a singular parabolic equation with logarithmic nonlinearity and l p initial data abstract we study a parabolic equation with logarithmic nonlinearity and homogeneous dirichlet condition in a smooth bounded domain we firstly prove a result about existence of positive solutions with initial data in l p spaces 1 p in order to overcome the singular logarithmic nonlinearity among other arguments we employ hardy s inequality after the life span of the obtained solutions is studied in particular we show a connection between global existence and solutions of the associated elliptic problem

on the generalized bi skew symmetric solutions of a linear matrix equation and its procrust problems in this paper the solvability conditions and the explicit expressions of the generalized bisymmetric and bi skew symmetric solutions of the matrix equation ax b are respectively established by applying two methods then the maximal and minimal ranks of the solutions are derived if the solvability conditions are not satisfied the generalized bisymmetric and bi skew symmetric least squares solutions of the matrix equation are considered and the generalized bisymmetric and bi skew symmetric least squares solutions with the minimum norm are also obtained in addition two algorithms are provided to compute the generalized bi skew symmetric least squares solution and some examples are given to illustrate that the algorithms are feasible

direct transcription solution of high index optimal control problems and regular euler lagrange equations direct transcription methods are used to solve optimal control problems in many industrial settings models for physical systems often take the form of differential algebraic equations daes the index of the dae traditionally is viewed as an important factor in deciding whether a particular numerical approach should be used recently it has been observed that what the user thinks is the index of the dae may not be the same as the index available to the optimization software an investigation of this fact is underway in order to develop guidelines to assist users of various numerical optimal control packages this paper develops some theoretical results that will be needed for this development

matrix pade approximation definitions and properties abstract some proposals are made to give a general definition of matrix pade approximants depending on the normalization of the denominator we define type i constant term is the unit matrix and type ii by conditions on the leading coefficient approximants existence and uniqueness are considered determinant expressions are given and relations among types i ii and left right approximants are considered

impulsive mode elimination for descriptor systems by a structured p d feedback the problem of impulsive mode elimination for descriptor systems via a structured output proportional and derivative p d feedback is investigated in this technical note the motivation of solving this problem arises from an observation that the impulse behavior for some descriptor systems can not be eliminated completely by a parameterized output p d feedback but it can be removed by a structured p d feedback elegantly designed in order to explain this phenomenon explicitly a concept of the structured p d feedback is first introduced then an explicit necessary and sufficient condition is constructively derived for the closed loop systems to be regular and impulse free by the structured p d feedback the main result is represented in terms of the original system parameters instead of decomposed subsystem parameters finally an example is provided to illustrate the effectiveness of the proposed result

regularization of descriptor systems by output feedback conditions are given under which a descriptor or generalized state space system can be regularized by output feedback it is shown that under these conditions proportional and derivative output feedback controls can be constructed such that the closed loop system is regular and has index at most one this property ensures the solvability of the resulting system of dynamic algebraic equations a reduced form is given that allows the system properties as well as the feedback to be determined the construction procedures used to establish the theory are based only on orthogonal matrix decompositions and can therefore be implemented in a numerically stable way

technical communique on impulsive algebraic multiplicities of linear time invariant singular systems under feedback impulsive algebraic multiplicities of linear time invariant singular systems under output feedback or decentralized output feedback are characterized in terms of system matrices directly the multiplicities describe the number of impulsive modes that can be eliminated and indicate the number of finite poles that can be assigned by the respective feedback these characterizations also lead to new regularizability conditions for singular systems

positive definite solutions of the nonlinear matrix equation x a h x 1 a i this paper is concerned with the positive definite solutions to the nonlinear matrix equation x a email protected 1a i where x is the unknown and a is a given complex matrix by introducing and studying a matrix operator on complex matrices it is shown that the existence of positive definite solutions of this class of nonlinear matrix equations is equivalent to the existence of positive definite solutions of the nonlinear matrix equation w b tw 1b i which has been extensively studied in the literature where b is a real matrix and is uniquely determined by a it is also shown that if the considered nonlinear matrix equation has a positive definite solution then it has the maximal and minimal solutions bounds of the positive definite solutions are also established in terms of matrix a finally some sufficient conditions and necessary conditions for the existence of positive definite solutions of the equations are also proposed

exact number of positive solutions for semipositone problems this paper is concerned with the exact number of positive solutions for the boundary value problem y p 2y f y 0 and y 1 y 1 0 where p 1 and 0 is a positive parameter we consider the case in which both f u and g u p 1 f u uf u change sign exactly once from negative to positive on 0

brief paper the open loop linear quadratic differential game for index one descriptor systems in this paper we consider the linear quadratic differential game for descriptor systems that have index one we derive both necessary and sufficient conditions for existence of an open loop nash equilibrium

a system of matrix equations with five variables in this paper we give some necessary and sufficient conditions for the consistence of the system of quaternion matrix equations a 1 x c 1 y b 1 d 1 a 2 w c 2 z b 2 d 2 a 3 v c 3 v b 3 c 4 a 4 v b 4 c 5 a 5 x y b 5 c 6 w z d 6 e 6 v f 6 g 6 and constitute an expression of the general solution to the system when it is solvable the outcomes of this paper encompass some recognized results in the collected works in addition we establish an algorithm and a numerical example to illustrate the theory constructed in the paper

a simultaneous decomposition for seven matrices with applications abstract let h m n be the set of all m n matrices over the quaternion algebra h in this paper we construct a simultaneous decomposition for seven matrices with compatible sizes a h m n b h m p 1 c h m p 2 d h m p 3 e h q 1 n f h q 2 n and g h q 3 n as applications of the simultaneous matrix decomposition we give some solvability conditions general solutions as well as the range of ranks of the general solutions to the following two generalized sylvester matrix equations b x e c y f d z g a and b x w e c y f d z g a where a b c d e f and g are given quaternion matrices moreover we provide some numerical examples to illustrate the results of this paper

on quenching with logarithmic singularity in this paper we study the quenching problem for the reaction diffusion equation ut uxx f u with cauchy dirichlet data in the case where we have only a logarithmic singularity i e f u ln u 0 1 we show that for sufficiently large domains of x quenching occurs and that under certain assumptions on the initial function the set of quenching points is finite the main result of this paper concerns the asymptotic behavior of the solution in a neighborhood of a quenching point this result gives the quenching rate for the problem we also obtain new blow up results for the equation vt vxx vev vx2 these concern the occurrence of blow up the blow up set and the asymptotics in a neighborhood of a blow up point the analysis is based on the equivalence between the quenching and the blow up for these two equations

feedback controller norm optimization for linear time invariant descriptor systems with pole region constraint an algorithm is proposed to compute a state feedback gain matrix for a linear time invariant regular descriptor system which ensures that i the closed loop system is impulse free and ii all the finite poles are placed within a pre defined stability region in the complex plane the associated design flexibilities are exploited in minimizing upper bounds of the frobenius norm of respective gain matrices a class of linear matrix inequality lmi regions in the complex plane are chosen as constraints for the closed loop finite poles by representing a subset of the nonsingular matrices through solution of an lmi and linearizing the set of matrix inequalities arise in the regional pole finite poles assignment the associated optimizations are formulated as semidefinite programs the effectiveness of the developed algorithm is demonstrated through numerical examples significant reduction in the norm of feedback gain matrix is achieved in a two generator six bus power system

inclusion principle for descriptor systems the purpose of this paper is to propose an expansion contraction framework for linear constant descriptor systems within the inclusion principle for dynamic systems our primary objective is to provide an explicit characterization of the expansion process whereby a given descriptor system is expanded into the larger space where all its solutions are reproducible by the expanded descriptor system if appropriate initial conditions are selected when a control law is formulated in the expanded space the proposed characterizations provide contractibility conditions for implementation of the control law in the original system a full freedom is provided for selecting appropriate matrices in the proposed expansion contraction control scheme in particular the derived theoretical framework serves as a flexible environment for expansion contraction control design of descriptor systems under overlapping information structure constraints

a tabular methodology to identify minimal row degrees for matrix pade approximants in this paper we propose a tabular procedure for identifying sets of minimal row degrees corresponding to matrix pade approximants the objective is to make it easier to interpret and properly apply them in various fields when considering formal matrix power series with kxm coefficients it is useful to examine the linearly dependent rows that are in the last k rows in certain hankel matrices corresponding to the coefficients of the series a list of properties and suitable examples are added not only by way of illustration but also because they are an original part of the method having been chosen to teach the procedure and to underscore certain precautions to be taken

minimum norm regularization of descriptor systems by mixed output feedback we study the regularization problem for linear constant coefficient descriptor systems ex ax bu y1 cx y2 x by proportional and derivative mixed output feedback necessary and sufficient conditions are given which guarantee that there exist output feedbacks such that the closed loop system is regular has index at most one and e bg has a desired rank i e there is a desired number of differential and algebraic equations to resolve the freedom in the choice of the feedback matrices we then discuss how to obtain the desired regularizing feedback of minimum norm and show that this approach leads to useful results in the sense of robustness only if the rank of e is decreased numerical procedures are derived to construct the desired feedback gains these numerical procedures are based on orthogonal matrix transformations which can be implemented in a numerically stable way

a three solution theorem for singular nonlinear elliptic boundary value problems abstract we establish a three solution theorem for singular elliptic boundary value problems of the form u f u u in u 0 on where is a bounded domain in r n n 1 with a smooth boundary here f 0 0 is a c 1 function in 0 with f 0 0 and 0 1 in particular if there exist two pairs of sub supersolutions 1 1 and 2 2 where 1 2 1 1 2 1 with 2 2 and 2 2 are strict sub and supersolutions then we establish existence of three solutions u 1 u 2 and u 3 for the above boundary value problem such that u 1 1 2 u 2 2 1 and u 3 1 1 1 2 2 1 our results extend the work in 1 and also in 17 where such multiplicity results were discussed in the non singular case 0 further our results strengthen the multiplicity results in 12 and 13 for such singular problems 0 namely establish an additional third solution we also establish a sub super solution theorem for infinite semipositone problems i e the case when f 0 0

on a singular nonlinear semilinear elliptic problem where k x c2 b v9 a p 0 1 and l is a real parameter such singular elliptic problems arise in the contexts of chemical heterogeneous catalysts nonnewtonian fluids and also the theory of heat conduction in electrically conducting materials see 3 5 8 9 for a detailed discussion obviously 1 1 cannot have a solution u c2 v9 if k x is not vanishing near v however under various appropriate assumptions on k x we will obtain classical solutions of 1 1 for l belonging to a certain range and we will also obtain some uniqueness criteria here a classical solution is a solution u of 1 1 which belongs to c2 v mc v9 with u 0 in v we also study the boundary behaviour of solutions of 1 1 and we will show that the solution u of 1 1 lies in a certain holder class the special case when k x is negative and l 0 has been studied by several authors the existence and uniqueness of the solution were established by crandall rabinowitz and tartar 6 del pino 7 gomes 10 and lazer and mckenna

technical communique regularizability of linear time invariant descriptor systems under decentralized control regularizability conditions for linear time invariant descriptor systems under decentralized output feedback are presented in this paper the regularizability is characterized by rank conditions that involve the original system matrices only new rank conditions for regularity of the system are also derived

branches of positive and free boundary solutions for some singular quasilinear elliptic problems we study the existence and multiplicity of solutions strictly positive or nonnegative having a free boundary the boundary of the set where the solution vanishes of some one dimensional quasilinear problems of eigenvalue type with possibly singular nonlinear terms

the p q skew symmetric extremal rank solutions to a system of quaternion matrix equations abstract let h m n denote the set of all m n matrices over the quaternion algebra h and p h m m q h n n be involutions we say that a h m n is p q symmetric or p q skewsymmetric if a paq or a paq we in this paper mainly investigate the p q skew symmetric maximal and minimal rank solutions to the system of quaternion matrix equations ax b xc d we present necessary and sufficient conditions for the existence of the maximal and minimal rank solutions with p q symmetry and p q skewsymmetry to the system the expressions of such solutions to this system are also given when the solvability conditions are satisfied a numerical example is presented to illustrate our results the findings of this paper extend some known results in this literature

regularity of an elliptic problem with a singular nonlinearity we study a semilinear boundary value problem with the feature that the vanishing boundary value makes the equation singular we prove that the positive solution is in general holder continuous up to the boundary and has even better regularity in some special cases

solutions to optimization problems on ranks and inertias of a matrix function with applications abstract consider the hermitian matrix function f x a 3 b 3 x b 3 x subject to a consistent system of matrix equations 0 1 a 1 x c 1 a 2 xb 2 c 2 where means conjugate transpose in this paper we first establish explicit expansion formulas to calculate the global maximal and minimal ranks and inertias of the hermitian matrix function f x then we use the derived formulas to give necessary and sufficient conditions for system 0 1 to have re nonnegative definite re nonpositive definite re positive definite and re negative definite solutions moreover as another application of the derived formulas we establish necessary and sufficient conditions for the solvability to the system of matrix equations 0 2 a 1 x c 1 a 2 xb 2 c 2 b 3 x b 3 x a 3 and provide an expression of the general solution to 0 2 when it is solvable the findings of this paper widely extend the known results in the literature

sublinear singular elliptic problems with two parameters abstract we establish several existence and nonexistence results for the boundary value problem u k x g u f x u h x in u 0 on where is a smooth bounded domain in r n and are positive parameters h is a positive function while f has a sublinear growth the main feature of this paper is that the nonlinearity g is assumed to be unbounded around the origin our analysis shows the importance of the role played by the decay rate of g combined with the signs of the extremal values of the potential k x on the proofs are based on various techniques related to the maximum principle for elliptic equations

a neumann problem with logarithmic nonlinearity in a ball abstract the nonlinear elliptic problem u log u h x u p in a ball b r r n with neumann boundary conditions u 0 on b r were studied by means of a double perturbation argument we obtain a positive radial solution u c 2 b r 0 c b r

a type of matrix pade approximant inspired by scalar component models in this paper we define a type of matrix pade approximant inspired by the identification stage of multivariate time series models considering scalar component models of course the formalization of certain properties in the matrix pade approximation framework can be applied to time series models and in other fields specifically we want to study matrix pade approximants as follows to find rational representations or rational approximations of a matrix formal power series with both matrix polynomials numerator and denominator satisfying three conditions a minimum row degrees for the numerator and denominator b an invertible denominator at the origin and c canonical representation without free parameters

minimax methods for singular elliptic equations with an application to a jumping problem abstract a jumping problem for a class of singular semilinear elliptic equations is considered minimax methods in the framework of nonsmooth critical point theory are applied

robust pole assignment in linear state feedback numerical methods are described for determining robust or well conditioned solutions to the problem of pole assignment by state feedback the solutions obtained are such that the sensitivity of the assigned poles to perturbations in the system and gain matrices is minimized it is shown that for these solutions upper bounds on the norm of the feedback matrix and on the transient response are also minimized and a lower bound on the stability margin is maximized a measure is derived which indicates the optimal conditioning that may be expected for a particular system with a given set of closed loop poles and hence the suitability of the given poles for assignment

the existence and multiplicity of classical positive solutions for a singular nonlinear elliptic problem with any growth exponents a b s t r a c t in this paper we consider a class of singular elliptic boundary problems u u p x u u 0 in u 0 on with 1 0 1 and p c 0 0 1 we emphasize that there is no restriction on the boundedness of from above by means of the approximating variational arguments and truncation two classical positive solutions are obtained 2014 the authors published by elsevier ltd this is an open access article under the cc by nc nd license http creativecommons org licenses by nc nd 3 0

the zero set of a solution of a parabolic equation on etudie l ensemble nul d une solution u t x de l equation u t a x t u xx b x t u x c x t u sous des hypotheses tres generales sur les coefficients a b et c

existence and regularity to an elliptic equation with logarithmic nonlinearity abstract we study the nonlinear elliptic problem u u 0 log u f x u in r n with u 0 on the function f 0 0 is nondecreasing sublinear and f u is continuous for every 0 we obtain a maximal solution u 0 and prove its global regularity c 1 there is a constant such that u vanishes on a set of positive measure for 0 and u 0 for if f is concave for we characterize u by its stability

an exact estimate result for a class of singular equations with critical exponents abstract we consider the singular boundary value problem u h x u u p 1 in u 0 on with p 2 n n 2 0 1 it is well known that there exists 0 such that the problem has a solution for all 0 and no solution for we obtain an exact result for p h

on dirichlet problems with singular nonlinearity of indefinite sign abstract let be a smooth bounded domain in r n n 1 let k m be two nonnegative functions and let 0 we study existence and nonexistence of positive solutions for singular problems of the form u k x u m x u in u 0 on where 0 is a real parameter we mention that as a particular case our results apply to problems of the form u m x u in u 0 on where m is allowed to change sign in

local behavior near blow up points for semilinear parabolic equations abstract this paper is concerned with solutions of u t u xx u p p 1 defined in a cylinder qr x t r 0 which blowup at t 1 more precisely we are interested in the asymptotic behaviour of such solutions near a blow up point as t 1 we show that whenever the blow up set can be confined to the interior of the interval r r asymptotics are the same as those corresponding to solutions defined in the whole line which assume bounded initial values however if the blow up set reaches the boundary of the interval r r a different behaviour may appear as indicated by theorem 2 in the introduction

stability analysis of discrete singular fuzzy systems abstract this work mainly develops a discrete singular fuzzy model and discusses its stability the standard discrete t s model is generalized into the so called discrete singular takagi sugeno dst s model which can represent a larger class of nonlinear systems and the stability criteria of this system are then investigated these criteria are described by non strict linear matrix inequalities and we thus involve the projection method for ensuring stability based on some derived projection operators a projection algorithm is proposed for solving a feasible solution an illustrating example demonstrates the validity and effectiveness of the proposed method

on the non normal pade table in a non commutative algebra abstract in the non commutative algebra the blocks in the table of orthogonal polynomials and therefore in the pade table are not square and generally it is impossible to say anything on the structure of these blocks except for infinite blocks this last case is extensively studied here for the non normal pade table the non normal table p of orthogonal polynomials and the non normal table some examples of illustration of different situations are given

on the quenching behavior of the solution of a semilinear parabolic equation where 0 e 0 0 6 u0 0 there exists an e b 0 such that u quenches at finite time t if e e b we say that a is a quenching point for u if there exists a sequence x t with x a and t r t such that u x t 1 as n co throughout this paper we assume that u quenches at finite time t and that u0 is smooth and satisfies