control of singularly perturbed hybrid stochastic systems we study a class of optimal stochastic control problems involving two different time scales the fast mode of the system is represented by deterministic state equations whereas the slow mode of the system corresponds to a jump disturbance process under a fundamental ergodicity property for a class of infinitesimal control systems associated with the fast mode we show that there exists a limit problem which provides a good approximation to the optimal control of the perturbed system both the finite and infinite discounted horizon cases are considered we show how an approximate optimal control law can be constructed from the solution of the limit control problem in the particular case where the infinitesimal control systems possess the so called turnpike property i e characterized by the existence of global attractors the limit control problem can be given an interpretation related to a decomposition approach

improved delay dependent stability criteria for uncertain lur e systems with sector and slope restricted nonlinearities and time varying delays abstract this paper deals with the absolute stability analysis for uncertain time delay lur e systems that have time varying delays and sector and slope restricted nonlinearities new delay dependent stability criteria are derived via linear matrix inequality lmi formulation that can be easily solved by various optimization techniques sector bounds and slope bounds are employed to the lyapunov krasovskii functional through convex representation of the nonlinearities so that less conservative stability conditions are obtained numerical examples show effectiveness of the proposed stability condition

spherical harmonics and integration in superspace the study of spherical harmonics in superspace introduced in de bie and sommen 2007 j phys a math theor 40 7193 is further elaborated a detailed description of spherical harmonics of degree k is given in terms of bosonic and fermionic pieces which also determines the irreducible pieces under the action of so m x sp 2n in the second part of the paper this decomposition is used to describe all possible integrations over the supersphere it is then shown that only one possibility yields the orthogonality of spherical harmonics of different degrees this is the so called pizzetti integral of which it was shown in de bie and sommen 2007 j phys a math theor 40 7193 that it leads to the berezin integral

q deformed harmonic and clifford analysis and the q hermite and laguerre polynomials we define a q deformation of the dirac operator inspired by the one dimensional q derivative this implies a q deformation of the partial derivatives by taking the square of this dirac operator we find a q deformation of the laplace operator this allows us to construct q deformed schrodinger equations in higher dimensions the equivalence of these schrodinger equations with those defined on q euclidean space in quantum variables is shown we also define the m dimensional q clifford hermite polynomials and show their connection with the q laguerre polynomials these polynomials are orthogonal with respect to an m dimensional q integration which is related to integration on q euclidean space the q laguerre polynomials are the eigenvectors of an suq 1 1 representation

exponential bounds for discrete time singularly perturbed markov chains abstract this paper develops exponential type upper bounds for scaled occupation measures of singularly perturbed markov chains in discrete time by considering two time scale in the markov chains asymptotic analysis is carried out the cases of the fast changing transition probability matrix is irreducible and that are divisible into l ergodic classes are examined first the upper bounds of a sequence of scaled occupation measures are derived then extensions to markov chains involving transient states and or nonhomogeneous transition probabilities are dealt with the results enable us to further our understanding of the underlying markov chains and related dynamic systems which is essential for solving many control and optimization problems

hermite and gegenbauer polynomials in superspace using clifford analysis the clifford hermite and the clifford gegenbauer polynomials of standard clifford analysis are generalized to the new framework of clifford analysis in superspace in a merely symbolic way this means that one does not a priori need an integration theory in superspace furthermore a lot of basic properties such as orthogonality relations differential equations and recursion formulae are proven finally an interesting physical application of the super clifford hermite polynomials is discussed thus giving an interpretation to the super dimension

realizing the berezin integral as a superspace contour integral integration on supermanifolds using contours and the covariant differential forms of kostant b kostant graded manifolds graded lie theory and prequantisation in lecture notes in mathematics vol 570 springer berlin 1977 is described the good properties that these integrals naturally acquire are considered it is then shown that the formal process of integration over even and odd variables introduced by berezin f a berezin the method of second quantization academic new york 1966 using partly covariant and partly contravariant forms can be regarded as a special case of these contour integrals over covariant forms

robust stabilization of discrete time nonlinear lur e systems with sector and slope restricted nonlinearities abstract in this paper we present a novel method for the robust control problem of uncertain nonlinear discrete time linear systems with sector and slope restrictions the nonlinear function considered in this paper is expressed as convex combinations of sector and slope bounds then the equality constraint is derived by using convex properties of the nonlinear function a stabilization criterion for the existence of the state feedback controller is derived in terms of linear matrix inequalities lmis by using finsler s lemma the proposed method is demonstrated by a system with saturation nonlinearity

numerical solution of complex modified korteweg de vries equation by petrov galerkin method abstract in this paper a petrov galerkin method is used to derive a numerical scheme for the complex modified korteweg de vries equation cmkdv where we have used cubic b splines as test functions and linear b splines as trial functions an implicit midpoint rule is used to discretize in time a block non linear pentadiagonal system is obtained we solve this system by newton s method the resulting scheme has a fourth order accuracy in space direction and second order in time direction it is unconditionally stable by von neumann method the exact soliton solution and the conserved quantities are used to assess the accuracy and to show the robustness of the scheme the interaction of two solitary waves for different parameters is discussed

on the efetov wegner terms by diagonalizing a hermitian supermatrix the diagonalization of hermitian supermatrices is studied such a change of coordinates is inevitable if one wishes to find certain structures in random matrix theory however it still poses serious problems since until now the calculation of all rothstein contributions known as efetov wegner terms in physics was quite cumbersome we derive the supermatrix bessel function with all efetov wegner terms for an arbitrary rotation invariant probability density function as applications we consider representations of generating functions for hermitian random matrices with and without an external field as integrals over eigenvalues of hermitian supermatrices all results are obtained with all efetov wegner terms which were previously unknown in such an explicit and compact representation

the control of a two level markov decision process by time aggregation the solution of markov decision processes mdps often relies on special properties of the processes for two level mdps the difference in the rates of state changes of the upper and lower levels has led to limiting or approximate solutions of such problems in this paper we solve a two level mdp without making any assumption on the rates of state changes of the two levels we first show that such a two level mdp is a non standard one where the optimal actions of different states can be related to each other then we give assumptions conditions under which such a specially constrained mdp can be solved by policy iteration we further show that the computational effort can be reduced by decomposing the mdp a two level mdp with m upper level states can be decomposed into one mdp for the upper level and m to m m 1 mdps for the lower level depending on the structure of the two level mdp the upper level mdp is solved by time aggregation a technique introduced in a recent paper cao x r ren z y bhatnagar s fu m marcus s 2002 a time aggregation approach to markov decision processes automatica 38 6 929 943 and the lower level mdps are solved by embedded markov chains

segal bargmann transforms associated with finite coxeter groups using a polarization of a suitable restriction map and heat kernel analysis we construct a generalized segal bargmann transform associated with every finite coxeter group g on n we find the integral representation of this transform and we prove its unitarity to define the segal bargmann transform we introduce a hilbert space open image in new window of holomorphic functions on open image in new window with reproducing kernel equal to the dunkl kernel the definition and properties of open image in new window extend naturally those of the well known classical fock space the generalized segal bargmann transform allows to exhibit some relationships between the dunkl theory in the schrodinger model and in the fock model further we prove a branching decomposition of open image in new window as a unitary open image in new window module and a general version of hecke s formula for the dunkl transform

a two factor stochastic production model with two time scales a two factor production model where one factor is characterized by a continuous state variable and corresponds to the fast mode via a controlled diffusion process whereas the second factor is characterized by a discrete variable and corresponds to the slow mode via a controlled jump process

fundamental solutions for the super laplace and dirac operators and all their natural powers the fundamental solutions of the super dirac and laplace operators and their natural powers are determined within the framework of clifford analysis

orthosymplectically invariant functions in superspace the notion of spherically symmetric superfunctions as functions invariant under the orthosymplectic group is introduced this leads to dimensional reduction theorems for differentiation and integration in superspace these spherically symmetric functions can be used to solve orthosymplectically invariant schrodinger equations in superspace such as the an harmonic oscillator or the kepler problem finally the obtained machinery is used to prove the funk hecke theorem and bochner s relations in superspace

absolute stability criteria for multiple slope restricted monotonic nonlinearities absolute stability criteria such as the classical popov criterion guarantee stability for a class of sector bounded nonlinearities although the sector restriction bounds the admissible class of the nonlinearities the local slope of the nonlinearity may be arbitrarily large in this paper the authors derive absolute stability criteria for multiple slope restricted time invariant monotonic nonlinearities like the popov criterion in the single input single output case the authors results provide a simple graphical interpretation involving a straight line in a modified popov plane

stability of linear discrete dynamics employing state saturation arithmetic this note is concerned with the stability of discrete time dynamical systems employing saturation arithmetic in the state space a matrix measure is introduced so that it can administer the proximity evaluation of a matrix to the set of diagonal matrices and the measure is utilized for making an additional condition to the lyapunov stein matrix inequality the solvability of the modified matrix inequality ensures not only the stability but also the absence of overflow oscillation under the state saturation arithmetic and this approach has the advantage of being free from auxiliary parameters as an application the obtained result is applied to the stability analysis of two dimensional dynamics numerical examples are given to illustrate the results

two time scale controlled markov chains a decomposition and parallel processing approach this correspondence deals with a class of ergodic control problems for systems described by markov chains with strong and weak interactions these systems are composed of a set of subchains that are weakly coupled using results already available in the literature one formulates a limit control problem the solution of which can be obtained via an associated nondifferentiable convex programming ndcp problem the technique used to solve the ndcp problem is the analytic center cutting plane method accpm which implements a dialogue between on one hand a master program computing the analytical center of a localization set containing the solution and on the other hand an oracle proposing cutting planes that reduce the size of the localization set at each main iteration the interesting aspect of this implementation comes from two characteristics i the oracle proposes cutting planes by solving reduced sized markov decision problems mdp via a linear program lp or a policy iteration method ii several cutting planes can be proposed simultaneously through a parallel implementation on processors the correspondence concentrates on these two aspects and shows on a large scale mdp obtained from the numerical approximation ldquoa la kushner dupuisrdquo of a singularly perturbed hybrid stochastic control problem the important computational speed up obtained

berezin integral as a limit of riemann sum berezin integration of functions of anticommuting grassmann variables is usually seen as a formal operation sometimes even defined via differentiation using the formalism of geometric algebra and geometric calculus in which the grassmann numbers are endowed with a second associative product coming from a clifford algebra structure we show how berezin integrals can be realized in the high dimensional limit as integrals in the sense of geometric calculus we then show how the concepts of spinors and superspace transform into this framework

lmi approach to stability of a class of discrete time systems using two s complement arithmetic abstract ystematic and computationally tractable i e linear matrix inequality based approach to the global asymptotic stability of a class of discrete time systems implemented with fixed point arithmetic utilizing a very strong nonlinearity which is produced by employing two s complement overflow arithmetic is presented application of the approach to second order digital filter is discussed two third order examples illustrating the effectiveness of the approach are given

hilbert space for quantum mechanics on superspace in superspace a realization of sl2 is generated by the super laplace operator and the generalized norm squared in this paper an inner product on superspace for which this representation is skew symmetric is considered this inner product was already defined for spaces of weighted polynomials see k coulembier h de bie and f sommen orthogonality of hermite polynomials in superspace and mehler type formulae proc london math soc accepted arxiv 1002 1118 in this article it is proven that this inner product can be extended to the super schwartz space but not to the space of square integrable functions subsequently the correct hilbert space corresponding to this inner product is defined and studied a complete basis of eigenfunctions for general orthosymplectically invariant quantum problems is constructed for this hilbert space then the integrability of the sl2 representation is proven finally the heisenberg uncertainty principle for the super fourier transform is constructed

paper optimal control of markov chains admitting strong and weak interactions this paper is devoted to the study of an optimal control problem for a markov chain with generator b ea where e is a small parameter it is shown that an approximate solution can be calculated by a policy improvement algorithm involving computations relative to an aggregated problem the dimension of which is given by n the number of ergodic sets for the b matrix together with a family of decentralized problems the dimensions of which are given by the number of elements in each ergodic set for the b matrix

novel criterion for stability of discrete time systems in a state space realization utilizing saturation nonlinearities abstract a new criterion for the global asymptotic stability of discrete time systems in a state space realization utilizing saturation nonlinearities is presented the form of the criterion is unconventional in that it offers the flexibility of using even an unsymmetric lyapunov matrix p a recently reported criterion is shown to be a special case of the present criterion an example shows the effectiveness of the new criterion

invariant integration on orthosymplectic and unitary supergroups the orthosymplectic supergroup osp m 2n and unitary supergroup u p q are studied following a new approach that starts from harish chandra pairs and links the sheaf theoretical supermanifold approach of berezin and others with the differential geometry approach of rogers and others the matrix elements of the fundamental representation of the lie supergroup are expressed in terms of functions on the product supermanifold with the underlying lie group and n the odd dimension of this product supermanifold is isomorphic to the supermanifold of this leads to a new expression for the standard generators of the corresponding lie superalgebra as invariant derivations on using these results a new and transparent formula for the invariant integrals on osp m 2n and u p q is obtained

stability criterion for a class of fixed point digital filters using two s complement arithmetic a new stability criterion for a class of digital filters using two s complement arithmetic is presented the effectiveness of the obtained results is shown by using a numerical example

perturbation and stability theory for markov control problems a unified approach to the asymptotic analysis of a markov decision process disturbed by an epsilon additive perturbation is proposed irrespective of whether the perturbation is regular or singular the underlying control problem that needs to be understood is the limit markov control problem the properties of this problem are studied

fischer decomposition for polynomials on superspace recently the fischer decomposition for polynomials on superspace m 2n that is polynomials in m commuting and 2n anti commuting variables has been obtained unless the superdimension m m 2n is even and non positive in this case it turns out that the fischer decomposition of polynomials into spherical harmonics is quite analogous as in m and it is an irreducible decomposition under the natural action of lie superalgebra m 2n in this paper we describe explicitly the fischer decomposition in the exceptional case when m 2 0 in particular we show that under the action of m 2n the fischer decomposition is not in general a decomposition into irreducible but just indecomposable pieces

stability criteria of sector and slope restricted lur e systems presents stability criteria of sector and slope restricted lur e systems in terms of linear matrix inequalities by fully exploiting inherent properties of sector and slope restrictions in the time domain interpreting the time domain criteria in the frequency domain furthermore supplies simpler expressions several examples show excellent performances of these criteria

the berezin integral as a contour integral abstract in superspace formulations of supersymmetric field theories use is made of the berezin integral a linear map which assigns a real number to each polynomial function of the grasmann coordinates however this map has not been shown to be an integral in the mathematical sense of a limit of sums or an inverse to differentiation consequently its mathematical basis is unclear and its generalization to arbituary curved supermanifolds is problematic i will show that certain normalized contour integrals in the space of possess the properties of berezin integrals including translation invariance and correct transformation properties under changes of coordinates these contour integrals are mathematically well defined and may generalize to arbitrary supermanifolds alice rogers definition of super space is used

applications of singular perturbation techniques to control problems this paper discusses typical applications of singular perturbation techniques to control problems in the last fifteen years the first three sections are devoted to the standard model and its time scale stability and controllability properties the next two sections deal with linear quadratic optimal control and one with cheap near singular control then the composite control and trajectory optimization are considered in two sections and stochastic control in one section the last section returns to the problem of modeling this time in the context of large scale systems the bibliography contains more than 250 titles

algorithms for singularly perturbed limiting average markov control problems a singularly perturbed markov decision process with the limiting average reward criterion is considered it is assumed that the underlying process is composed of n separate irreducible processes and that the small perturbation is such that it unites these processes into a single irreducible process two algorithms for the solution of the underlying limit markov control problem are presented the first of these is a linear program possessing the wolfe dantzig structure inherited from the ergodic nearly decomposable assumption in the model the second is an aggregation disaggregation policy improvement algorithm

the harmonic transvector algebra in two vector variables the decomposition of polynomials of one vector variable into irreducible modules for the orthogonal group is a crucial result in harmonic analysis which makes use of the howe duality theorem and leads to the study of spherical harmonics the aim of the present paper is to describe a decomposition of polynomials in two vector variables and to obtain projection operators on each of the irreducible components to do so a particular transvector algebra will be used as a new dual partner for the orthogonal group leading to a generalisation of the classical howe duality the results are subsequently used to obtain explicit projection operators and formulas for integration of polynomials over the associated stiefel manifold

a reduction process for perturbed markov chains given a perturbed markov chain with generator a varepsilon a 0 varepsilon a 1 varepsilon 2 a 2 cdots it is shown that to each time scale t t varepsilon t varepsilon 2 cdots is associated a reduced markov chain this process of reduction has only finitely many steps it is given in a recursive way and each reduced chain admits as state space the recurrent classes of the reduced chain defined at the previous stage it is shown that if lambda is an eigenvalue different from zero of the generator of the kth reduced chain with multiplicity p then a varepsilon admits p eigenvalues lambda varepsilon such that lambda varepsilon varepsilon k lambda o varepsilon k a basis of ker a 0 resp ker a 0 which is robust to the perturbation is constructed

multiple time scale decomposition of discrete time markov chains abstract the multiple time scale decomposition of discrete time finite state markov chains is addressed in 1 2 the behavior of a continuous time markov chain is approximated using a fast time scale independent continuous time process and a reduced order perturbed process the procedure can then be iterated to obtain a complete multiple time scale decomposition in the discrete time case presented in this paper the basic approximation has a hybrid form in this form the fast time scale behavior is approximated using an independent discrete time markov chain and the slow behavior is captured by a perturbed continuous time process further time scale decomposition then involves the continuous time procedure in 1 2 this extension to discrete time chains bridges previous multiple time scale decomposition results which have dealt exclusively with either continuous time or discrete time processes and provides a uniform framework for the analysis of both types of systems

distributions and integration in superspace distributions in superspace constitute a very useful tool for establishing an integration theory in particular distributions have been used to obtain a suitable extension of the cauchy formula to superspace and to define integration over the superball and the supersphere through the heaviside and dirac distributions respectively in this paper we extend the distributional approach to integration over more general domains and surfaces in superspace the notions of domain and surface in superspace are defined by smooth bosonic phase functions g this allows us to define domain integrals and oriented as well as non oriented surface integrals in terms of the heaviside and dirac distributions of the superfunction g it will be shown that the presented definition for the integrals does not depend on the choice of the phase function g defining the corresponding domain or surface in addition some examples of integration over a super paraboloid and a super hyperboloid will be presented finally a new distributional cauchy pompeiu formula will be obtained which generalizes and unifies the previously known approaches distributions in superspace constitute a very useful tool for establishing an integration theory in particular distributions have been used to obtain a suitable extension of the cauchy formula to superspace and to define integration over the superball and the supersphere through the heaviside and dirac distributions respectively in this paper we extend the distributional approach to integration over more general domains and surfaces in superspace the notions of domain and surface in superspace are defined by smooth bosonic phase functions g this allows us to define domain integrals and oriented as well as non oriented surface integrals in terms of the heaviside and dirac distributions of the superfunction g it will be shown that the presented definition for the integrals does not depend on the choice of the phase function g defining the corresponding domain or surface in addition some examples of integration over a super paraboloid and a super hyperboloid will be presented finally a new dist

a stability inequality for nonlinear feedback systems with slope restricted nonlinearity using lyapunov s direct method a stability inequality in frequency domain is derived for nonlinear feedback systems with slope restricted nonlinearity in the present approach a transformed system that involves the slope of the nonlinearity is considered thus leading to a stability inequality that incorporates the slope information but not the sector information of the nonlinearity the approach is essentially an extension of a recent idea of the author

graded manifolds supermanifolds and infinite dimensional grassmann algebras a new approach to superdifferentiable functions of grassmann variables is developed which avoids ambiguities in odd derivatives this is used to give an improved definition of supermanifold over a finite dimensional grassmann algebra a natural embedding of super manifolds over grassmann algebras with increasing number l of generators is developed and thus a limit asl tends to infinity is possible a correspondence between graded manifolds and supermanifolds is constructed extending results of 5 and 8

fourier transform and related integral transforms in superspace abstract in this paper extensions of the classical fourier fractional fourier and radon transforms to superspace are studied previously a fourier transform in superspace was already studied but with a different kernel in this work the fermionic part of the fourier kernel has a natural symplectic structure derived using a clifford analysis approach several basic properties of these three transforms are studied using suitable generalizations of the hermite polynomials to superspace see h de bie f sommen hermite and gegenbauer polynomials in superspace using clifford analysis j phys a 40 2007 10441 10456 an eigenfunction basis for the fourier transform is constructed

a singular perturbation approach to modeling and control of markov chains finite state continuous time markov processes with weak interactions are modeled as singularly perturbed systems aggregate states are obtained using a grouping algorithm two time scale expansions simplify cost equations and lead to decentralized optimization algorithms

integration in superspace using distribution theory in this paper a new class of cauchy integral formulae in superspace is obtained using formal expansions of distributions this allows us to solve five open problems in the study of harmonic and clifford analysis in superspace

technical communique modified criterion for global asymptotic stability of fixed point state space digital filters using two s complement arithmetic a criterion for the global asymptotic stability of fixed point state space digital filters using two s complement arithmetic is presented the criterion is a modified form of a well known criterion due to mills mullis and roberts the criterion is in the form of linear matrix inequality and hence computationally tractable an example shows the effectiveness of the modified criterion