reduced order feedback control of distributed parameter systems via singular perturbation methods abstract implementable feedback control of distributed parameter systems must often be based on reduced order models due to the infinite dimensional nature of the actual open loop system the behavior of controllers designed via reduced order models obtained with singular perturbation techniques is analyzed when such controllers are used in the actual distributed parameter system the closed loop stability is in question the results presented here provide bounds on the smallness of the singular perturbation parameter to ensure stable operation such a priori bounds may be used to evaluate potential reduced order controllers for distributed parameter systems

clustered model reduction of positive directed networks this paper proposes a clustered model reduction method for semistable positive linear systems evolving over directed networks in this method we construct a set of clusters i e disjoint sets of state variables based on a notion of cluster reducibility defined as the uncontrollability of local states by aggregating the reducible clusters with aggregation coefficients associated with the frobenius eigenvector we obtain an approximate model that preserves not only a network structure among clusters but also several fundamental properties such as semistability positivity and steady state characteristics furthermore it is found that the cluster reducibility can be characterized for semistable systems based on a projected controllability gramian that leads to an a priori h 2 error bound of the state discrepancy caused by aggregation the efficiency of the proposed method is demonstrated through an illustrative example of enzyme catalyzed reaction systems described by a chemical master equation this captures the time evolution of chemical reaction systems in terms of a set of ordinary differential equations

a new approach to the perfect regulation and the bounded peaking in linear multivariable control systems this paper presents a new approach to the perfect regulation p r and the bounded peaking b p in linear multivariable control systems based on the pole and the eigenvector assignment technique roughly speaking the p r represents an ideal control action which reduces the settling time to 0 while the b p simply means the bounded overshoot in this ideal situation simple frequency domain characterizations of the p r and the b p are derived they reveal some invariance properties of the p r and the b p that provide a powerful tool for achieving the p r and the b p the existence condition for the p r is derived which turns out to be identical to the result obtained in the optimal regulator theory this result is extended to a more general control objective of attaining the p r for one output while keeping the b p for another output a condition on which the p r is realized by an output feedback is also derived a simple design algorithm is proposed for achieving the p r which is essentially the eigenvector assignment procedure finally an application to a real system is discussed

perfect and subperfect regulation in linear multivariable control systems this paper is concerned with the synthesis of a high gain feedback control called perfect regulation p r for linear multivariable systems with external signals the existence condition for p r is derived in which the minimum phase property plays an essential role the use of an observer is discussed for achieving p r under restricted state observation some asymptotically ideal feedback properties are demonstrated such as the complete desensitization the complete servo performance with decoupling and the complete disturbance rejection which formulate the loop tightness for multivariable systems a design method for applying p r to non minimum phase systems called subperfect regulation s p r is proposed based on the factorization of the plant transfer function matrix into the minimum phase part and the totally non minimum phase part it is a multivariable extension of a well known design technique for scalar systems to overcome the difficulty of phase non minimality computational procedures for p r and s p r are discussed an illustrative example is shown

semi invariants and their use for stability analysis of planar systems semi invariants and relative characteristic functions extend to nonlinear systems the concept of eigenvector eigenvalue pair for linear systems and are therefore very useful to depict the behaviour of the system in this article semi invariants are used to construct explicitly lyapunov functions useful for studying the stability of the origin for continuous time systems moreover using well known tools from differential geometry such as orbital symmetries it is shown how semi invariants can be found for several classes of systems important connections with centre manifold theory are pointed out by using the proposed general techniques a new proof of a result claimed to bendixson bendixson i 1901 sur les courbes definies par des equations differentielles acta mathematics 24 1 88 is given

stabilization of discrete systems with multiple time scales a two stage method is developed for the stabilization of linear time invariant discrete systems with multiple time scales it is shown that the feedback gains are completely independent the closed loop system is asymptotically stable for all sufficiently small singular perturbation parameters which are constrained to be in a bounded set

novel gramians for linear semistable systems in this paper the notions of pseudo gramians are introduced for linear time invariant semistable systems which allow multiple semisimple poles at the origin the proposed gramians are the generalizations of standard gramian matrices defined for asymptotically stable systems and they can be computed by a set of lyapunov equations furthermore it is shown that the controllability and observability of a semistable system are indicated by the ranks of the pseudo gramians and the controllability and observability energy functions are also characterized using the pseudo gramians additionally the h2 norm and h norm of a semistable system are analyzed and then the results are used for the model reduction of semistable systems finally the effectiveness of the methods is illustrated by an example of a gene regulation network

common diagonal lyapunov function for third order linear switched system consider the stability problem for the following linear switched system differential inclusion x a x a a 1 a 2 a n here a i i 1 2 n are n i n dimensional hurwitz stable real matrices in this study for this system we investigate the problem of the existence and construction of a common diagonal lyapunov function of the form v x x t d x where d is a positive diagonal matrix in the case of n 3 i e third order system we suggest a simple elimination algorithm which gives a common d in the case of existence highlights we consider n dimensional linear switched systems construction of a common diagonal lyapunov function for the systems is considered a simple elimination algorithm for third order system is given for the general case we apply kelly s cutting plane method to the given problem

order and the index of singular time invariant linar systems various concepts of order have been proposed in the literature it is shown that many of these are directly related to the index of a system

sufficient conditions for schur and hurwitz diagonal stability of complex interval matrices abstract let c a j b be a complex interval matrix where a b r n n are real interval matrices and j 2 1 denote by d the set of positive definite diagonal n n matrices the schur respectively hurwitz diagonal stability of c abbreviated as sds respectively hds is defined in terms of the stein respectively lyapunov inequality associated with a generic matrix c c the inequality must have a common solution q d for all c c the paper develops two types of results for the sds respectively hds analysis which exploit different properties of c each type of results presents i a sufficient condition relying on the solvability of a matrix norm inequality sds analysis or a matrix measure inequality hds analysis ii a construction procedure for a concrete matrix that satisfies the inequality at i and concomitantly yields a common solution q d for some particular cases both necessary and sufficient conditions can be derived the schur and hurwitz approaches are developed in parallel for concision reasons as well as for emphasizing the similarities numerical examples illustrate the applicability of the theoretical results

a singular perturbation canonical form of invertible systems determination of multivariate root loci a new canonical form of multivariable invertible systems is given this form is suitable for the singular perturbation analysis of such singular problems as multi variable asymptotic root loci under high gain feedback cheap control state variable estimation with weak measurement noise etc appropriate selection and decomposition of state variables into slow and fast changing groups is a key feature that loads to well defined singular perturbation models the selection of state variables is accomplished without explicitly calculating any eigenvalues of the given system but by knowing the direct relationship of the output and its differential coefficients with the input the use of the canonical form to characterize the multivariable root loci under high gain feedback is demonstrated

on the robustness of output feedback control methods to modeling errors singular perturbation analysis is employed to study the robustness of output feedback control schemes for linear time invariant systems the effect of designing the control strategy using a model that neglects unknown parasitic elements is considered it is shown that such a strategy may in general be destabilizing even when the neglected modes are asymptotically stable special structures that will not permit this situation and will guarantee asymptotic stability are illustrated

structured and simultaneous lyapunov functions for system stability problems abstract it is shown that many system stability and robustness problems can be reduced to the question of when there is a quadratic lyapunov function of a certain structure which establishes stability of x ax for some appropriate a the existence of such a lyapunov function can be determined by solving a convex program we present several numerical methods for these optimization problems a simple numerical example is given

a numerical method for the stability analysis of quasi polynomial vector fields this paper shows the sufficient conditions for the existence of a lyapunov function in the class of quasi polynomial dynamical systems we focus on the cases where the system s parameters are numerically specified a numerical algorithm to analyze this problem is presented which involves the resolution of a linear matrix inequality lmi this lmi is collapsed to a linear programming problem from the numerical viewpoint this computational method is very useful to search for sufficient conditions for the stability of non linear systems of odes the results of this paper greatly enlarge the scope of applications of a method previously presented by the authors

on the determination of a diagonal solution of the lyapunov equation in this note for a given matrix a the solution of the lyapanov equation a p pa q 0 considering p diagonal or both p and q diagonal matrices is investigated for this purpose a simple algorithm based on nonlinear programming is presented

subsystems time scales and multimodeling through a couple of naive examples the control theorists are invited to re examine the role of modeling in the study of large scale dynamic systems instead of assuming the existence of n diagonally dominant blocks it is possible to justify one strongly coupled slow core and n weakly coupled fast subsystems this structure is exhibited with a physically meaningful choice of state variables the controls are introduced following the recent concept of multimodeling

on quadratic lyapunov functions a topological structure as a subset of 0 2 spl pi sup l spl times spl ropf sub sup n 1 is proposed for the set of quadratic lyapunov functions qlfs of a given stable linear system a necessary and sufficient condition for the existence of a common qlf of a finite set of stable matrices is obtained as the positivity of a certain integral the structure and the conditions are considerably simplified for planar systems it is also proved that a set of block upper triangular matrices share a common qlf if each set of diagonal blocks share a common qlf

stabilization of switched linear systems with mode dependent time varying delays abstract in this paper we investigate the problem of stabilization via state feedback and or state based switching for switched linear systems with mode dependent time varying delays by using the multiple lyapunov functional method we establish sufficient conditions that guarantee the switched system is stabilizable via state feedback and or switching under time varying delays with appropriate upper bounds the main results are presented in terms of linear matrix inequalities lmis which generalize some known results and can be easily tested by using the matlab s lmi tool box

brief paper positivity preserving h model reduction for positive systems this paper is concerned with the model reduction of positive systems for a given stable positive system our attention is focused on the construction of a reduced order model in such a way that the positivity of the original system is preserved and the error system is stable with a prescribed h performance based upon a system augmentation approach a novel characterization on the stability with h performance of the error system is first obtained in terms of linear matrix inequality lmi then a necessary and sufficient condition for the existence of a desired reduced order model is derived accordingly furthermore iterative lmi approaches with primal and dual forms are developed to solve the positivity preserving h model reduction problem finally a compartmental network is provided to show the effectiveness of the proposed techniques

a note on the discrete lyapunov and riccati matrix equations abstract inequalities which are satisfied by the determinants of the positive definite solutions of the discrete algebraic riccati and lyapunov matrix equations are presented the results give lower bounds for the product of the eigenvalues of the matrix solutions also for a discrete lyapunov equation an algorithm is presented to determine under what conditions a positive diagonal solution will exist if all the conditions are satisfied the algorithm also provides such a diagonal solution

multi scaling mix and non universality between population and facility density the distribution of facilities is closely related to our social economic activities recent studies have reported a scaling relation between population and facility density with the exponent depending on the type of facility in this paper we show that generally this exponent is not universal for a specific type of facility instead by using chinese data we find that it increases with per capita gross domestic product gdp thus our observed scaling law is actually a mixture of several multi scaling relations this result indicates that facilities may change their public or commercial attributes according to the outside environment we argue that this phenomenon results from an unbalanced regional economic level and suggest a modification of a previous model by introducing the consuming capacity the modified model reproduces most of our observed properties c 2012 elsevier b v all rights reserved

direct singular perturbation analysis of high gain and cheap control problems a new and direct approach to the analysis and design of high gain feedback and cheap control problems is developed transformation of such singular problems to singular perturbation models of conventional type early at the onset of analysis or design allows a direct application of the readily available singular perturbation literature the approach permits a simple characterization of a multivariable root loci under a high gain feedback and b asymptotic behavior of optimal closed loop poles state and control trajectories performance index and optimal transfer function as the control cost coefficient in the performance index goes to zero

bifurcation in the presence of small noise the methods of deterministic bifurcation arc sensitive to the addition of small amounts of white noise thus for example in systems whose macroscopic deterministic description arises from an aggregation of microscopically fluctuating dynamics the predictions of deterministic bifurcation may be incorrect here we use laplace s method of steepest descent to study bifurcation in the presence of small noise motivation for this study arises from maxwell s equal area rule for phase transitions in van der waals gases the theory is then applied to the study of the dynamics of noisy constrained or implicitly defined dynamical systems

on the solutions of linear differential equations with singular coefficients abstract equations of the form e ax u with e and a square matrices and e singular are considered the controversy that exists in the literature concerning the solutions of such equations is investigated solutions are arrived at through an application of singular perturbation theory

matrix theoretic conditions for the realizability of sliding manifolds systems of the type s ksgn s s rm where sgn is the vector sign function usually determine the realizability of sliding modes in control systems with high frequency gain k a new condition on k is established for the global stability of this system using a nonsmooth liapunov function this condition is used to establish necessary and sufficient stability conditions for second order systems

convergence in the boundary layer for singularly perturbed equations a singularly perturbed linear finite dimensional ordinary differential equation is considered on the half line 0 the reduced system is assumed to have a unique solution in the sense of distributions it is proved that under reasonable conditions the solution of the full system converges to that of the reduced in the distributional sense and the order of the limiting distribution is computed an application to the optimal linear quadratic regulator with cheap control is described

application of lyapunov functionals to studying stability of linear hyperbolic systems the lyapunov functional method is used to prove the stability conditions for cauchy problems and initial boundary value problems if the system is described by a set of linear first order partial differential equations of the hyperbolic type the application of the lyapunov functional method to stability of linear hyperbolic systems with more than two equations leads to the search for functionals with diagonal matrices the question of whether or not there exists a positive diagonal matrix g such that d sup t g gd

a spatial weighted network model based on optimal expected traffic abstract we propose a spatial weighted network model based on the optimal expected traffic the expected traffic represents the prediction of the flow created by two vertices and is calculated by the improved gravity equation the model maximizes the total expected traffic of the network by changing two parameters which control the fitness and the geographical constraints the model can vary its topology and give rise to a variety of statistical properties observed in the real world network notably our study shows that a linear and a nonlinear strength degree correlation can emerge when considering and neglecting the transport effect respectively

on structural properties of the lyapunov matrix equation for optimal diagonal solutions we revisit the classical problem of finding a positive diagonal solution p to the lyapunov equation at p pa 0 that minimizes the lyapunov exponent the maximum eigenvalue of at p pa for a isin rnxn with the aim of identifying structural properties of the lyapunov matrix equation at the optimum using eigenvalue sensitivity notions together with optimization machinery we are able to obtain an explicit characterization of the minimum lyapunov exponent that provides such structural insight

model reduction and clusterization of large scale bidirectional networks this paper proposes two model reduction methods for large scale bidirectional networks that fully utilize a network structure transformation implemented as positive tridiagonalization first we present a krylov based model reduction method that guarantees a specified error precision in terms of the h norm positive tridiagonalization allows us to derive an approximation error bound for the input to state model reduction without computationally expensive operations such as matrix factorization second we propose a novel model reduction method that preserves network topology among clusters i e node sets in this approach we introduce the notion of cluster uncontrollability based on positive tridiagonalization and then derive its theoretical relation to the approximation error this error analysis enables us to construct clusters that can be aggregated with a small approximation error the efficiency of both methods is verified through numerical examples including a large scale complex network

on the existence of positive diagonal p such that pa a t p l 0 given a real square matrix a we address the question does there exist a positive diagonal matrix p such that pa a t p is negative definite we present an algorithm which when applied to a gives a definite yes or no answer to this question if the answer is yes the algorithm will provide one such p

on a class of optimal control problems with an almost cost free solution for a class of linear singular optimal control problems with a nonunique singular arc the solution of the corresponding nearly singular problem is analyzed and a limit solution based on formal singular perturbations is derived a rigorous proof of the correctness of the result is given by constructing a convergent power series satisfying the riccati equation of the nearly singular problem

growing networks under geographical constraints inspired by the structure of technological weblike systems we discuss network evolution mechanisms which give rise to topological properties found in real spatial networks thus we suggest that the peculiar structure of transport and distribution networks is fundamentally determined by two factors these are the dependence of the spatial interaction range of vertices on the vertex attractiveness or importance within the network and on the inhomogeneous distribution of vertices in space we propose and analyze numerically a simple model based on these generating mechanisms which seems for instance to be able to reproduce known structural features of the internet

perturbation analysis of a system of quasi variational inequalities for optimal stochastic scheduling optimal scheduling of continuously evolving dynamical systems involves the analysis of certain partial differential inequalities on solution dependent domains these are the quasi variatlonai inequalities qvi s introduced for such problems by a bensoussan and j l lions they arise naturally in such applications as inventory control and unit commitment scheduling in electric energy systems qvi s are in these contexts the analog of the dynamic programming partial differential equations of r bellman for continuous control problems analytical treatment of qvi s involves not only the solution of partial differential inequalities but also the simultaneous analysis of the free boundaries of the domain of the solution in effect the optimal return function is the solution of the inequalities and the optimal control switching policy is defined by the free boundaries this paper considers a system of two singularly perturbed second order quasi variational inequalities with a turning point in a bounded domain in r1 asymptotic approximations for the free boundaries are derived using standard perturbation analysis for boundary value problems also local regularity properties of the limit solution are obtained the results are illustrated by a parametric study of a simple model of commitment scheduling for a single unit in a power generation system

singular perturbational model reduction in the frequency domain singular perturbational approximations for linear continuous time and discrete time systems are developed in the frequency domain it is shown that the familiar singular perturbational result is an approximation at the origin in the complex plane however if the system has multiple time scale effects other approximations can be obtained at different locations on the negative real axis to emphasize such behavior the relationship between singular perturbational approximations and direct subsystem elimination is also investigated

brief paper singular perturbation analysis of a receding horizon controller a new receding horizon approach to the linear regulator problem is presented at each instant of time the control strategy is based on the minimization of a quadratic performance index extending over an interval of fixed length into the future connections with standard results are discussed and under general assumptions the closed loop system is shown to be asymptotically stable a potential advantage of our method lies in the fact that a singular perturbation analysis is well posed this indicates that for plants containing slow and fast modes the design procedure can be split up into two stages the resulting composite control exhibits attractive engineering properties use and effectiveness of the proposed techniques are demonstrated by investigating a practical example

on two classes of matrices with positive diagonal solutions to the lyapunov equation abstract we characterize real indecomposable quasi jacobi matrices of class d i e those which satisfy the lyapunov equation pa a p q with p diagonal and both p and q positive definite the subclass d 2 of class d when also q is diagonal is also characterized in the case of general indecomposable real matrices

brief paper krylov projection framework for fourier model reduction this paper analyzes the fourier model reduction fmr method from a rational krylov projection framework and shows how the fmr reduced model which has guaranteed stability and a global error bound can be computed in a numerically efficient and robust manner by monitoring the rank of the krylov subspace that underlies the fmr model the projection framework also provides an improved criterion for determining the number of fourier coefficients that are needed and hence the size of the resulting reduced order model the advantages of applying fmr in the rational krylov projection framework are demonstrated on a simple example

stability properties of a general class of nonlinear dynamical systems we establish sufficient conditions for the boundedness of the trajectories and the stability of the fixed points in a class of general nonlinear systems the so called quasi polynomial vector fields with the help of a natural embedding of such systems in a family of generalized lotka volterra lv equations a purely algebraic procedure is developed to determine such conditions we apply our method to obtain new results for lv systems by a reparametrization in time variable and to study general nonlinear vector fields originally far from the lv format

scaling a canonical design problem for networks motivated by decentralized control and numerical methods applications we explore the scaling problem that is we consider the design of a diagonal matrix k so as to i place the eigenvalues of the product kg in the olhp and ii optimize a scalar performance measure c kg of interest for a square topology matrix g we first illustrate our methodology for scaling design which combines optimization machinery with graph algebra notions for a class of topology matrices and a specific performance measure in particular for symmetric positive definite topologies and for a dominant eigenvalue ratio performance measure we find that the optimally scaled system kg has a special structure the right eigenvectors associated with the largest and smallest eigenvalues have corresponding entries of identical magnitude and hence differ only in sign pattern using this structure we develop both explicit and algorithmic methods for computing the optimal scaling and cost after illustrating our methodology with this

asymptotic stability of nonlinear multiparameter singularly perturbed systems sufficient conditions are derived to guarantee the asymptotic stability of a class of nonlinear singularly perturbed systems with several perturbation parameters of the same order estimates of the region of attraction and bounds on the small parameters are obtained

the finite state projection approach for the analysis of stochastic noise in gene networks in order to capture important subcellular dynamics researchers in computational biology have begun to turn to mesoscopic models in which molecular interactions at the gene level behave as discrete stochastic events while the trajectories of such models cannot be described with deterministic expressions the probability distributions of these trajectories can be described by the set of linear ordinary differential equations known as the chemical master equation cme until recently it has been believed that the cme could only be solved analytically in the most trivial of problems and the cme has been analyzed almost exclusively with kinetic monte carlo kmc algorithms however concepts from linear systems theory have enabled the finite state projection fsp approach and have significantly enhanced our ability to solve the cme without resorting to kmc simulations in this paper we review the fsp approach and introduce a variety of systems theory based modifications and enhancements to the fsp algorithm notions such as observability controllability and minimal realizations enable large reductions and increase efficiency with little to no loss in accuracy model reduction techniques based upon linear perturbation theory allow for the systematic projection of multiple time scale dynamics onto a slowly varying manifold of much smaller dimension we also present a powerful new reduction approach in which we perform computations on a small subset of configuration grid points and then interpolate to find the distribution on the full set the power of the fsp and its various reduction approaches is illustrated on few important models of genetic regulatory networks

reduction of second order network systems with structure preservation this paper proposes a general framework for structure preserving model reduction of a second order network system based on graph clustering in this approach vertex dynamics are captured by the transfer functions from inputs to individual states and the dissimilarities of vertices are quantified by the mathcal h 2 norms of the transfer function discrepancies a greedy hierarchical clustering algorithm is proposed to place those vertices with similar dynamics into same clusters then the reduced order model is generated by the petrov galerkin method where the projection is formed by the characteristic matrix of the resulting network clustering it is shown that the simplified system preserves an interconnection structure i e it can be again interpreted as a second order system evolving over a reduced graph furthermore this paper generalizes the definition of network controllability gramian to second order network systems based on it we develop an efficient method to compute mathcal h 2 norms and derive the approximation error between the full order and reduced order models finally the approach is illustrated by the example of a small world network

krylov subspace methods for the sylvester equation abstract we describe galerkin and minimal residual algorithms for the solution of sylvester s equation ax xb c the algorithms use krylov subspaces forwhich orthogonal bases are generated by the arnoldi process for certain choices of krylov subspaces the computation of the solution splits into the solution of many independent subproblems this makes the algorithms suitable for implementation on parallel computers

on the set of points of smoothness for the value function of affine optimal control problems we study the regularity properties of the value function associated with an affine optimal control problem with quadratic cost plus a potential for a fixed final time and initial point without assuming any condition on singular minimizers we prove that the value function is continuous on an open and dense subset of the interior of the attainable set as a byproduct we obtain that it is actually smooth on a possibly smaller set still open and dense

modeling and simulating chemical reactions many students are familiar with the idea of modeling chemical reactions in terms of ordinary differential equations however these deterministic reaction rate equations are really a certain large scale limit of a sequence of finer scale probabilistic models in studying this hierarchy of models students can be exposed to a range of modern ideas in applied and computational mathematics this article introduces some of the basic concepts in an accessible manner and points to some challenges that currently occupy researchers in this area short downloadable matlab codes are listed and described

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

asymptotic stability of systems with saturation constraints in the present paper the authors establish new sufficient conditions for the global asymptotic stability of linear systems operating on the unit hypercube in r sup n and of linear systems with partial state saturation constraints they also determine a set of necessary and sufficient conditions for the global asymptotic stability of second order linear systems with saturation constraints on both states using the above results systems of the type considered herein are widely used in several areas of applications including control systems signal processing and artificial neural networks

an algorithm for rescaling a matrix positive definite abstract for a given square real matrix m we present a general algorithm which decides the existence of a positive diagonal matrix d such that dm is positive definite and which constructs the d if it exists it is shown that solving this matrix rescaling problem is equivalent to finding a solution of an infinite system of linear inequalities the algorithm solves the infinite system of linear inequalities by generating and solving a sequence of linear programs keywords eigenvalues eigenvectors

optimal transportation under nonholonomic constraints we study monge s optimal transportation problem where the cost is given by an optimal control cost we prove the existence and uniqueness of an optimal map under certain regularity conditions on the lagrangian absolute continuity of the measures with respect to lebesgue and most importantly the absence of sharp abnormal minimizers in particular this result is applicable in the case of subriemannian manifolds with a 2 generating distribution and cost given by d 2 where d is the subriemannian distance also we discuss some properties of the optimal plan when abnormal minimizers are present finally we consider some examples of displacement interpolation in the case of the grushin plane

state decomposition for singular perturbation order reduction a modal approach singular perturbation techniques for order reduction are based on decomposing the high order system into two parts approximation of the high order system dynamics by the low order model is mainly dependent on this partition this paper introduces a measure for the degree of dependence of the state variables on the high order dominant eigenvalues by means of this measure it is not only possible to find an appropriate decomposition yielding a good low order model but also to get an indication of the order of the reduced model two illustrative examples demonstrate the main results

on the complexity of some common geometric location problems given n demand points in the plane the p center problem is to find p supply points anywhere in the plane so as to minimize the maximum distance from a demand point to its respective nearest supply point the p median problem is to minimize the sum of distances from demand points to their respective nearest supply points we prove that the p center and the p median problems relative to both the euclidean and the rectilinear metrics are np hard in fact we prove that it is np hard even to approximate the p center problems sufficiently closely the reductions are from 3 satisfiability

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

a dynamical approach to the sard problem in carnot groups abstract we introduce a dynamical systems approach for the study of the sard problem in sub riemannian carnot groups we show that singular curves can be obtained by concatenating trajectories of suitable dynamical systems as an application we positively answer the sard problem in some classes of carnot groups

a multimodel approach to stochastic nash games a decentralized filtering and control scheme is presented for obtaining low order nash equilibrium strategies for decision makers using different models of the same large scale system multiparameter singular perturbations are employed to capture the multimodel nature of the fast dynamic subsystems interconnected through slow dynamic variables the small parameters are appropriately scaled so that the variables in both time scales are well defined the decision makers have decentralized information structures and are constrained to use only finite dimensional compensators of a specified dimension it is shown that the proposed scheme is in fact the asymptotic limit of the exact solution as the small parameters go to zero

morse sard type results in sub riemannian geometry let m g be a sub riemannian manifold and x0 m assuming that chow s condition holds and that m endowed with the sub riemannian distance is complete we prove that there exists a dense subset n1 of m such that for every point x of n1 there is a unique minimizing path steering x0 to x this trajectory admitting a normal extremal lift if the distribution is everywhere of corank one we prove the existence of a subset n2 of m of full lebesgue measure such that for every point x of n2 there exists a minimizing path steering x0 to x which admits a normal extremal lift is nonsingular and the point x is not conjugate to x0 in particular the image of the sub riemannian exponential mapping is dense in m and in the case of corank one is of full lebesgue measure in m

singular perturbations and time scale methods in control theory survey 1976 1983 recent progress in the use of singular perturbation and two time scale methods of modeling and design for control systems is reviewed over 350 references are organized into major problem areas representative issues and results are discussed with a view to outlining research directions and indicating potential areas of application the survey is aimed at engineers and applied mathematicians interested in model order reduction separation of time scales and allied simplified methods of control system analysis and design the exposition does not assume prior knowledge of singular perturbation methods