mitter conjecture for low dimensional estimation algebras in non linear filtering it is well known that a systematical way to construct finite dimensional filter is to classify all finite dimensional estimation algebras mitter conjecture which states that all functions in any finite dimensional estimation algebra are necessarily degree one polynomial plays a crucial role in classifying all finite dimensional estimation algebras the purpose of this paper is to prove the mitter conjecture for estimation algebra of dimension at most 5 with arbitrary state space dimension dedicated to professor han fu chen on the occasion of his 70th birthday

a note on estimation algebras on nonlinear filtering theory abstract the idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed by brockett and mitter independently it turns out that the concept of estimation algebra plays a crucial role in the investigation of finite dimensional nonlinear filters in his talk at the international congress of mathematicians in 1983 brockett proposed to classify all finite dimensional estimation algebras in this paper we consider some filtering systems in a special filtering system 1 we have some structure results 2 for any arbitrary finite dimensional state space under the condition that the drift term is a linear vector field plus a gradient vector field we classify all finite dimensional estimation algebras with maximal rank 3 we classify all finite dimensional estimation algebras with maximal rank if the dimension of the state space is less than or equal to three a more general filtering system is considered the above three results can be used locally therefore from the algebraic point of view we have now understood generically some finite dimensional filters

on a necessary and sufficient condition for finite dimensionality of estimation ever since the technique of the kalman bucy filter was popularized there has been an intense interest in finding new classes of finite dimensional recursive filters in the late seventies the concept of the estimation algebra of a filtering system was introduced it has proven to be an invaluable tool in the study of nonlinear filtering problems in this paper a simple algebraic necessary and sufficient condition is established for an estimation algebra of a special class of filtering systems to be finite dimensional also presented is a rigorous proof of the wei norman program which allows one to construct finite dimensional recursive filters from finite dimensional estimation algebras

optimal control for partially observed diffusions stochastic control problems are considered in which a state process x t and an observation process y t are governed by ito sense stochastic differential equations driven by independent brownian motions the control u t enters linearly in the dynamics of x t a separated control problem is introduced in which the state at any time t is a measure lambda t representing an unnormalized conditional distribution for x t given y s u s for s leqq t the method depends on introducing a pathwise version of lambda t which depends continuously on observation and control trajectories y u existence of an optimal control is obtained in a suitable class larger than the usual class of controls admissible in the strict sense that u t is measurable on the sigma algebra mathcal f t y generated by observations y s s leqq t the dynamics of lambda t are studied using a method of forward and backward partial differential equations under a suitable nondegeneracy condition the m

modular properties of the hard hexagon model the physical quantities or powers thereof in the hard hexagon model that were computed exactly by baxter are shown to be modular functions with respect to the number theoretic group 1 n this allows us to determine the analytic structure of the partition function per site in the thermodynamic limit and the density as functions of the activityz

parallel computations for yau filters in the early 1990s yau developed a new class of nonlinear filters called the yau filterswhich contain the kalman bucy filters and the benes filters as special cases it has been shown that from the lie algebraic point of view the yau filters are the most general finite dimensional filters yau and hu proved that the dmz equation for a yau filter can be reduced to a kolmogorov type partial differential equation and a system of linear differential equations they noticed that the pde is independent of the observed data and hence can be solved off line an efficient parallel algorithm for the system of odes would lead to fast solutions of yau filters and hence their suitability for real life applications in this paper we have proposed several parallel methods suitable for this system of odes

mitter conjecture and structure theorem for six dimensional estimation algebras the problem of classification of finite dimensional estimation algebras was formally proposed by brockett in his lecture at international congress of mathematicians in 1983 due to the difficulty of the problem in the early 1990s brockett suggested that one should understand the low dimensional estimation algebras first in this article we extend yau and his coauthors work of the mitter conjecture for low dimensional estimation algebras in nonlinear filtering problem and we apply the results to give classification of estimation algebras of dimension six

direct method for yau filtering system with nonlinear observations for all known finite dimensional filters one always assumes that the observation terms be degree one polynomials however in practice the observation terms may be nonlinear e g tracking proble

lie algebraic methods in optimal control of stochastic systems with exponential of integral cost abstract the purpose of this paper is to formulate and study the optimal control of partially observed stochastic systems with exponential of integral sample cost known as risk sensitive problems using lie algebraic tools this leads to the introduction of the sufficient statistic algebra l s through which one can determine a priori the maximum order of the controller when dim l s the construction of the control laws is addressed through extensions of the wei norman method as in nonlinear filtering problems aside from specific known finite dimensional examples which are studied in order to delineate the application of the lie algebraic tools new classes of finite dimensional controllers are identified as well in addition relations with minimax dynamic games are explored to best assess the importance and generality of the finite dimensional control systems

a comparison principle for hamilton jacobi equations related to controlled gradient flows in infinite dimensions we develop new comparison principles for viscosity solutions of hamilton jacobi equations associated with controlled gradient flows in function spaces as well as the space of probability measures our examples are optimal control of ginzburg landau and fokker planck equations they arise in limit considerations of externally forced non equilibrium statistical mechanics models or through the large deviation principle for interacting particle systems our approach is based on two key ingredients an appropriate choice of geometric structure defining the gradient flow and a free energy inequality resulting from such gradient flow structure the approach allows us to handle hamiltonians with singular state dependency in the nonlinear term as well as hamiltonians with a state space which does not satisfy the radon nikodym property in the case where the state space is a hilbert space the method simplifies existing theories by avoiding the perturbed optimization principle

optimal control for a mixed flow of hamiltonian and gradient type in space of probability measures abstract in this paper we investigate an optimal control problem in the space of measures on r2 the problem is motivated by a stochastic interacting particle model which gives the 2 d navier stokes equations in their vorticity formulation as a mean field equation we prove that the associated hamiltonjacobi bellman equation in the space of probability measures is well posed in an appropriately defined viscosity solution sense

wei norman equations for a unitary evolution the wei norman technique allows one to express the solution of a system of linear non autonomous differential equations in terms of product of exponentials with time dependent exponents being solutions of a system of nonlinear differential equations we show that in the unitary case i e when the solution of the linear system is given by a unitary evolution operator the nonlinear system by an appropriate choice of ordering can be reduced to a hierarchy of matrix riccati equations to this end we consider a general linear non autonomous dynamical system on the special linear group the unitary case of particular significance for quantum optimal control problems is then obtained by restriction to anti hermitian generators we also point to the connections of the obtained results with the theory of the so called lie systems

a software package for lie algebraic computations the paper presents a computer algebra package that facilitates lie algebraic symbolic computations required in the solution of a variety of problems such as the solution of right invariant differential equations evolving on lie groups lie theory is a powerful tool helpful in the analysis and design of modern nonlinear control laws nonlinear filters and the study of particle dynamics the practical application of lie theory often results in highly complex symbolic expressions that are difficult to handle efficiently without the aid of a computer software tool the aim of the package is to facilitate and encourage further research relying on lie algebraic computations

explicit solution of dmz equation in nonlinear filtering via solution of odes in this note we develop a real time and accurate solution for nonlinear filtering problems based on the gaussian distribution specifically we present an explicit solution of the duncan mortensen zakai dmz equation of the yau filtering system which includes the linear filtering system and exact filtering system the solution is given in terms of a solution of a system of ordinary differential equations in particular our method can be implemented in hardware the complexity of our algorithms is the same as those of kalman bucy filters in the case of linear filtering systems

new explicit filters and smoothers for diffusions with nonlinear drift and measurements the optimal least squares filtering of a diffusion x t from its noisy measurements y 0 t is given by the conditional mean e x t y 0 t when x t satisfies the stochastic diffusion equation dx t f x t dt dw t and y t 0tx s ds b t where f is a global solution of the riccati equation xf x f x 2 f x 2 x2 x for some r3 and w b are independent brownian motions benes gave an explicit formula for computing the conditional mean this paper extends benes results to measurements y t 0tx s ds 0t dx s b t and its multidimensional version without imposing additional conditions on f analogous results are also derived for the optimal least squares smoothed estimate e x s y 0 t s t the methodology relies on girsanov s measure transformations gauge transformations function space integrations lie algebras and the duncan mortensen zakai equation

equation of state and isothermal compressibility for the hard hexagon model in the disordered regime the equation of state for the hard hexagon model in the disordered regime is computed the isothermal compressibility is given as a rational function of k and p the hard hexagon model is a lattice gas on a triangular lattice with the rule that if a site is occupied then the six neighbouring sites are necessarily vacant baxter 1980 1981 has shown that this model is exactly solvable by the method of commuting transfer matrices baxter 1972 1982 in the disordered regime baxter s exact results are

on the relation of zakai s and mortensen s equations the problem of optimal control for partially observable diffusion processes is studied by the dynamic programming in function space approach first proposed by r e mortensen the density of the conditional distribution of the unobservable signal given past and present observations which satisfies the zakai equation of nonlinear filtering is viewed as the new state of the system a verification result is established for the corresponding bellman hamilton jacobi equation known as mortensen s equation

certain nonlinear partially observable stochastic optimal control problems with explicit control laws equivalent to leqg lqg problems this paper is concerned with partially observed stochastic optimal control problems when nonlinearities enter the dynamics of the unobservable state and the observations as gradients of potential functions explicit representations for the information state are derived in terms of a finite number of sufficient statistics consequently the partially observed problem is recast as one of complete information with a new state generated by a modified version of the kalman filter when the terminal cost is quadratic in the unobservable state and includes the integral of the nonlinearities the optimal control laws are explicitly computed similar to linear exponential quadratic gaussian leqg and linear quadratic gaussian lqg tracking problems the results are applicable to filtering and control of hamiltonian systems

finite dimensional filters with nonlinear drift viii classification of finite dimensional estimation algebras of maximal rank with state space dimension 4 the idea of using estimation algebras to construct finite dimensional nonlinear filters was first proposed by brockett and mitter independently it turns out that the concept of estimation algebra plays a crucial role in the investigation of finite dimensional nonlinear filters in his talk at the international congress of mathematics in 1983 brockett proposed a classification of all finite dimensional estimation algebras chiou and yau classify all finite dimensional estimation algebras of maximal rank with dimension of the state space less than or equal to two in this paper we succeed in classifying all finite dimensional estimation algebras of maximal rank with state space dimension equal to three thus from the lie algebraic point of view we have now understood generically all finite dimensional filters with state space dimension less than four

rogers ramanujan identities in the hard hexagon model the hard hexagon model in statistical mechanics is a special case of a solvable class of hard square type models in which certain special diagonal interactions are added the sublattice densities and order parameters of this class are obtained and it is shown that many rogers ramanujan type identities naturally enter the working

exact results for the activity and isothermal compressibility of the hard hexagon model the klein and fricke theory of modular functions is used to derive closed form algebraic expressions for the activity z rho and isothermal compressibility kappa t rho of the hard hexagon lattice gas model in the disordered regime where rho is the dimensionless number density similar results which are valid in the ordered regime are also presented

classification of finite dimensional estimation algebras of maximal rank with arbitrary state space dimension and mitter conjecture in the late seventies the concept of the estimation algebra of a filtering system was introduced it was proven to be an invaluable tool in the study of non linear filtering problems in the early eighties brockett proposed to classify finite dimensional estimation algebras and mitter conjectured that all functions in finite dimensional estimation algebras are necessarily polynomials of total degree at most one despite the massive effort in understanding the finite dimensional estimation algebras the 20 year old problem of brockett and mitter conjecture remains open in this paper we give a classification of finite dimensional estimation algebras of maximal rank and solve the mitter conjecture affirmatively for finite dimensional estimation algebras of maximal rank in particular for an estimation algebra e of maximal rank we give a necessary and sufficient conditions for e to be finite dimensional in terms of the drift fi x and observation hj x as an important corollary we show that the numb

hamiltonian odes in the wasserstein space of probability measures in this paper we consider a hamiltonian h on p 2 r 2d the set of probability measures with finite quadratic moments on the phase space r 2d r d r d which is a metric space when endowed with the wasserstein distance w 2 we study the initial value problem d t dt j d v t t 0 where j d is the canonical symplectic matrix 0 is prescribed and v t is a tangent vector to p 2 r 2d at t belonging to h t the subdifferential of h at t two methods for constructing solutions of the evolutive system are provided the first one concerns only the case where 0 is absolutely continuous it ensures that fit remains absolutely continuous and v t h t is the element of minimal norm in h t the second method handles any initial measure 0 if we further assume that h is convex proper and lower semicontinuous on p 2 r 2d we prove that the hamiltonian is preserved along any solution of our evolutive system h t h 0

finite dimensional filters with non linear drift ix construction of finite dimensional estimation algebras of non maximal rank the idea of using estimation algebra to construct finite dimensional nonlinear filters was first proposed by brockett and mitter independently it has proven to be an invaluable tool in the study of nonlinear filtering problem in 1983 brockett proposed to classify all finite dimensional estimation algebras in this paper we give the construction of finite dimensional estimation algebras of non maximal rank these non maximal rank finite dimensional estimation algebras play an important role in brockett s classification problem

partially observable nonlinear risk sensitive control problems dynamic programming and verification theorems in this paper we consider continuous time partially observable optimal control problems with exponential of integral cost criteria we derive a rigorous verification theorem when the state and control enter nonlinear in the dynamics in addition we show that the quadratic sensor problem is estimation solvable with respect to a certain cost criterion the framework relies on dynamic programming and the hamilton jacobi theory

on a new class of finite dimensional estimation algebras abstract we introduce a new class of solvable finite dimensional estimation algebras that contains both the linear and the benes cases the condition for finite dimensionality is expressed in terms of two matrices they play an important role in understanding the structure of estimation algebras

a finite dimensional risk sensitive control problem a partially observed stochastic control problem with exponential running cost is considered the dynamics are linear and the running cost is quadratic although the control may enter nonlinearly explicit solutions are found to a modified zakai equation and a backward adjoint equation this enables the problem to be expressed in terms of observable finite dimensional dynamics and a separation principle to be applied

value functions in the wasserstein spaces finite time horizons abstract we study analogs of value functions arising in classical mechanics in the space of probability measures endowed with the wasserstein metric w p for 1 p our main result is that each of these generalized value functions is a type of viscosity solution of an appropriate hamilton jacobi equation completing a program initiated by gangbo nguyen and tudorascu of particular interest is a formula we derive for a generalized value function when the associated potential energy is of the form v r d v x d x this formula allows us to make rigorous a well known heuristic connection between euler poisson equations and classical hamilton jacobi equations further results are presented which suggest there is a rich theory to be developed of deterministic control in the wasserstein spaces

classes of nonlinear partially observable stochastic optimal control problems with explicit optimal control laws this paper introduces certain nonlinear partially observable stochastic optimal control problems which are equivalent to completely observable control problems with finite dimensional state space in some cases the optimal control laws are analogous to linear exponential quadratic gaussian and linear quadratic gaussian tracking problems the problems discussed allow nonlinearities to enter the unobservable dynamics as gradients of potential functions the methodology is based on explicit solutions of a modified duncan mortensen zakai equation

monte carlo approximations for 2d navier stokes equations with measure initial data we are interested in proving monte carlo approximations for 2d navier stokes equations with initial data u 0 belonging to the lorentz space l 2 and such that curl u 0 is a finite measure giga miyakawaand osada 7 proved that a solution u exists and that u k curl u where k is the biot savartkernel and v curl u is solution of a nonlinear equation in dimension one called the vortex equation in this paper we approximate a solution v of this vortex equationby a stochastic interacting particlesystem and deduce a monte carlo approximation for a solution of the navier stokesequation that gives in this case a pathwise proofof the vortex algorithm introducedby chorin and consequently generalizes the works ofmarchioro pulvirenti 12 and meleardv 15 obtained in the case of a vortex equation with bounded density initial data

a smooth variational principle with applications to subdifferentiability and to differentiability of convex functions we show that typically lower semicontinuous functions on a banach space densely inherit lower subderivatives of the same degree of smoothness as the norm in particular every continuous convex function on a space with a g teaux weak hadamard frechet smooth renorm is densely g teaux weak hadamard frechet differentiable our technique relies on a more powerful analogue of ekeland s variational principle in which the function is perturbed by a quadratic like function this smooth variational principle has very broad applicability in problems of nonsmooth analysis

the novel classes of finite dimensional filters with non maximal rank estimation algebra on state dimension four and rank of one ever since the technique of kalman bucy filter was popularised finding new classes of finite dimensional recursive filters has drawn much concern the idea of using estimation algebra to construct

the role of information state and adjoint in relating nonlinear output feedback risk sensitive control and dynamic games this paper employs logarithmic transformations to establish relations between continuous time nonlinear partially observable risk sensitive control problems and analogous output feedback dynamic games the first logarithmic transformation is introduced to relate the stochastic information state to a deterministic information state the second logarithmic transformation is applied to the risk sensitive cost function using the laplace varadhan lemma in the small noise limit this cost function is shown to be logarithmically equivalent to the cost function of an analogous dynamic game

on a class of first order hamilton jacobi equations in metric spaces abstract we establish well posedness of a class of first order hamilton jacobi equation in geodesic metric spaces the result is then applied to solve a hamilton jacobi equation in the wasserstein space of probability measures which arises from the variational formulation of a compressible euler equation

structure theorem for five dimensional estimation algebras abstract the problem of classification of finite dimensional estimation algebras was formally proposed by brockett in his lecture at international congress of mathematicians in 1983 due to the difficulty of the problem in the early 1990s brockett suggested that one should understand the low dimensional estimation algebras first in this paper we give classification of estimation algebras of dimension at most five although the classification of finite dimensional estimation algebra of maximal rank was completed by yau and his coworkers chen chiou hu wong and wu the problem of classification of non maximal rank finite dimensional estimation algebra is still wide open except for the case of state space dimension 2 hopefully the result of this paper will shed some light on the non maximal rank estimation algebras

hydrodynamics in two dimensions and vortex theory we consider the navier stokes equation for a viscous and incompressible fluid inr2 we show that such an equation may be interpreted as a mean field equation vlasov like limit for a system of particles called vortices interacting via a logarithmic potential on which in addition a stochastic perturbation is acting more precisely we prove that the solutions of the navier stokes equation may be approximated in a suitable way by finite dimensional diffusion processes with the diffusion constant related to the viscosity as a particular case when the diffusion constant is zero the finite dimensional theory reduces to the usual deterministic vortex theory and the limiting equation reduces to the euler equation

finite dimensional filters with nonlinear drift x explicit solution of dmz equation in this note we consider the explicit solution of duncan mortensen zakai dmz equation for the finite dimensional filtering system we show that yau s 1990 1994 filtering system spl delta f sub j spl delta x sub i spl delta f sub i spl delta x sub i c sub ij constant for all i j can be solved explicitly with an arbitrary initial condition by solving a system of ordinary differential equations and a kolmogorov type equation let n be the dimension of state space we show that we need only n sufficient statistics in order to solve the dmz equation

finite dimensional filters with nonlinear drift xi explicit solution of the generalized kolmogorov equation in brockett mitter program abstract ever since the technique of the kalman bucy filter was popularized there has been an intense interest in finding new classes of finite dimensional recursive filters in the late 1970s the concept of the estimation algebra of a filtering system was introduced brockett clark and mitter proposed to use the wei norman approach to solve the nonlinear filtering problem in 1990 tam wong and yau presented a rigorous proof of the brocket mitter program which allows one to construct finite dimensional recursive filters from finite dimensional estimation algebras later yau wrote down explicitly a system of ordinary differential equations and generalized kolmogorov equation to which the robust duncan mortenser zakai equation can be reduced thus there remains three fundamental problems in brockett mitter program the first is the problem of finding explicit solution to the generalized kolmogorov equation the second is the problem of finding real time solution of a system of odes the third is the brockett s problem of classification of finite dimensional estimation algebras in this paper we solve the first problem

complete classification of finite dimensional estimation algebras of maximal rank the idea of using estimation algebras to construct finite dimensional non linear filters was first proposed by brockett and clark and mitter independently in his famous talk at the international congress of mathematics in 1983 brockett proposed to classify all finite dimensional estimation algebras in this paper we explain why the theory of estimation algebras plays an important role in non linear filtering we show how to use the wei norman approach to construct finite dimensional filters from finite dimensional estimation algebras we survey some results in estimation algebras after 1984 we give a self contained proof of complete classification of finite dimensional estimation algebras of maximal rank in one place the proof given here is simpler than those proofs scattered in several papers this provides the readers with a complete coherent view of the important topic of the classification of finite dimensional estimation algebras

solution of filtering problem with nonlinear observations for all known finite dimensional filters one always needs the conditon that the observation terms are degree one polynomial on the other hand in many practical examples e g tracking problem the observation terms may be nonlinear our new method in this paper can treat filtering problems with nonlinear observation terms in the first time which includes kalman bucy filter as a special case

state space formulae for all stabilizing controllers that satisfy and h norm bound and relations to risk sensitivity abstract given a linear system all stabilizing controllers such that a specified closed loop transfer function has h norm less than a given scalar are parametrized this characterization involves the solution to two algebraic riccati equations each with the same order as the system and further gives gives feasible controllers also with this order the relationship to the risk sensitive lqg stochastic control problem is established giving an equivalence between robust and stochastic control

real time solution of the nonlinear filtering problem without memory ii it is well known that the nonlinear filtering problem has important applications in both military and commercial industries the central problem of nonlinear filtering is to solve the duncan mortensen zakai dmz equation in real time and in a memoryless manner the purpose of this paper is to show that under very mild conditions which essentially say that the growth of the observation h is greater than the growth of the drift f the dmz equation admits a unique nonnegative weak solution u which can be approximated by a solution u r of the dmz equation on the ball b r with u r pat b r 0 the error of this approximation is bounded by a function of r which tends to zero as r goes to infinity the solution u r can in turn be approximated efficiently by an algorithm depending only on solving the observation independent kolmogorov equation on b r in theory our algorithm can solve basically all engineering problems in real time specifically we show that the solution obtained from ou

brockett s problem of classification of finite dimensional estimation algebras for nonlinear filtering systems in his talk at the international congress of mathematicians in 1982 brockett proposed to classify all finite dimensional estimation algebras arising in nonlinear filters recently chen yau and leung siam j control optim 35 1997 pp 1132 1141 claimed that they had classified all finite dimensional estimation algebras of maximal rank when the dimension of the state space is less than or equal to four in this paper we introduce a series of completely new computations about the estimation algebra and we find two sets of new equations about the omega matrix as a consequence we can prove without any dimension assumption on the state space that the omega matrix is a constant matrix and thus we succeed in classifying all finite dimensional estimation algebras of maximal rank with the state space dimension being arbitrary

classification of four dimensional estimation algebras of central importance in nonlinear filtering theory is the classification of finite dimensional estimation algebras through which we can construct recursive filters we classify all t dimensional estimation algebras t spl les 4 for arbitrary state space dimension

general finite dimensional risk sensitive problems and small noise limits for a risk sensitive partially observed stochastic control problem the modified zakai equation includes an extra term related to the exponential running cost the finite dimensional solutions of this modified zakai equation are obtained these are analogs of the kalman and benes filters the small noise limits of the finite dimensional risk sensitive problems are then obtained these lead to differential games with deterministic disturbances

information states in stochastic control and filtering a lie algebraic theoretic approach the purpose of the paper is two fold i to introduce the sufficient statistic algebra which is responsible for propagating the sufficient statistics or information state in the optimal control of stochastic systems and ii to apply certain lie algebraic methods and gauge transformations widely used in nonlinear control theory and quantum physics to derive new results concerning finite dimensional controllers this enhances our understanding of the role played by the sufficient statistics the sufficient statistic algebra enables us to determine a priori whether there exist finite dimensional controllers it also enables us to classify all finite dimensional controllers relations to minimax dynamic games are delineated

viscosity solutions for a class of hamilton jacobi equations in hilbert spaces we study hamilton jacobi equations with an unbounded term in hilbert spaces we introduce a new variant of the notion of viscosity solution for a class of hamilton jacobi equations and obtain comparison and existence results for viscosity solutions we also examine the usefulness of the notion of viscosity solution in optimal control

swarming patterns in a two dimensional kinematic model for biological groups we construct a continuum model for the motion of biological organisms experiencing social interactions and study its pattern forming behavior the model takes the form of a conservation law in two spatial dimensions the social interactions are modeled in the velocity term which is nonlocal in the population density and includes a parameter that controls the interaction length scale the dynamics of the resulting partial integrodifferential equation may be uniquely decomposed into incompressible motion and potential motion for the purely incompressible case the model resembles one for fluid dynamical vortex patches there exist solutions which have constant population density and compact support for all time numerical simulations produce rotating structures which have circular cores and spiral arms and are reminiscent of naturally observed phenomena such as ant mills the sign of the social interaction term determines the direction of the rotation and the interaction length scale affects the degree o

a counterexample of comments on stability margin evaluation for uncertain linear systems in the original paper c gong and s thompson ibid vol 39 p 548 50 1994 there is given what is claimed to be the tightest bound for all unstructured pertubations such that a specified system keeps its asymptotic stability j h su ibid vol 39 p 2523 4 1994 showed that it is not the tightest bound and gives another bound that bound is shown by a counterexample to be incorrect

finite dimensional estimation algebras with state dimension 3 and rank 2 mitter conjecture in this paper we study the structure of finite dimensional estimation algebras with state dimension 3 and rank 2 arising from a nonlinear filtering system by using the theories of the euler operat