positive radial solutions and non radial bifurcation for semilinear elliptic equations in annular domains abstract we discuss the existence and multiplicity of positive radial solutions and the non radial bifurcation of u f u 0 in and u 0 on where is an annular domain of r n n 2 we prove that if f u 0 for u 0 and lim u f u u then there exists 0 such that there are at least two positive radial solutions for each 0 at least one for and none for if f 0 0 lim u 0 f u u 1 and uf u 1 e f u for u 0 e 0 then there exists a variational solution for 0 1 where 1 is the least eigenvalue of if f 0 0 lim u 0 f u u 0 and lim u f u u then there exists at least one positive radial solution for any 0 we obtain some precise multiplicity results for narrow annulus and show that the non radial bifurcation occurs if the growth of f u is rapid enough as u

frequency weighted spl lscr sub spl infin norm and optimal hankel norm model reduction a new relative error model reduction method is proposed using frequency weighted balanced realization and explicit spl lscr sub spl infin norm error bounds are also derived for the relative error and multiplicative error the method only needs to solve two lyapunov equations it is further shown that this method is equivalent to the balanced stochastic truncation if the plant is square and minimum phase this paper also gives a complete solution to the frequency weighted hankel norm approximation with antistable weighting these results are then applied to spl lscr sub spl infin norm model reduction and several numerically effective algorithms are proposed it is shown through many numerical examples that these algorithms work very well and in many cases produce almost optimal solutions

uniqueness and nonuniqueness for positive radial solutions of δu f u r 0 on considere le probleme de l unicite pour les solutions positives du probleme de dirichlet non lineaire u f u x 0 dans u 0 est une boule ou un anneau de r n et f 0 est superlineaire en u et satisfaisait f 0 x 0

on the stability of projections of balanced realizations abstract this paper generalizes a stability property concerning the state matrix of a balanced realization established by pernebo and silverman it is shown that stability is preserved under a general projection of the state matrix provided that the hankel singular values of the realization are distinct a necessary and sufficient condition for the case with nondistinct hankel singular values is also established

convergent algorithms for frequency weighted l 2 model reduction abstract this paper is concerned with computing an l 2 optimal reduced order model for a given stable multivariable linear system in the presence of input and output frequency weightings by parametrizing a class of reduced order models in terms of an orthogonal projection and using manifold techniques as tools both continuous and iterative algorithms are derived and their convergence properties are established as an application we show that an l 2 optimal reduced order filter in the closed loop sense can be computed using these algorithms

l sub 2 optimal filter reduction a closed loop approach this paper is concerned with the problem of order reduction of a full order kalman filter for a stable linear signal model so that the steady state filtering error variance associated with the reduced order filter is minimized by an orthogonal parameterization the above problem is formulated to minimize the filtering error variance over a set of orthogonal matrices both continuous and iterative algorithms are derived to compute an optimal reduced order filter the algorithms are shown to possess good properties including the desirable convergence property the proposed algorithms are simple and effective numerical examples are presented to demonstrate the effectiveness and the significant advantages of the proposed algorithms over the existing open loop methods such as the well known balanced truncation method

switched scalar quantizers for hidden markov sources an algorithm for designing switched scalar quantizers for hidden markov sources is described the design problem is cast as a nonlinear optimization problem the optimization variables are the thresholds and reproduction levels for each quantizer and the parameters defining the next quantizer map the cost function is the average distribution incurred by the system in steady state the next quantizer map is treated as a stochastic map so that all of the optimization variables are continuous valued allowing the use of a gradient based optimization procedure this approach solves a major problem in the design of switched scalar quantizing systems namely that of determining an optimal next quantizer decision rule details are given for computing the cost function and its gradient for weighted squared error distortion simulation results which compare the new system to current systems show that the present system performs better it is observed that the optimal system can in fact have a next quantizer map with stochastic components

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

balancing and model reduction for second order form linear systems model reduction of second order form linear systems is considered where a second order form reduced model is desired the focus is on reduction methods that employ or mimic moore s balance and truncate 1981 first we examine second order form model reduction by conversion to first order form and obtain a complete solution for this problem then new gramians and input output i o invariants for second order systems are motivated and defined based on these direct second order balancing methods are developed this leads naturally to direct second order form analogs for the well known first order form balance and truncate model reduction method explicit algorithms are given throughout the paper

component cost analysis of large scale systems abstract a brief summary of component cost analysis cca is given cca provides a means to determine the contribution of each dynamical element of the system in the overall performance metric using cca a theory of cost equivalent realizations is developed cost equivalent realizations are realizations that have the same value of the performance metric as the original system and they can be of lower order than kalman s minimal realization

principal component analysis in linear systems controllability observability and model reduction kalman s minimal realization theory involves geometric objects controllable unobservable subspaces which are subject to structural instability specifically arbitrarily small perturbations in a model may cause a change in the dimensions of the associated subspaces this situation is manifested in computational difficulties which arise in attempts to apply textbook algorithms for computing a minimal realization structural instability associated with geometric theories is not unique to control it arises in the theory of linear equations as well in this setting the computational problems have been studied for decades and excellent tools have been developed for coping with the situation one of the main goals of this paper is to call attention to principal component analysis hotelling 1933 and an algorithm golub and reinsch 1970 for computing the singular value decompositon of a matrix together they form a powerful tool for coping with structural instability in dynamic systems as developed in this paper principal component analysis is a technique for analyzing signals singular value decomposition provides the computational machinery for this reason kalman s minimal realization theory is recast in terms of responses to injected signals application of the signal analysis to controllability and observability leads to a coordinate system in which the internally balanced model has special properties for asymptotically stable systems this yields working approximations of x c x bar o the controllable and unobservable subspaces it is proposed that a natural first step in model reduction is to apply the mechanics of minimal realization using these working subspaces

the optimal projection equations for model reduction and the relationships among the methods of wilson skelton and moore first order necessary conditions for quadratically optimal reduced order modeling of linear time invariant systems are derived in the form of a pair of modified lyapunov equations coupled by an oblique projection which determines the optimal reduced order model this form of the necessary conditions considerably simplifies previous results of wilson 1 and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of moore 2 the possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation type algorithm is proposed for computing these local extrema a component cost analysis of the model error criterion similar to the approach of skelton 3 is utilized at each iteration to direct the algorithm to the global minimum

an approximate approach to h sup 2 optimal model reduction deals with the problem of computing an h sub 2 optimal reduced order model for a given stable multivariable linear system by way of orthogonal projection the problem is formulated as that of minimizing the h sub 2 model reduction cost over the stiefel manifold so that the stability constraint on reduced order models is automatically satisfied and thus totally avoided in the new problem formulation the closed form expression for the gradient of the cost over the manifold is derived from which a gradient flow results as an ordinary differential equation ode a number of nice properties about such a flow are established furthermore two explicit iterative convergent algorithms are developed from the flow one has a constant step size and the other has a varying step size and is much more efficient both of them inherit the properties that the iterates remain on the manifold starting from any orthogonal initial point and that the model reduction cost is decreasing to minima along the iterates a procedure for closing the gap between the original and modified problem is proposed in the symmetric case the two problems are shown to be equivalent numerical examples are presented to illustrate the effectiveness of the proposed algorithms as well as convergence

the global solution set for a class of semilinear problems we use bifurcation theory to study positive negative and sign changing solutions for several classes of boundary value problems depending on a real parameter we show the existence of infinitely many points of pitchfork bifurcation and study global properties of the solution curves

realisation and approximation of linear infinite dimensional systems with error bounds the class of linear infinite dimensional systems with finite dimensional inputs and outputs whose impulse response h satisfies h in l 1 cap l 2 0 infty mathbb c p times m and induces a nuclear hankel operator is said to be of nuclear type for this class of systems it is shown that balanced or output normal realisations always exist and their truncations converge to the original system in various topologies furthermore explicit l infty bounds on the transfer function errors l 1 and l 2 bounds on the impulse response errors and hilbert schmidt and nuclear bounds on the hankel operator errors are obtained these truncations also generate an approximating sequence to the optimal hankel norm approximations to the original system and various error bounds of these approximants are deduced

exact multiplicity results for a class of boundary value problems with cubic nonlinearities abstract we give an exact multiplicity result for a class of boundary value problems for cubic nonlinearities with an explicit x dependence moreover we provide a detailed analysis of global solution curves

model reduction via balanced state space representations a model reduction procedure based on balanced state space representations is studied in this paper the reduced order model is examined from the point of view of stability controllability and observability both continuous time and discrete time systems are considered

exact multiplicity results for boundary value problems with nonlinearities generalizing cubic using techniques of bifurcation theory we present two exact multiplicity results for boundary value problems of the type the first result concerns the case when the nonlinearity is independent of x and behaves like a cubic in u the second one deals with a class of nonlinearities with explicit x dependence

exact multiplicity and ordering properties of positive solutions of a p laplacian dirichlet problem and their applications we study the exact multiplicity and ordering properties of positive solutions of the p laplacian dirichlet problem

balanced gains and their significance for l 2 model reduction the singular values do not give enough information for model reduction in the l2sense other invariants are introduced which help describe the contribution of each state to the l2magnitude of the impulse response this leads to another suboptimal balanced truncation criterion

identification and rational l 2 approximation a gradient algorithm this paper deals with the identification of linear constant dynamical systems when formalized as a rational approximation problem the criterion is the l2 norm of the transfer function which is of interest in a stochastic context the problem can be expressed as nonlinear optimization in a hilbert space but standard algorithms are usually not well adapted we present a generic recursive procedure to find a local optimum of the criterion in the case of scalar systems our methods are borrowed from differential theory mixed with a bit of classical complex analysis to our knowledge the algorithm described in this paper is the first that ensures convergence to a local minimum