an error estimate for changing the stefan problem it is shown that the solution of a particular nonlinear parabolic pde approaches in l2 norm the solution of the stefan problem as a parameter goes to zero since the solution to the new problem is smoother than the solution to the stefan problem it is much easier to approximate numerically this result thus has implications for computational work 1 introduction this paper concerns the solution of a nonlinear parabolic equa tion and its relation to the solution of the stefan problem the classic free boundary problem of heat flow in particular we show that the solution of the parabolic equation is close in l2 norm to the solution of the stefan problem the precise error depending on a parameter of the parabolic equation since the parabolic equation yields a somewhat smoother solution it is very attractive for numerical work our results here show that it can be used in place of the stefan problem when doing computations with only a small error being introduced to define our notation let ql c r n 1 be our spatial domain with boundary f we assume that f is of class c2 although this restriction can be relaxed somewhat qt fl x 0 t will be the space time cylinder corresponding to ql with 0 um 3 h u j u j ua u 0 is the latent heat offusion and um is the melting temperature the function h u is called the enthalpy and represents the thermal energy of the material

numerical solution of projected algebraic riccati equations we consider the numerical solution of projected algebraic riccati equations using newton s method such equations arise for instance in model reduction of descriptor systems based on positive real and bounded real balanced truncation we also discuss the computation of low rank cholesky factors of the solutions of projected riccati equations numerical examples are given that demonstrate the properties of the proposed algorithms

analytic solutions for a stefan problem with gibbs thomson correction we provide existence of a unique smooth solution for a class of oneand two phase stefan problems with gibbs thomson correction in arbitrary space dimensions in addition it is shown that the moving interface depends analytically on the temporal and spatial variables of crucial importance for the analysis is the property of maximal lpregularity for the linearized problem which is fully developed in this paper as well

cal l _ infty norm computation for continuous time descriptor systems using structured matrix pencils in this technical note we discuss an algorithm for the computation of the l norm of transfer functions related to descriptor systems we show how one can achieve this goal by computing the eigenvalues of certain skew hamiltonian hamiltonian matrix pencils and analyze arising problems we also formulate and prove a theoretical result which serves as a basis for testing a transfer function matrix for properness finally we illustrate our results using a descriptor system related to mechanical engineering

existence of multidimensional traveling waves in the transport of reactive solutes through periodic porous media we prove the existence of multidimensional traveling wave solutions to the scalar equation for the transport of solutes contaminants with nonlinear adsorption and spatially periodic convection diffusion adsorption coefficients under the assumption that the nonlinear adsorption function satisfies the lax and oleinik entropy conditions in the nondegenerate case we also prove the uniqueness of the traveling waves these traveling waves are analogues of viscous shock profiles they propagate with effective speeds that depend on the periodic porous media only up to their mean states and are given by an averaged rankine hugoniot relation this is a direct consequence of the fact that the transport equation is in conservation form we use the sliding domain method the continuation method spectral theory maximum principles and a priori estimates in the degenerate case the traveling waves are weak solutions of a degenerate parabolic equation and are only holder continuous we obtain them by taking suitable limits on the non degenerate traveling waves the uniqueness of the degenerate traveling waves is open

orthogonality conditions and asymptotic stability in the stefan problem with surface tension we prove nonlinear asymptotic stability of steady spheres in the two phase stefan problem with surface tension our method relies on the introduction of appropriate orthogonality conditions in conjunction with a high order energy method

discounted cost linear quadratic gaussian control for descriptor systems we consider the linear quadratic gaussian control problem with a discounted cost functional for descriptor systems on the infinite time horizon based on recent results from the deterministic frame

on the generalized algebraic riccati equation for continuous time descriptor systems abstract in this paper we consider the generalized algebraic riccati equation gare for a continuous time descriptor system necessary and sufficient conditions for the existence of stabilizing solutions of the gare are derived based on the hamiltonian matrix pencil approach a parametrization of all stabilizing solutions is also provided the main result has a potential applicability to a wide class of control problems for a descriptor system including h 2 h controls

the robustness properties of the linear quadratic regulators for singular systems the lqr linear quadratic regulator robustness properties of singular systems are investigated it is shown that the optimal regulators for singular systems with an appropriate selection of the feedback gain have the same robustness properties as those in regular systems namely at least 60 degrees guaranteed phase margin infinite gain margin and 50 gain reduction margin

formulas for calculating the extremum ranks and inertias of a four term quadratic matrix valued function and their applications abstract this paper studies the quadratic matrix valued function x dxax d dxb b x d c through some expansion formulas for ranks and inertias of hermitian matrices where a b c and d are given complex matrices with a and c hermitian x is a variable matrix and denotes the conjugate transpose of a complex matrix we first introduce an algebraic linearization method for studying this matrix valued function and establish a group of explicit formulas for calculating the global maximum and minimum ranks and inertias of this matrix valued function with respect to the variable matrix x we then use these rank and inertia formulas to derive i necessary and sufficient conditions for the matrix equation x 0 to have a solution as well as the four matrix inequalities x 0 in the lowner partial ordering to be feasible respectively ii necessary and sufficient conditions for the four matrix inequalities x 0 in the lowner partial ordering to hold for all matrices x respectively iii the two matrices x and x such that the inequalities x x and x x hold for all matrices x in the lowner partial ordering respectively an application of the quadratic matrix valued function in control theory is also presented

continuity of the temperature in the two phase stefan problem the classical two phase stefan problem a model for ice water melting gives rise to a singular nonlinear partial differential equation which admits a unique weak solution here we prove that this solution and therefore the temperature in the stefan problem are continuous

interior and boundary continuity of the solution of the singular equation β u t lu abstract we extend some results of dibenedetto and vespri arch rational mech anal 132 3 1995 247 proving the interior and boundary continuity of bounded solutions of the singular equation u t l u where l is a second order elliptic operator with bounded measurable coefficients that depend both on space and time in a proper way

concerning the regularity of the anisotropic porous medium equation we show that a locally bounded nonnegative weak solution of the anisotropic porous media equation is locally continuous the proof is based on dibenedetto s technique called intrinsic scaling by choosing an appropriate geometry one can deduce energy and logarithmic estimates from which one can implement an iterative method to obtain the regularity result

the semi stabilizing solution of generalized algebraic riccati equation for descriptor systems this paper considers the generalized algebraic riccati equation gare associated with a non standard j spectral factorization problem ns j sfp for descriptor systems of which system matrices have full column rank we derive necessary and sufficient conditions for the existence of a semi stabilizing solution of the gare based on the hamiltonian matrix pencil approach and gives a parametrization of all the semi stabilizing solutions then we provide a nonstandard j spectral factor ns j sf in terms of a particular semi stabilizing solution of the gare a simple numerical result is also included

regularity of the free boundary in parabolic phase transition problems in this paper we start the study of the regularity properties of the free boundary for parabolic two phase free boundary problems may be the best known example of a parabolic two phase free boundary problem is the stefan problem a simplified model describing the melting or solidification of a material with a solid liquid interphase the concept of solution can be stated in several ways classical solution weak so lution on divergence form or viscosity solution and as usual one would like to prove that the weak solutions that may be constructed are in fact as smooth and classical as possible locally a classical solution of the stefan problem may be described as following on the unit cylinder q1 b1 1 1 we have two complementary domains and qi separated by a smooth surface s oi2 nq1 in fl and qi we have two smooth solutions ul and u2 of the heat equations

on monotone and geometric convergence of schwarz methods for two sided obstacle problems convergence of schwarz methods is discussed for certain discretizations of two sided obstacle problems monotone and especially geometrical convergence are established if the stiffness matrix is a kind of m matrix and an h independent convergence rate is proved for uniformly overlapping decomposition

riesz potentials and pointwise estimates of solutions to anisotropic porous medium equation abstract for an anisotropic porous medium equation with l 1 regular lower order terms we prove the local boundedness and continuity of weak solutions in terms of the linear riesz potential of the right hand side of equation

monotonic iterative algorithms for an implicit two sided obstacle problem abstract in this paper we present some iterative algorithms mainly schwarz algorithms for an implicit two sided obstacle problem the monotonic convergence of the algorithms is proved

linear quadratic gaussian balancing for model reduction of differential algebraic systems we generalise the model reduction method of linear quadratic gaussian balanced truncation to linear time invariant systems governed by differential algebraic equations the presented method relies on the solution of generalised algebraic riccati equations a simple a priori error bound in the gap metric is given

on the double degenerate equation ut diva x t u u b x t u u abstract we show that a locally bounded nonnegative weak solution of the general double degenerate parabolic equation u t div a x t u u b x t u u satisfying the structure conditions a x t u u u u u 2 0 x t a x t u u u u 1 2 u 1 x t b x t u u u u 2 x t being a continuous function vanishing in u 0 and u 1 is locally continuous generalizing therefore the study of the local regularity theory for the saturation in the flow of two immiscible fluids in a porous medium presented in dibenedetto et al in press 16

global stability and decay for the classical stefan problem the classical one phase stefan problem describes the temperature distribution in a homogeneous medium undergoing a phase transition such as ice melting to water this is accom plished by solving the heat equation on a time dependent domain whose boundary is transported by the normal derivative of the temperature along the evolving and a priori unknown free boundary we establish a global in time stability result for nearly spherical geometries and small tempera tures using a novel hybrid methodology which combines energy estimates decay estimates and hopf type inequalities 1 1 the problem formulation we consider the problem of global existence and asymptotic stability of classical solutions to the classical stefan problem describing the evolving free boundary between the liquid and solid phases the temperature of the liquid p t x and the a priori unknown moving phase boundary t must satisfy the following system of equations

the linear quadratic optimal regulator for descriptor systems in this paper we investigate the linear quadratic optimal regulator problem for the continuous time descriptor system e dot x ax bu where e is in general a singular matrix we solve first a general finite horizon problem by applying the calculus of variations to derive the optimal trajectory of the vector consisting of the concatenated descriptor codescriptor and control vectors from this trajectory the optimal feedback gain relating the control and descriptor variable can be computed by transforming to a coordinate system which can be computed by performing a singular value decomposition of e we derive several riccati differential equations all of which have the same solution this solution gives the optimal cost the steady state optimal feedback gain can be computed by solving an eigenvalue eigenvector problem formulated from the untransformed system parameters in general there does not exist a unique optimal feedback gain but rather the gain is constrained to lie in a linear variety whose dimension is equal to the number of inputs times the rank deficiency of e

existence of analytic solutions for the classical stefan problem we prove that under mild regularity assumptions on the initial data the two phase classical stefan problem admits a unique solution that is analytic in space and time

schwarz algorithm for the solution of variational inequalities with nonlinear source terms numerical solution of variational inequalities of obstacle type associated with some free boundary problems with nonlinear source terms is considered iterative methods based on the combination of the domain decomposition technique and the newton like approach are proposed for solving discrete obstacle problems arising from the continuous piecewise linear finite element approximation of the problems the calculation processes are easily to be parallelized the convergence and specially the convergent rate of the algorithms are obtained

h2 output feedback control for descriptor systems this paper solves the h 2 output feedback control problem for continuous time descriptor systems an existence condition of the h 2 optimal controller is derived based on the standard model matching technique and a characterization of the optimal controller is given in terms of the solutions of two generalized algebraic riccati equations it turns out that the optimal controller has an observer based structure well known in the state space case so that the results in this paper are natural extensions of the state space results to the descriptor systems a numerical example is also included to show the applicability of the proposed results

continuity of the temperature in boundary heat control problems abstract motivated by the boundary heat control problems formulated in the book of duvaut and lions we study a boundary stefan problem and a boundary porous media problem we prove continuity of the solution with the appropriate modulus we also extend the results to the fractional order case and to the anomalous diffusion problems

the kalman yakubovich popov inequality for differential algebraic systems abstract in this paper we revisit the kalman yakubovich popov lemma for differential algebraic control systems this lemma relates the positive semi definiteness of the popov function on the imaginary axis to the solvability of a linear matrix inequality on a certain subspace further emphasis is placed on the lur e equation whose solution set consists loosely speaking of the rank minimizing solutions of the kalman yakubovich popov inequality we show that there is a correspondence between the solution set of the lur e equation and the deflating subspaces of certain even matrix pencils finally we show that under certain conditions the lur e equation admits stabilizing anti stabilizing and extremal solutions we note that for our results we neither assume impulse controllability nor make any assumptions on the index of the system

general matrix pencil techniques for the solution of algebraic riccati equations a unified approach we present a unified theory of matrix pencil techniques for solving both continuous and discrete time algebraic riccati equations ares under fairly general conditions on the coefficient matrices the theory applies to a large class of ares and riccati like equations arising from the singular h sup spl infin and h sup 2 control problems singular linear quadratic control the 4 block nehari problem or from singular j spectral factorizations the underlying concept is the so called proper deflating subspace of a possibly singular matrix pencil in terms of which necessary and sufficient conditions for the solvability of riccati equations are given it is shown that these conditions can be checked and the solutions computed by a numerically sound algorithm

on the schwarz alternating method with more than two subdomains for nonlinear monotone problems in a hilbert space an iterative process is defined which generalizes the schwarz alternating method for nonlinear monotone problems to more than two subdomains and the convergence of this process is proved in the case of the problems with the solution in a convex of h 0 1 omega it is proved that the method converges geometrically numerical results are given concerning the problem of elastic plastic torsion of prismatic bars and the problem of seepage of fluids through porous media

input estimation over frequency region in presence of disturbances a new formulation of the unknown input estimation problem in presence of unknown disturbances over a frequency region is presented necessary and sufficient conditions for existence of a solution are given and three estimation filters are given the minimal attenuation is equal to the norm of a rational matrix over the frequency region as a result the minimum can be computed by solving linear matrix inequalities an example is given and the estimation with the proposed filters is illustrated by a numerical simulation and is compared with some state of the art estimation filters

a quantitative modulus of continuity for the two phase stefan problem we derive the quantitative modulus of continuity omega r left p ln left frac r 0 r right right alpha n p which we conjecture to be optimal for solutions of the p degenerate two phase stefan problem even in the classical case p 2 this represents a twofold improvement with respect to the early 1980 s state of the art results by caffarelli evans arch rational mech anal 81 3 199 220 1983 and dibenedetto ann mat pura appl 103 4 131 176 1982 in the sense that we discard one logarithm iteration and obtain an explicit value for the exponent n p

rate of convergence of some space decomposition methods for linear and nonlinear problems convergence of a space decomposition method is proved for a class of convex programming problems a space decomposition refers to a method that decomposes a space into a sum of subspaces which could be a domain decomposition or a multilevel method when applied to partial differential equations two algorithms are proposed both can be used for linear as well as nonlinear elliptic problems and they reduce to the standard additive and multiplicative schwarz methods for linear elliptic problems

well posedness for the classical stefan problem and the zero surface tension limit we develop a framework for a unified treatment of well posedness for the stefan problem with or without surface tension in the absence of surface tension we establish well posedness in sobolev spaces for the classical stefan problem we introduce a new velocity variable which extends the velocity of the moving free boundary into the interior domain the equation satisfied by this velocity is used for the analysis in place of the heat equation satisfied by the temperature solutions to the classical stefan problem are then constructed as the limit of solutions to a carefully chosen sequence of approximations to the velocity equation in which the moving free boundary is regularized and the boundary condition is modified in a such a way as to preserve the basic nonlinear structure of the original problem with our methodology we simultaneously find the required stability condition for well posedness and obtain new estimates for the regularity of the moving free boundary finally we prove that solutions of the stefan problem with positive surface tension sigma converge to solutions of the classical stefan problem as sigma to 0

approximation of degenerate parabolic problems using numerical integration a class of multidimensional degenerate parabolic equations is considered the two phase stefan problem and the porous medium equation are analyzed as models of singular parabolic equations nonstat

continuity estimates for porous medium type equations with measure data abstract we consider parabolic equations of porous medium type of the form u t div a x t u d u in e t in some space time cylinder e t the most prominent example covered by our assumptions is the classical porous medium equation u t u m in e t with m 1 we establish a sufficient condition for the continuity of u in terms of a natural riesz potential of the right hand side measure as an application we come up with a borderline condition ensuring the continuity of u more precisely if l n 2 2 1 then u is continuous in e t