controlled invariance for discrete time nonlinear systems with an application to the disturbance decoupling problems invariant distributions are defined for discrete time nonlinear control systems and necessary and sufficient conditions are given for their controlled invariance this extends to discrete time systems the basic tool which has been so important in solving the various synthesis problems for continuous time systems to indicate their utility in the discrete time setting they are used to locally solve the disturbance decoupling problem

radon nikodym derivatives of quantum instruments a convenient representation for radon nikodym derivatives of completely positive c p instruments on b h with respect to a scalar measure is suggested similar to the stinespring kraus representation for c p maps but involving possibly nonclosable unbounded operators the structure of covariant c p instruments is studied in detail in particular an exhaustive description is given to instruments covariant with respect to shifts or rotations corresponding to position or angle measurements

generalized simulated annealing we discuss and illustrate a new stochastic algorithm generalized simulated annealing for computationally finding the global minimum of a given not necessarily convex energy cost function defined in a continuous d dimensional space this algorithm recovers as particular cases the so called classical boltzmann machine and fast cauchy machine simulated annealings and turns out to be quicker than both

quantum probabilities for time extended alternatives we study the probability assignment for the outcomes of time extended measurements we construct the class operator that incorporates the information about a generic time smeared quantity these class operators are employed for the construction of positive operator valued measures for the time averaged quantities the scheme highlights the distinction between velocity and momentum in quantum theory propositions about velocity and momentum are represented by different class operators hence they define different probability measures we provide some examples we study the classical limit and we construct probabilities for generalized time extended phase space variables

cooling schedules for optimal annealing a monte carlo optimization technique called simulated annealing is a descent algorithm modified by random ascent moves in order to escape local minima which are not global minima the level of randomization is determined by a control parameter t called temperature which tends to zero according to a deterministic cooling schedule we give a simple necessary and sufficient condition on the cooling schedule for the algorithm state to converge in probability to the set of globally minimum cost states in the special case that the cooling schedule has parametric form tt c log1 t the condition for convergence is that c be greater than or equal to the depth suitably defined of the deepest local minimum which is not a global minimum state

survey paper research on probabilistic methods for control system design a novel approach based on probability and randomization has emerged to synergize with the standard deterministic methods for control of systems with uncertainty the main objective of this paper is to provide a broad perspective on this area of research known as probabilistic robust control and to address in a systematic manner recent advances the focal point is on design methods based on the interplay between uncertainty randomization and convex optimization and on the illustration of specific control applications

ergodic theorems for discrete time stochastic systems using a stochastic lyapunov function sufficient conditions are established under which the law of large numbers and related ergodic theorems hold for nonlinear stochastic systems operating under feedback it is shown that these conditions hold whenever a moment condition is satisfied this may be interpreted as a generalizaton of the martingale property if in addition a stochastic controllability condition holds then it is shown that the underlying distributions governing the system converge to an invariant probability at a geometric rate the key assumption used is that a markov chain with stationary transition probabilities exists that serves as a state process for the closed loop system

on a special class of primitive words when representing dna molecules as words it is necessary to take into account the fact that a word u encodes basically the same information as its watson crick complement q u where q denotes the watson crick complementarity function thus an expression which involves only a word u and its complement can be still considered as a repeating sequence in this context we define and investigate the properties of a special class of primitive words called pseudo primitive words relative to q or simply q primitive words which cannot be expressed as such repeating sequences for instance we prove the existence of a unique q primitive root of a given word and we give some constraints forcing two distinct words to share their q primitive root also we present an extension of the well known fine and wilf theorem for which we give an optimal bound

randomized algorithms for robust controller synthesis using statistical learning theory by now it is known that several problems in the robustness analysis and synthesis of control systems are np complete or np hard these negative results force us to modify our notion of solving a given problem an approach that is recently gaining popularity is that of using randomized algorithms which can be used to solve a problem approximately most of the time we begin with the premise that many problems in robustness analysis and synthesis can be formulated as the minimization of an objective function with respect to the controller parameters it is argued that in order to assess the performance of a controller as the plant varies over a prespecified family it is better to use the average performance of the controller as the objective function to be minimized rather than its worst case performance as the worst case objective function usually leads to rather conservative designs then it is shown that a property from statistical learning theory known as uniform convergence of empirical means ucem plays an important role in allowing us to construct efficient randomized algorithms for a wide variety of controller synthesis problems in particular whenever the ucem property holds there exists an efficient i e polynomial time randomized algorithm using very recent results in statistical learning theory it is shown that the ucem property holds in any problem in which the satisfaction of a performance constraint can be expressed in terms of a finite number of polynomial inequalities in particular several problems such as robust stabilization and weighted h 2 h norm minimization are amenable to the randomized approach

minimal realization of nonlinear systems described by input output difference equations we consider a general class of nonlinear discrete time systems described by their input output i o relationship we first present the necessary and sufficient conditions for existence of a local observable state space realization of the system and an explicit algorithm for computing it in cases where the i o map and its resulting realization are nonminimal we formulate an algorithm for extracting its minimal realization whenever possible the developed theory and algorithms are illustrated by means of several examples

on the computational complexity and generalization properties of multi stage and stage wise coupled scenario programs abstract we discuss the computational complexity and feasibility properties of scenario sampling techniques for uncertain optimization programs we propose an alternative way of dealing with a special class of stage wise coupled programs and compare it with existing methods in the literature in terms of feasibility and computational complexity we identify trade offs between different methods depending on the problem structure and the desired probability of constraint satisfaction to illustrate our results an example from the area of approximate dynamic programming is considered

on uncontrollable discrete time bilinear systems which are nearly controllable in this note for a class of uncontrollable discrete time bilinear systems it is shown that the controllable region nearly covers the whole space while the uncontrollable region is only a hypersurface as a result for almost any initial state and any terminal state of the system the former can be transferred to the latter in addition the two dimensional controllability counterexamples in are generalized to arbitrary finite dimensional cases

application of statistical mechanics to np complete problems in combinatorial optimisation recently developed techniques of the statistical mechanics of random systems are applied to the graph partitioning problem the averaged cost function is calculated and agrees well with numerical results the problem bears close resemblance to that of spin glasses the authors find a spin glass transition in the system and the low temperature phase space has an ultrametric structure this sheds light on the nature of hard computation problems

on near controllability and stabilizability of a class of discrete time bilinear systems in this paper the near controllability and stabilizability of a class of discrete time bilinear systems are studied a necessary and sufficient condition for the near controllability of the systems is first derived then on the basis of the corresponding near controllability criterion the stabilizability problem of the systems is solved in turn

fine and wilf s theorem for three periods and a generalization of sturmian words we extend the theorem of fine and wilf to words having three periods we then define the set 3 per of words of maximal length for which such result does not apply we prove that the set 3 per and the sequences of complexity 2n 1 introduced by arnoux and rauzy to generalize sturmian words have the same set of factors

stochastic programming approach to optimization under uncertainty in this paper we discuss computational complexity and risk averse approaches to two and multistage stochastic programming problems we argue that two stage say linear stochastic programming problems can be solved with a reasonable accuracy by monte carlo sampling techniques while there are indications that complexity of multistage programs grows fast with increase of the number of stages we discuss an extension of coherent risk measures to a multistage setting and in particular dynamic programming equations for such problems

quantum stochastic processes as models for state vector reduction an elementary introduction of quantum state valued markovian stochastic processes qsp for n state quantum systems is given it is pointed out that a so called master constraint must be fulfilled for a given master equation a continuous and as a new alternative possibility a discontinuous qsp are derived both are discussed as possible models for state reduction during measurement

stochastic optimization on continuous domains with finite time guarantees by markov chain monte carlo methods we introduce bounds on the finite time performance of markov chain monte carlo mcmc algorithms in solving global stochastic optimization problems defined over continuous domains it is shown that mcmc algorithms with finite time guarantees can be developed with a proper choice of the target distribution and by studying their convergence in total variation norm this work is inspired by the concept of finite time learning with known accuracy and confidence developed in statistical learning theory

on controllability and near controllability of discrete time inhomogeneous bilinear systems without drift in this paper controllability and near controllability of discrete time inhomogeneous bilinear systems without drift are studied necessary and sufficient conditions for controllability and near controllability of the systems are respectively presented in particular algorithms for computing the control inputs to achieve the transition of a given pair of states for the controllable systems and nearly controllable systems are also provided examples are shown to demonstrate the conceptions and results of the paper

the scenario approach to robust control design this paper proposes a new probabilistic solution framework for robust control analysis and synthesis problems that can be expressed in the form of minimization of a linear objective subject to convex constraints parameterized by uncertainty terms this includes the wide class of np hard control problems representable by means of parameter dependent linear matrix inequalities lmis it is shown in this paper that by appropriate sampling of the constraints one obtains a standard convex optimization problem the scenario problem whose solution is approximately feasible for the original usually infinite set of constraints i e the measure of the set of original constraints that are violated by the scenario solution rapidly decreases to zero as the number of samples is increased we provide an explicit and efficient bound on the number of samples required to attain a priori specified levels of probabilistic guarantee of robustness a rich family of control problems which are in general hard to solve in a deterministically robust sense is therefore amenable to polynomial time solution if robustness is intended in the proposed risk adjusted sense

time orthogonal unitary dilations and noncommutative feynman kac formulae an analysis of feynman kac formulae reveals that typically the unperturbed semigroup is expressed as the expectation of a random unitary evolution and the perturbed semigroup is the expectation of a perturbation of this evolution in which the latter perturbation is effected by a cocycle with certain covariance properties with respect to the group of translations and reflections of the line we consider generalisations of the classical commutative formalism in which the probabilistic properties are described in terms of non commutative probability theory based on von neumann algebras examples of this type are generated by means of second quantisation from a unitary dilation of a given self adjoint contraction semigroup called the time orthogonal unitary dilation whose key feature is that the dilation operators corresponding to disjoint time intervals act nontrivially only in mutually orthogonal supplementary hilbert spaces

invariant codistributions and the feedforward form for discrete time nonlinear systems the goal of this paper is two fold first given an arbitrary n dimensional discrete time nonlinear dynamical system necessary and sufficient conditions for the existence of a one dimensional invariant codistribution are obtained second it is shown that the previous conditions can be used iteratively to obtain a nested sequence of n invariant codistributions with the properties that each codistribution contains the previous one and the last one coincides with the cotangent bundle of the state manifold as a byproduct necessary and sufficient conditions are obtained for a discrete time nonlinear dynamical system to be equivalent to the so called feedforward form

optimal bayesian design by inhomogeneous markov chain simulation we consider decision problems defined by a utility function and an underlying probability model for all unknowns the utility function quantifies the decision maker s preferences over consequences the optimal decision maximizes the expected utility function where the expectation is taken with respect to all unknowns that is future data and parameters in many problems the solution is not analytically tractable for example the utility function might involve moments that can be computed only by numerical integration or simulation also the nature of the decision space i e the set of all possible actions might have a shape or dimension that complicates the maximization the motivating application for this discussion is the choice of a monitoring network when the optimization is performed over the high dimensional set of all possible locations of monitoring stations possibly including choice of the number of locations we propose an approach to optimal bayesian design based on inhomogeneous markov

stabilizing feedback controls for quantum systems no quantum measurement can give full information on the state of a quantum system hence any quantum feedback control problem is necessarily one with partial observations and can generally be converted into a completely observed control problem for an appropriate quantum filter as in classical stochastic control theory here we study the properties of controlled quantum filtering equations as classical stochastic differential equations we then develop methods using a combination of geometric control and classical probabilistic techniques for global feedback stabilization of a class of quantum filters around a particular eigenstate of the measurement operator

observability of autonomous discrete time non linear systems a geometric approach in this paper the observability of autonomous discrete time systems is studied from a purely differential geometric point of view as with continuous time systems this approach leads to a local canonical form for an observable system a proposal for the generalization of an invariant subspace is made

a continuous counting observation and posterior quantum dynamics a stochastic model for a continuous photon counting measurement is proposed an equation for the generating functional of the statistics of counts is found a stochastic linear equation for the unnormalised posterior state vector of atoms continuously collapsing without mixing is derived by using quantum stochastic calculus methods

a periodicity lemma for partial words abstract we investigate the function l h p q called here the length function such that l h p q is the minimum length which guarantees that a natural extension of the periodicity lemma is valid for partial words with h holes and so called strong periods p q in a series of papers the formulae for the length function in terms of p and q were provided for each fixed h 7 we demystify the generic structure of such formulae and give a complete characterization of the length function for any parameter h expressed in terms of a piecewise linear function with o h pieces we also show how to evaluate the length function in o log p log q time which is an improvement upon the best previously known o p q time algorithm

annealed feynman kac models we analyze the concentration properties of an annealed feynman kac model in distribution space we characterize the concentration regions in terms of a variational problem involving a competition between the potential function and the mutation kernel when the temperature parameter is evanescent with time and under appropriate hypotheses the probability mass tends to concentrate on regions with minimal potential values we give a precise description of these areas using non linear semi group contractions and large deviation techniques we illustrate this annealed model with two physical interpretations related respectively to markov motions in absorbing media and interacting measure valued processes

on controllability near controllability controllable subspaces and nearly controllable subspaces of a class of discrete time multi input bilinear systems abstract this paper considers a class of discrete time multi input inhomogeneous bilinear systems the structure of such systems is most close to linear time invariant systems but they own a strong property that is if the systems are uncontrollable they can still be nearly controllable necessary and sufficient conditions for controllability and near controllability of the systems are established by using a classical decomposition furthermore a geometric characterization is given for the systems such that controllable subspaces and nearly controllable subspaces are derived and characterized similar results on controllability are also obtained for the continuous time counterparts of the systems finally examples are provided to demonstrate the conceptions and results of this paper

analysis still matters a surprising instance of failure of runge kutta felberg ode solvers this paper provides a nice example to illustrate that without supporting analysis a numerical simulation may lead to incorrect conclusions we explore a pedagogical example of failure of runge kutta felberg rkf algorithms for a simple dynamical system that models the coupling of two oscillators although the system appears to be well behaved the explicit rkf solvers provide erratic numerical solutions the mode of failure is based in a period doubling route to chaos due to the existence of stable linear solutions in the problem

a model for an objective description of macroscopic state variables in quantum mechanics abstract a generalised stochastic process is obtained for the boltzmann distribution function in quantum mechanics this is an example of an objective state space description of a quantum system

pseudopalindrome closure operators in free monoids we consider involutory antimorphisms of a free monoid a and their fixed points called palindromes or pseudopalindromes a palindrome reduces to a usual palindrome when is the reversal operator for any word w a the right resp left palindrome closure of w is the shortest palindrome having w as a prefix resp suffix we prove some results relating palindrome closure operators with periodicity and conjugacy and derive some interesting closure properties for the languages of finite sturmian and episturmian words in particular a finite word w is sturmian if and only if both its palindromic closures are so moreover in such a case both the palindromic closures of w share the same minimal period of w a new characterization of finite sturmian words follows in terms of periodicity and special factors of their palindromic closures some weaker results can be extended to the episturmian case by using the right palindrome closure we extend the construction of standard episturmian words via directive words in this way one obtains a family of infinite words called standard words which are morphic images of episturmian words as well as a wider family of infinite words including the thue morse word on two symbols

minorization conditions and convergence rates for markov chain monte carlo abstract general methods are provided for analyzing the convergence of discrete time general state space markov chains such as those used in stochastic simulation algorithms including the gibbs sampler the methods provide rigorous a priori bounds on how long these simulations should be run to give satisfactory results results are applied to two models of the gibbs sampler a bivariate normal model and a hierarchical poisson model with gamma conditionals the methods use the notion of minorization conditions for markov chains

relative frequencies of generalized simulated annealing we consider a class of nonhomogeneous markov chains arising in simulated annealing and related stochastic search algorithms using only elementary first principles we analyze the convergence and rate of convergence of the relative frequencies of visits to states in the markov chain we describe in detail three examples including the standard simulated annealing algorithm to show how our framework applies to specific stochastic search algorithmsthese examples have not previously been recognized to be sufficiently similar to share common analytical grounds our analysis though elementary provides the strongest sample path convergence results to date for simulated annealing type markov chains our results serve to illustrate that by taking a purely sample path view surprisingly strong statements can be made using only relatively elementary tools

controllability and near controllability of a class of discrete time inhomogeneous bilinear systems in this paper a class of discrete time inhomogeneous bilinear systems whose controllability cannot be checked by the classical results is considered by using an algebraic technique it is shown t

quantum stochastic calculus operation valued stochastic processes and continual measurements in quantum mechanics the physical idea of a continual observation on a quantum system has been recently formalized by means of the concept of operation valued stochastic process ovsp in this article it is shown how the formalism of quantum stochastic calculus of hudson and parthasarathy allows in a simple way for constructing a large class of ovsp s that in particular contains the quantum counting processes of davies and srinivas and continual gaussian measurements this result is obtained by means of a stochastic dilation of the ovsp s at the level of the enlarged system probabilities turn out to be expressed in terms of projection valued measures associated with certain time dependent commuting self adjoint operators

commutative algebraic methods for controllability of discrete time polynomial systems abstractin this paper we consider controllability of discrete time polynomial systems first we present a forward accessibility local reachability condition that can be verified in finite time in contrast to conventional conditions second we give a backward accessibility local controllability condition for an invertible system and a condition to verify invertibility finally we derive sufficient conditions to test whether the forward accessible system is reachable and to test the backward accessible system is controllable

optimal pointers for joint measurement of σx and σz via homodyne detection we study a model of a qubit in interaction with the electromagnetic field by means of homodyne detection the field quadrature at a t is observed continuously in time due to the interaction information about the initial state of the qubit is transferred to the field thus influencing the homodyne measurement results we construct random variables pointers on the probability space of homodyne measurement outcomes having distributions close to the initial distributions of x and z using variational calculus we find the pointers that are optimal these optimal pointers are very close to hitting the bound imposed by heisenberg s uncertainty relation on joint measurement of two non commuting observables we close the paper by giving the probability densities of the pointers

on controllability of discrete time bilinear systems by near controllability abstract in this paper controllability of discrete time bilinear systems is studied by applying a recent result on near controllability a new sufficient condition for controllability of the systems is presented where controllability is proved by approximation with near controllability the new condition is algebraically verifiable and is hence easy to apply compared with a classical result on controllability of discrete time bilinear systems which can be effective even when the classical result does not work furthermore the control inputs to achieve the transition of the systems between any given pair of states are approximately computable according to near controllability therefore near controllability can be used to not only better characterize the system properties but also prove controllability with computable control inputs the new condition is then generalized to derive similar results on controllability and near controllability of the systems finally examples are given to illustrate the results of this paper

from chronological calculus to exponential representations of continuous and discrete time dynamics a lie algebraic approach in this paper formal exponential representations of the solutions to nonautonomous nonlinear differential equations are derived it is shown that the chronological exponential admits an ordinary exponential representation the exponent being given by an explicitly computable lie series expansion the results are then used to describe controlled dynamics dynamics under sampling and forced discrete time dynamics the study emphasizes the role of lie algebra techniques in nonlinear control theory and specifies structural similarities between nonautonomous differential equations dynamics under sampling and forced discrete time dynamics up to hybrid ones

on the solvability of polynomial systems arising in control we consider a compactification of a general system of polynomial equations in weighted projective space and give some sufficient conditions about the solvability of a polynomial system over c and r we also prove that for generic linear subspaces the inverse eigenvalue problem with perturbations in the linear subspace always has n solutions

on small controllability and controllability of a class of nonlinear systems in this paper small controllability and controllability of a class of nonlinear systems are studied both the discrete time case and continuous time case are considered by proposing an implicit function approach sufficient conditions for the systems respectively to be small controllable and controllable are obtained examples are also provided to illustrate the results of the paper

sequential monte carlo samplers summary we propose a methodology to sample sequentially from a sequence of probability distributions that are defined on a common space each distribution being known up to a normalizing constant these probability distributions are approximated by a cloud of weighted random samples which are propagated over time by using sequential monte carlo methods this methodology allows us to derive simple algorithms to make parallel markov chain monte carlo algorithms interact to perform global optimization and sequential bayesian estimation and to compute ratios of normalizing constants we illustrate these algorithms for various integration tasks arising in the context of bayesian inference

on controllability of homogeneous and inhomogeneous discrete time multi input bilinear systems in dimension two abstractin this paper the controllability problem of two dimensional discrete time multi input bilinear systems is completely solved the homogeneous and the inhomogeneous cases are studied separately and necessary and sufficient conditions for controllability are established by using a linear algebraic method which are easy to apply moreover for the uncontrollable systems near controllability is considered and similar necessary and sufficient conditions are also obtained finally examples are provided to demonstrate the results of this paper

confidence level solutions for stochastic programming we propose an alternative approach to stochastic programming based on monte carlo sampling and stochastic gradient optimization the procedure is by essence probabilistic and the computed solution is a random variable we propose a solution concept in which the probability that the random algorithm produces a solution with an expected objective value departing from the optimal one by more than e is small enough we derive complexity bounds on the number of iterations of this process we show that by repeating the basic process on independent samples one can significantly reduce the number of iterations

randomized strategies for probabilistic solutions of uncertain feasibility and optimization problems in this paper we study two general semi infinite programming problems by means of a randomized strategy based on statistical learning theory the sample size results obtained with this approach are generally considered to be very conservative by the control community the first main contribution of this paper is to demonstrate that this is not necessarily the case utilizing as a starting point one sided results from statistical learning theory we obtain bounds on the number of required samples that are manageable for ldquoreasonablerdquo values of probabilistic confidence and accuracy in particular we show that the number of required samples grows with the accuracy parameter epsiv as 1 epsivln 1 epsiv and this is a significant improvement when compared to the existing bounds which depend on 1 epsiv2ln 1 epsiv2 secondly we present new results for optimization and feasibility problems involving boolean expressions consisting of polynomials in this case when the accuracy parameter is sufficiently small an explicit bound that only depends on the number of decision variables and on the confidence and accuracy parameters is presented for convex optimization problems we also prove that the required sample size is inversely proportional to the accuracy for fixed confidence thirdly we propose a randomized algorithm that provides a probabilistic solution circumventing the potential conservatism of the bounds previously derived

fine and wilf words for any periods ii given positive integers n and p 1 p r we establish a fast word combinatorial algorithm for constructing a word w w 1 w n of length n with periods p 1 p r and on the maximal number of distinct letters moreover we show that the constructed word which is unique up to word isomorphism is a pseudo palindrome i e it is a fixed point of an involutory antimorphism

comments on a controllability counterexample and the continuation lemma a technical note in this journal vol 50 no 6 pp 840 841 june 2005 by elliott gives a bilinear example showing that the euler discretization of a noncontrollable continuous time system can be controllable the example is correct but there was a flaw in a result of the tn lemma 1 for discrete time systems local controllability implies controllability that has independent interest in this note the lemma is reformulated as a conjecture for continuous in state systems and it is also proved under additional conditions for a class of two dimensional bilinear systems the euler discretization is shown directly to be small controllable a fortiori controllable

quadratic forms and approximate feed back linearization in discrete time quadratic feedback linearization is discussed for discrete time systems i e approximate feedback linearization up to the second order with respect to input and state variables after introducing in discrete time two equivalence classes invariant under quadratic static state feedback and diffeomorphism two canonical quadratic representations are proposed then it is shown that any nonlinear discrete time dynamics which are controllable in their first approximations can be linearized up to the second order under dynamic feedback the results stated here can be considered as the general discrete time analogues of the continuous time ones kang and krener 1992

a quantum analogue of hunt s representation theorem for the generator of convolution semigroups on lie groups in quantum mechanics certain operator valued measures are introduced called instruments which are an analogue of the probability measures of classical probability theory as in the classical case it is interesting to study convolution semigroups of instruments on groups and the associated semigroups of probability operators in this paper the case is considered of a finite dimensional hilbert space n level quantum system and of instruments defined on a finite dimensional lie group then the generator of a continuous semigroup of quantum probability operators is characterized in this way a quantum analogue of hunt s representation theorem for the generator of convolution semigroups on lie groups is obtained

rates of convergence of the hastings and metropolis algorithms we apply recent results in markov chain theory to hastings and metropolis algorithms with either independent or symmetric candidate distributions and provide necessary and sufficient conditions for the algorithms to converge at a geometric rate to a prescribed distribution pi in the independence case in mathbb r k these indicate that geometric convergence essentially occurs if and only if the candidate density is bounded below by a multiple of pi in the symmetric case in mathbb r only we show geometric convergence essentially occurs if and only if pi has geometric tails we also evaluate recently developed computable bounds on the rates of convergence in this context examples show that these theoretical bounds can be inherently extremely conservative although when the chain is stochastically monotone the bounds may well be effective

on different generalizations of episturmian words in this paper we study some classes of infinite words generalizing episturmian words and analyse the relations occurring among such classes in each case the reversal operator r is replaced by an arbitrary involutory antimorphism q of the free monoid a in particular we define the class of q words with seed whose standard elements q standard words with seed are constructed by an iterative q palindrome closure process starting from a finite word u 0 called the seed when the seed is empty one obtains q words episturmian words are exactly the r words one of the main theorems of the paper characterizes q words with seed as infinite words closed under q and having at most one left special factor of each length n n where n is some nonnegative integer depending on the word when n 0 we call such words q episturmian further results on the structure of q episturmian words are proved in particular some relationships between q words with or without seed and q episturmian words are shown

identification and control of a nonlinear discrete time system based on its linearization a unified framework this paper presents a unified theoretical framework for the identification and control of a nonlinear discrete time dynamical system in which the nonlinear system is represented explicitly as a sum of its linearized component and the residual nonlinear component referred to as a higher order function this representation substantially simplifies the procedure of applying the implicit function theorem to derive local properties of the nonlinear system and reveals the role played by the linearized system in a more transparent form under the assumption that the linearized system is controllable and observable it is shown that 1 the nonlinear system is also controllable and observable in a local domain 2 a feedback law exists to stabilize the nonlinear system locally and 3 the nonlinear system can exactly track a constant or a periodic sequence locally if its linearized system can do so with some additional assumptions the nonlinear system is shown to have a well defined relative degree delay and zero dynamics if the zero dynamics of the linearized system is asymptotically stable so is that of the nonlinear one and in such a case a control law exists for the nonlinear system to asymptotically track an arbitrary reference signal exactly in a neighborhood of the equilibrium state the tracking can be achieved by using the state vector for feedback or by using only the input and the output in which case the nonlinear autoregressive moving average narma model is established and utilized these results are important for understanding the use of neural networks as identifiers and controllers for general nonlinear discrete time dynamical systems

a controllability counterexample simulation by time discretizations can be qualitatively misleading as counterexamples a class of noncontrollable single input two dimensional bilinear control systems is presented whose euler discretizations are controllable on the punctured plane

quantum measuring processes of continuous observables the purpose of this paper is to provide a basis of theory of measurements of continuous observables we generalize von neumann s description of measuring processes of discrete quantum observables in terms of interaction between the measured system and the apparatus to continuous observables and show how every such measuring process determines the state change caused by the measurement we establish a one to one correspondence between completely positive instruments in the sense of davies and lewis and the state changes determined by the measuring processes we also prove that there are no weakly repeatable completely positive instruments of nondiscrete observables in the standard formulation of quantum mechanics so that there are no measuring processes of nondiscrete observables whose state changes satisfy the repeatability hypothesis a proof of the wigner araki yanase theorem on the nonexistence of repeatable measurements of observables not commuting conserved quantities is given in our framework we also discuss the implication of these results for the recent results due to srinivas and due to mercer on measurements of continuous observables

probability operators and convolution semigroups of instruments in quantum probability in quantum measurement theory a central notion is that of instrument which is a certain kind of operator valued measure in this paper instruments on locally compact groups are studied and as in classical probability theory probability operators associated with instruments are introduced then the generator of a norm continuous semigroup of probability operators is characterized

markovian limit for a reduced operation valued stochastic process operation valued stochastic processes give a formalization of the concept of continuous in time measurements in quantum mechanics in this article a first stage m of a measuring apparatus coupled to the system s is explicitly introduced and continuous measurement of some observables of m is considered one can speak of an indirect continuous measurement on s when the degrees of freedom of the measuring apparatus m are eliminated and the weak coupling limit is taken it is shown that an operation valued stochastic process describing a direct continuous observation of the system s is obtained

reachability and steering of rolling polyhedra a case study in discrete nonholonomy rolling a ball on a plane is a standard example of nonholonomy reported in many textbooks and the problem is also well understood for any smooth deformation of the surfaces for nonsmoothly deformed surfaces however much less is known although it may seem intuitive that nonholonomy is conserved think e g to polyhedral approximations of smooth surfaces current definitions of nonholonomy are inherently referred to systems described by ordinary differential equations and are thus inapplicable to such systems in this paper we study the set of positions and orientations that a polyhedral part can reach by rolling on a plane through sequences of adjacent faces we provide a description of such reachable set discuss conditions under which the set is dense or discrete or has a compound structure and provide a method for steering the system to a desired reachable configuration robustly with respect to model uncertainties based on ideas and concepts encountered in this case study and in some other examples we provide we turn back to the most general aspects of the problem and investigate the possible generalization of the notion of kinematic nonholonomy to nonsmooth discrete and hybrid dynamical systems to capture the essence of phenomena commonly regarded as nonholonomic at least two irreducible concepts are to be defined of internal and external nonholonomy which may coexist in the same system these definitions are instantiated by examples

the n city travelling salesman problem statistical mechanics and the metropolis algorithm in any n city travelling salesman problem there are n 1 mathord left vphantom 2 right kern nulldelimiterspace 2 possible tours we use the metropolis algorithm to generate a sequence of such tours this sequence may be viewed as the random evolution of a physical system in contact with a heat bath as the temperature is lowered the tours generated approach the optimal tour it appears that for large n one arrives within a few percent of the optimal solution in better than quadratic time

quantum ito s formula and stochastic evolutions using only the boson canonical commutation relations and the riemann lebesgue integral we construct a simple theory of stochastic integrals and differentials with respect to the basic field operator processes this leads to a noncommutative ito product formula a realisation of the classical poisson process in fock space which gives a noncommutative central limit theorem the construction of solutions of certain noncommutative stochastic differential equations and finally to the integration of certain irreversible equations of motion governed by semigroups of completely positive maps the classical ito product formula for stochastic differentials with respect to brownian motion and the poisson process is a special case

feedback linearization of sampled data systems it is shown that feedback linearizability of a continuous time system can be destroyed through the introduction of the usual sample and hold devices a method whereby a feedback linearizable discrete time model can be recovered is given

generalised fine and wilf s theorem for arbitrary number of periods the well known fine and wilf s theorem for words states that if a word has two periods and its length is at least as long as the sum of the two periods minus their greatest common divisor then the word also has as period the greatest common divisor we generalise this result for an arbitrary number of periods our bound is strictly better in some cases than previous generalisations moreover we prove it optimal we show also that any extrenal word is unique up to letter renaming and give an algorithm to compute both the bound and a corresponding extremal word

on graphs of central episturmian words episturmian sequences are a natural extension of sturmian sequences to the case of finite alphabets of arbitrary cardinality in this paper we are interested in central episturmian words or simply epicentral words i e the palindromic prefixes of standard episturmian sequences an epicentral word admits a variety of faithful representations including as a directive word as a certain type of period vector as a parikh vector as a certain type of fine and wilf extremal word as a suitable modular matrix and as a labeled graph various interconnections between the different representations of an epicentral word are analyzed in particular we investigate the structure of the graphs of epicentral words proving some curious and surprising properties

generalized stochastic processes and continual observations in quantum mechanics we give here a mathematically rigorous form to an earlier work by barchielli lanz and prosperi in which it was found that a generalized stochastic process describes the results of continual observations of the position of a quantum particle with the help of albeverio and ho egh krohn s theory of feynman path integrals we define the characteristic functional of this process and demonstrate that it possesses the necessary properties of normalization continuity and positive definiteness an explicit calculation of the feynman path integral which defines the functional allows an analysis of the process to be made

new insights on the analysis of nonlinear time delay systems application to the triangular equivalence in this paper we present recent advances towards an algebraic framework for the analysis and control of nonlinear systems with delays we show the usefulness of these results by applying them to obtain a constructive characterization of the equivalence of a given system to the so called triangular form

critical exponents of infinite balanced words abstract over an alphabet of size 3 we construct an infinite balanced word with critical exponent 2 2 2 over an alphabet of size 4 we construct an infinite balanced word with critical exponent 5 5 4 over larger alphabets we give some candidates for balanced words found computationally having small critical exponents we also explore a method for proving these results using the automated theorem prover walnut

controllability of nonlinear discrete time systems a lie algebraic approach this paper presents a geometric study of controllability for discrete time nonlinear systems various accessibility properties are characterized in terms of lie algebras of vector fields some of the results obtained are parallel to analogous ones in continuous time but in many respects the theory is substantially different and many new phenomena appear

on the road between robust optimization and the scenario approach for chance constrained optimization problems we propose a new method for solving chance constrained optimization problems that lies between robust optimization and scenario based methods our method does not require prior knowledge of the underlying probability distribution as in robust optimization methods nor is it based entirely on randomization as in the scenario approach it instead involves solving a robust optimization problem with bounded uncertainty where the uncertainty bounds are randomized and are computed using the scenario approach to guarantee that the resulting robust problem is solvable we impose certain assumptions on the dependency of the constraint functions with respect to the uncertainty and show that tractability is ensured for a wide class of systems our results lead immediately to guidelines under which the proposed methodology or the scenario approach is preferable in terms of providing less conservative guarantees or reducing the computational cost