on the motion of non homogeneous fluids in the presence of diffusion in this paper we study the motion of a non viscous fluid consisting of two components both incompressible say saturated salt water and water taking into consideration the diffusion among these components the equations of the model are obtained for example in frank and kamenetskii 6 and the viscous case is studied in ignat ev and kuznetsov 8 and kazhikhov and smagulov 9 lo for a presentation of this kind of problems see also kuznetsov 111 observe that in s lo the problem is solved only for fluids whose viscosity d is sufficiently large i e

on the solutions in the large of the two dimensional flow of a nonviscous incompressible fluid abstract the euler equations 1 1 for the motion of a nonviscous imcompressible fluid in a plane domain are studied let e be the banach space defined in 1 4 let the initial data v 0 belong to e and let the external forces f t belong to l loc 1 r e in theorem 1 1 the strong continuity and the global boundedness of the unique solution v t are proved and in theorem 1 2 the strong continuous dependence of v on the data v 0 and f is proved in particular the vorticity rot v t is a continuous function in gw for every t r if and only if this property holds for one value of t in theorem 1 3 some properties for the associated group of nonlinear operators s t are stated finally in theorem 1 4 a quite general sufficient condition is given on the data in order to get classical solutions

on stabilization and the existence of coprime factorizations shows that any transfer function matrix whose elements belong to the quotient field of h sub infinity and which is stabilizable has a matrix fraction representation over h sub infinity which is coprime in the sense that a matrix bezout identity can be satisfied

feedback stabilization over commutative rings the matrix case this paper provides a solution of the feedback stabilization problem over commutative rings for matrix transfer functions stabilizability of a transfer matrix is realised as local stabilizability over the entire spectrum of the ring for stabilizable plants certain modules generated by its fractions and that of the stabilizing controller are shown to be projective compliments of each other in the case of rings with irreducible spectrum this geometric relationship shows that the plant is stabilizable if and only if the above modules of the plant are projective of ranks equal to the number of inputs and the outputs if the maxspectrum of the ring is noetherian and of zero krull dimension then this result shows that the stabilizable plants have doubly coprime fractions over unique factorization domains the above stabilizability condition is interpreted in terms of matrices formed by the fractions of the plant certain invariants of these matrices known as elementary factors are defined and it is shown that the plant is stabilizable if and only if these elementary factors generate the whole ring this condition thus provides a generalization of the coprime factorizability as a condition for stabilizabilty a formula for the class of all stabilizing controllers is then developed that generalizes the previous well known formula in factorization theory for multidimensional transfer functions these results provide concrete conditions for stabilizabilty finally it is shown that the class of polynomial rings over principal ideal domains is an additional class of rings over which stabilizable plants always have doubly coprime fractions

nonlinear evolution equations and the euler flow abstract a new theorem on abstract nonlinear equations of evolution is proved as an application the existence uniqueness regularity and continuous dependence on the data are proved for the solution of the euler equation for incompressible fluids in a bounded domain in r m

on a general structure of the stabilizing controllers based on stable range in this paper we prove that some stabilizing controllers of a plant which admits a left right coprime factorization have a special form where their stable and unstable parts are separated the dimension of the unstable part depends on the algebraic concept of stable range of the ring a of siso stable plants moreover we prove that if the stable range of a is equal to 1 then every plant defined by a transfer matrix with entries in the quotient field of a and admitting a left right coprime factorization can be stabilized by a stable controller strong stabilization in particular using a result of treil proving that the stable range of h infty mathbb d is equal to 1 we show that every stabilizable plant defined by a transfer matrix with entries in the quotient field of h infty mathbb d or h infty mathbb c is strongly stabilizable and equivalently every couple of stabilizable plants can be simultaneously stabilized by a controller simultaneous stabilization finally using the fact that the topological stable range of h infty mathbb d is equal to 2 a result due to suarez we show that every unstabilizable siso plant defined by a transfer function with entries in the quotient field of h infty mathbb d is as close as we want to a stabilizable plant in the product topology

a well posedness theorem for non homogeneous inviscid fluids via a perturbation theorem on applique un theoreme de perturbation pour montrer le caractere bien pose d equations hyperboliques non lineaires application a une equation d ecoulement

a review on some contributions to perturbation theory singular limits and well posedness abstract in the beginning of the 1990s we devoted a sequence of papers to perturbation theory singular limits and well posedness problems in particular the strong well posedness of the initial boundary value problem for the compressible euler equations was demonstrate for the first time our method also allowed singular limit results in the strong norm even under assumptions weaker than the current ones in the literature where the strong norm is not reached it is worth noting that until now the above method and results have not been substantially improved hence an introduction to it still looks timely actually in a forthcoming paper by returning to this method we improve in a very substantial way some important results recently appeared in the literature

algebraic geometric aspects of feedback stabilization this paper develops a theory of feedback stabilization for siso transfer functions over a general integral domain which extends the well known coprime factorization approach to stabilization necessary and sufficient conditions for stabilizability of a transfer function in this general setting are obtained these conditions are then refined in the special cases of unique factorization domains ufds noetherian rings and rings of fractions it is shown that these conditions can be naturally interpreted geometrically in terms of the prime spectrum of the ring this interpretation provides a natural generalization to the classical notions of the poles and zeros of a plant the set of transfer functions is topologized so as to restrict to the graph topology of vidyasagar ieee trans automatic control ac 29 1984 pp 403 418 when the ring is a bezout domain it is shown that stability of a feedback system is robust in this topology when the ring is a ufd this theory is then applied to the problem of sta

feedback stabilization over commutative rings further study of the coordinate free approach this paper is concerned with the coordinate free approach to control systems the coordinate free approach is a factorization approach but does not require the coprime factorizations of the plant we present two criteria for feedback stabilizability for multi input multi output mimo systems in which transfer functions belong to the total rings of fractions of commutative rings both of them are generalizations of sule s results in siam j control optim 32 1994 pp 1675 1695 the first criterion is expressed in terms of modules generated from a causal plant and does not require the plant to be strictly causal it shows that if the plant is stabilizable the modules are projective the other criterion is expressed in terms of ideals called generalized elementary factors this gives the stabilizability of a causal plant in terms of the coprimeness of the generalized elementary factors as an example a discrete finite time delay system is considered

strict stationarity of generalized autoregressive processes in this paper we consider the multivariate equation x n 1 a n 1 x n b n 1 with i i d coefficients which have only a logarithmic moment we give a necessary and sufficient condition for existence of a strictly stationary solution independent of the future as an application we characterize the multivariate arma equations with general noise which have such a solution

on a generalization of the youla kucera parametrization part i the fractional ideal approach to siso systems abstract in this paper we show how to use the theory of fractional ideals in order to study the fractional representation approach to analysis and synthesis problems for siso systems within this mathematical framework we give necessary and sufficient conditions so that a plant is internally strongly bistably stabilizable or admits a weak coprime factorization moreover we show how to generalize the youla kucera parametrization of the stabilizing controllers to any stabilizable plant which does not necessarily admit a coprime factorization this parametrization is generally affine in two free parameters

on the vanishing viscosity limit of 3d navier stokes equations under slip boundary conditions in general domains we consider the vanishing viscosity limit for the navier stokes equations with certain slip without friction boundary conditions in a bounded domain with non flat boundary in particular we are able to show convergence in strong norms for a solution starting with initial data belonging to the special subclass of data with vanishing vorticity on the boundary the proof is obtained by smoothing the initial data and by a perturbation argument with quite precise estimates for the equations of the vorticity and for that of the curl of the vorticity

elementary proof of controller parametrization without coprime factorizability recently a controller parametrization method without coprime factorizability was presented by mori in this note we present this method using elementary matrix operations unlike in the earlier paper good knowledge of commutative algebra and module theory is not required

the stabilization problem for discrete time varying linear systems in this paper we consider the strong stabilization problem for a time varying linear system which admits a coprime factorization we also study the simultaneous and robust stabilization problems for this class of systems

convergence of two dimensional branching recursions abstract the asymptotic distribution of branching type recursions ln of the form l n d a l n 1 b l n 1 is investigated in the two dimensional case here l n 1 is an independent copy of ln 1 and a b are random matrices jointly independent of l n 1 l n 1 the asymptotics of ln after normalization are derived by a contraction method the limiting distribution is characterized by a fixed point equation the assumptions of the convergence theorem are checked in some examples using eigenvalue decompositions and computer algebra

corrigendum feedback stabilization over commutative rings the matrix case in a previous paper 1 certain arguments in the proofs of proposition 1 proposition 4 and subsection 4 4 are erroneous these errors are highly regretted by this author and are corrected as follows these corrections however do not affect the main results of the paper 1 in definition 1 of a strictly causal matrix the second sentence should be corrected as a causal matrix m is called strictly causal if all entries of m are in j 2 in the sufficiency part of the proof of proposition 1 the two sentences starting in line 3 of the paragraph since it n j using units of r 1a should be replaced by the following from ny i x d and the assumption that all entries of p are in j and hence that of n since d is nonzerodivisor it follows that all entries of i x are in j hence detx det i i x is of the form 1 j where j is in j this implies that detx is a unit of r 1a as j is the jacobson s radical 3 the proof of the necessity part of proposition 4 is erroneous since this result is not needed further in the paper we replace it with the following version stating only the sufficient condition proposition 4 a strictly causal p nd 1 in f n is stabilizable if a b a

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part ii internal stabilization in this second part of the paper a quadrat siam j control optim 40 2003 pp 266 299 we show how to reformulate the fractional representation approach to synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions for internal stabilizability moreover we characterize all the integral domains a of siso stable plants such that every mimo plant defined by means of a transfer matrix whose entries belong to the quotient field k q a of a is internally stabilizable finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 s shankar and v r sule siam j control optim 30 1992 pp 11 30 v r sule siam j control optim 32 1994 pp 1675 1695 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

on stabilization and the existence of coprime factorizations let k be an integral domain and f its quotient field we show there are plants over f which have no stable coprime factorizations but can be stabilized in a stable closed loop this answers a question posed by vidyasagar schneider and francis

the fractional representation approach to synthesis problems an algebraic analysis viewpoint part i weakly doubly coprime factorizations in this paper we show how to reformulate the fractional representation approach to analysis and synthesis problems within an algebraic analysis framework in terms of modules we give necessary and sufficient conditions so that a system admits weakly left right doubly coprime factorizations moreover we explicitly characterize the integral domains a such that every plant defined by means of a transfer matrix whose entries belong to the quotient field of a admits weakly doubly coprime factorizations finally we show that this algebraic analysis approach allows us to recover on the one hand the approach developed in m c smith ieee trans automat control 34 1989 pp 1005 1007 and on the other hand the ones developed in k mori and k abe siam j control optim 39 2001 pp 1952 1973 v r sule siam j control optim 32 1994 pp 1675 1695 and 36 1998 pp 2194 2195 m vidyasagar h schneider and b a francis ieee trans automat control 27 1982 pp 880 894 m vidyasagar control system synthesis a factorization approach mit press cambridge ma 1985

stationarity of garch processes and of some nonnegative time series abstract two time series models are considered garch processes and generalized multivariate autoregressive equations x n 1 a n 1 x n b n 1 with nonnegative i i d coefficients in each case a necessary and sufficient condition ensuring the existence of a strictly stationary solution is given