suboptimal stochastic linear feedback control of linear systems with state and control dependent noise the incomplete information case the stochastic regulation problem for linear systems with state and control dependent noise and a noisy linear output equation is considered the optimal quadratic cost output feedback control law in a class of linear controllers is found this problem was first addressed in the early 1970s and solved in the complete information case by wonham in this paper we give the solution of the problem in the incomplete information case that is for a linear output equation corrupted by gaussian noise moreover a different method is used here giving the solution in a more direct way even in the complete information case

curvature weighted gradient based shape orientation determining the orientation of a shape is a common task in many image processing applications it is usually part of the image preprocessing stages and further processing may rely on an adequate method to determine the orientation there are several methods for computing the orientation of a shape each of them with its own strengths and weaknesses a method which performs outstandingly for one application may have a poor performance for a different application in this paper we present a new method for computing shape orientation based on the projection of the tangent vectors of a shape onto a line and weighting them using a function of the curvature some of the results from zunic 2008 14 are particular cases of the results presented here

symmetry detection by generalized complex gc moments a close form solution this paper presents a unified method for detecting both reflection symmetry and rotation symmetry of 2d images based on an identical set of features i e the first three nonzero generalized complex gc moments this method is theoretically guaranteed to detect all the axes of symmetries of every 2d image if more nonzero gc moments are used in the feature set furthermore we establish the relationship between reflectional symmetry and rotational symmetry in an image which can be used to check the correctness of symmetry detection this method has been demonstrated experimentally using more than 200 images

probability inequalities for decomposition integrals recently in mathematical economics even and lehrer introduced the decomposition integral even and lehrer 2014 in this paper general versions of some well known probabilistic inequalities for the decomposition integrals and the superdecomposition integrals are discussed that are still open for research the main results of this paper generalize some previous results for particular integral inequalities obtained by several researchers in generalized probability theory

local symmetries of digital contours from their chain codes in this work symmetry is evaluated as a numeric feature for each point of a contour using only the positions of a local vicinity of points a measurement is defined named as local symmetric deficiency lsd so that the lower this quantity is the higher the symmetry will be in the local region considered this approach is very simple and it is based on a suitable manipulation of the chain code of the curve its computational cost is very low and it has the advantages of a parallel algorithm since values for lsd can be computed for each point independently

boundary based shape orientation the computation of a shape s orientation is a common task in the area of computer vision and image processing being used for example to define a local frame of reference and is helpful for recognition and registration robot manipulation etc it is usually an initial step or a part of data preprocessing in many image processing and computer vision tasks thus it is important to have a good solution for shape orientation because an unsuitable solution could lead to a big cumulative error at the end of the computing process there are several approaches to the problem most of them could be understood as the area based ones or at least they do not take into account all the boundary points if a shape orientation measure is based on its encasing rectangle only the convex hull points count for example thus the demand for a pure boundary based method where the orientation of the shape is dependent on the boundary points seems to be very reasonable such a method is presented in this paper we are initially focused on the shapes having polygonal boundaries we define the orientation of a polygonal shape by the line that maximises the total sum of squared lengths of all the boundary edge projections onto this line the advantages and limitations of the new method are analysed next we suggested how the method can be adapted in order to be applicable to a wider class than the initial method is finally we introduced another modification of the method in such a way that the modified method can be applied to shapes with arbitrary boundaries several illustrative experiments are provided

universal principal axes an easy to construct tool useful in defining shape orientations for almost every kind of shape abstract a convenient tool for defining shape orientations called universal principal axes is introduced these axes are formed of half lines starting from the shape centroid with their directional angles expressed as functions of the polar angle of the first non zero complex number encountered in the sequence x iy 1 d a with 1 2 3 4 provided that the integration domain is taken to be the given two dimensional 2d shape and the origin is assumed to be the shape centroid universal principal axes are shown to be independent of the translation scaling and rotation of the coordinate system used and they are therefore qualified in defining shape orientations a major benefit of using the proposed universal principal axes is that they exist for almost every kind of shape another benefit is that there is no need to judge in advance whether a given shape is mirror symmetric rotationally symmetric irregular etc moreover in the case of rotationally symmetric shapes the universal principal axes make the preprocessing procedure for detecting the number of folds contained in the given shape unnecessary although such procedure is essential for many existing tools designed to define orientations of rotationally symmetric shapes defining shape orientations by universal principal axes is therefore quite convenient an algorithm describing the construction of universal principal axes and several examples showing the detected universal principal axes for some shapes of distinct types are given the relationship between universal and conventional principal axes is also discussed

generalized affine invariant image normalization we provide a generalized image normalization technique which basically solves all problems in image normalization the orientation of any image can be uniquely defined by at most three non zero generalized complex moments the correctness of our method is demonstrated theoretically as well as in practice by applying them to a number of degenerate images which have failed other previously reported techniques for image normalization

detection of partial symmetry using correlation with rotated reflected images abstract symmetry is one of the important structural properties of a figure both in visual psychology and in computer vision symmetrical properties of planar two dimensional figures are considered and a new method for the extraction of rotational symmetry and reflectional symmetry is presented a figure is said to have symmetry if it is invariant with some congruent transformation which consists of translation rotation and reflection the approach is to obtain the congruent transformations which make the transformed image a good match with the original image the directional correlation of edge features is used to evaluate how well the original and transformed images match this method does not require any segmentation nor knowledge of the centroid position and the symmetrical properties of partially symmetric figures can also be detected the approach is described and this method is applied to synthetic and real images

the family of universal axes the family of universal axes ua is introduced in this paper this family has infinitely many members each of which generates a method that is universal in the sense that each generated method can define shape orientation for almost any kind of shape and once the numerical computing system has been set up the users of this orientation detection system need not supply the system the information about whether a given shape is mirror symmetric rotationally symmetric irregular etc some mathematical properties of the proposed ua are investigated and numerical techniques for implementing ua are discussed the method of universal principal axes upa introduced recently is just one of the infinitely many members of the proposed family of ua the upa method yields at least two orientations for each shape and thus must be combined with another non universal single orientation method in order to reduce the time needed to match irregular shapes most members of the proposed ua family do not need to be supplemented by a single orientation method in this way

reversed version of a generalized sharp holder s inequality and its applications in this paper we give a reversed version of a generalized sharp holder s inequality which is due to wu the results are then used to improve beckenbach s inequality and minkowski s inequality moreover as an application in information theory we present a refinement of singh s inequality which is one generalized shannon s inequality

brief h performance of preview control systems performance of the preview control is investigated in terms of h criterion in the infinite horizon time setting the limitation of h performance in preview control action is characterized with the hamiltonian matrix by employing the property of hamiltonian matrix the solvability and analytic solution for the preview full information control problem are clarified based on an auxiliary introduced matrix riccati equation the property of resulting control law is illustrated with a numerical example

linear quadratic regulation for linear time varying systems with multiple input delays this paper studies the classic linear quadratic regulation lqr problem for both continuous time and discrete time systems with multiple input delays for discrete time systems the lqr problem for systems with single input delay has been studied in existing literature whereas a solution to the multiple input delay case is not known to our knowledge for continuous time systems with multiple input delays the lqr problem has been tackled via an infinite dimensional system theory approach and a frequency time domain approach the objective of the present paper is to give an explicit solution to the lqr problem via a simple and intuitive approach the main contributions of the paper include a fundamental result of duality between the lqr problem for systems with multiple input delays and a smoothing problem for an associated backward stochastic system the duality allows us to obtain a solution to the lqr problem via standard projection in linear space the lqr controller is simply constructed by the solution of one backward riccati difference for the discrete time case or differential for the continuous time case equation of the same order as the plant ignoring the delays

karhunen loeve decomposition in the presence of symmetry i the karhunen loeve kl decomposition is widely used for data which very often exhibit some symmetry afforded by a group action for a finite group we derive an algorithm using group representation theory to reduce the cost of determining the kl basis we demonstrate the technique on a lorenz type ode system for a compact group such as tori or so 3 r the method also applies and we extend results to these cases as a short example we consider the circle group s sup 1

state feedback control of systems with multiplicative noise via linear matrix inequalities we consider lti systems perturbed by parametric uncertainties modeled as white noise disturbances we show how to maximize via state feedback control the smallest norm of the noise intensity vector producing instability in the mean square sense using convex optimization over linear matrix inequalities we also show how to maximize performance robustness where performance is measured by expected output energy with either bounded initial conditions and zero inputs classical lqg cost or zero initial conditions and deterministic inputs of bounded energy a generalization of the h norm

robust kalman filters for linear time varying systems with stochastic parametric uncertainties we present a robust recursive kalman filtering algorithm that addresses estimation problems that arise in linear time varying systems with stochastic parametric uncertainties the filter has a one step predictor corrector structure and minimizes an upper bound of the mean square estimation error at each step with the minimization reduced to a convex optimization problem based on linear matrix inequalities the algorithm is shown to converge when the system is mean square stable and the state space matrices are time invariant a numerical example consisting of equalizer design for a communication channel demonstrates that our algorithm offers considerable improvement in performance when compared with conventional kalman filtering techniques

generalizations of holder s and some related inequalities in this paper we establish some new generalizations and refinements of holder s inequality and some related inequalities we also show that many existing inequalities related to the holder inequality are special cases of the inequalities presented

frequency domain classification of cyclic and dihedral symmetries of finite 2 d patterns this paper presents a very simple and effective algorithm for classifying cyclic and dihedral symmetries from images of finite two dimensional patterns it consists in an extension of a frequency domain algorithm proposed by the author for estimating two dimensional rotation of rigid objects within this framework the estimation of rotation is conveniently turned into the problem of detecting two orthogonal lines within the locus of the zero crossings of a certain function represented in orthogonal cartesian coordinates the main contribution of this paper is the use and interpretation of the ambiguities that arise in such a locus in the form of additional pairs of orthogonal lines when applied to symmetric patterns and the development of other algorithmic steps which allow a simple and fast discrimination between cyclic and dihedral symmetries unlike any other available method this algorithm does not require any conversion from cartesian to polar image representations besides classification reliability and consistency observed through several dozen testing experiments the nice features of this new method also include robustness to noise and ease of implementation with fast fourier transform algorithms which makes it amenable to almost real time pattern classifications applications several examples representative of the performance of the algorithm through extensive testing are reported and discussed in the paper

on symmetry detection a straight line is an axis ofsymmetry of a planar figure if the figure is invariant to reflection with respect to that line the purpose of this correspondence is to describe an o n log n time algorithm for enumerating all the axes of symmetry of a planar figure which is made up of possibly intersecting segments circles points etc the solution involves a reduction of the problem to a combinatorial question on words our algorithm is optimal since we can establish an n log n time lower bound for this problem

automatic symmetry determination and normalization for rotationally symmetric 2d shapes and 3d solid objects abstract determination of the rotational symmetry of a shape is useful for object recognition and shape analysis in computer vision application however the usual normalization methods are inapplicable when the shape is n fold rotationally symmetric n 3 in this paper an efficient and reliable new method using modified fourier transform is introduced to normalize such a 2d rotationally symmetric shape furthermore we extend this new method to the normalization of a n fold axial rotationally symmetric 3d object there are two obvious advantages when using this new method 1 it prevents the high computation load and the difficulty in normalizing the shape s orientation which occurred in the old methods using full matching or correlation functions and 2 the weakness introduced by the old methods using data reduction from 2d data to 1d data is remedied experiments will show the validity of our method

revisiting complex moments for 2 d shape representation and image normalization when comparing 2 d shapes a key issue is their normalization translation and scale are easily taken care of by removing the mean and normalizing the energy however defining and computing the orientation of a 2 d shape is not so simple in fact although for elongated shapes the principal axis can be used to define one of two possible orientations there is no such tool for general shapes as we show in the paper previous approaches fail to compute the orientation of even noiseless observations of simple shapes we address this problem and show how to uniquely define the orientation of an arbitrary 2 d shape in terms of what we call its principal moments we start by showing that a small subset of these moments suffices to describe the underlying 2 d shape i e that they form a compact representation which is particularly relevant when dealing with large databases then we propose a new method to efficiently compute the shape orientation principal moment analysis finally we discuss how this method can further be applied to normalize gray level images besides the theoretical proof of correctness we describe experiments demonstrating robustness to noise and illustrating with real images

brief h control and filtering of discrete time stochastic systems with multiplicative noise linear discrete time systems with stochastic uncertainties in their state space matrices are considered the problems of finite horizon filtering and output feedback control are solved taking into account possible cross correlations between the uncertain parameters in both problems a cost function is defined which is the expected value of the relevant standard h performance index with respect to the uncertain parameters a solution to the filtering problem is obtained first by applying the adjoint system and deriving a bounded real lemma for this system this solution guarantees a prescribed estimation level of accuracy while minimizing an upper bound on the covariance of the estimation error the solution of the filtering problem is also extended to the infinite horizon case the results of the filtering problem are used to solve the corresponding output feedback problem a filtering example is given where a comparison is made with the results obtained using bounded uncertainty design techniques

skew symmetry detection via invariant signatures we propose a new approach to skew symmetry detection based on the theory of invariant signatures for planar objects invariant signatures associated to object boundaries are generalizations of the curvature versus arclength description of curves invariant under geometric transformations more complex than the euclidean ones we show that symmetries of objects and hence of closed boundaries translate into simple structures in the invariant signature functions and are therefore in principle readily detectable

lq control for ito type stochastic systems with input delays this paper considers the linear quadratic lq control problem for the ito type stochastic system with input delays due to simultaneous appearances of diffusion terms dependent on the state and the control as well as delays in the dynamic system the problem is very involved and remains to be solved we not only provide the solvable condition of the problem but also the explicit expression of the causal and adapted controller for a kind of lq problems all of these are based on a stochastic riccati equation the key technique is to pursue the explicit cost value of the lq problem by fbsde and derive the analytical controller via the interplay between the original problem and its equivalent abstract description

full and partial symmetries of non rigid shapes symmetry and self similarity are the cornerstone of nature exhibiting themselves through the shapes of natural creations and ubiquitous laws of physics since many natural objects are symmetric the absence of symmetry can often be an indication of some anomaly or abnormal behavior therefore detection of asymmetries is important in numerous practical applications including crystallography medical imaging and face recognition to mention a few conversely the assumption of underlying shape symmetry can facilitate solutions to many problems in shape reconstruction and analysis traditionally symmetries are described as extrinsic geometric properties of the shape while being adequate for rigid shapes such a description is inappropriate for non rigid ones extrinsic symmetry can be broken as a result of shape deformations while its intrinsic symmetry is preserved in this paper we present a generalization of symmetries for non rigid shapes and a numerical framework for their analysis addressing the problems of full and partial exact and approximate symmetry detection and classification

a new measure of symmetry and its application to classification of bifurcating structures this paper presents a new measure of symmetry for bifurcating structures which relies not only on topology and ordering but also on quantitative properties e g length of branches this measure is based on a specific biological mechanism and on the concept of minimum energy the effectiveness of the approach is demonstrated in a classification test where leaves taken from plants growing under different stress conditions are classified results show that the proposed measure improves classification performance compared to classification based on other leading measures

similarity and symmetry measures for convex shapes using minkowski addition this paper is devoted to similarity and symmetry measures for convex shapes whose definition is based on minkowski addition and the brunn minkowski inequality this means in particular that these measures are region based in contrast to most of the literature where one considers contour based measures all measures considered in this paper are invariant under translations furthermore they can be chosen to be invariant under rotations multiplications reflections or the class of affine transformations it is shown that the mixed volume of a convex polygon and a rotation of another convex polygon over an angle spl theta is a piecewise concave function of spl theta this and other results of a similar nature form the basis for the development of efficient algorithms for the computation of the given measures various results obtained in this paper are illustrated by experimental data although the paper deals exclusively with the two dimensional case many of the theoretical results carry over almost directly to higher dimensional spaces

h_ infty control of discrete time systems with multiple input delays this paper is concerned with the finite horizon hinfin full information control for discrete time systems with multiple control and exogenous input delays we first establish a duality between the hinfin full information control and the h2 smoothing of a stochastic backward system in krein space like the duality between the linear quadratic regulation lqr of linear systems without delays and the kalman filtering the established duality allows us to address complicated multiple input delay problems in a simple way indeed by applying innovation analysis and standard projection in krein space in this paper we derive conditions under which the hinfin full information control is solvable an explicit controller is constructed in terms of two standard riccati difference equations of the same order as the original plant ignoring the delays as special cases solutions to the hinfin state feedback control problem for systems with delays only in control inputs and the hinfin control with preview are obtained an example is given to demonstrate the effectiveness of the proposed hinfin control design

testing for image symmetries with application to confocal microscopy statistical tests are introduced for checking whether an image function f x y defined on the unit disc d x y x 2 y 2 les 1 is invariant under certain symmetry transformations of d given that discrete and noisy data are observed invariance under reflections or under rotations by rational angles is considered as well as rotational invariance these symmetry relations can be naturally expressed as restrictions for the zernike moments of f x y therefore the test statistics are based on the l 2 distance between zernike series estimates of the image function itself and its version obtained after applying the symmetry transformation the asymptotic distribution of the test statistics under both the hypothesis of symmetry as well as under fixed alternatives is derived furthermore the quality of the asymptotic approximations via simulation studies is investigated the usefulness of our theory is verified by examining an important problem in confocal microscopy i e possible imprecise alignments in the optical path of the microscope are investigated for optical systems with rotational symmetry the theoretical point spread function psf is reflection symmetric with respect to two orthogonal axes and rotationally invariant if the detector plane matches the optical plane of the microscope the tests are used to investigate whether the required symmetries can indeed be detected in the empirical psf

an alternative approach to computing shape orientation with an application to compound shapes we consider the method that computes the shape orientation as the direction that maximises the integral of the length of projections taken to the power of 2n of all the straight line segments whose end points belong to the shape to a line that has the slope we show that for n 1 such a definition of shape orientation is consistent with the shape orientation defined by the axis of the least second moment of inertia for n 1 this is not the case and consequently our new method can produce different results as an additional benefit our approach leads to a new method for computation of the orientation of compound objects

detection of generalized principal axes is rotationally symmetric shapes abstract automatic detection of the principal axes of given shapes with known symmetry properties is studied in this paper the inapplicability of a well known equation which is used to compute the direction of the principal axis of a given shape using moment functions to a class of so called degenerate shapes is first pointed out rotationally symmetric shapes often encountered in real applications are shown to belong to this class this problem is solved by extending the notion of principal axis to higher order ones in terms of higher order moment functions analytic equations for computing the direction of high order principal axes are derived they include the well known equation for computing the direction of the second order principal axis as a special case some experimental results are included finally to show the effectiveness of the derived analytic equations

application of elliptic fourier descriptors to symmetry detection under parallel projection in this paper the method of elliptic fourier descriptors using arc length parameterization is applied to tackle the problem of detection and recovery of symmetry under parallel projection a simple and fast iteration algorithm together with the invariants of symmetry provides sufficient information for the detection and recovery of symmetry under parallel projection the proposed method has been extensively tested using symmetric figures under different parallel projections simulation results of the algorithm are presented the extension of this method for planar object recognition under parallel projection is also addressed

3 d object recognition with symmetric models symmetry extraction and encoding object recognition systems which employ solid models and range data have been a topic of interest for several years model databases have the potential to become large in some environments this paper proposes a pair of techniques for incorporating knowledge of the symmetries of object models into the recognition process the effects of symmetric models on the speed of an object recognition system is examined in the context of an implemented system employing invariant feature indexing as a correspondence building mechanism groups of model surfaces are enumerated and examined to yield a list of segment label permutations which summarize the model s symmetry this symmetry extraction process is followed by a symmetry encoding procedure which replaces groups of features which are indistinguishable because of symmetry with a single prototype feature group experiments with a large model database demonstrate the utility of these symmetry extraction and encoding techniques

on the detection of the axes of symmetry of symmetric and almost symmetric planar images a simple algorithm is presented for finding all the axes of symmetry of symmetric and almost symmetric planar images having nonuniform gray level intensity images the technique which can also be used in conjunction with planar curves is based on the identification of the centroids of the given image and other related sets of points followed by a maximization of a specially defined coefficient of symmetry owing to the nature of the required procedures which are strictly based on the computation of mean values the method is not particularly affected by the presence of noise and is especially suited for application in conjunction with digitized figures

notes on shape orientation where the standard method does not work in this paper we consider some questions related to the orientation of shapes with particular attention to the situation where the standard method does not work there are irregular and non symmetric shapes whose orientation cannot be computed in a standard way but in the literature the most studied situations are those where the shape under consideration has more than two axes of symmetry or where it is an n fold rotationally symmetric shape with n 2 the basic reference for our work is w h tsai s l chou detection of generalized principal in rotationally symmetric shapes pattern recognition 24 1991 95 104 we give a very simple proof of the main result from w h tsai s l chou detection of generalized principal in rotationally symmetric shapes pattern recognition 24 1991 95 104 and suggest a modification of the proposal on how the principal axes of rotationally symmetric shapes should be computed we show some desirable property in defining the orientation of such shapes if the modified approach is applied also we give some comments on the problems that arise when computing shape elongation

stabilization of discrete time systems with multiplicative noise and multiple delays in the control variable this paper focuses on the stabilization problem of discrete time systems with multiplicative noise and multiple delays in the control variable first a transformation is constructed to convert the original system into a delay free system with its input matrix containing future noise next the finite horizon linear quadratic regulation problem for the delay free system subject to a constraint on the control is investigated we provide the necessary and sufficient condition for the problem having a unique solution and the optimal feedback control using riccati type difference equations finally we show that the original system can be stabilized in the mean square sense if and only if the set of solutions to the riccati type difference equations is convergent this further yields the fact that the system under consideration can be stabilized if and only if the algebraic riccati type equations have a unique solution such that a specific matrix is positive definite

detection of rotationally symmetric shape orientations by fold invariant shape specific points abstract two new types of shape specific points called fold invariant centroid fic and fold invariant radius weighted mean firwm are introduced for detecting the orientations of rotationally symmetric shapes a given rotationally symmetric shape is first transformed into a new fold invariant shape the centroid and the radius weighted mean of the new shape are then transformed back to obtain the proposed fic and firwm respectively fic and firwm points are proved to be invariant to translation rotation and scaling and so can be used for matching rotationally symmetric shapes using the proposed fold invariant shape specific points to detect the orientations of rotationally symmetric shapes is easier to implement than that of using the fold principal axes some illustrative examples and a method to avoid the case where the proposed fold invariant shape specific points are too close to the centroid are also given

symmetry as a continuous feature symmetry is treated as a continuous feature and a continuous measure of distance from symmetry in shapes is defined the symmetry distance sd of a shape is defined to be the minimum mean squared distance required to move points of the original shape in order to obtain a symmetrical shape this general definition of a symmetry measure enables a comparison of the amount of symmetry of different shapes and the amount of different symmetries of a single shape this measure is applicable to any type of symmetry in any dimension the symmetry distance gives rise to a method of reconstructing symmetry of occluded shapes the authors extend the method to deal with symmetries of noisy and fuzzy data finally the authors consider grayscale images as 3d shapes and use the symmetry distance to find the orientation of symmetric objects from their images and to find locally symmetric regions in images

automatic rotational symmetry determination for shape analysis abstract the determination of the rotational symmetry of a shape is useful for object recognition and shape analysis in computer vision applications in this paper a simple but effective algorithm to analyse the rotational symmetry of a given closed curve shape s is proposed a circle c with the centroid of s as the circle center and the average radius of s as the circle radius is superimposed on s resulting in the intersection of c and s at a set of points by theoretical analysis the relationship between the order n s of the rotational symmetry of s and the number of intersection points between c and s is established all the possible values for n s are determined and finally n s is determined by evaluating the similarity between s and its rotated versions in the proposed algorithm only simple pixel operations and second order moment function computation are involved several problems caused by the use of discrete image coordinates are analysed and solved for correct decision making in the algorithm good experimental results are also included

irregular shape symmetry analysis theory and application to quantitative galaxy classification this paper presents a set of imperfectly symmetric measures based on a series of geometric transformation operations for quantitatively measuring the amount of symmetry of arbitrary shapes the definitions of both bilateral symmetricity and rotational symmetricity give new insight into analyzing the geometrical property of a shape and enable characterizing arbitrary shapes in a new way we developed a set of criteria for quantitative galaxy classification using our proposed irregular shape symmetry measures our study has demonstrated the effectiveness of the proposed method for the characterization of the shape of the celestial bodies the concepts described in the paper are applicable to many fields such as mathematics artificial intelligence digital image processing robotics biomedicine etc

brief paper stochastic linear quadratic regulation for discrete time linear systems with input delay this paper considers the stochastic lqr problem for systems with input delay and stochastic parameter uncertainties in the state and input matrices the problem is known to be difficult due to the presence of interactions among the delayed input channels and the stochastic parameter uncertainties in the channels the key to our approach is to convert the lqr control problem into an optimization one in a hilbert space for an associated backward stochastic model and then obtain the optimal solution to the stochastic lqr problem by exploiting the dynamic programming approach our solution is given in terms of two generalized riccati difference equations rdes of the same dimension as that of the plant