convergence of time stepping schemes for passive and extended linear complementarity systems generalizing recent results in m k camlibel complementarity methods in the analysis of piecewise linear dynamical systems ph d thesis center for economic research tilburg university tilburg the netherlands 2001 m k camlibel w p m h heemels and j m schumacher ieee trans circuits systems i fund theory appl 49 2002 pp 349 357 and j s pang and d stewart math program ser a 113 2008 pp 345 424 this paper provides an in depth analysis of time stepping methods for solving initial value and boundary value non lipschitz linear complementarity systems lcss under passivity and broader assumptions the novelty of the methods and their analysis lies in the use of least norm solutions in the discrete time linear complementarity subproblems arising from the numerical scheme these subproblems are not necessarily monotone and are not guaranteed to have convex solution sets among the principal results it is shown that using such least norm solutions of the discrete time subproblems an implicit euler scheme is convergent for passive initial value lcss generalizations under a strict copositivity assumption and for boundary value lcss are also established

strong detectability and observers conditions are given for the existence of observers that estimate unmeasured outputs on the basis of partial information on the input and the state the concepts of strong detectability and strong observability introduced before in the literature for discrete time systems only are defined and studied for continuous time systems it is shown that there are two different concepts of strong detectability which coincide for discrete time systems algebraic conditions for either concept are given it is shown that these concepts are intimately related to the existence of strong observers i e observers that only use the output of the plant

a note on stabilization by dynamic high gain output feedback in this note we study the problem of stabilization by dynamic output feedback for a certain class of linear systems we establish the existence of pairs of subspaces im upsilon a where im is a minimum phase input subspace contained in an almost stabilizability snbspace upsilon a by applying a recent theorem by sehumacber this result leads to the existence of stabilizing high gain compensators for the linear systems under consideration in particular it will be shown that invertible minimum phase systems can always be stabilized by dynamic compensators of dynamic order equal to the excess of poles over zeros minus the number of inputs

solvability of variational inequalities on hilbert lattices this paper provides a systematic solvability analysis for generalized variational inequalities on separable hilbert lattices by contrast to a large part of the existing literature our approach is lattice theoretic and is not based on topological fixed point theory this allows us to establish the solvability of certain types of generalized variational inequalities without requiring the involved set valued maps be hemicontinuous or monotonic some of our results generalize those obtained in the context of nonlinear complementarity problems in earlier work and appear to have scope for applications this is illustrated by means of several applications to fixed point theory optimization and game theory

almost invariant subspaces an approach to high gain feedback design part ii almost conditionally invariant subspaces in part i of this paper almost controlled invariant subspaces were studied in this part we will consider their duals the almost conditionally invariant subspaces these concepts give immediately by dualization of the almost disturbance decoupling control by state feedback the solution of the almost disturbance decoupled estimation problem finally we consider the problem of approximate disturbance decoupling by measurement feedback and it is shown that this problem is solvable to any arbitrary degree of accuracy if and only if 1 almost disturbance decoupling by state feedback and 2 almost disturbance decoupled estimation of the to be controlled output are both possible

strongly regular differential variational systems a differential variational system is defined by an ordinary differential equation ode parameterized by an algebraic variable that is required to be a solution of a finite dimensional variational inequality containing the state variable of the system this paper addresses two system theoretic topics for such a nontraditional nonsmooth dynamical system namely non zenoness and local observability of a given state satisfying a blanket strong regularity condition for the former topic which is of contemporary interest in the study of hybrid systems we extend the results in our previous paper where we have studied zeno states and switching times in a linear complementarity system lcs as a special case of the differential variational inequality dvi the lcs consists of a linear time invariant ode and a linear complementarity problem the extension to a nonlinear complementarity system ncs with analytic inputs turns out to be non trivial as we need to use the lie derivatives of analytic functions in order to arrive at an expansion of the solution trajectory near a given state further extension to a differential variational inequality is obtained via its equivalent karush kuhn tucker formulation for the second topic which is classical in system theory we use the non zenoness result and the recent results in a previous paper pertaining to the b differentiability of the solution operator of a nonsmooth ode to obtain a sufficient condition for the short time local observability of a given strongly regular state of an ncs refined sufficient conditions and necessary conditions for local observability of the lcs satisfying the p property are obtained

system structure and singular control the general continuous time linear quadratic control problem is considered it is shown that recently developed linear system theoretic properties and algorithms play an important role in solving this singular control problem

entropy and boundary conditions in random rhombus tilings the tilings of rhombi in two dimensions and of rhomboedra in three dimensions are studied when they are constrained by fixed boundary conditions we establish a link between those conditions and free or periodic boundary ones the entropy is written as a functional integral which is treated via a saddle point method we can exhibit the dominant states of the statistical ensemble of tilings and show that they can display a strong structural inhomogeneity caused by the boundary this inhomogeneity is responsible for a difference of entropy between the studied fixed boundary tilings and free boundary ones this method uses a representation of tilings by membranes embedded in a higher dimensional hypercubic lattice it is illustrated in the case of 60 degree rhombus tilings

geometric insight into discrete time cheap and singular linear quadratic riccati lqr problems the hamiltonian system related to discrete time cheap linear quadratic riccati lqr problems is analyzed in a purely geometric context with the twofold purpose of getting a useful insight into its structural features and deriving a numerically implementable solution for the infinite horizon case by only using the standard geometric approach routines that are available

uniqueness of solutions of linear relay systems conditions are given for uniqueness of solutions of linear time invariant systems under relay feedback from a hybrid dynamical point of view this entails the deterministic specification of the discrete transition rules the results are based on the formulation of relay systems as complementarity systems and use the constructive theory of the linear complementarity problem

the square triangle random tiling model in the thermodynamic limit the thermodynamic limit for the square triangle random tiling model is considered the analytic solution of the bethe ansatz equations found recently by widom 1993 is obtained the analytic expression for entropy density as a function of the fraction of the plane occupied by triangles alpha t is derived for the case alpha t or 1 2 i e for the phason gradients having 6 fold symmetry including 12 fold symmetric phase as the limiting case the exact values for phason elastic constants are also found

some p properties for linear transformations on euclidean jordan algebras a real square matrix is said to be a p matrix if all its principal minors are positive it is well known that this property is equivalent to the nonsign reversal property based on the componentwise product of vectors the order p property based on the minimum and maximum of vectors uniqueness property in the standard linear complementarity problem lipschitzian homeomorphism property of the normal map corresponding to the nonnegative orthant in this article we extend these notions to a linear transformation defined on a euclidean jordan algebra we study some interconnections between these extended concepts and specialize them to the space s n of all n n real symmetric matrices with the semidefinite cone s n and to the space r n with the lorentz cone

optimal solutions to variational inequalities on banach lattices abstract we study the existence of maximum and minimum solutions to generalized variational inequalities on banach lattices the main tools of analysis are the variational characterization of the generalized metric projection operator and order theoretic fixed point theory

construction and parameterization of all static and dynamic h 2 optimal state feedback solutions for discrete time systems abstract this paper considers an h 2 optimization problem via state feedback for discrete time systems the class of problems dealt with here has a left invertible transfer matrix function from the control input to the controlled output the paper constructs and parameterizes all the static and dynamic h 2 optimal state feedback solutions moreover all the eigenvalues of an optimal closed loop system are characterized all optimal closed loop systems share a set of eigenvalues which are termed the optimal fixed modes every h 2 optimal controller must assign among the closed loop eigenvalues the set of optimal fixed modes this set of optimal fixed modes includes a set of optimal fixed decoupling zeros which shows the minimum absolutely necessary number and locations of pole zero cancellations present in any h 2 optimal design most of the results presented here are analogous to but not quite the same as those for continuous time systems in fact there are some fundamental differences between the continuous and discrete time systems reflecting mainly the inherent nature and characteristics of these systems

solvability conditions consistency and weak consistency for linear differential algebraic equations and time invariant singular systems the general case we present several solvability concepts for linear differential algebraic equations daes with constant coefficients on the positive time axis as well as for the associated singular systems and investigate under which conditions these concepts are met next we derive necessary and sufficient conditions for global consistency of initial conditions for the dae as well as for the system and generalize these conditions with respect to our concept of weak consistency our distributional approach enables us to generalize results in an earlier paper where singular systems are assumed to have a regular pencil in the sense of gantmacher in particular we establish that global weak consistency in the system sense is equivalent to impulse controllability

passivity and complementarity this paper studies the interaction between the notions of passivity of systems theory and complementarity of mathematical programming in the context of complementarity systems these systems consist of a dynamical system given in the form of state space representation and complementarity relations we study existence uniqueness and nature of solutions for this system class under a passivity assumption on the dynamical part a complete characterization of the initial states and the inputs for which a solution exists is given these initial states are called consistent states for the inconsistent states we introduce a solution concept in the framework of distributions

exceptional family of elements for general order complementarity problems abstract the notion of exceptional family of elements for general order complementarity problems in banach spaces will be introduced it will be shown that for general order complementarity problems defined by completely continuous fields the problem has either a solution or an exceptional family of elements finite dimensional examples and an application to integral operators will be given

a subspace approach to linear dynamical systems abstract in this paper we present an approach to linear dynamical systems which combines the positive features of two well known formulations namely standard state space theory see for e g wonham 1978 9 and behavioural systems theory see polderman and willems 1997 4 our development is also geometric in the tradition of wonham and others but instead of using explicit linear maps we work with linear relations implicitly which amounts to working with subspaces one of our primary motivations is computational efficiency all our computations can be performed on the system as it is without elimination of variables and further unlike the behaviourists who manipulate matrices with polynomial entries we work only with real matrices using our formulation we derive the standard vector space results on controlled and conditioned invariant subspaces of linear dynamical systems duality which is a distinctive feature of state space theory but not of the behavioural view point comes out naturally in our approach too through the use of the adjoint we illustrate our ideas for an important class of dynamical systems viz electrical networks the theory proposed in this paper gives a unified description of both the standard linear dynamical systems and the linear singular systems or the linear descriptor systems see for e g f gantmacher 1959 1 and lewis 1986 2 therefore the algorithms described for the invariant spaces in this paper are also applicable to linear singular systems

phase diagram of a random tiling quasicrystal we study the phase diagram of a two dimensional random tiling model for quasicrystals at proper concentrations the model has 8 fold rotational symmetry landau theory correctly gives most of the qualitative features of the phase diagram which is in turn studied in detail numerically using a transfer matrix approach we find that the system can enter the quasicrystal phase from many other crystalline and incommensurate phases through first order or continuous transitions exact solutions are given in all phases except for the quasicrystal phase and for the phase boundaries between them we calculate numerically the phason elastic constants and entropy density and confirm that the entropy density reaches its maximum at the point where phason strains are zero and the system possesses 8 fold rotational symmetry in addition to the obvious application to quasicrystals this study generalizes certain surface roughening models to two dimensional surfaces in four dimensions

an extension of tarski s fixed point theorem and its application to isotone complementarity problems tarski s fixed point theorem is extended to the case of set valued mappings and is applied to a class of complementarity problems defined by isotone set valued operators in a complete vector lattice

squaring down lti systems a geometric approach in this paper the problem of reducing a given lti system into a left or right invertible one is addressed and solved with the standard tools of the geometric control theory first it will be shown how an lti system can be turned into a left invertible system thus preserving key system properties like stabilizability phase minimality right invertibility relative degree and infinite zero structure moreover the additional invariant zeros introduced in the left invertible system thus obtained can be arbitrarily assigned in the complex plane by duality the scheme of a right inverter will be derived straightforwardly moreover the squaring down problem will be addressed in fact when the left and right reduction procedures are applied together a system with an unequal number of inputs and outputs is turned into a square and invertible system furthermore as an example it will be shown how these techniques may be employed to weaken the standard assumption of left invertibility of the plant in many optimization problems

brief on solution concepts and well posedness of linear relay systems in this paper we study the well posedness existence and uniqueness of solutions of linear relay systems with respect to two different solution concepts filippov solutions and forward solutions we derive necessary and sufficient conditions for well posedness in the sense of filippov of linear systems of relative degree one and two in closed loop with relay feedback to be precise uniqueness of filippov and also forward solutions follows in this case if the first non zero markov parameter is positive by means of an example it is shown that this intuitively clear condition is not true for systems with relative degree larger than two the influence of the zeno phenomenon an infinite number of relay switching times in a finite length time interval on well posedness is highlighted and although linear relay systems form a rather limited subclass of hybrid dynamical systems the consequences of the presence of the zeno behaviour is typical for many other classes of non smooth and hybrid systems

solution of the dimer problem on a hexagonal lattice with boundary the dimer problem is solved exactly for a hexagonal lattice with general boundary using a known generating function from the theory of partitions it is shown that the leading term in the entropy depends on the shape of the boundary

equivalence of nonlinear complementarity problems and least element problems in banach lattices a class of not necessarily linear operators a v v is introduced where the banach space v and its dual v carry dual vector lattice orderings these operators called z maps generalize the n n real matrices with nonnegative off diagonal elements if a is a strictly monotone z map with certain regularity and growth conditions and if f denotes the set of all vectors v v for which v 0 and av 0 then it is shown that the complementarity problem to find v f such that 0 and the least element problem to find v f with v w for all w f have the same unique solution some other problems equivalent to these and some examples are discussed

new results and application of singular control we reconsider the singular control problem with marginal stability of the closed loop system when the transfer matrix from the input to the output can have linearly dependent columns and zeros on the extended imaginary axis we present a new theoretical existence result and a new numerical algorithm based on finding an orthogonal transformation of the matrix pencil associated to the euler lagrange differential equations into a block triangular form we present an application in linear quadratic control of descriptor systems under the constraints of physical realizability of the control and impulse free and marginally stable closed loop system

multiple time scale decomposition in cheap control problems singular control a linear quadratic regulator problem where a small parameter 2multiplies the control cost is considered as mu rightarrow 0 it exhibits a dynamic behavior with multiple time scales whereas for mu 0 it is a singular control problem at first a multiparameter singularly perturbed model of the given problem is constructed by transferring the singularity that exists in the performance index to the dynamic equations then the problem is decomposed into several subproblems of minimal order each pertaining to only one time scale the method developed permits under a single framework the characterization of asymptotic behavior of optimal closed loop poles state and control trajectories performance index and optimal transfer function as mu rightarrow 0 moreover the asymptotic solution brings into focus 1 the nature of singular control and its connection to transmission zeros and 2 the presence of various types of impulses and other higher order distributions in both state and control the approach unifies and extends significantly several ideas previously somewhat disjoint

the linear order complementarity problem the classical complementarity problem in euclidean space can be viewed alternatively as a variational inequality or as a lattice orthogonality problem generalizations of the former have been extensively studied but infinite dimensional analogues of the latter have been largely ignored moreover as we show many well known results about the classical complementarity problem are more appropriately viewed order theoretically this is particularly true of least element solutions which are central to the present study of order complementarity in vector lattices we emphasize that the lattice theoretic descriptions we employ are very useful even in the standard finite dimensional setting

an enhanced gap source model an enhanced gap source model is introduced by combination with rao wilton glisson rwg basis functions for decades the stability of the delta gap source model remains a big problem the rwg basis with triangulation and the feeding edge model have been widely used in numerical analysis of antennas since the feeding edge model uses a single rwg edge as the driving source and neglects the gap width effect the resulting current undergoes sharp transition across the gap and thus the model is highly sensitive to the choice of the feeding edge consequently the feeding edge model suffers instability in calculations of current and input impedance to circumvent these pitfalls this enhanced gap source model uses multiple rwg edges and impulsive function to formulate the feeding behavior this new gap source model exhibits significant improvements on both accuracy and stability numerical results are included to illustrate improved accuracy and stability of this new model by virtue of several typical antennas

variational calculus for descriptor problems the first order necessary condition for optimality is derived from a variational argument that involves an ad hoc modification of the bliss method resulting in a hamiltonian characterization in terms of edx dt rather than dx dt the former being smoother than the latter this approach sidesteps the regularity conditions of the lagrange multiplier theory under some mild assumptions the necessary condition for optimality is also sufficient and the optimal control exists the numerically relevant result is a generalized eigenvector inverse free characterization of optimality

z transformations on proper and symmetric cones z transformations motivated by the similarities between the properties of z matrices on r n and lyapunov and stein transformations on the semidefinite cone mathcal s n we introduce and study z transformations on proper cones we show that many properties of z matrices extend to z transformations we describe the diagonal stability of such a transformation on a symmetric cone by means of quadratic representations finally we study the equivalence of q and p properties of z transformations on symmetric cones in particular we prove such an equivalence on the lorentz cone

driving variable realizations and the non negative realization problem for controllable behaviours non negative linear systems traditionally investigated within the state space framework have been recently analysed within the behavioural setting in a couple of recent papers j w nieuwenhuis when to call a linear system nonnegative linear algebra its appl 281 1998 pp 43 58 m e valcher nonnegative linear systems in the behavioural approach the autonomous case linear algebra and its appl 2000 319 pp 147 162 several definitions and results about non negative behaviours and in particular about non negative autonomous behaviours have been derived moreover the non negative realization problem for autonomous behaviours has been fully explored in m e valcher non negative realization of autonomous systems in the behavioral approach siam j on control and opt 2001 40 pp 540 556 thus deriving an extended set of necessary and sufficient conditions for an autonomous behaviour to admit a non negative realization here we focus our attention on the non negative re

fundamentals of thin wire integral equations with the finite gap generator part ii we discuss certain fundamental properties of hallen s and pocklington s equations with the finite gap generator fgg which has long been used as a feed model for cylindrical antennas for the case of the exact kernel we focus on whether important quantities are finite or infinite and compare to corresponding quantities obtained using the delta function generator dfg for the approximate kernel we focus on the unsolvability of the two equations and the relation of unsolvability to the important phenomenon of unphysical oscillations we also extend to the fgg the so called effective current method this is an easy to apply remedy for the oscillatory solutions that allows one to obtain reasonable answers for important quantities such as current distributions and the input impedance it is shown here that these answers are close to those obtained with the exact kernel we show that in a certain sense the method is more advantageous to apply in the present fgg case than in the dfg case our discussions especially in part ii of this paper bring out similarities to 1 the phenomenon of superdirectivity and 2 recently discussed numerical phenomena occurring within the method of auxiliary sources

technical communique on uniqueness of solutions to relay feedback systems this note deals with the uniqueness of solutions to relay feedback systems and presents an improved criterion over that in lootsma et al automatica 35 3 1999 467

rigorous analysis of a circular patch antenna excited by a microstrip transmission line boundary conditions are enforced on a portion of the microstrip feed line as well as the patch antenna the integral equation for the unknown currents on the antenna and feed is solved by applying the galerkin method of moments in the fourier transform domain the validity of the solution is tested by comparison of computed results with experimental data the theoretical treatment proves to be applicable to the most common feeding arrangements namely the direct edge feed and proximity coupling excitation in the latter case two layer substrates having distinct dielectric constants are studied the purpose of the study is to deduce for a given overall substrate thickness the smallest line ground plane separation for which a match of the radiator to the feed line is still possible the advantages of such a configuration are discussed

a formula for the number of tilings of an octagon by rhombi we propose the first algebraic determinantal formula to enumerate tilings of a centrosymmetric octagon of any size by rhombi this result uses the gessel viennot technique and generalizes to any octagon a formula given by elnitsky in a special case

controllability and elimination of impulsive modes in descriptor systems as a refined notion impulsive mode controllability is suggested for the general class of linear continuous descriptor systems this notion reveals rich properties of the general descriptor systems in terms of admissibility of initial conditions admissibility of feedback control and impulsive mode elimination the condition for impulsive mode controllability is derived and proved to be necessary and sufficient for the existence of admissible feedback controls eliminating impulsive modes under impulsive mode controllability admissible feedback controls are constructed for eliminating impulsive modes on the basis of numerically reliable decompositions of the system s coefficient matrices

inverse m matrices abstract this is an attempt at a comprehensive expository study of those nonnegative matrices which happen to be inverses of m matrices and is aimed at an audience conversant with basic ideas of matrix theory a theme is the parallels anddifferences between the class of m matrices and the class of inverse m matrices among the primary tools used are diagonal multiplications the neumann expansion and the form of the inverse of a partitioned matrix some of the results seem not to have appeared before and several unsolved problems are mentioned

a uniqueness theorem for the generalized order linear complementary problem associated with m matrices abstract in this paper we prove a uniqueness theorem for a class of complementarity problems involving m matrices

the rational complementarity problem abstract an extension of the linear complementarity problem lcp of mathematical programming is the so called rational complementarity problem rcp this problem occurs if complementarity conditions are imposed on input and output variables of linear dynamical input state output systems the resulting dynamical systems are called linear complementarity systems since the rcp is crucial both in issues concerning existence and uniqueness of solutions to complementarity systems and in time simulation of complementarity systems it is worthwhile to consider existence and uniqueness questions of solutions to the rcp in this paper necessary and sufficient conditions are presented guaranteeing existence and uniqueness of solutions to the rcp in terms of corresponding lcps using these results and proving that the corresponding lcps have certain properties we can show uniqueness and existence of solutions to linear mechanical systems with unilateral constraints electrical networks with diodes and linear dynamical systems subject to relays and or coulomb friction

enumeration of octagonal tilings random tilings are interesting as idealizations of atomistic models of quasicrystals and for their connection to problems in combinatorics and algorithms of particular interest is the tiling entropy density which measures the relation of the number of distinct tilings to the number of constituent tiles tilings by squares and 45 rhombi receive special attention as presumably the simplest model that has not yet been solved exactly in the thermodynamic limit however an exact enumeration formula can be evaluated for tilings in finite regions with fixed boundaries we implement this algorithm in an efficient manner enabling the investigation of larger regions of parameter space than previously were possible our new results appear to yield monotone increasing and decreasing lower and upper bounds on the fixed boundary entropy density that converge toward s 0 36021 3

lyapunov stability of complementarity and extended systems a linear complementarity system lcs is a piecewise linear dynamical system consisting of a linear time invariant ordinary differential equation ode parameterized by an algebraic variable that is required to be a solution to a finite dimensional linear complementarity problem lcp whose constant vector is a linear function of the differential variable continuing the authors recent investigation of the lcs from the combined point of view of system theory and mathematical programming this paper addresses the important system theoretic properties of exponential and asymptotic stability for an lcs with a c 1 state trajectory the novelty of our approach lies in our employment of a quadratic lyapunov function that involves the auxiliary algebraic variable of the lcs when expressed in the state variable alone the lyapunov function is piecewise quadratic and thus nonsmooth the nonsmoothness feature invalidates standard stability analysis that is based on smooth lyapunov functions in addition to providing sufficient conditions for exponential stability we establish a generalization of the well known lasalle invariance theorem for the asymptotic stability of a smooth dynamical system to the lcs which is intrinsically a nonsmooth system sufficient matrix theoretic copositivity conditions are introduced to facilitate the verification of the stability properties properly specialized the latter conditions are satisfied by a passive like lcs and certain hybrid linear systems having common quadratic lyapunov functions we provide numerical examples to illustrate the stability results we also develop an extended local exponential stability theory for nonlinear complementarity systems and differential variational inequalities based on a new converse theorem for odes with b differentiable right hand sides the latter theorem asserts that the existence of a b differentiable lyapunov function is a necessary and sufficient condition for the exponential stability of an equilibrium of such a differential system

convergence of a time stepping scheme for rigid body dynamics and resolution of painleve s problem this paper gives convergence theory for a new implicit time stepping scheme for general rigid body dynamics with coulomb friction and purely inelastic collisions and shocks an important consequence of this work is the proof of existence of solutions of rigid body problems which include the famous counterexamples of painleve the mathematical basis for this work is the formulation of the rigid body problem in terms of measure differential inclusions of moreau and monteiro marques the implicit time stepping method is based on complementarity problems and is essentially a particular case of the algorithm described in anitescu potra 2 which in turn is based on the formulation in stewart trinkle 47

linear complementarity systems zeno states a linear complementarity system lcs is a hybrid dynamical system defined by a linear time invariant ordinary differential equation coupled with a finite dimensional linear complementarity problem lcp the present paper is the first of several papers whose goal is to study some fundamental issues associated with an lcs specifically this paper addresses the issue of zeno states and the related issue of finite number of mode switches in such a system the cornerstone of our study is an expansion of a solution trajectory to the lcs near a given state in terms of an observability degree of the state on the basis of this expansion and an inductive argument we establish that an lcs satisfying the p property has no strongly zeno states we next extend the analysis for such an lcs to a broader class of problems and provide sufficient conditions for a given state to be weakly non zeno while related mode switch results have been proved by brunovsky and sussmann for more general hybrid systems our analysis exploits the special structure of the lcs and yields new results for the latter that are of independent interest and complement those by these two and other authors

linear complementarity systems we introduce a new class of dynamical systems called linear complementarity systems the time evolution of these systems consists of a series of continuous phases separated by events which cause a change in dynamics and possibly a jump in the state vector the occurrence of events is governed by certain inequalities similar to those appearing in the linear complementarity problem of mathematical programming the framework we describe is suitable for certain situations in which both differential equations and inequalities play a role for instance in mechanics electrical networks piecewise linear systems and dynamic optimization we present a precise definition of the solution concept of linear complementarity systems and give sufficient conditions for existence and uniqueness of solutions

stochastic linear quadratic regulators with indefinite control weight costs ii in part i of this paper s chen x li and x zhou siam j control optim 36 1998 pp 1685 1702 an optimization model of stochastic linear quadratic regulators lqrs with indefinite control cost weighting matrices is proposed and studied in this sequel the problem of solving lqr models with system diffusions dependent on both state and control variables which is left open in part i is tackled first the solvability of the associated stochastic riccati equations sres is studied in the normal case namely all the state and control weighting matrices and the terminal matrix in the cost functional are nonnegative definite with at least one positive definite which in turn leads to an optimal state feedback control of the lqr problem in the general indefinite case the problem is decomposed into two optimal lqr problems one with a forward dynamics and the other with a backward dynamics the well posedness and solvability of the original lqr problem are then obtained by solving these two subproblems and an optimal control is explicitly constructed examples are presented to illustrate the results

brief paper a new approach to the cheap lq regulator exploiting the geometric properties of the hamiltonian system the cheap lq regulator is reinterpreted as an output nulling problem which is a basic problem of the geometric control theory in fact solving the lq regulator problem is equivalent to keep the output of the related hamiltonian system identically zero the solution lies on a controlled invariant subspace whose dimension is characterized in terms of the minimal conditioned invariant of the original system and the optimal feedback gain is computed as the friend matrix of the resolving subspace this study yields a new computational framework for the cheap lq regulator relying only on the very basic and simple tools of the geometric approach namely the algorithms for controlled and conditioned invariant subspaces and invariant zeros

a new formulation of pocklington s equation for thin wires using the exact kernel pocklington s integro differential equation for thin wires with the exact kernel is reformulated using a second derivative formula for improper integrals this allows for analytical evaluation of the second derivatives resulting in a pure integral equation in a similar manner to what is done when using the approximate kernel however as opposed to using the approximate kernel the resulting integral equation developed here is numerically stable even with a simple pulse function point matching solution good convergence for the current is obtained using pulse functions and the severe unphysical oscillations of the current that are encountered when using pulse functions with the approximate kernel are avoided

all optimal controls for the singular linear quadratic problem without stability a new interpretation of the optimal cost abstract the singular linear quadratic control problem without stability is solved by means of a generalized dual structure algorithm in order to generate all optimal inputs furthermore it is shown that the optimal cost can be interpreted as the smallest nonnegative rank minimizing solution of a certain matrix inequality the so called dissipation inequality

continuous time singular linear quadratic control necessary and sufficient conditions for the existence of regular solutions abstract the purpose of this paper is to provide a full understanding of the role that the constrained generalized continuous algebraic riccati equation plays in singular linear quadratic lq optimal control indeed in spite of the vast literature on lq problems only recently a sufficient condition for the existence of a non impulsive optimal control has for the first time connected this equation with the singular lq optimal control problem in this paper we establish four equivalent conditions providing a complete picture that connects the singular lq problem with the constrained generalized continuous algebraic riccati equation and with the geometric properties of the underlying system

three modeling paradigms in mathematical programming celebrating the sixtieth anniversary since the zeroth international symposium on mathematical programming was held in 1949 this paper discusses several promising paradigms in mathematical programming that have gained momentum in recent years but have yet to reach the main stream of the field these are competition dynamics and hierarchy the discussion emphasizes the interplay between these paradigms and their connections with existing subfields including disjunctive equilibrium and nonlinear programming and variational inequalities we will describe the modeling approaches mathematical formulations and recent results of these paradigms and sketch some open mathematical and computational challenges arising from the resulting optimization and equilibrium problems our goal is to elucidate the need for a systematic study of these problems and to inspire new research in the field

existence and uniqueness of solutions for a class of piecewise linear dynamical systems we consider the class of dynamical systems that arises when inputs and outputs of a linear system are connected pairwise by means of piecewise linear algebraic relations it is not assumed that these relations define inputs in terms of outputs or vice versa in particular the relations need not be lipschitzian we obtain conditions for existence and uniqueness of solutions of such dynamical systems in the class of piecewise bohl functions

fixed boundary octagonal random tilings a combinatorial approach some combinatorial properties of fixed boundary rhombus random tilings with octagonal symmetry are studied a geometrical analysis of their configuration space is given as well as a description in terms of discrete dynamical systems thus generalizing previous results on the more restricted class of codimension one tilings in particular this method gives access to counting formulas which are directly related to questions of entropy in these statistical systems methods and tools from the field of enumerative combinatorics are used

convergence of singular perturbations in singular linear systems abstract a convergence criterion for singular perturbations in linear systems is established the criterion is useful for regular and singular systems it is assumed that the matrices of the system depend analytically on e it is proved that if the real parts of the divergent eigenvalues tend to the solution of the system converges to the solution for e 0 uniformly in compact subsets of 0 on the other hand if there is a divergent eigenvalue whose real part does not tend to then there are initial values such that the solution does not converge at some points the criterion is valid both for the homogeneous equation and for the nonhomogeneous equation if functions are sufficiently smooth the use of distributions allows the analysis with inconsistent initial conditions

on the existence of pareto efficient points we prove some general theorems on the existence of minimal points of compact sets in linear spaces with respect to closed convex orderings these results are compared to the axiom of choice