non constant positive steady states of a prey predator system with cross diffusions abstract in this paper we study a strongly coupled elliptic system arising from a lotka volterra prey predator system where cross diffusions are included in such a way that the prey runs away from the predator and the predator moves away from a large group of preys we establish the existence and non existence of its non constant positive solutions our results show that if m 1 b a 2 m 1 b 1 m 1 m 2 when 0 m 1 m 2 1 or a m 1 b when m 1 m 2 1 0 d 1 m 1 v u 1 d 2 0 d 3 0 and d 4 1 m 1 v u then there exists d 1 d 2 d 3 d 4 such that the stationary problem admits non constant positive solutions otherwise the stationary problem has no non constant positive solution in particular the results indicate that its non constant positive solutions are mainly created by the cross diffusion d 4

explicit solutions to the singular discrete finite time linear estimation problem the problem of optimal filtering of linear discrete time processes is considered it is assumed that the driving and measurement noise signals are stationary white and that the number of measurements is finite explicit expressions for the filtering error covariance and the corresponding kalman gain matrices are obtained for the limiting case where the measurement noise intensity tends uniformly to zero similar expressions are derived for the limiting singular smoothing problem with finite number of measurements

the optimal projection equations for reduced order state estimation the singular measurement noise case the optimal projection equations for reduced order state estimation are generalized to allow for singular i e colored measurement noise the noisy and noise free measurements serve as inputs to dynamic and static estimators respectively the optimal solution is characterized by necessary conditions which involve a pair of oblique projections corresponding to reduced estimator order and singular measurement noise intensity

explicit solution to the singular discrete time stationary linear filtering problem a closed form solution to the stationary discrete time linear filtering problem is obtained explicitly in terms of the system state space matrices in the limiting singular case where the measurement noise tends to zero simple expressions in closed form are obtained for the kalman gain matrix both for uniform and nonuniform rank systems and the explicit eigenstructure of the kalman filter closed loop matrix is derived the minimum error covariance matrices of the a priori and a posteriori filtered estimates are obtained using this special eigenstructure and a remarkably different behavior of the solution in the minimum and nonminimum phase cases is found

riccati equation solution of the minimum time output control problem by taking a geometrical approach and by using a type of orthogonality in constructing bases for the k th controllable subspaces a general solution is derived for the multivariable minimum time output control problem the solution is easier to implement than solutions derived by traditional approaches to output control and in fact it is shown that the optimal control sequence may be constructed from the solution to a specialized riccati equation this new approach thus relates geometrical approaches to the problem to the previously known result that output control is just a special case of the singular linear quadratic optimal control problem

riccati equation computation af supremal f g invariant and reachability subspaces we consider two riccati equations and associate with each riccati equation three recursions for generating subspace sequences it is shown that with appropriate choice of parameters and initial conditions several of these recursions generate suhspace sequences which are important in system theory particular examples are the supremal unobservable suhspace the supremal teachability subspace and the smallest reachable subspace obtained by output injection the recursions may be implemented by solving the associated riccati equations and thus we are able to derive methods for computing the above mentioned subspaces in addition we derive a feedback which makes a system maximally unobservable and also present nonrecursive expressions for the supremal unobservable subspace and for the subspace of k predictable directions

loop transfer recovery techniques for discrete time optimal regulators using prediction estimators for discrete time lqg optimal regulators using current estimators it has been shown that perfect recoveries of the loop transfer matrices at the input or at the output are possible for square minimum phase systems however current estimators cannot be used when the processing time of the controller is significant in this note the loop transfer recoveries using prediction estimators for square minimum phase systems are considered it is shown that although the perfect recoveries are impossible the feedback properties obtained by the recovery techniques are those that can be recovered best in the presence of the delay in the controller

analysis of a parabolic cross diffusion population model without self diffusion abstract the global existence of non negative weak solutions to a strongly coupled parabolic system arising in population dynamics is shown the cross diffusion terms are allowed to be arbitrarily large whereas the self diffusion terms are assumed to disappear the last assumption complicates the analysis since these terms usually provide h 1 estimates of the solutions the existence proof is based on a positivity preserving backward euler galerkin approximation discrete entropy estimates and l 1 weak compactness arguments furthermore employing the entropy entropy production method we show for special stationary solutions that the transient solution converges exponentially fast to its steady state as a by product we prove that only constant steady states exist if the inter specific competition parameters disappear no matter how strong the cross diffusion constants are

a generalized inverse solution to the discrete time singular riccati equation a closed form solution to the discrete time singular riccati equation is presented which is based on the moore penrose matrix inverse applications to deadbeat control and stabilizability are discussed

brief limiting performance of optimal linear discrete filters the limiting variance of the estimation error is obtained for the optimal h 2 filter for discrete time systems when the intensity of the measurement noise tends to zero in particular an explicit expression is obtained for the lowest achievable mean square error when the measurement noise is slowly varying and has intensity which tends to zero this limitation can be characterized completely by the system order the dimension of the process noise the number of the system s transmission zeros and the location of the system s unstable transmission zeros

explicit solution to the singular lq regulation problem an explicit solution to the multivariable discrete linear quadratic lq regulation problem is obtained in the limiting singular case where the input weighting matrix tends to zero such a solution follows from a suitable spectral factorization of the input spectrum density matrix under the assumption that the system is stabilizable and detectable and that its transfer function matrix is of full rank the suitable spectral factor is shown to be the product of the system s minimum phase image and its unitary interactor matrix the unitary interactor matrix defined is a special case of the nilpotent interactor matrix defined by m w rogozinski et al 1987

the design of optimal reduced order stochastic observers for discrete time linear systems the minimum variance state estimation of linear discrete time systems with random white noise input and partially noisy measurements is investigated an observer of minimal order is found which attains the minimum variance estimation error the structure of this observer is shown to depend strongly on the geometry of the system this geometry dictates the length of the delays that are applied on the measurements in order to obtain the optimal estimate the transmission properties of the observer are investigated for systems that are left invertible and free of measurement noise an explicit expression is found for the transfer function matrix of this observer from which a simple solution to the linear discrete time singular optimal filtering problem is obtained

cheap optimal control of discrete single input single output systems this paper presents a solution to the linear discrete quadratic optimal control problem for a single input single output plant with a control weighting that tends to zero explicit expressions are derived for the optimal gain vector in terms of the first significant powers of the control weighting and the limiting values of these expressions are considered the optimal index of performance is obtained both for the minimum and non minimum phase plants the clear distinction between these two types of plants is discussed and the non vanishing value of the index of performance in the latter case is expressed in terms of the zero directions of the system extensions to the single input multi output case and the case where the index of performance is a function of dynamic operators are introduced

a closed form solution to the discrete time kalman filter and its applications abstract this paper presents a closed form solution to the discrete time kalman filter and its applications we first represent the kalman filter in terms of model parameters without using the riccati equation and requiring any artificial conditions such as invertibility of a system matrix and no system noises replacing the initial time with the fixed lag time to achieve the finite memory with respect to inputs and outputs and choosing the proper initial covariances on the recent finite horizon we easily obtain the minimum variance finite memory filter and then propose its iterative computation algorithm as another application a closed form solution to the difference riccati equation on the finite horizon is utilized to obtain a stabilizing gain matrix of a luenberger type filter as in ackermann s formula

sensitivity properties of a class of discrete time lqg controllers with computation delays sensitivity properties of the discrete time lqg controller which utilizes the optimal state predictor to compensate l step computation delays are discussed we obtain multiplicative decompositions of the sensitivity matrices at the input and output of a plant which clarify the effect of computation delays and imperfect observation furthermore these decompositions are used to show that duality between the sensitivity properties at the input and those at the output holds for l 1 and to identify the asymptotic sensitivity matrices obtained by loop transfer recovery procedures

on prediction type reduced order observers for discrete time systems we define the combination of the standard reduced order observer with one step ahead state predictor as the prediction type reduced order observer we first obtain a simple and significant relation that ensures the equivalence of the reduced and the full order observers in the steady state we show the fact that a prediction type reduced order does not always provide a reduced order controller we also discuss the implications of the equivalence of the observers on state space representation of the doubly coprime factorization and yjb parametrization we use the equivalence relation to clarify the feedback properties achieved by the loop transfer recovery technique using the prediction type reduced order observer at both the input and output of a square and minimum phase plant in addition we discuss the computational requirements for the observer based controllers

singular filtering via spectral interactor matrix in this paper the notion of spectral interactor matrix sim for a given square transfer matrix is introduced and used in the singular filtering problem a sim is defined as an interactor matrix preserving the spectral properties of the underlying system after studying the class of all spectral interactor matrices associated with a given transfer matrix an algorithm for the computation of a sim is proposed by resorting to the sim it is possible to point out the role played by delays zeros at infinity and nonminimum phase zeros in the solution of the singular filtering problem

a riccati equation approach to the singular lqg problem abstract the problem of optimal fixed order dynamic compensation for the singular lqg problem is considered necessary conditions characterizing the optimal compensator for the case involving both singular measurement noise and singular control weighting are given the solution consists of a set of two algebraic riccati equations and two lyapunov equations coupled by three projection matrices one projection is the standard order reduction projection while the other two projections reflect the two types of singularity that exist in the system the three projections are shown to satisfy disjointness conditions in addition to order reduction an advantage of the fixed structure approach is that differentiation which is often undersirable from a practical point of view and which may exist in the unconstrained optimal control can be avoided it is shown that the fixed order compensator agrees with the unconstrained solution when the latter possesses the same number of differentiations as are included in the prespecified controller structure and when the order is selected appropriately

a new method for evaluating the log likelihood gradient the hessian and the fisher information matrix for linear dynamic systems a method is presented for evaluating the log likelihood gradient score the hessian and the fisher information matrix of the parameters of linear dynamic stochastic systems the method incorporates the optimal kalman smoothing equations and is therefore ideal for simultaneous state estimation and parameter identification the result can be used for efficient implementation of gradient based algorithms for maximum likelihood identification of the unknown system parameters and for assessing the mean square estimation accuracy

on the convergence of splittings for semidefinite linear systems abstract recently lee et al young ju lee jinbiao wu jinchao xu ludmil zikatanov on the convergence of iterative methods for semidefinite linear systems siam j matrix anal appl 28 2006 634 641 introduce new criteria for the semi convergence of general iterative methods for semidefinite linear systems based on matrix splitting the new conditions generalize the classical notion of p regularity introduced by keller h b keller on the solution of singular and semidefinite linear systems by iterations siam j numer anal 2 1965 281 290 in view of their results we consider here stipulations on a splitting a m n which lead to fixed point systems such that the iterative scheme converges to a weighted moore penrose solution to the system ax b our results extend the result of lee et al to a more general case and we also show that it requires less restrictions on the splittings than keller s p regularity condition to ensure the convergence of iterative scheme

control under stochastic multiplicative uncertainties part ii optimal design for performance this paper studies the optimal control design problem for linear discrete time systems with stochastic multiplicative uncertainties these uncertainties are assumed to be present in the control inputs and modeled as independent and identically distributed i i d random processes the optimal performance under study is defined in the mean square sense referred to as the mean square optimal h 2 performance it is shown that the mean square optimal h 2 control problem via state feedback can be solved using a mean square stabilizing solution to a modified algebraic riccati equation mare a necessary and sufficient condition for the existence of this solution is presented which constitutes a generalization of the solution to the classic optimal h 2 state feedback design problem whereas the latter can be obtained by solving an algebraic riccati equation are it is also proven that the optimal control design problem can be cast as an eigenvalue problem evp for the output feedback case with possible input delays we show that the mean square optimal h 2 control problem also amounts to solving an mare when the plant has no nonminimum phase zeros from the inputs to the measurement outputs that is the global optimal solution is obtained by solving an mare incorporating the delays the implication is that in this case a separation principle still holds

existence and uniqueness of splittings for stationary iterative methods with applications to alternating methods given a nonsingular matrix a and a matrix t of the same order under certain very mild conditions there is a unique splitting a b c such that t b 1 c moreover all properties of the splitting are derived directly from the iteration matrix t these results do not hold when the matrix a is singular in this case given a matrix t and a splitting a b c such that t b 1 c there are infinitely many other splittings corresponding to the same matrices a and t and different splittings can have different properties for instance when t is nonnegative some of these splittings can be regular splittings while others can be only weak splittings analogous results hold in the symmetric positive semidefinite case given a singular matrix a not for all iteration matrices t there is a splitting corresponding to them necessary and sufficient conditions for the existence of such splittings are examined as an illustration of the theory developed the convergence of certain alternating iterations is analyzed different cases where the matrix is monotone singular and positive semi definite are studied

explicit solutions to the linear optimal estimation problem of discrete time invariant linear processes the linear optimal estimation problem of linear discrete time invariant processes is considered using a mixed z and time domain approach explicit expressions am obtained in the case where the measurement record is infinitely long for the constant kalman gain and the minimum error covariance matrix of the filtered estimate in terms of the zero structure of the measurement power spectrum density matrix these expressions are simple to obtain both conceptually and computationally and they provide a geometric insight to the structure of the solution they are also applied to the filtering problems with finite number of measurements and to the optimal discrete smoothing problems where similar explicit results are obtained

positive steady states for a prey predator model with some nonlinear diffusion terms abstract this paper discusses a prey predator system with strongly coupled nonlinear diffusion terms we give a sufficient condition for the existence of positive steady state solutions our proof is based on the bifurcation theory some a priori estimates for steady state solutions will play an important role in the proof

stationary patterns of strongly coupled prey predator models abstract we study some elliptic systems arising from 3 component predator prey models where cross diffusions are included in such a way that predator chases the prey and the prey runs away from the predator we establish the existence and non existence of non constant positive solutions our results show that the cross diffusions can create the stationary patterns

a strongly coupled predator prey system with non monotonic functional response abstract the predator prey system with non monotonic functional response is an interesting field of theoretical study in this paper we consider a strongly coupled partial differential equation model with a non monotonic functional response a holling type iv function in a bounded domain with no flux boundary condition we prove a number of existence and non existence results concerning non constant steady states patterns of the underlying system in particular we demonstrate that cross diffusion can create patterns when the corresponding model without cross diffusion fails

discrete time lqg ltr technique for systems with time delays loop transfer recovery ltr is applied for lqg controllers with prediction estimators for a class of square minimum phase discrete time linear systems with time delays we consider the class of plants described by the triple a b c whose time delay structure is characterized by the following hypothesis on the plant impulse response cb cab cad 2b 0 det cad 1b 0 when the estimator gain is fixed by recovery technique it is shown that the loop transfer matrix at the plant input is equal to the transfer matrix obtained with the optimal digital regulator having d sample controller delays described in 6 the dual result corresponding to the situation where the feedback gain is fixed by recovery technique is also analysed

stability of a reaction diffusion alcohol model with the impact of tax policy abstract to understand the impact of tax policy on the persistence of a drinking behavior a reaction diffusion alcohol model with the impact of tax policy is studied with the focus on the positivity and boundedness of solutions and particularly the asymptotic profile of the equilibria the basic reproduction number r 0 is calculated and existence of a drinking free equilibrium and a unique drinking present equilibrium is established for r 0 above the threshold value 1 for both of ode system and reaction diffusion model it is shown that the drinking free equilibrium is globally asymptotically stable if r 0 1 and if r 0 1 we obtain a sufficient condition for global asymptotically stability of the drinking present equilibrium numerical simulations are also provided to illustrate our analytical results

bifurcation and spatiotemporal patterns in a homogeneous diffusive predator prey system abstract a diffusive predator prey system with holling type ii predator functional response subject to neumann boundary conditions is considered hopf and steady state bifurcation analysis are carried out in details in particular we show the existence of multiple spatially non homogeneous periodic orbits while the system parameters are all spatially homogeneous our results and global bifurcation theory also suggest the existence of loops of spatially non homogeneous periodic orbits and steady state solutions these results provide theoretical evidences to the complex spatiotemporal dynamics found by numerical simulation

coexistence of three species in a strongly coupled elliptic system abstract a strongly coupled elliptic system which describes three interacting species with homogeneous dirichlet boundary conditions is considered it is shown that there is no coexistence state if diffusion rates are strong or if the intrinsic growth rates are slow making use of the schauder fixed point theory we derive some sufficient conditions to have a semi coexistence or a coexistence state for the strongly coupled elliptic problem moreover our results reveal that this problem possesses at least one coexistence state if cross diffusions and intra specific competitions are weak

strategy and stationary pattern in a three species predator prey model abstract in this paper we study a strongly coupled system of partial differential equations which models the dynamics of a two predator one prey ecosystem in which the prey exercises a defense switching mechanism and the predators collaboratively take advantage of the prey s strategy we demonstrate the emergence of stationary patterns for this system and show that it is due to the cross diffusion that arises naturally in the model as far as the authors are aware this is the first example of stationary patterns in a predator prey system arising solely from the effect of cross diffusion

multiple coexistence states for a prey predator system with cross diffusion abstract we study the multiple existence of positive solutions for the following strongly coupled elliptic system 1 v u u a u cv 0 in 1 u v v b du v 0 in u v 0 on where a b c d are positive constants and is a bounded domain in r n this is the steady state problem associated with a prey predator model with cross diffusion effects and u resp v denotes the population density of preys resp predators in particular the presence of represents the tendency of predators to move away from a large group of preys assuming that is small and that is large we show that the system admits a branch of positive solutions which is s or shaped with respect to a bifurcation parameter so that the system has two or three positive solutions for suitable range of parameters our method of analysis uses the idea developed by du lou j differential equations 144 1998 390 and is based on the bifurcation theory and the lyapunov schmidt procedure

cross diffusion induced turing instability for a three species food chain model abstract in this paper we study a strongly coupled reaction diffusion system describing three interacting species in a food chain model where the third species preys on the second one and simultaneously the second species preys on the first one we first show that the unique positive equilibrium solution is globally asymptotically stable for the corresponding ode system the positive equilibrium solution remains linearly stable for the reaction diffusion system without cross diffusion hence it does not belong to the classical turing instability scheme we further proved that the positive equilibrium solution is globally asymptotically stable for the reaction diffusion system without cross diffusion by constructing a lyapunov function but it becomes linearly unstable only when cross diffusion also plays a role in the reaction diffusion system hence the instability is driven solely from the effect of cross diffusion our results also exhibit some interesting combining effects of cross diffusion intra species competitions and inter species interactions

positive steady state solutions of a competing reaction diffusion system with large cross diffusion coefficients abstract this paper deals with a coupled system of reaction diffusion equations modeling the competition of two species with cross diffusion we discuss the existence of positive steady state solutions in relation to the intrinsic birth rates a 1 a 2 of species and the cross diffusion coefficients 12 21 our results show that positive steady state solutions exist if a 1 and a 2 lie in certain range or if 12 and 21 are sufficiently large

a diffusive predator prey model with a protection zone in this paper we study the effects of a protection zone 0 for the prey on a diffusive predator prey model with holling type ii response and no flux boundary condition we show the existence of a critical patch size described by the principal eigenvalue d 0 of the laplacian operator over 0 with homogeneous dirichlet boundary conditions if the protection zone is over the critical patch size i e if d 0 is less than the prey growth rate then the dynamics of the model is fundamentally changed from the usual predator prey dynamics in such a case the prey population persists regardless of the growth rate of its predator and if the predator is strong then the two populations stabilize at a unique coexistence state if the protection zone is below the critical patch size then the dynamics of the model is qualitatively similar to the case without protection zone but the chances of survival of the prey species increase with the size of the protection zone as generally expected our mathematical approach is based on bifurcation theory topological degree theory the comparison principles for elliptic and parabolic equations and various elliptic estimates

on necessary conditions for convergence of stationary iterative methods for hermitian semidefinite linear systems abstract in an earlier paper frommer et al 2008 5 we gave sufficient conditions in terms of an energy seminorm for the convergence of stationary iterations for solving linear systems whose coefficient matrix is hermitian and positive semidefinite in this paper we show in which cases these conditions are also necessary and show that they are not necessary in others

guaranteed stability margins for the discrete time linear quadratic optimal regulator the stability properties of the linear quadratic deterministic optimal regulator are investigated in the discrete time case guaranteed phase and gain margins are obtained for the regulator in terms of the singular values of the matrices of the regulator state space description these margins are compared to their continuous time counterparts and their implication on the regulator robustness is discussed

on the geometric convergence of optimized schwarz methods with applications to elliptic problems the schwarz method can be used for the iterative solution of elliptic boundary value problems on a large domain one subdivides into smaller more manageable subdomains and solves the differential equation in these subdomains using appropriate boundary conditions optimized schwarz methods use robin conditions on the artificial interfaces for information exchange at each iteration and for which one can optimize the robin parameters while the convergence theory of classical schwarz methods with dirichlet conditions on the artificial interface is well understood the overlapping optimized schwarz methods still lack a complete theory in this paper an abstract hilbert space version of the optimized schwarz method osm is presented together with an analysis of conditions for its geometric convergence it is also shown that if the overlap is relatively uniform these convergence conditions are met for optimized schwarz methods for two dimensional elliptic problems for any positive robin parameter in the discrete setting we obtain that the convergence factor h varies like a polylogarithm of h numerical experiments show that the methods work well and that the convergence factor does not appear to depend on h

bifurcation of positive solutions for a three species food chain model with diffusion abstract in this paper we consider a reaction diffusion system describing a three species lotka volterra food chain model with homogeneous dirichlet boundary conditions by regarding the birth rate of prey r 1 as a bifurcation parameter the global bifurcation of positive steady state solutions from the semi trivial solution set is obtained via the bifurcation theory the results show that if the birth rate of mid level predator and top predator are located in the regions 0 r 2 1 a 23 u 3 r 3 and r 3 1 respectively then the three species can co exist provided the birth rate of prey exceeds a critical value moreover an explicit expression of coexistence steady state solutions is constructed by applying the implicit function theorem it is demonstrated that the explicit coexistence steady state solutions is locally asymptotically stable

extended discrete time ltr synthesis of delayed control systems abstract this paper addresses compensation of dealys within a multi input multi output discrete time feedback loop for application to real time distributed control systems the delay compensation algorithm formulated in this paper is an extension of the standard loop transfer recovery ltr procedure from one step prediction to the general case of p step prediction p 1 it is shown that the steady state minimum variance filter gain is the h 2 minimization solution of the relative error between the target sensitivity matrix and the actual sensitivity matrix for p step prediction p 1 this concept forms the basis for synthesis of robust p step delay compensators p 1 the proposed control synthesis procedure for delay compensation is demonstrated via simulation of the flight control system of an advanced aircraft

discrete time loop transfer recovery for systems with nonminimum phase zeros and time delays abstract the purpose of this paper is to study what happens when the discrete time loop transfer recovery ltr procedure is applied to plants with nonminimum phase zeros and with uniform time delay in all channels explicit expressions are given for the asymptotic behavior of the resulting sensitivity function and loop transfer function these results yield a better understanding of the mechanism of the discrete time loop transfer recovery procedure and the design limitations due to nonminimum phase zeros and time delays

turing pattern formation in a three species model with generalist predator and cross diffusion abstract in a natural ecosystem specialist predators feed almost exclusively on one species of prey but generalist predators feed on many types of species consequently their dynamics is not coupled to the dynamics of a specific prey population however the defense of prey formed by congregating made the predator tend to move in the direction of lower concentration of prey species this is described by cross diffusion in a generalist predator prey model first the positive equilibrium solution is globally asymptotically stable for the ode system and for the reaction diffusion system without cross diffusion respectively hence it does not belong to the classical turing instability scheme but it becomes linearly unstable only when cross diffusion also plays a role this implies that cross diffusion can lead to the occurrence and disappearance of the instability our results exhibit some interesting combining effects of cross diffusion predations and intra species interactions furthermore we consider the existence and non existence results concerning non constant positive steady states patterns of the system we demonstrate that cross diffusion can create non constant positive steady state solutions

decoupling properties of singular lq regulation problem some properties of a limiting solution derived from a standard lq regulator problem are discussed in the case where the input weighting matrix tends to zero for a discrete time system the explicit limiting solution is given in some literature it is shown that the limiting solution is closely connected with a disturbance decoupling problem moreover it is pointed out that the limiting feedback gain with the help of feedforward gain constructs the decoupled system with the maximum number of blocks

solution in the z domain to the discrete time stationary kalman filtering problem for systems with perfect measurements the discrete time stationary kalman filtering problem is solved in the z domain for general right invertible linear systems whose measurements are all noise free a simple expression in closed form is obtained for the transfer function of the optimal filter both for the uniform and the general nonuniform rank cases it provides an optimal solution also for systems with uniform and nonuniform time delays the minimum variance error of the estimate is analyzed and the independent contributions to this error of the system zeros outside the unit circle finite and infinite and the excess of the number of inputs over the number of measurements are investigated

optimal nearly singular estimation of continuous linear stationary uniform rank systems the problem of optimal estimation of linear continuous systems is considered for the case where all the noise signals are stationary white and the intensity of the measurement noise tends uniformly to zero explicit expressions for the estimation procedures and the corresponding minimum error covariance matrices of the estimate are obtained in terms of the measurement noise intensity and the zero properties of the system these expressions are derived for the case where the first non zero markov parameter of the system is of full rank for the infinite and the finite time filtering problems they provide a simple geometric interpretation to the structure of the error covariance matrices and they distinctly show the difference between the estimation of minimum and non minimum phase systems the theory is developed both for square and non square systems

from singular to nonsingular filtering of periodic systems filling the gap with the spectral interactor matrix with reference to periodic discrete time systems the concept of the periodic spectral interactor matrix operator psimo is introduced such an interactor enables one to transform the singular system into a nonsingular one with identical spectral properties this concept is useful to characterize the family of periodic solutions of the singular periodic riccati equation arising from the singular filtering problem the psimo is also a synthetic tool to represent the invariant zeros structure at infinity delays of periodic systems in an operational fashion

ltr design of discrete time proportional integral observers applies the proportional integral pi observer in connection with loop transfer recovery ltr design for discrete time systems both the prediction and the filtering versions of the pi observer are considered we show that a pi observer makes it possible to obtain time recovery i e exact recovery for t spl rarr spl infin under mild conditions two systematic ltr design methods one based on an extension of the linear quadratic gaussian loop transfer recovery lqg ltr and the other based on linear matrix inequality lmi are derived for the discrete time pi observer case explicit expressions for the recovery error when exact recovery is not achievable for all frequencies are also given

a general riccati equation solution to the deadbeat control problem it is well known that the state of a controllable system can be driven to zero in minimum time for arbitrary initial condition by the time varying kalman gain sequence computed via a singular riccati equation this paper shows how a constant deadbeat control feedback which will accomplish the same result in the same minimum time can always be found via the same riccati equation

allee effect and bistability in a spatially heterogeneous predator prey model a spatially heterogeneous reaction diffusion system modelling predator prey interaction is studied where the interaction is governed by a holling type ii functional response existence of multiple positive steady states and global bifurcation branches are examined as well as related dynamical behavior it is found that while the predator population is not far from a constant level the prey population could be extinguished persist or blow up depending on the initial population distributions the various parameters in the system and the heterogeneous environment in particular our results show that when the prey growth is strong the spatial heterogeneity of the environment can play a dominant role for the presence of the allee effect our mathematical analysis relies on bifurcation theory topological methods various comparison principles and elliptic estimates we combine these methods with monotonicity arguments to the system through the use of some new auxiliary scalar equations though the system itself does not keep an order structure as the competition system does among other things this allows us to obtain partial descriptions of the dynamical behavior of the system

splittings of symmetric matrices and a question of ortega a complete answer is given to a problem posed in 1988 by ortega concerning convergent splittings of symmetric matrices

exhibiting cross diffusion induced patterns for reaction diffusion systems on evolving domains and surfaces the aim of this manuscript is to present for the first time the application of the finite element method for solving reaction diffusion systems with cross diffusion on continuously evolving domains and surfaces furthermore we present pattern formation generated by the reaction diffusion systemwith cross diffusion on evolving domains and surfaces a two component reaction diffusion system with linear cross diffusion in both u and v is presented the finite element method is based on the approximation of the domain or surface by a triangulated domain or surface consisting of a union of triangles for surfaces the vertices of the triangulation lie on the continuous surface a finite element space of functions is then defined by taking the continuous functions which are linear affine on each simplex of the triangulated domain or surface to demonstrate the role of cross diffusion to the theory of pattern formation we compute patterns with model kinetic parameter values that belong only to the cross diffusion parameter space these do not belong to the standard parameter space for classical reaction diffusion systems numerical results exhibited show the robustness flexibility versatility and generality of our methodology the methodology can deal with complicated evolution laws of the domain and surface and these include uniform isotropic and anisotropic growth profiles as well as those profiles driven by chemical concentrations residing in the domain or on the surface