existence and large time behavior to coupled chemotaxis fluid equations in besov morrey spaces abstract the paper deals with the cauchy problem of coupled chemotaxis fluid equations by taking advantage of a coupling structure of the equations and using the semigroup approach we show existence and asymptotic stability with small initial data and external force u 0 n 0 c 0 c 0 n r 1 1 n r 2 2 n r 3 3 l m n for certain technical assumptions for the embedded relationship between function spaces we show that our initial data class is larger than that of kozono et al 2016 11 and covers physical cases of initial aggregation at points diracs and on filaments as an application we obtain a class of asymptotically existence of a basin of attraction for each self similar solutions with homogeneous initial data

global regular and singular solutions for a model of gravitating particles the existence of solutions of a nonlinear parabolic equation describing the gravitational interaction of particles is studied for the initial data in spaces of generalized pseudomeasures this approach permits us to relax regularity assumptions on the initial conditions and to prove asymptotic stability results for the above problem

persistency of excitation for nonminimal models of systems having purely deterministic disturbances all results to date on the question of persistency of excitation depend on the model being minimal thus the results do not apply to systems having purely deterministic disturbances since these automatically give rise to uncontrollable modes on the unil circle this note presents new results on persistency of excitation applicable to nonminimal models of systems having purely deterministic disturbances

discrete painleve equations coalescences limits and degeneracies abstract starting from the standard form of the five discrete painleve equations we show how one can obtain through appropriate limits a host of new equations which are also the discrete analogues of the continuous painleve equations a particularly interesting technique is the one based on the assumption that some simplification takes place in the autonomous form of the mapping following which the deautonomization leads to a new n dependence and introduces more new discrete painleve equations

simultaneous parameter tracking and state estimation in a linear system abstract the task of simultaneous tracking of time varying parameters and estimation of the state is treated for a linear system described by a time varying input output armax or delta model with known c noise parameters first a bayesian approach based conceptual solution is presented then it is shown that utilizing the properties of the observer canonical state model algebraic recursion operating on the joint parameter and state mean and covariance matrix can be obtained with no approximation involved several illustrative examples are included

fake algebraic riccati techniques and stability conditions sufficient and necessary for monotonic behavior of the solutions of the riccati differential equation and riccati difference equation are derived for the optimal filtering respectively control equation these results are derived without the usual requirement of detectability respectively stabilizability the monotonic behavior allows proof of stabilizing properties of the solutions subject only to requirements on the initial conditions

riccati differential equation in optimal filtering of periodic non stabilizable systems this paper discusses the periodic solutions of the matrix riccati differential equation in the optimal filtering of periodic systems special emphasis is given to non stabilizable systems and the question addressed is the existence and uniqueness of a steady state periodic non negative definite solution of the periodic riccati differential equation which gives rise to an asymptotically stable steady state filter the results presented show that the stabilizability is not a necessary condition for the existence of such a periodic solution the convergence of the general solution of the periodic riccati differential equation to a periodic equilibrium solution is also investigated the results are extensions of existing time invariant systems results to the case of periodic systems

brief constrained linear state estimation a moving horizon approach this article considers moving horizon strategies for constrained linear state estimation additional information for estimating state variables from output measurements is often available in the form of inequality constraints on states noise and other variables formulating a linear state estimation problem with inequality constraints however prevents recursive solutions such as kalman filtering and consequently the estimation problem grows with time as more measurements become available to bound the problem size we explore moving horizon strategies for constrained linear state estimation in this work we discuss some practical and theoretical properties of moving horizon estimation we derive sufficient conditions for the stability of moving horizon state estimation with linear models subject to constraints on the estimate we also discuss smoothing strategies for moving horizon estimation our framework is solely deterministic

on discrete algebraic riccati equations a rank characterization of solutions abstract we give a rank characterization of the solution set of discrete algebraic riccati equations dares involving real matrices our characterization involves the use of invariant subspaces of the system matrix we observe that a dare can be considered as a specific type of a non symmetric continuous time algebraic riccati equation we show how to extract the low rank solutions from the full rank solution of a homogeneous dare we also study the eigen structure of various feedback closed loop matrices associated with solutions of a dare we show as a consequence of our observations that a solution x of a dare for controllable systems and certain specific types of uncontrollable systems satisfies the inequality x m i n x x m a x we characterize conditions under which the solution set of dare is bounded unbounded we use the method of invariant subspaces to identify some invariant sets of a difference riccati equation

maximal versus strong solution to algebraic riccati equations arising in infinite markov jump linear systems abstract we deal with a perturbed algebraic riccati equation in an infinite dimensional banach space which appears for instance in the optimal control problem for infinite markov jump linear systems from now on imjls infinite or finite here has to do with the state space of the markov chain being infinite countable or finite see e g m d fragoso j baczynski optimal control for continuous time lq problems with infinite markov jump parameters siam j control optim 40 1 2001 270 297 by using a certain concept of stochastic stability a sort of l 2 stability we have proved in j baczynski m d fragoso maximal solution to algebraic riccati equations linked to infinite markov jump linear systems internal report lncc no 6 2006 existence and uniqueness of maximal solution for this class of equations as it is noticed in this paper unlike the finite case including the linear case we cannot guarantee anymore that maximal solution is a strong solution in this setting via a discussion on the main mathematical hindrance behind this issue we devise some mild conditions for this implication to hold specifically our main result here is that under stochastic stability along with a condition related with convergence in the infinite dimensional scenario and another one related to spectrum weaker than spectral continuity we ensure the maximal solution to be also a strong solution these conditions hold trivially in the finite case allowing us to recover the result of strong solution of c e de souza m d fragoso on the existence of maximal solution for generalized algebraic riccati equations arising in stochastic control systems control lett 14 1990 233 239 set for mjls the issue of whether the convergence condition is restrictive or not is brought to light and together with some counterexamples unveil further differences between the finite and the infinite countable case

global classical solutions in chemotaxis navier stokes system with rotational flux term the coupled chemotaxis fluid system nt n ns x n c c u n x t 0 t ct c nc u c x t 0 t ut u u u p n x t 0 t u 0 x t 0 t is considered under the no flux boundary conditions for n c and the dirichlet boundary condition for u on a bounded smooth domain rn n 2 3 0 1 we assume that s x n c is a matrix valued sensitivity under a mild assumption such that s x n c s0 c0 with some non decreasing function s0 c2 0 it contrasts with the related scalar sensitivity case that does not possess the natural gradient like functional structure associated estimates based on the natural functional seem no longer available in the present work a global classical solution is constructed under a smallness assumption on c0 l and moreover we obtain boundedness and large time convergence for the solution meaning that small initial concentration of chemical forces stabilization

an extension of the algebraic riccati equation for the stationary control problem without stabilizability condition existence of stationary constant gain optimal control is established and some optimality conditions are derived for the linear quadratic control problem under the following condition weaker than stabilizability there exists a control strategy which makes the sequence of cost bounded for example a system with an unassignable eigenvalue on the unit circle satisfies this condition although the associated algebraic riccati equation usually has no solution for the extended class the existence of the stationary control which minimizes the average cost or the average expected cost per unit time is proved a complete optimality condition is given by part of the algebraic riccati equation weakened conditions of detectability are also introduced

eventual smoothness of generalized solutions to a singular chemotaxis stokes system in 2d abstract we study the chemotaxis fluid system n t u n n n c c x t 0 c t u c c n c x t 0 u t p u n x t 0 u 0 x t 0 under homogeneous neumann boundary conditions for n and c and homogeneous dirichlet boundary conditions for u where r 2 is a bounded domain with smooth boundary and c 2 from recent results it is known that for suitable regular initial data the corresponding initial boundary value problem possesses a global generalized solution we will show that for small initial mass n 0 these generalized solutions will eventually become classical solutions of the system and obey certain asymptotic properties moreover from the analysis of certain energy type inequalities arising during the investigation of the eventual regularity we will also derive a result on global existence of classical solutions under assumption of certain smallness conditions on the size of n 0 in l 1 and in l log l u 0 in l 4 and of c 0 in l 2

strong solutions of the discrete time algebraic riccati equation abstract conditions are given under which a solution of the dare is positive semidefinite if and only if all the eigenvalues of its associated closed loop matrix are in the closed unit disc

convergence properties of the riccati difference equation in optimal filtering of nonstabilizable systems this paper studies the convergence and properties of the solutions of the riccati difference equation special emphasis is given to systems which are not necessarily stabilizable in the filtering sense particularly those having uncontrollable roots on the unit circle besides generalizing and unifying previous work the results have application to a number of important problems including filtering and control of systems with purely deterministic disturbances such as sinusoids and drift components

a max plus based fundamental solution for a class of discrete time linear regulator problems abstract efficient riccati equation based techniques for the approximate solution of discrete time linear regulator problems are restricted in their application to problems with quadratic terminal payoffs where non quadratic terminal payoffs are required these techniques fail due to the attendant non quadratic value functions involved in order to compute these non quadratic value functions it is often necessary to appeal directly to dynamic programming in the form of grid or element based iterations for the value function these iterations suffer from poor scalability with respect to problem dimension and time horizon in this paper a new max plus based method is developed for the approximate solution of discrete time linear regulator problems with non quadratic payoffs this new method is underpinned by the development of new fundamental solutions to such linear regulator problems via max plus duality in comparison with a typical grid based approach a substantial reduction in computational effort is observed in applying this new max plus method a number of simple examples are presented that illustrate this and other observations

on the design of a stable adaptive filter for state estimation in high dimensional systems design of specific parametrized gain structures provides a new tool for ensuring stability of reduced order filter for high dimensional systems which is optimized adaptively in the minimum prediction error framework

global existence and boundedness in a keller segel stokes system involving a tensor valued sensitivity with saturation the 3d case abstract in this paper we continue to deal with the initial boundary value problem for the coupled keller segel stokes system n t u n n n s x n c c x t 0 t c t u c c c n x t 0 t u t p u n x t 0 t u 0 x t 0 t where r d is a bounded domain with smooth boundary and the chemotactic sensitivity s is not a scalar function but rather attains values in r d d and satisfies s x n c c s 1 n with some c s 0 and 0 when d 2 our previous work j differential equations 2015 has established the existence of global bounded classical solutions under the subcritical assumption 0 which is consistent with the corresponding results of the fluid free system but the method seems to be invalid in the three dimensional setting in this paper for the case d 3 we develop a new method to establish the existence and boundedness of global classical solutions for arbitrarily large initial data under the assumption 1 2 which is slightly stronger than the corresponding subcritical assumption 1 3 on the fluid free system where such an assumption is essentially necessary and sufficient for the existence of global bounded solutions the key idea here is to establish the general l p regularity of u from a rather low l p regularity of n which will be obtained by a new combinational functional

riccati equations in optimal filtering of nonstabilizable systems having singular state transition matrices until recently it was believed that a necessary and sufficient condition for convergence of the riccati difference equation of optimal filtering was that the system be both delectable and stabilizable recently it has been shown that the stabilizability condition can be removed but convergence has only established under restrictive assumptions including the requirement that the state transition matrix be nonsingular the present paper generalizes these results in several directions first properties of the algebraic riccati equation are established for the case of singular state transition matrix second several assumptions previously imposed in establishing convergence of the riccati difference equation for systems with unreachable modes on the unit circle are relaxed including replacing observability by detectability weakening the conditions on the initial covariance and allowing the state transition matrix to be singular third results on the convergence and properties of the riccati equations are expressed as both necessary and sufficient conditions whereas previous results were only sufficient these extensions mean that the results have wider applicability including fixed lag smoothing problems and filtering for systems with time delays the implications of the results in the dual problem of optimal control are also studied

criterion for the convergence of the solution of the riccati differential equation the optimal control problem for a linear system with a quadratic cost function leads to the matrix riccati differential equation the convergence of the solution of this equation for increasing time interval is investigated as a function of the final state penalty matrix a necessary and sufficient condition for convergence is derived for stabilizable systems even if the output in the cost function is not detectable an algorithm is developed to determine the limiting value of the solution which is one of the symmetric positive semidefinite solutions of the algebraic riccati equation examples for convergence and nonconvergence are given a discussion is also included of the convergence properties of the solution of the riccati differential equation to any real symmetric not necessarily positive semidefinite solution of the algebraic riccati equation

receding horizon mimo lq controller design with guaranteed stability to design an lq optimal receding horizon controller with guaranteed stability the initialization of the underlying riccati equation corresponding to an lq problem with a fixed terminal state can be used it is shown that the result can be generalized to mimo systems and that the initialization is particularly simple if input multirate sampling is used

finite time behavior of inner systems in this paper we investigate how nonminimum phase characteristics of a dynamical system affect its controllability and tracking properties for the class of linear time invariant dynamical systems these characteristics are determined by transmission zeros of the inner factor of the system transfer function the relation between nonminimum phase zeros and hankel singular values of inner systems is studied and it is shown how the singular value structure of a suitably defined operator provides relevant insight about system invertibility and achievable tracking performance the results are used to solve various tracking problems both on finite as well as on infinite time horizons a typical receding horizon control scheme is considered and new conditions are derived to guarantee stabilizability of a receding horizon controller

infinite horizon linear quadratic control with end point state penalty term the discrete time case abstract for a given discrete time system consider an infinite horizon linear quadratic control problem with positive semidefinite cost criterion and an extra penalty term for the state variable at infinity our central result shows that this problem is structurally equivalent to the associated problem where the state penalty term is required to vanish at infinity provided only that the latter problem has finite optimal cost everywhere for this case the optimal cost is represented by a unique solution of the possibly singular algebraic riccati equation and if in addition the underlying system is left invertible then optimal inputs for either problem are implementable as state feedback laws expressed in terms of the original system coefficients only even when the control weighting matrix in the cost criterion is not invertible

asymptotic optimality of the minimum variance fixed interval smoother this correspondence investigates the asymptotic performance of the discrete time and continuous time time varying minimum variance fixed interval smoothers comparison theorems are generalized to provide sufficient conditions for the monotonic convergence of the underlying riccati equations under these conditions the energy of the estimation errors asymptotically approach a lower bound and attain lscr2 l2 stability

meromorphic solutions of difference equations integrability and the discrete painleve equations the painlev e property is closely connected to differential equations that are integrable via related iso monodromy problems many apparently integrable discrete analogues of the painlev e equations have appeared in the literature the existence of sufficiently many finite order meromorphic solutions appears to be a good analogue of the painlev e property for discrete equations in which the independent variable is taken to be complex a general introduction to nevanlinna theory is presented together with an overview of recent applications to meromorphic solutions of difference equations and the difference and q difference operators new results are presented concerning equations of the form w z 1 w z 1 r z w where r is rational in w with meromorphic coefficients

the small convection limit in a two dimensional chemotaxis navier stokes system this paper deals with an initial boundary value problem for the chemotaxis navier stokes system begin aligned left begin array lcll n t u cdot nabla n delta n nabla cdot n nabla c qquad quad x in omega t 0 c t u cdot nabla c delta c nc qquad quad x in omega t 0 u t kappa u cdot nabla u delta u nabla p n nabla phi qquad quad x in omega t 0 nabla cdot u 0 qquad quad x in omega t 0 end array right end aligned in a bounded convex domain omega subset mathbb r 2 with smooth boundary with kappa in mathbb r and a given smooth potential phi omega rightarrow mathbb r it is known that for each kappa in mathbb r and all sufficiently smooth initial data this problem possesses a unique global classical solution n kappa c kappa u kappa the present work asserts that these solutions stabilize to n 0 c 0 u 0 uniformly with respect to the time variable more precisely it is shown that there exist mu 0 and c 0 such that whenever kappa in 1 1 begin aligned big vert n kappa cdot t n 0 cdot t big vert l infty omega big vert c kappa cdot t c 0 cdot t big vert l infty omega quad big vert u kappa cdot t u 0 cdot t big vert l infty omega le c kappa e mu t end aligned for all t 0 this result thereby provides an example for a rigorous quantification of stability properties in the stokes limit process as frequently considered in the literature on chemotaxis fluid systems in application contexts involving low reynolds numbers

global well posedness and asymptotic behavior in besov morrey spaces for chemotaxis navier stokes fluids in this work we consider the keller segel system coupled with navier stokes equations in rn for n 2 we prove the global well posedness with small initial data in besov morrey spaces our initial data class extends previous ones found in the literature such as that obtained by kozono miura and sugiyama j funct anal 270 5 1663 1683 2016 it allows us to consider initial cell density and fluid velocity concentrated on smooth curves or at points depending on the spatial dimension self similar solutions are obtained depending on the homogeneity of the initial data and considering the case of a chemical attractant without the degradation rate moreover we analyze the asymptotic stability of solutions at infinity and obtain a class of asymptotically self similar ones in this work we consider the keller segel system coupled with navier stokes equations in rn for n 2 we prove the global well posedness with small initial data in besov morrey spaces our initial data class extends previous ones found in the literature such as that obtained by kozono miura and sugiyama j funct anal 270 5 1663 1683 2016 it allows us to consider initial cell density and fluid velocity concentrated on smooth curves or at points depending on the spatial dimension self similar solutions are obtained depending on the homogeneity of the initial data and considering the case of a chemical attractant without the degradation rate moreover we analyze the asymptotic stability of solutions at infinity and obtain a class of asymptotically self similar ones

on a fallacious conjecture about the stabilizability properties of solutions of the riccati difference equation abstract in 1 among other results some conjectures concerning the monotonicity and stabilizing properties of solutions of the difference riccati equation were proven to be fallacious by means of suitable counterexamples moreover a possibly fallacious conjecture was formulated for which no counterexample had been found in this letter such a counterexample is provided together with an interpretation of the somewhat counterintuitive behaviour of the riccati equation

symmetrization techniques on unbounded domains application to a chemotaxis system on rn the authors study the parabolic elliptic system on rn u t u u v 0 v v u u 0 u0 a version of the mathematical model of chemotaxis proposed by keller and segel a differential inequality for the quantity s0u t d where u is the decreasing rearrangement of the solution u t with respect to the spatial variable is obtained as a consequence they obtain e g lp bounds of the solution u v on r2 and global in time existence of solutions under the condition r2u0 8 then the solution u v blows up in a finite time compared to the previous work of diaz diaz and nagai adv math sci appl 5 1995 no 2 659 680 mr1361010 96j 35246 where this problem has been considered on bounded domains of rn there are some additional technical difficulties connected with the regularity of the derivative u t

existence and uniqueness theorem on mild solutions to the keller segel system coupled with the navier stokes fluid abstract we consider the keller segel system coupled with the navier stokes fluid in the whole space and prove the existence of global mild solutions with the small initial data in the scaling invariant space our method is based on the implicit function theorem which yields necessarily continuous dependence of solutions for the initial data as a byproduct we show the asymptotic stability of solutions as the time goes to infinity since we may deal with the initial data in the weak l p spaces the existence of self similar solutions provided the initial data are small homogeneous functions

properties of the solutions of rational matrix difference equations abstract we prove a comparison theorem for the solutions of a rational matrix difference equation generalizing the riccati difference equation and existence and convergence results for the solutions of this equation moreover we present conditions ensuring that the corresponding algebraic matrix equation has a stabilizing or almost stabilizing solution

global strong solution to the semi linear keller segel system of parabolic parabolic type with small data in scale invariant spaces abstract we shall show existence of global strong solution to the semi linear keller segel system in r n n 3 of parabolic parabolic type with small initial data u 0 h n r 2 r r n and v 0 h n r r r n for max 1 n 4 r n 2 our method is based on the perturbation of linealization together with the l p l q estimates of the heat semigroup and the fractional powers of the laplace operator as a by product of our method we shall prove the decay property of solutions as the time goes to infinity

simple method for solving the constant gains of kalman filters with single output a simple method is described for the calculation of the optimal gains for steady state kalman filters in continuous and discrete time systems with single output the procedure consists of comparing the optimal closed loop characteristic polynomial with a determinant expansion containing the unknown constant gains it is then shown that for the case when a computer program is not used for finding the roots of a polynomial the optimal closed loop characteristic polynomial in the discretetime problem is not as readily obtainable as that in the continuous time problem even though the system under consideration is of lower order this drawback however is shown to be overcome by invoking the bilinear transformation

a comparison theorem for matrix riccati difference equations abstract difference equations of the form x t f t x t 1 f t f t x t 1 g t i g t x t 1 g t 1 g t x t 1 f t q t and their associated hermitian matrices h t f q f gg t are studied solution of different riccati equations can be compared if the difference of their corresponding hermitian matrices is semidefinite for all t an application to the discrete time lq optimal control problem is given

an adaptive control algorithm for linear systems having unknown time delay abstract this paper presents an algorithm for adaptive control of linear systems having unknown time delay a rational approximation is used for the delay and it is argued that within this framework the analysis of the properties of the corresponding adaptive control algorithm essentially reduces to the issue of robustness to unmodelled dynamics a global convergence theory is presented using recent methods for analyzing robustness properties of adaptive control algorithms the analysis is supported by simulation results

convergence of the dre solution to the are strong solution in this paper we use the boundary solutions to a linear matrix inequality lmi associated with kalman filtering to investigate the convergence of the solution of a differential riccati equation dre to the so called strong solution of an algebraic riccati equation are we furthermore extend our results to kalman filtering with indefinite input noise covariances

global existence and asymptotic stability of the fractional chemotaxis fluid system in r3 abstract in this paper we consider a model arising from biology consisting of fractional chemotaxis equations coupled to viscous incompressible fluid equations throughout transport and external force in r 3 we establish the existence and uniqueness of global solution with small initial data by combining the local existence and a priori estimates as well as continuation argument moreover the decay rates of the solutions and their higher order spatial derivatives are established towards the equilibrium by introducing the negative sobolev space h s 0 s 3 2

numerical solution of the discrete time periodic riccati equation in this paper we present a method for the computation of the periodic nonnegative definite stabilizing solution of the periodic riccati equation this method simultaneously triangularizes by orthogonal equivalences a sequence of matrices associated with a cyclic pencil formulation related to the euler lagrange difference equations in doing so it is possible to extract a basis for the stable deflating subspace of the extended pencil from which the riccati solution is obtained this algorithm is an extension of the standard qz algorithm and retains its attractive features such as quadratic convergence and small relative backward error a method to compute the optimal feedback controller gains for linear discrete time periodic systems is dealt with

discrete minimax linear quadratic regulation of continuous time systems abstract in this paper the statistical cyclostationary characteristics of a controlled continuous time system which result as a consequence of implementing a digital controller are investigated a minimax quadratic cost function which takes into account the periodic nature of the statistics is defined and optimized the resulting linear state feedback law improves the intersample behavior of the controlled system when compared to discrete time optimization in particular the minimax variance regulator which is proposed as a new design method for regulating the output variance offers a significant improvement over the classical minimum variance regulator

global stability of homogeneous steady states in scaling invariant spaces for a keller segel navier stokes system abstract in this paper we study the global stability of homogeneous equilibria in keller segel navier stokes equations in scaling invariant spaces we prove that for any given 0 m 1 1 with 1 being the first eigenvalue of neumann laplacian the initial boundary value problem of the keller segel navier stokes system has a unique globally bounded classical solution provided that the initial datum is chosen sufficiently close to m m 0 in the norm of l d 2 w 1 d l d and satisfies a natural average mass condition our proof is based on the perturbation theory of semigroups and certain delicate exponential decay estimates for the linearized semigroup our result suggests a new observation that nontrivial classical solution for keller segel navier stokes equation can be obtained globally starting from suitable initial data with arbitrarily large total mass provided that volume of the bounded domain is large correspondingly

cyclomonotonicity and stabilizability properties of solutions of the difference periodic riccati equation the kalman filter associated with a discrete time linear t periodic system is tested the problem considered is that of selecting an initial covariance matrix such that the periodic filter based on the first t values of the kalman filter gain is stabilizing sufficient conditions are given that hinge on the cyclomonotonicity of the solution of the periodic riccati equation potential applications are found in filter design quasi linearization techniques for the periodic riccati equation and the design of receding horizon control strategies for periodic and multirate systems when specialized to time invariant systems the results give rise to new sufficient conditions for the cyclomonotonicity of the solutions of the time invariant riccati equation and the existence of periodic stabilizing feedback

self duality and schlesinger chains for the asymmetric d pii and q piii equations we analyse two asymmetric discrete painleve equations namely d pii and q piii which are known to be discrete forms of piii and pvi respectively we show that both equations are self dual this means that the same equation governs the evolution along the discrete independent variable and the transformations under the action of the schlesinger transforms along the parameters of the discrete painleve a bilinear formulation of the self dual equation is given as a system of nonautonomous hirota miwa equations

convergence of the kalman filter gain for a class of nondetectable signal extraction problems convergence of the filter gain is proved for a class of problems with undetectable modes on the unit circle this situation arises when the signal is an unstable arma process corrupted by arma noise with the same unstable modes since modes common to signal and noise are unobservable

a miscellany of results of an equation of count j f riccati a collection of results on the riccati equation is presented the questions addressed are the existence of strong solutions of the algebraic riccati equation and the convergence of solutions of the riccati difference equation to those of the algebraic equation the results derived utilize detestability conditions only

global existence and convergence rates to a chemotaxis fluids system with mixed boundary conditions abstract in this paper we investigate the large time behavior of strong solutions to a chemotaxis fluids system in an unbounded domain with mixed boundary conditions based on the anisotropic l p technique the elliptic estimates and stokes estimates we first establish the global existence of strong solution around the equilibrium state 0 c satn 0 with the help of the continuity arguments where c satn is the saturation value of oxygen inside the fluid then we use de giorgi s technique and energy method to show that such a solution will converge to 0 c satn 0 with an explicit convergence rate in the chemotaxis free case our assumptions and results are consistent with the experimental descriptions and the numerical analysis the novelty here consists of deriving some new elliptic estimates and stokes estimates and choosing a suitable weight in de giorgi s technique to deal with the mixed boundary conditions

singularity analysis and integrability for discrete dynamical systems abstract evidence that suggests a necessary consequence of integrability for discrete dynamical systems is presented the evidence is analogous to that provided by the painleve test of integrability for continuous dynamical systems autonomous discretizations of several integrable continuous systems are studied it is shown that those discrete systems exhibiting no chaotic behaviour in the complex space of the iterates possess solutions that have a property analogous to the painleve property of their corresponding continuous dynamical system

monotonicity and stabilizability properties of solutions of the riccati difference equation propositions lemmas theorems fallacious conjectures and counterexamples abstract the problem considered is that of selecting an initial covariance matrix for the kalman filter to ensure that the closed loop filter at every subsequent time instant is exponentially asymptotically stable as a time invariant filter sufficient conditions are derived based on monotonicity properties of the solution of the riccati difference equation the results have application in observer design and the cases of filtering for nonstabilizable systems and systems with singular system matrices are included

a note on the maximal solution of the periodic riccati equation the periodic symmetric solutions of the periodic riccati differential equation associated with the filtering problem are considered by the authors it is proven that under the sole assumption of detectability there exists a maximal solution moreover such a solution turns out to be strong i e the characteristic multipliers of the associated closed loop system belong to the closed unit disk the proof relies on an iterative linearization technique which calls for a sequence of periodic lyapunov equations similar results are given for the minimal solution

difference and differential riccati equations a note on the convergence to the strong solution the convergence of the solutions of the differential difference riccati equations to the strong solution of the corresponding algebraic riccati equation are is addressed detectability only is required in the analysis and no assumption is made on the eigenvalues on the real imaginary axis on the unit circle in the discrete time case in particular from the results it follows that under the sole assumption of detectability a positive definite initial condition guarantees convergence to the strong solution even in the presence of unreachable eigenvalues on the imaginary axis or on the unit circle

cyclomonotonicity riccati equations and periodic receding horizon control abstract in this paper periodic receding horizon control of periodic and time invariant systems is considered this type of receding horizon control is based on the periodic extension of the control law that solves a finite horizon optimal control problem it is shown that cyclomonotonicity of the solution of the differential riccati equation entails stability of the closed loop system stability analysis hinges on the so called fake two point boundary riccati equation a differential riccati equation whose solution is constrained to satisfy a two point boundary condition in the case of time invariant systems cyclomonotonicity is always achievable by a proper scaling of the terminal condition of the riccati equation it is also proven that cyclomonotonic solutions of the riccati equation not only yield stabilizing controllers but also constant and stabilizing control laws

continuous symmetries of difference equations lie group theory was originally created more than 100 years ago as a tool for solving ordinary and partial differential equations in this article we review the results of a much more recent program the use of lie groups to study difference equations we show that the mismatch between continuous symmetries and discrete equations can be resolved in at least two manners one is to use generalized symmetries acting on solutions of difference equations but leaving the lattice invariant the other is to restrict them to point symmetries but to allow them to also transform the lattice

the road to the discrete analogue of the painleve property nevanlinna meets singularity confinement the question of integrability of discrete systems is analysed in the light of the recent findings of ablowitz et al who have conjectured that a fast growth of the solutions of a differ ence equation is an indication of nonintegrability the study of the behaviour of the solutions of a mapping is based on the theory of nevanlinna in this paper we show how this approach can be implemented in the csse of second order mappings which include the discrete painlevk equations since the nevanlinna approach does offer only a necessary condition which is not restrictive enough we complement it by the singularity confinement requirement first in an autonomous setting and then for deautonomisation we believe that this three tiered approach is the closest one can get to a discrete analogue of the painlevb property 2003 elsevier science ltd all rights reserved

time global existence and finite time blow up criterion for solutions to the keller segel system coupled with the navier stokes fluid abstract we will deal with the chemotaxis model under the effect of the navier stokes fluid i e the incompressible viscous fluid we shall show the existence of a local mild solution for large initial data and a global mild solution for small initial data in the scale invariant class demonstrating that n 0 l 1 r 2 and u 0 l 2 r 2 our method is based on the perturbation of linearization together with the l p l q estimates of the heat semigroup as a by product of our method we shall prove the smoothing effect and uniqueness of our mild solution in addition we shall show a blow up criterion which almost covers the well known threshold number 8 of the size n 0 l 1 r 2 under the rest state of the fluid motion furthermore the blow up rate will be also discussed

some properties of meromorphic solutions of q difference equations abstract we investigate the growth of transcendental meromorphic solutions of some complex q difference equations and find lower bounds for nevanlinna lower order for meromorphic solutions of such equations we also obtain a q difference version of tumura clunie theorem

singularity confinement and chaos in discrete systems we present a number of second order maps which pass the singularity confinement test commonly used to identify integrable discrete systems but which nevertheless are non integrable as a more sensitive integrability test we propose the analysis of the complexity algebraic entropy of the map using the growth of the degree of its iterates integrability is associated with polynomial growth while the generic growth is exponential for chaotic systems

existence and uniqueness of mild solutions to the incompressible nematic liquid crystal flow abstract in this paper we consider the incompressible nematic liquid crystal flow in the whole space and obtain the existence of global mild solutions with small initial data in the scaling invariant space the main tool we have used is the implicit function theorem furthermore we establish the asymptotic stability of solutions when the time goes to infinity and in view of our construction of solutions in the weak l p spaces we finally derive the existence of self similar solutions provided the initial data are small homogeneous functions

rational mappings arborescent iterations and the symmetries of integrability we describe a class of nonlinear birational representations of groups generated by a finite number of involutions these groups are symmetries of the yang baxter equations and their higher dimensional generalizations they provide discrete dynamical systems with a variety of behaviors from chaotic to integrable according to the number of invariants of the representation

on stabilizing properties of solutions of the riccati difference equation monotonicity and stabilizing properties of solutions of the riccati difference equation rde are discussed the author considers the problem of selecting an initial condition for the rde in such a way that the update of the kalman filter gain can be stopped at any time and the resulting frozen filter is asymptotically stable the author also considers the case in which the initial condition of the rde may be less than the asymptotic solution the results are relevant to control and observer design including the stability of finite time horizon discrete time predictive control

a kalman filtering approach to short time fourier analysis the problem of estimating time varying harmonic components of a signal measured in noise is considered the approach used is via state estimation two methods are proposed one involving pole placement of a state observer the other using quadratic optimization techniques the result is the development of a new class of filters akin to recursive frequency sampling filters for inclusion in a parallel bank to produce sliding harmonic estimates kalman filtering theory is applied to effect the good performance in noise and the class of filters is parameterized by the design tradeoff between noise rejection and convergence rate these filters can be seen as generalizing the dft

how to detect the integrability of discrete systems several integrability tests for discrete equations will be reviewed all tests considered can be applied directly to a given discrete equation and do not rely on the a priori knowledge of the existence of related structures such as lax pairs specifically singularity confinement algebraic entropy nevanlinna theory diophantine integrability and discrete systems over finite fields will be described

on the growth of logarithmic differences difference quotients and logarithmic derivatives of meromorphic functions a crucial ingredient in the recent discovery by ablowitz halburd herbst and korhonen 2000 2007 that a connection exists between discrete painleve equations and finite order nevanlinna theory is an estimate of the integrated average of log f z 1 f z on z r we obtained essentially the same estimate in our previous paper 2008 independent of halburd et al 2006 we continue our study in this paper by establishing complete asymptotic relations amongst the logarithmic differences difference quotients and logarithmic derivatives for finite order meromorphic functions in addition to the potential applications of our new estimates in integrable systems they are also of independent interest in particular our findings show that there are marked differences between the growth of meromorphic functions with nevanlinna order less than and greater than one we have established a difference analogue of the classical wiman valiron type estimates for meromorphic functions with order less than one which allow us to prove that all entire solutions of linear difference equations with polynomial coefficients of order less than one must have positive rational order of growth we have also established that any entire solution to a first order algebraic difference equation with polynomial coefficients must have a positive order of growth which is a difference analogue of a classical result of polya

global existence in critical besov spaces for the coupled chemotaxis fluid equations abstract in this paper we are concerned with a system coupling the parabolic parabolic keller segel system to the viscous incompressible navier stokes equations in spatial dimensions three and prove the global existence of solutions with small initial data in critical besov spaces with minimal regularity order by using the elementary paradifferential calculus and a special type of iteration scheme

boundedness in a 2d chemotaxis stokes system with general sensitivity and nonlinear diffusion abstract this paper deals with the following chemotaxis stokes system n t u n n m n s x n c c x t 0 c t u c c n f c x t 0 u t u p n x t 0 u 0 x t 0 in a bounded domain r 2 with smooth boundary w 1 f and s are given sufficiently smooth functions with values in 0 and r 2 2 respectively here s satisfies s x n c s 0 c n with 0 and some nondecreasing nonnegative function s 0 it is showed that when m 1 and 0 the corresponding system possesses a global bounded weak solution for any sufficiently regular initial data n 0 c 0 u 0 satisfying n 0 0 and c 0 0

meromorphic solutions of difference painleve equations we show that under certain conditions some difference painleve equations have nontrivial meromorphic solutions in the whole complex plane these meromorphic solutions are obtained by analytic continuation of asymptotic solutions given in sectors of zero opening angle the existence of these asymptotic solutions is shown by using fundamental matrix solutions of the associated linear systems with double unit characteristic roots and with coefficients expressed by asymptotic series in fractional powers

on the solution of discrete time markovian jump linear quadratic control problems abstract a necessary and sufficient condition for the existence of a positive semidefinite solution of the coupled algebraic discrete time riccati like equation occurring in markovian jump control problems is derived by verifying a simple matrix inequality it is shown that such a solution exists and can be obtained as a limit of a monotonic sequence this leads to a straightforward numerical algorithm for the computation of the solution an example is given to illustrate the proposed method

a stable iterative adaptive control an adaptive control algorithm that carries out the online controller design iteratively is formulated a set of conditions is provided under which the closed loop system is stable in particular the controller iterations must satisfy certain convergence and continuity assumptions but only one step iteration is required for each parameter update resulting in a considerable reduction in computation time the iteration algorithm is applied to an adaptive linear quadratic regulator

global existence and stabilization in a degenerate chemotaxis stokes system with mildly strong diffusion enhancement abstract a class of chemotaxis stokes systems generalizing the prototype n t u n n m 1 n n c c t u c c n c u t p u n u 0 is considered in bounded convex three dimensional domains where w 2 is given the paper develops an analytical approach which consists in a combination of energy based arguments and maximal sobolev regularity theory and which allows for the construction of global bounded weak solutions to an associated initial boundary value problem under the assumption that 0 1 m 9 8 moreover the obtained solutions are shown to approach the spatially homogeneous steady state 1 n 0 0 0 in the large time limit this extends previous results which either relied on different and apparently less significant energy type structures or on completely alternative approaches and thereby exclusively achieved comparable results under hypotheses stronger than 0 1

detectability and stabilizability of time varying discrete time linear systems the concepts of detectability and stabilizability are explored for time varying systems we study duality invariance under feedback an extended version of the lemma of lyapunov existence of stabilizing feedback laws linear quadratic filtering and control and the existence of approximate canonical forms

time varying discrete riccati equation some monotonicity results abstract using a fre chet derivative based approach some monotonicity and comparison results concerning the solutions of the time varying discrete time riccati equation are obtained connections with the existence of the semistabilizing solution are made explicit as well

estimates of asymptotic trajectory bounds in digital implementations of linear feedback control systems estimates are established for trajectory bounds which are easy to calculate applicable to general controller topologies and dependent on parameters i e eigenvalues and eigenvectors of the idealized linear system i e the control system model neglecting quantizers the approach utilizes the comparison principle in the lyapunov theory and the properties of minimal matrix norms the applicability of the results are demonstrated by means of a specific example

the set of positive semidefinite solutions of the algebraic riccati equation of discrete time optimal control in this paper the algebraic riccati equation are of the discrete time linear quadratic lq optimal control problem and its set of positive semidefinite solutions is studied under the most general assumption which is output stabilizability with respect to an appropriate basis the discrete time algebraic riccati equation dare decomposes into a lyapunov equation and an irreducible riccati equation the focus is on the riccati part which amounts to studying a dare where all unimodular modes are controllable a bijection between positive semidefinite solutions and certain well defined sets of f invariant subspaces is established which together with its inverse is order reversing as an application issues concerning positive definite or strong solutions are clarified analogous results for negative semidefinite solutions are valid only under an additional assumption on the unobservable subspace