asymptotic efficiency and distributions of principal components associated projections and the explained variance is developed a test of dimensionality is also proposed finally multilinear principal component analysis is shown to improve conventional principal component analysis in analysing the olivetti faces dataset which is achieved by extracting a more modularly oriented basis set in reconstructing the test faces

existence and stability of perturbed transonic shocks for compressible steady potential flows abstract we establish an existence theorem for perturbed transonic steady potential flows where the boundary of the subsonic region consists of a transonic shock and thus the governing equation becomes a free boundary problem we establish gradient estimates and use the fixed point theorem of caristi s which does not require a compactness of the map to establish the existence result we provide new techniques to handle the gradients and show that the flow is strictly elliptic inside of the subsonic region

observer design for a class of nonlinear ode pde cascade systems abstract the problem of state observation is addressed for nonlinear systems that can be modeled by an ode pde series association the ode subsystem assumes a triangular structure while the pde element is of heat diffusion type the aim is to accurately estimate online the state vector of the ode subsystem and the distributed state of the pde element one major difficulty is that the state observation must only rely on the global system output i e the pde state at the terminal boundary in particular the connection point between the ode and the pde blocks is not accessible to measurements the observation problem is dealt with by designing a high gain type observer sufficient conditions involving the pde domain length are formally established that ensure the observer exponential convergence

tensor products and statistics abstract a dictionary between operator based and matrix based languages in multivariate statistical analysis is proposed then this formulary is applied to asymptotic factorial analyses especially for giving asymptotic covariance matrices and operators in an explicit form finally we present the mathematical foundations on which are based the functional tools i e tensor products of linear spaces of vectors and of operators

subsonic solutions to compressible transonic potential problems for isothermal self similar flows and steady flows abstract we establish boundary and interior gradient estimates and show that no supersonic bubble appears inside of a subsonic region for transonic potential flows for both self similar isothermal and steady problems we establish an existence result for the self similar isothermal problem and improve the hopf maximum principle to show that the flow is strictly elliptic inside of the subsonic region for the steady problem

exponential stability analysis of sampled data ode pde systems and application to observer design a small gain approach is presented for analyzing exponential stability of a class of dynamical hybrid systems the systems considered in the paper are composed of finite dimensional dynamics represented by a linear ordinary differential equation ode and infinite dimensional dynamics described by a parabolic partial differential equation pde exponential stability is established under conditions involving the maximum allowable sampling period masp this new stability result is shown to be useful in the design of sampled output exponentially convergent observers for linear systems that are described by an ode pde cascade the new stability result also proves to be useful in designing practical approximate observers involving no pdes

local estimates for subsolutions and supersolutions of oblique derivative problems for general second order elliptic equations we consider solutions and subsolutions and supersolutions of the boundary value problem au x u du diju u x u du 0 in 2 s x diu y x u g x on3fi for a lipschitz domain s a positive definite matrix valued function u j and a vector field s which points uniformly into q without making any continuity assumptions on the known functions we prove harnack and holder estimates for u near 3i2 in addition we bound the l norm of u near 3i2 in terms of an appropriate lp norm and the known functions our approach is based on that for the corresponding interior estimates of trudinger this paper is concerned with analogs for solutions of oblique boundary value problems of the results in 10 our results will extend previous work of lieberman and trudinger 5 and of nadirashvili 9 specifically we consider second order differential operators q of the form

subsonic solutions for compressible transonic potential flows abstract we establish an existence theorem for transonic isentropic potential flows where the subsonic region is bounded by the sonic line and thus the governing equation may become degenerate on the boundary partly or entirely it has been conjectured by experiments and numerical studies that the self similar multidimensional flow changes its type namely hyperbolic far from the origin supersonic region and elliptic near the origin subsonic region furthermore the potential equation has a different nonlinearity compared to other transonic problems such as the unsteady transonic small disturbance equation the nonlinear wave equation and the pressure gradient equation namely the coefficients of the potential equation depend on the gradients while others are independent of the gradients we provide techniques to handle the gradients establish interior and boundary gradient estimates for the potential flow in a convex region and answer the conjecture that is the flow is strictly elliptic and the region is subsonic

sampled data adaptive observer for a class of state affine output injection nonlinear systems the problem of observer design is addressed for output injection nonlinear systems a major difficulty with this class of systems is that the state equation involves an output dependent term that is explicitly dependent on unknown parameters as the output is only accessible to measurement at sampling times the output dependent term turns out to be almost all time subject to a double uncertainty making previous adaptive observers inappropriate presently a new hybrid adaptive observer is designed and shown to be exponentially convergent under ad hoc conditions

mach configuration in pseudo stationary compressible flow when a shock front in unsteady compressible flow is moving along an inclined ramp the incident shock will be reflected by the ramp as it moves forward in the process of reflection various reflected wave pattern may occur depending on the slope of the ramp and the strength of the incident shock 14 23 this problem is one of the most important multidimensional prototype problems in gas dynamics generally if the angle of inclination is larger than a critical value then the reflected shock starting from the intersection of the incident shock with the ramp forms a smooth bubble which expands when time goes on such a reflection is called regular reflection on the other hand if the angle of inclination is small then the intersection of the incident shock and the reflected shock will not touch the ramp and an additional shock front called mach stem appears which connects the ramp with the intersection of the incident shock and the reflected shock in this case three shock fronts meet at one point and form a triple shock configuration an important fact is that the structure solely containing three shock fronts separating the neighborhood of the triple intersection into three zones with different continuous states does not exist 14 27 many physical experiments indicate that there is a slip line issuing from the triple intersection such a local wave pattern is called mach configuration or mach structure correspondingly the whole process of reflection is called mach reflection because of the complexity of the whole picture of mach reflection it is natural and necessary to start work on getting a clear understanding of the local mach configuration mach reflection also occurs in various other cases for compressible flow for instance in the study on oblique shock reflection in stationary compressible flow von neumann indicated that see 14 25 if the incident shock hit a wall with an angle larger than a critical value then the reflection must be mach reflection many physical experiments approximate analysis and numerical computations verified the appearance of mach configuration and a tremendous amount of literature is devoted to study such problems see 3 4 11 19 20 24 and the references therein however for the rigorous proof on the stability of mach configuration we can only mention a related result on stationary flow in 13

an elliptic problem arising from the unsteady transonic small disturbance equation abstract we prove a theorem on existence of a weak solution of the dirichlet problem for a quasilinear elliptic equation with a degeneracy on one part of the boundary the degeneracy is of a type keldysh type associated with singular behavior blow up of a derivative at the boundary we define an associated operator which is continuous pseudo monotone and coercive and show that a weak solution displaying singular behavior at the boundary exists

instability of strong regular reflection and counterexamples to the detachment criterion we consider a particular instance of reflection of shock waves in self similar compressible flow we prove that local self similar regular reflection rr cannot always be extended into a global flow therefore the detachment criterion is not universally correct more precisely consider the following angle condition the tangent of the strong type reflected shock meets the opposite wall at a sharp or right downstream side angle in cases where the condition is violated and the weak type reflected shock is transonic we show that global rr does not exist combined with earlier work we have shown that none of the classical criteria for rr rightarrow mr transition is universally correct a new criterion is proposed moreover we have shown that strong type rr is unstable in the sense that global rr cannot persist under perturbations to one side this yields a definite answer to the weak strong problem because earlier work shows stability of weak rr in the same sense

asymptotic inference for eigenvectors abstract asymptotic procedures are given for testing certain hypotheses concerning eigenvectors and for constructing confidence regions for eigenvectors these asymptotic procedures are derived under fairly general conditions on the estimates of the matrix whose eigenvectors are of interest applications of the general results to principal components analysis and canonical variate analysis are given author

optimal rank based testing for principal components this paper provides parametric and rank based optimal tests for eigenvectors and eigenvalues of covariance or scatter matrices in elliptical families the parametric tests extend the gaussian likelihood ratio tests of anderson 1963 and their pseudo gaussian robustifications by davis 1977 a nd tyler 1981 1983 the rank based tests address a much broader class of problems where covariance matrices need not exist and principal components are associated with more general scatter matrices the proposed tests are shown to outperform daily practice both from the point of view of validity as from the point of view of efficiency this is achieved by utilizing the le cam theory of locally asymptotically normal experiments in the nonstandard context however of a curved parametrization the results we derive for curved experiments are of independent interest and likely to apply in other contexts 1 introduction this fairly detailed introduction aims at providing a comprehensive and nontechnical overview of the paper including its asymptotic theory aspects and a rough description of some of the rank based test statistics to be derived it is expected to be accessible to a broad readership it should be sufficiently informative for the reader not interested in the technical aspects of asymptotic theory to proceed to sections 5 gaussian and pseudo gaussian tests and 6 rank based tests where the proposed testing procedures are described and for the reader mainly interested in asymptotics to decide whether he she is interested in the treatment of a lan family with curved parametrization developed in sections 3 and 4

on the asymptotics of maronna s robust pca the eigenvalue decomposition evd parameters of the second order statistics are ubiquitous in statistical analysis and signal processing notably the evd of the m estimators of the scatter matrix is a popular choice to perform robust probabilistic pca or other dimension reduction related applications towards the goal of characterizing this process this paper derives new asymptotics for the evd parameters i e eigenvalues eigenvectors and principal subspace of m estimators in the context of complex elliptically symmetric distributions first their gaussian asymptotic distribution is obtained by extending standard results on the sample covariance matrix in the gaussian context second their convergence towards the evd parameters of a gaussian core wishart equivalent is derived this second result represents the main contribution in the sense that it quantifies when it is acceptable to directly rely on well established results on the evd of wishart distributed matrix for characterizing the evd of m estimators finally some examples intrinsic bias analysis rank estimation and low rank adaptive filtering illustrate where the obtained results can be leveraged

on the instability problem of a 3 d transonic oblique shock wave abstract in this paper we are concerned with the instability problem of a 3 d transonic oblique shock wave for the steady supersonic flow past an infinitely long sharp wedge the flow is assumed to be isentropic and irrotational it was indicated on p 317 of 7 that if a steady supersonic flow comes from minus infinity and hits a sharp symmetric wedge then it follows from the rankine hugoniot conditions and the physical entropy condition that there possibly appears a weak shock or a strong shock attached at the edge of the sharp wedge which corresponds to a supersonic shock or a transonic shock respectively the question arises which of the two actually occurs it has frequently been stated that the strong one is unstable and that therefore only the weak one could occur however a convincing proof of this instability has apparently never been given the aim of this paper is to understand such a longstanding open question we will show that the attached 3 d transonic oblique shock problem is overdetermined with respect to the periodic perturbation which implies that the 3 d transonic shock is unstable in general

stability of transonic shock fronts in two dimensional euler systems we study the stability of stationary transonic shock fronts under two dimensional perturbation in gas dynamics the motion of the gas is described by the full euler system the system is hyperbolic ahead of the shock front and is a hyperbolic elliptic composed system behind the shock front the stability of the shock front and the downstream flow under two dimensional perturbation of the upstream flow can be reduced to a free boundary value problem of the hyperbolic elliptic composed system we develop a method to deal with boundary value problems for such systems the crucial point is to decompose the system to a canonical form in which the hyperbolic part and the elliptic part are only weakly coupled in their coefficients by several sophisticated iterative processes we establish the existence and uniqueness of the solution to the described free boundary value problem our result indicates the stability of the transonic shock front and the flow field behind the shock

spectral analysis for categorical time series scaling and the spectral envelope summary many studies produce categorical time series in which harmonic analysis is of interest although there exist time domain approaches for the analysis of categorical time series such as markov chains or link function based regression models there is apparently little statistical theory or methodology for analyzing qualitative valued time series in the frequency domain the purpose of this paper is to initiate the development of a general framework for the frequency domain analysis of categorical time series in doing so we discuss the scaling of categorical time series and introduce a new concept that we call the spectral envelope of a categorical time series we demonstrate our methodology on a data set from a problem in molecular biology

a numerical study of riemann problems for the two dimensional unsteady transonic small disturbance equation we study a two parameter family of riemann problems for the unsteady transonic small disturbance utsd equation also called the two dimensional burgers equation the two parameters a and b which define oblique shock initial data correspond to the slopes of the initial shock waves in the upper half plane for each a and b the three constant states in the upper half plane satisfy the rankine hugoniot conditions across the shocks this leads to a two parameter family of oblique shock interaction problems in this paper we present a numerical study of global solution behavior for the values of a and b in a previously obtained bifurcation diagram our study supplements the related theoretical results and conjectures recently obtained by s canic and b l keyfitz we employ a high resolution numerical method which reveals fine solution structures our findings confirm theoretical results and conjectures about the solution patterns and deepen the understanding of the structure of several intricate wave in

higher regularity for free boundary and tangential derivative problems and global solutions to regular reflections for the unsteady transonic small disturbance utsd equations abstract we establish c 2 estimates for solutions of a class of quasilinear elliptic equations with free boundary and tangential derivative boundary problems using this regularity result we show the existence of global solutions to regular shock reflections for the unsteady transonic small disturbance utsd equation we also present lipschitz estimates near the degenerate dirichlet boundary the sonic boundary for the utsd equation

transonic shocks in 3 d compressible flow passing a duct with a general section for euler systems this paper is devoted to the study of a transonic shock in three dimensional steady compressible flow passing a duct with a general section the flow is described by the steady full euler system which is purely hyperbolic in the supersonic region and is of elliptic hyperbolic type in the subsonic region the upstream flow at the entrance of the duct is a uniform supersonic one adding a three dimensional perturbation while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference the problem to determine the transonic shock and the flow behind the shock is reduced to a free boundary value problem of an elliptic hyperbolic system the new ingredients of our paper contain the decomposition of the elliptic hyperbolic system the determination of the shock front by a pair of partial differential equations coupled with the three dimensional euler system and the regularity analysis of solutions to the boundary value problems introduced in our discussion

sonic supersonic solutions for the two dimensional steady full euler equations this paper focuses on the structure of classical sonic supersonic solutions near sonic curves for the two dimensional full euler equations in gas dynamics in order to deal with the parabolic degeneracy near the sonic curve a novel set of dependent and independent variables are introduced to transform the euler equations into a new system of governing equations which displays a clear regularity singularity structure with the help of technical characteristic decompositions the existence of a local smooth solution for the new system is first established in a weighted metric space by using the iteration method and then expressed in terms of the original physical variables this is the first construction of a classical sonic supersonic solution near a sonic curve for the full euler equations

iss with respect to boundary disturbances for 1 d parabolic pdes due to unbounded input operators in partial differential equations pdes with boundary inputs there has been a long held intuition that input to state stability iss properties and finite gains cannot be established with respect to disturbances at the boundary this intuition has been reinforced by many unsuccessful attempts as well as by the success in establishing iss only with respect to the derivative of the disturbance contrary to this intuition we establish such a result for parabolic pdes our methodology does not rely on the transformation of the boundary disturbance to a distributed input and the stability analysis is performed in time varying subsets of the state space the obtained results are used for the comparison of the gain coefficients of transport pdes with respect to inlet disturbances and for the establishment of the iss property with respect to control actuator errors for parabolic systems under boundary feedback control

a linear approximation for the regular reflection of a weak shock at a wedge satisfying sonic condition abstract the problem of shock reflection by a wedge which the flow is dominated by the unsteady potential flow equation is a important problem in weak regular reflection the flow behind the reflected shock is immediately supersonic and becomes subsonic further downstream the reflected shock is transonic its position is a free boundary for the unsteady potential equation which is degenerate at the sonic line in self similar coordinates applying the special partial hodograph transformation used in zhouping xin huicheng yin transonic shock in a nozzle i 2 d case comm pure appl math 57 2004 1 51 zhouping xin huicheng yin transonic shock in a nozzle ii 3 d case ims preprint 2003 we derive a nonlinear degenerate elliptic equation with nonlinear boundary conditions in a piecewise smooth domain when the angle which between incident shock and wedge is small we can see that weak regular reflection as the disturbance of normal reflection as in shuxing chen linear approximation of shock reflection at a wedge with large angle comm partial differential equations 21 78 1996 1103 1118 by linearizing the resulted nonlinear equation and boundary conditions with above viewpoint we obtain a linear degenerate elliptic equation with mixed boundary conditions and a linear degenerate elliptic equation with oblique boundary conditions in a curved quadrilateral domain by means of elliptic regularization techniques delicate a priori estimate and compact arguments we show that the solution of linearized problem with oblique boundary conditions is smooth in the interior and lipschitz continuous up to the degenerate boundary

indirect control to stabilise reaction diffusion equation this paper considers stabilisation of a reaction diffusion process by indirect control the reaction diffusion equation is cascaded at one boundary with an ordinary control system by introducing s

a global subsonic solution to an interacting transonic shock for the self similar nonlinear wave equation abstract we present the existence of the subsonic solution to a two dimensional riemann problem governed by a self similar nonlinear wave equation where the boundary of the subsonic region consists of a transonic shock and the sonic circle thus the governing equation becomes a free boundary problem on the transonic shock and degenerates on the sonic circle by utilizing the barrier methods and iterative methods we show the well posedness of the transonic shock in the entire subsonic region and thus establish the global solution this result does not rely on any smallness of riemann data

the asymptotic inadmissibility of the spatial sign covariance matrix for elliptically symmetric distributions the asymptotic efficiency of the spatial sign covariance matrix relative to affine equivariant estimators of scatter is studied in particular the spatial sign covariance matrix is shown to be asymptotically inadmissible i e the asymptotic covariance matrix of the consistency corrected spatial sign covariance matrix is uniformly larger than that of its affine equivariant counterpart namely tyler s scatter matrix although the spatial sign covariance matrix has often been recommended when one is interested in principal components analysis its inefficiency is shown to be most severe in situations where principal components are of greatest interest simulation shows that the inefficiency of the spatial sign covariance matrix also holds for small sample sizes and that the asymptotic relative efficiency is a good approximation to the finite sample efficiency for relatively modest sample sizes

global transonic conic shock wave for the symmetrically perturbed supersonic flow past a cone in this paper we are concerned with the global existence and stability of a steady transonic conic shock wave for the symmetrically perturbed supersonic flow past an infinitely long conic body the flow is assumed to be polytropic isentropic and described by a steady potential equation theoretically as indicated in r courant k o friedrichs supersonic flow and shock waves interscience publishers inc new york 1948 it follows from the rankine hugoniot conditions and the entropy condition that there will appear a weak shock or a strong shock attached at the vertex of the sharp cone in terms of the different pressure states at infinity behind the shock surface which correspond to the supersonic shock and the transonic shock respectively in the references shuxing chen zhouping xin huicheng yin global shock wave for the supersonic flow past a perturbed cone comm math phys 228 2002 47 84 dacheng cui huicheng yin global conic shock wave for the steady supersonic flow past a cone polytropic case preprint 2006 dacheng cui huicheng yin global conic shock wave for the steady supersonic flow past a cone isothermal case pacific j math 233 2 2007 257 289 and zhouping xin huicheng yin global multidimensional shock wave for the steady supersonic flow past a three dimensional curved cone anal appl 4 2 2006 101 132 the authors have established the global existence and stability of a supersonic shock for the perturbed hypersonic incoming flow past a sharp cone when the pressure at infinity is appropriately smaller than that of the incoming flow at present for the supersonic symmetric incoming flow we will study the global transonic shock problem when the pressure at infinity is appropriately large

analysis of linear combinations with extreme ratios of variance abstract if the covariance matrices 1 and 2 of two multivariate populations are not identical insight into the differences between 1 and 2 can often be gained by analyzing the linear combinations with extreme ratios of variances that is those defined by the eigenvectors associated with the extreme roots of 1 1 2 this article gives a descriptive method similar to variable selection procedures in regression for screening and simplifying these linear combinations a hypothesis of redundancy of variables is defined and a statistic for testing this hypothesis is derived the method is illustrated by an example

fuzzy observer based control for nonlinear coupled hyperbolic pde ode systems in this work the fuzzy observer based control problem is investigated for a class of nonlinear coupled systems which consists of a hyperbolic partial differential equation pde containing nonlinearities and a nonlinear ordinary differential equation ode the nonlinear coupled system is represented as a takagi sugeno t s fuzzy coupled hyperbolic pde ode model based on the t s fuzzy model a novel lyapunov functional approach is proposed to design a fuzzy observer based control strategy more specifically a fuzzy observer is presented to estimate the state variables of the fuzzy coupled pde ode system with the measurements of the pde and the exponential convergence of the observer error is proved then a fuzzy controller is given utilizing the estimated states as feedback variables and it is proved that the evolution profiles of the pde and the trajectory of the ode in the closed loop fuzzy system converge exponentially to the desired values respectively the sufficient existence conditions of the fuzzy observer based controller are formulated in terms of a set of space differential linear matrix inequalities sdlmis a recursive algorithm based on the finite difference approximation and the linear matrix inequality techniques are provided to solve the sdlmis finally the results are applied to case study of a predator prey system and the simulations are performed to illustrate the effectiveness of the proposed observer based control law

a mixed boundary value problem for chaplygin s hodograph equation abstract in this paper we will prove existence uniqueness and regularity of a classical solution to a mixed boundary value problem for chaplygin s hodograph equation which is degenerate elliptic on a part of the boundary this problem is derived from the study of detached bow shock ahead of a straight ramp in uniform supersonic flows in the hodograph plane the proof depends on perron s method and some techniques from linear elliptic equations

from continuous time design to sampled data design of observers in this work a sampled data nonlinear observer is designed using a continuous time design coupled with an inter sample output predictor the proposed sampled data observer is a hybrid system it is shown that under certain conditions the robustness properties of the continuous time design are inherited by the sampled data design as long as the sampling period is not too large the approach is applied to linear systems and to triangular globally lipschitz systems

stability of reflection and refraction of shocks on interface abstract in this paper we study the stability of the nonlinear wave structure caused by the attack of an incident shock on an interface of two different kinds of media the attack will produce a reflected wave and a refracted wave and also let the interface deflected in this paper we will mainly study the case when the reflected wave is a shock and the flow between the reflected wave and the refracted shock is relatively subsonic our result indicates that the wave structure and the flow field for the reflection refraction problem in this case is conditionally stable to describe the motion of the fluid we use the inviscid euler system as the mathematical model the reflection refraction problem can be reduced to a free boundary value problem where the unknown reflected shock and refracted shock are free boundaries and the deflected interface is also to be determined in the proof of the existence and the stability of the corresponding wave structure we apply the lagrange transformation to fix the interface and the decoupling technique to decouple the elliptic hyperbolic composite system in its principal part meanwhile some efficient weighted sobolev estimates are established to derive the existence for corresponding nonlinear problems

small gain stability analysis of certain hyperbolic parabolic pde loops abstract this work provides stability results in the spatial sup norm for hyperbolic parabolic loops in one spatial dimension the results are obtained by an application of the small gain stability analysis two particular cases are selected for the study because they contain challenges typical of more general systems to which the results are easily generalizable but at the expense of less pedagogical clarity and more notational clutter i the feedback interconnection of a parabolic pde with a first order zero speed hyperbolic pde with boundary disturbances and ii the feedback interconnection by means of a combination of boundary and in domain terms of a parabolic pde with a first order hyperbolic pde the first case arises in the study of the movement of chemicals underground and includes the wave equation with kelvin voigt damping as a subcase the second case arises when we apply backstepping to a pair of hyperbolic pdes that is obtained by ignoring diffusion phenomena moreover the second case arises in the study of parabolic pdes with distributed delays in the first case we provide sufficient conditions for iss in the spatial sup norm with respect to boundary disturbances in the second case we provide delay independent sufficient conditions for exponential stability in the spatial sup norm

transonic shocks in compressible flow passing a duct for three dimensional euler systems in this paper we study the transonic shock in steady compressible flow passing a duct the flow is a given supersonic one at the entrance of the duct and becomes subsonic across a shock front which passes through a given point on the wall of the duct the flow is governed by the three dimensional steady full euler system which is purely hyperbolic ahead of the shock and is of elliptic hyperbolic composed type behind the shock the upstream flow is a uniform supersonic one with the addition of a three dimensional perturbation while the pressure of the downstream flow at the exit of the duct is assigned apart from a constant difference the problem of determining the transonic shock and the flow behind the shock is reduced to a free boundary value problem in order to solve the free boundary problem of the elliptic hyperbolic system one crucial point is to decompose the whole system to a canonical form in which the elliptic part and the hyperbolic part are separated at the level of the principal part due to the complexity of the characteristic varieties for the three dimensional euler system the calculus of symbols is employed to complete the decomposition the new ingredient of our analysis also contains the process of determining the shock front governed by a pair of partial differential equations which are coupled with the three dimensional euler system