finite approximations of switched homogeneous systems for controller synthesis in this note we demonstrate the use of a control oriented notion of finite state input output approximation to synthesize correct by design controllers for hybrid plants under sensor limitations specifically we consider the problem of designing stabilizing switching controllers for a pair of unstable homogeneous second order systems with binary output feedback in addition to yielding a deterministic finite state approximate model of the hybrid plant our approach allows one to efficiently establish a useable upper bound on the quality of approximation and leads to a discrete optimization problem whose solution immediately provides a certified finite state controller for the plant the resulting controller consists of a deterministic finite state observer and a corresponding full state feedback control law

brief equivalence of hybrid dynamical models this paper establishes equivalences among five classes of hybrid systems mixed logical dynamical mld systems linear complementarity lc systems extended linear complementarity elc systems piecewise affine pwa systems and max min plus scaling mmps systems some of the equivalences are established under rather mild additional assumptions these results are of paramount importance for transferring theoretical properties and tools from one class to another with the consequence that for the study of a particular hybrid system that belongs to any of these classes one can choose the most convenient hybrid modeling framework

approximate reduction of dynamic systems abstract the reduction of dynamic systems has a rich history with many important applications related to stability control and verification reduction of nonlinear systems is typically performed in an exact manner as is the case with mechanical systems with symmetry which unfortunately limits the type of systems to which it can be applied the goal of this paper is to consider a more general form of reduction termed approximate reduction in order to extend the class of systems that can be reduced using notions related to incremental stability we give conditions on when a dynamic system can be projected to a lower dimensional space while providing hard bounds on the induced errors i e when it is behaviourally similar to a dynamic system on a lower dimensional space these concepts are illustrated on a series of examples

finite uniform bisimulations for linear systems with finite input alphabets we consider discrete time systems with linear dynamics and a finite input alphabet we propose a sufficient condition for the existence of finite uniform bisimulations and an algorithm that terminates when the condition holds and constructs the corresponding equivalence classes we derive a stronger sufficient condition for the construction of arbitrarily many equivalence classes together with a corresponding constructive algorithm finally we derive a necessary condition for the existence of finite uniform bisimulations all our conditions are topological in nature

approximately bisimilar symbolic models for nonlinear control systems control systems are usually modeled by differential equations describing how physical phenomena can be influenced by certain control parameters or inputs although these models are very powerful when dealing with physical phenomena they are less suited to describe software and hardware interfacing with the physical world for this reason there is a growing interest in describing control systems through symbolic models that are abstract descriptions of the continuous dynamics where each symbol corresponds to an aggregate of states in the continuous model since these symbolic models are of the same nature of the models used in computer science to describe software and hardware they provide a unified language to study problems of control in which software and hardware interact with the physical world furthermore the use of symbolic models enables one to leverage techniques from supervisory control and algorithms from game theory for controller synthesis purposes in this paper we show that every incrementally globally asymptotically stable nonlinear control system is approximately equivalent bisimilar to a symbolic model the approximation error is a design parameter in the construction of the symbolic model and can be rendered as small as desired furthermore if the state space of the control system is bounded the obtained symbolic model is finite for digital control systems and under the stronger assumption of incremental input to state stability symbolic models can be constructed through a suitable quantization of the inputs

approximation metrics for discrete and continuous systems established system relationships for discrete systems such as language inclusion simulation and bisimulation require system observations to be identical when interacting with the physical world modeled by continuous or hybrid systems exact relationships are restrictive and not robust in this paper we develop the first framework of system approximation that applies to both discrete and continuous systems by developing notions of approximate language inclusion approximate simulation and approximate bisimulation relations we define a hierarchy of approximation pseudo metrics between two systems that quantify the quality of the approximation and capture the established exact relationships as zero sections our approximation framework is compositional for a synchronous composition operator algorithms are developed for computing the proposed pseudo metrics both exactly and approximately the exact algorithms require the generalization of the fixed point algorithms for computing simulation and bisimulation relations or dually the solution of a static game whose cost is the so called branching distance between the systems approximations for the pseudo metrics can be obtained by considering lyapunov like functions called simulation and bisimulation functions we illustrate our approximation framework in reducing the complexity of safety verification problems for both deterministic and nondeterministic continuous systems

supervisory control of a class of discrete event processes this paper studies the control of a class of discrete event processes i e processes that are discrete asynchronous and possibly nondeter ministic the controlled process is described as the generator of a formal language while the controller or supervisor is constructed from the grammar of a specified target language that incorporates the desired closed loop system behavior the existence problem for a supervisor is reduced to finding the largest controllable language contained in a given legal language two examples are provided

modular feedback logic for discrete event systems we examine a modular approach to the synthesis of state feedback controls for the problem of maintaining a predicate on the state set of a discrete dynamic system invariant dynamical systems are modeled by automata together with a mechanism for enabling and disabling a subset of state transitions the basic problem of interest is to ensure by appropriate control action that a given predicate on the state set of the process remains invariantly true whenever it is initially satisfied assuming the predicate can be decomposed into the conjunction or disjunction of component predicates we determine conditions under which it is possible to synthesize the appropriate control in a modular fashion

an approximate simulation approach to symbolic control this paper introduces a methodology for the symbolic control of nonlinear systems based on an approximate notion of simulation relation this notion generalizes existing exact notions of simulation and is completely characterized in terms of known stabilizability concepts equipped with this notion we show how under certain stabilizability assumptions we can construct finite or symbolic models for nonlinear control systems synthesizing controllers for the original control system can then be done by using supervisory control techniques on the finite models and by refining the resulting finite controllers to hybrid controllers enforcing the specification on the original continuous control system the proposed design methodology can be seen as a correct by design way of obtaining both the feedback control laws as well as the control software responsible for deciding which law is executed and when

verification of infinite state dynamic systems using approximate quotient transition systems this paper deals with computational methods for verifying properties of labeled infinite state transition systems using quotient transition system qts a qts is a conservative approximation to the infinite state transition system based on a finite partition of the infinite state space for universal specifications positive verification for a qts implies the specification is true for the infinite state transition system we introduce the approximate qts or aqts the paper presents a sufficient condition for an aqts to be a bisimulation of the infinite state transition system an aqts bisimulation is essentially equivalent to the infinite state system for the purposes of verification it is well known however that finite state bisimulations do not exist for most hybrid systems of practical interest therefore the use of the aqts for verification of universal specifications is proposed and illustrated with an example this approach has been implemented in a tool for computer aided verification of a general class of hybrid systems

algebraic decompositions of dp problems with linear dynamics abstract we consider dynamic programming dp problems in which the dynamics are linear the cost is a function of the state and the state space is finite dimensional but defined over an arbitrary field starting from the natural decomposition of the state space into a direct sum of invariant subspaces consistent with the rational canonical form and assuming the cost functions exhibit an additive structure compatible with this decomposition we extract from the original problem two distinct families of smaller dp problems associated with the invariant subspaces each family constitutes a decomposition of the original problem when the optimal policy and value function can be reconstructed from the optimal policies and value functions of the smaller problems we derive necessary and sufficient conditions for these decompositions to exist propose a readily verifiable sufficient condition for the first decomposition and establish a hierarchy relating the two notions of decomposition

an input output construction of finite state rho mu approximations for control design we consider discrete time plants that interact with their controllers via fixed discrete alphabets for this class of systems and in the absence of exogenous inputs we propose a general methodical approach for constructing a sequence of finite state approximate models starting from finite length sequences of input and output signal pairs we explicitly derive conditions under which the proposed construct used in conjunction with a particular generalized structure satisfies desirable properties of approximations thereby leading to nominal deterministic finite state machine models that can be used in certified by design controller synthesis we also show that the cardinality of the min imal disturbance alphabet that can be used in this setting equals that of the sensor output alphabet finally we show that the pro posed construct satisfies a relevant semi completeness property

global qualitative description of a class of nonlinear dynamical systems in this paper we propose a methodology to derive a qualitative description of the behavior of a system from an incompletely known nonlinear dynamical model the model is written as an algebraic structure with unknown parameters and or functions under some hypotheses we obtain a graph describing the possible transitions between regions defined by the trends of the state variables and their relative positions a qualitative simulation of the model can be compared with on line data for fault detection purpose we give the example of a nonlinear biological model in dimension three for the growth of cells in a bioreactor

supervisory control of hybrid systems within a behavioural framework abstract this contribution addresses the synthesis of supervisory control for hybrid systems with discrete external signals such systems are in general neither l complete nor can they be represented by finite state machines we find an l complete approximation abstraction l for represent it by a finite state machine and investigate the control problem for the approximation if a solution exists we synthesize the maximally permissive supervisor for l we show that it also solves the control problem for the hybrid system if no solution exists approximation accuracy can be increased by computing a k complete abstraction k k l this paper is entirely set within the framework on willems behavioural systems theory

probabilistic reachability for multi parameter bifurcation analysis of cardiac alternans abstract using a probabilistic reachability based approach we present a multi parameter bifurcation analysis of electrical alternans in the two current mitchell schaeffer ms cardiac cell model electrical alternans is a phenomenon characterized by a variation in successive action potential durations generated by a cardiac cell or tissue alternans are known to initiate re entrant waves and are an important physiological indicator of an impending life threatening arrhythmia such as ventricular fibrillation the multi parameter bifurcation analysis we perform identifies a bifurcation hypersurface in the ms model parameter space such that a small perturbation to this region results in a transition from highly likely alternans to highly likely non alternans behavior our approach to this problem rests on encoding alternans like behavior in the ms model as a five mode multinomial hybrid automaton to perform multi parameter bifurcation analysis of cardiac alternans we first treat the parameters in question as bounded random variables we then apply a sophisticated guided search based probabilistic reachability analysis to compute a bounded bifurcation region possibly very tight that contains the bifurcation hypersurface bh our probabilistic reachability analysis uses a technique that combines a decision procedure with statistical tests in the process of computing the bifurcation region we further partition the parameter space into two more regions such that any valuation chosen from one of the regions will either produce alternans or non alternans behavior with a probability greater than a user defined threshold

on unbounded positive solutions of second order difference equations with a singular nonlinear term abstract in this paper several necessary and or sufficient conditions are given for the existence of unbounded positive solutions of nonlinear difference equations of the form r n x n f n x n 0 n n n 0 where n n0 n0 n0 1 n0 is a fixed nonnegative integer rn 0 for n n0 such that n n0 1 rn and f n n0 r r is a real valued continuous function such that f n x 0 for all n x n n0 0

corona problem and flows abstract using a version of ambrose s theorem we generalize the corona theorem to the setting of bounded analytic functions induced by flows the structure of the maximal ideal space is also investigated in particular an open dense subset just like the open unit disc in the classical case is determined

nonoscillation oscillation and growth of solutions of nonlinear difference equations of second order abstract sufficient conditions are obtained which guarantee that all solutions or all bounded solutions of a class of second order nonlinear difference equations are nonoscillatory oscillation criteria and theorems concerning the growth of solutions are also given

necessary and sufficient condition for oscillation of higher order nonlinear delay difference equations this paper is concerned with the oscillation of solutions of higher order nonlinear delay difference equations with forcing terms of the form nx t f t x t x t h t t i 0 1 where is the forward difference operator defined by x t x t 1 x t and mx t m 1x t m 1 a necessary and sufficient condition is established under which every solution x t is oscillatory when n is even and is either oscillatory or strongly monotone when n is odd a sufficient condition for the oscillation of solutions of neutral type delay difference equations is also obtained

a fully automated framework for control of linear systems from temporal logic specifications we consider the following problem given a linear system and a linear temporal logic ltl formula over a set of linear predicates in its state variables find a feedback control law with polyhedral bounds and a set of initial states so that all trajectories of the closed loop system satisfy the formula our solution to this problem consists of three main steps first we partition the state space in accordance with the predicates in the formula and construct a transition system over the partition quotient which captures our capability of designing controllers second using a procedure resembling model checking we determine runs of the transition system satisfying the formula third we generate the control strategy illustrative examples are included

formal analysis of piecewise affine systems through formula guided refinement we present a computational framework for identifying a set of initial states from which all trajectories of a piecewise affine pwa system with additive uncertainty satisfy a linear temporal logic ltl formula over a set of linear predicates in its state variables our approach is based on the construction and refinement of finite abstractions of infinite systems we derive conditions guaranteeing the equivalence of an infinite system and its finite abstraction with respect to a specific ltl formula and propose a method for the construction of such formula equivalent abstractions while provably correct the overall method is conservative and expensive a tool for pwa systems implementing the proposed procedure using polyhedral operations and analysis of finite graphs is made available examples illustrating the analysis of pwa models of gene networks are included

symbolic reachability analysis of genetic regulatory networks using discrete abstractions we use hybrid systems techniques for the analysis of reachability properties of a class of piecewise affine pa differential equations that are particularly suitable for the modeling of genetic regulatory networks more specifically we introduce a hyperrectangular partition of the state space that forms the basis for a discrete abstraction preserving the sign of the derivatives of the state variables the resulting discrete transition system provides a qualitative description of the network dynamics that is well adapted to available experimental data and that can be efficiently computed in a symbolic manner from inequality constraints on the parameters

conditional expectations and the corona problem of ergodic hardy spaces abstract using the regular corona theorem for the upper half plane we solve the coronatype problem for generalized analytic functions associated with flows

synthesis of reactive switching protocols from temporal logic specifications we propose formal means for synthesizing switching protocols that determine the sequence in which the modes of a switched system are activated to satisfy certain high level specifications in linear temporal logic ltl the synthesized protocols are robust against exogenous disturbances on the continuous dynamics and can react to possibly adversarial events both external and internal finite state approximations that abstract the behavior of the underlying continuous dynamics are defined using finite transition systems such approximations allow us to transform the continuous switching synthesis problem into a discrete synthesis problem in the form of a two player game between the system and the environment where the winning conditions represent the high level temporal logic specifications restricting to an expressive subclass of ltl formulas these temporal logic games are amenable to solutions with polynomial time complexity by construction existence of a discrete switching strategy for the discrete synthesis problem guarantees the existence of a switching protocol that can be implemented at the continuous level to ensure the correctness of the nonlinear switched system and to react to the environment at run time

comparison theorems for second order nonlinear difference equations abstract new oscillation results are obtained for the second order nonlinear difference equation r n f x n 1 g n x n 0 and its functional form r n f x n 1 g n x n 0 the role played by the argument n on the oscillation of the functional equation is explored in particular we characterize a class of sequences n which have a harmless effect on the oscillation of this type of equations some of our results generalize improve or unify known fundamental oscillation results for several particular cases of the above equations

the asymptotic behavior of the solutions of a volterra difference equation the aim of the present paper is to study the asymptotic behavior of the sum difference equation of volterra type an xn bn xn cn n l 0n 1gl h n xn xn as n

riccati type transformations for second order linear difference equations abstract oscillation and comparison theorems for a linear homogeneous second order difference equation are proved by employing various equivalent non linear equations obtained by means of transformations analogous to the riccati transformation for ordinary differential equations

discrete approximation and supervisory control of continuous systems addresses the following hybrid control problem a continuous plant its state evolving in euclidean space is to be controlled via symbolic output feedback both measurement and control signal live on finite sets of symbols we adopt the following approach the hybrid problem is first translated into a purely discrete problem by approximating the continuous plant model by a nondeterministic finite state machine by taking into account past measurement and control symbols approximation accuracy can be improved and adjusted to the specification requirements supervisory control theory for discrete event systems des is then applied to find the optimal controller which enforces the specifications as the behavior of the approximating automaton is guaranteed to contain the behavior of the underlying continuous plant model the controller also forces the latter to obey the specifications

oscillation of second order perturbed nonlinear difference equations using the riccati transformation techniques we will establish some new oscillation criteria for the second order perturbed nonlinear difference equation d a n 1 dx n 1 c f n x n g n x n dx n n 1where c 0 is a quotient of odd positive integers some comparison between our theorems and those previously known results are indicated examples are interested in the text to illustrate the relevance of our results

on persistent excitations for the identification of switched linear dynamical systems over finite fields this paper discusses the issue of the persistent excitation pe conditions in the context of identification for dynamical systems defined over a finite field the work is motivated by the fact that the asymptotical property of the pe conditions for dynamical systems defined over the field of real numbers is no longer valid in the case of systems defined over finite fields the special class of switched linear discrete time systems for which the mode is assumed to be unknown is considered a necessary and sufficient condition that provides the minimum amount of data required for the identification is first proposed next a necessary condition is derived that gives the structural condition the system must satisfy regardless of the availability of data finally some computational aspects are discussed and examples are given to illustrate the validity of the proposed results

supervisory control of timed discrete event systems the ramadge wonham framework for control of discrete event systems is augmented with timing features by use of ostroff s semantics for timed transition models it is shown that the rw concept of controllability and the existence of maximally permissive supervisory controls can be suitably generalized the enhanced setting admits subsystem composition and the concept of forcible event as an event that preempts the tick of a global clock an example of a simple manufacturing cell illustrates how the new framework can be used to solve synthesis problems which may include logic based temporal and quantitative optimality specifications

oscillation theorems for second order neutral difference equations abstract oscillation criteria for the second order difference equations of the form 2xn qnf xn r 0 and the neutral difference equation 2 xn pxn k qnf xn r 0 are established here xn xn 1 xn is the forward difference operator 1 p is a real number and k are nonnegative integers qn is a sequence of nonnegtive real numbers and f r r is continuous

classification and existence of positive solutions of systems of volterra nonlinear difference equations a classification scheme for the eventually positive solutions of a class of two dimensional volterra nonlinear difference equations is given in terms of their asymptotic magnitudes and necessary as well as sufficient conditions for the existence of such solutions are provided

a hybrid bounding method for computing an over approximation for the reachable set of uncertain nonlinear systems in this paper we show how to compute an over approximation for the reachable set of uncertain nonlinear continuous dynamical systems by using guaranteed set integration we introduce two ways to do so the first one is a full interval method which handles whole domains for set computation and relies on state of the art validated numerical integration methods the second one relies on comparison theorems for differential inequalities in order to bracket the uncertain dynamics between two dynamical systems where there is no uncertainty since the derived bracketing systems are piecewise ck differentiable functions validated numerical integration methods cannot be used directly hence our contribution resides in the use of hybrid automata to model the bounding systems we give a rule for building these automata and we show how to run them and address mode switching in a guaranteed way in order to compute the over approximation for the reachable set the computational cost of our method is also analyzed and shown to be smaller that the one of classical interval techniques sufficient conditions are given which ensure the epsiv practical stability of the enclosures given by our hybrid bounding method two examples are also given which show that the performance of our method is very promising

analysis and generic properties of gene regulatory networks with graded response functions the mass of new genomic data has lead to a growing interest in gene regulatory models describing the regulatory aspects of gene activity in these models the rates of change of gene product concentrations are expressed as a sum of regulatory switches in terms of sums of products of sigmoid functions mimicking complex logical on off functions turning gene activity on and off the customary way to analyse such equations is to replace the sigmoid functions by step functions and disregard models with autoregulation this leads to discontinuous equations of motion but in models without autoregulation continuous solutions can easily be obtained with effective autoregulation this simple approach breaks down as real regulators have finite gain and autoregulation is ubiquitous in biological systems we propose a generalised gene regulatory model framework admitting autoregulation and graded sigmoid functions with different steepnesses using singular perturbation analysis methods we derive a set of reformulated continuous equations for the limit solution when the sigmoids approach step functions and show that the solution for steep sigmoids approaches this limit solution uniformly in a finite time interval a conspicuous feature of the limit solution is so called sliding motionduring which the solution slides along a threshold hyperplane or the intersection of such hyperplanes a particular mapping leads to a dual picture of the flow where the fast parts are magnified at the expense of the slow parts combining the two dual pictures a simple and powerful method to analyse the flow is obtained we show that for steep sigmoid functions the flow may be highly sensitive to the relative steepnesses of the sigmoids this has important consequences when stochastic effects are taken into account also it implies that the observation by glass and

control invariance of hybrid systems with forcible events this paper considers controlled hybrid automata with forcible events first we introduce two semantics of the controlled hybrid automata using labeled transition systems next we extend a concept of control invariance in discrete event systems and show necessary and sufficient conditions for a predicate to be control invariant finally we show that there always exists the supremal control invariant subpredicate for any predicate

robustness analysis of nonlinear feedback systems an input output approach this paper presents an approach to robustness analysis for nonlinear feedback systems we pursue a notion of model uncertainty based on the closeness of input output trajectories which is not tied to a particular uncertainty representation such as additive parametric structured etc the basic viewpoint is to regard systems as operators on signal spaces we present two versions of a global theory where stability is captured by induced norms or by gain functions we also develop local approaches over bounded signal sets and give a treatment for systems with potential for finite time escape we compute the relevant stability margin for several examples and demonstrate robustness of stability for some specific perturbations e g small time delays we also present examples of nonlinear control systems which have zero robustness margin and are destabilized by arbitrarily small gap perturbations the paper considers the case where uncertainty is present in the controller as well as the plant and the generalization of the approach to the case where uncertainty occurs in several subsystems in an arbitrary interconnection

oscillation theorems for perturbed nonlinear second order difference equations abstract sufficient conditions for the oscillation of all solutions of perturbed difference equations of the form a n 1 y n 1 q n y n p n y n y n n n are obtained examples are inserted to illustrate the results

analysis of continuous time switching networks abstract models of a number of biological systems including gene regulation and neural networks can be formulated as switching networks in which the interactions between the variables depend strongly on thresholds an idealized class of such networks in which the switching takes the form of heaviside step functions but variables still change continuously in time has been proposed as a useful simplification to gain analytic insight these networks called here glass networks after their originator are simple enough mathematically to allow significant analysis without restricting the range of dynamics found in analogous smooth systems a number of results have been obtained before particularly regarding existence and stability of periodic orbits in such networks but important cases were not considered here we present a coherent method of analysis that summarizes previous work and fills in some of the gaps as well as including some new results furthermore we apply this analysis to a number of examples including surprising long and complex limit cycles involving sequences of hundreds of threshold transitions finally we show how the above methods can be extended to investigate aperiodic behaviour in specific networks though a complete analysis will have to await new results in matrix theory and symbolic dynamics

formal analysis of discrete time piecewise affine systems in this technical note we study temporal logic properties of trajectories of discrete time piecewise affine pwa systems specifically given a pwa system and a linear temporal logic formula over regions in its state space we attempt to find the largest region of initial states from which all trajectories of the system satisfy the formula our method is based on the iterative computation and model checking of finite quotients we illustrate our method by analyzing pwa models of two synthetic gene networks

bounded and monotone properties of solutions of second order quasilinear forced difference equations abstract the authors consider the quasilinear difference equation p n 1 y n 1 q n y n r n and obtain results on the asymptotic behavior of solutions of including sufficient conditions for all solutions to be bounded or unbounded some results on the existence and behavior of nonincreasing solutions of are also obtained examples are inserted to illustrate the results

bisimilar linear systems the notion of bisimulation in theoretical computer science is one of the main complexity reduction methods for the analysis and synthesis of labeled transition systems bisimulations are special quotients of the state space that preserve many important properties expressible in temporal logics and in particular reachability in this paper the framework of bisimilar transition systems is applied to various transition systems that are generated by linear control systems given a discrete time or continuous time linear system and a finite observation map we characterize linear quotient maps that result in quotient transition systems that are bisimilar to the original system interestingly the characterizations for discrete time systems are more restrictive than for continuous time systems due to the existence of an atomic time step we show that computing the coarsest bisimulation which results in maximum complexity reduction corresponds to computing the maximal controlled or reachability invariant subspace inside the kernel of the observations map these results establish strong connections between complexity reduction concepts in control theory and computer science

synthesis of supervisory controllers for hybrid systems based on approximating automata the paper concerns the synthesis of supervisory controllers for a class of continuous time hybrid systems with discrete valued input signals that select differential inclusions for continuous valued state trajectories and event valued output signals generated by threshold crossings in the continuous state space the supervisor is allowed to switch the input signal value when threshold events are observed the objective is to synthesize a nonblocking supervisor such that the set of possible sequences of control and threshold event pairs for the closed loop system lies between given upper and lower bounds in the sense of set containment we show how this problem can be converted into a supervisor synthesis problem for a standard controlled discrete event system des a finite representation may not exist for the exact des model of the hybrid system however to circumvent this difficulty we present an algorithm for constructing finite state muller automata that accept outer approximations to the exact controlled threshold event language and we demonstrate that supervisors that solve the synthesis problem for the approximating automata achieve the control specifications when applied to the original hybrid system

a comprehensive method for reachability analysis of uncertain nonlinear hybrid systems reachability analysis of nonlinear uncertain hybrid systems i e continuous discrete dynamical systems whose continuous dynamics guard sets and reset functions are defined by nonlinear functions can be decomposed in three algorithmic steps computing the reachable set when the system is in a given operation mode computing the discrete transitions i e detecting and localizing when and where the continuous flowpipe intersects the guard sets and aggregating the multiple trajectories that result from an uncertain transition once the whole flow pipe has transitioned so that the algorithm can resume this paper proposes a comprehensive method that provides a nicely integrated solution to the hybrid reachability problem at the core of the method is the concept of mspb i e geometrical object obtained as the minkowski sum of a parallelotope and an axes aligned box mspb are a way to control the over approximation of the taylor s interval integration method as they happen to be a specific type of zonotope they articulate perfectly with the zonotope bounding method that we propose to enclose in an optimal way the set of flowpipe trajectories generated by the transition process the method is evaluated both theoretically by analyzing its complexity and empirically by applying it to well chosen hybrid nonlinear examples

flows in fibers let h a be the algebra of all bounded analytic functions on the open unit disc a and let 9j h a be the maximal ideal space of h a using a flow we represent a reasonable portion of a fiber in 9j h a this indicates a relation between the corona theorem and the individual ergodic theorem as an application we answer a question of forelli 3 by showing that there exists a minimal flow on which the induced uniform algebra is not a dirichlet algebra the proof rests on the fact that the closure of a nonhomeomorphic part in 9 h a may contain a homeomorphic copy of 9 h a taking suitable factors we may construct a lot of minimal flows which are not strictly ergodic

hybrid abstractions of affine systems abstract this paper considers the problem of building a set of hybrid abstractions for affine systems in order to compute over approximations of the reachable space each abstraction is based on a decomposition of the continuous state space that is defined by hyperplanes generated by linear combinations of two vectors the choice of these vectors is based on consideration of the dynamics of the system and uses for example the left eigenvectors of the matrix that defines these dynamics we show that the reachability calculus can then be performed on a combination of such abstractions and how its accuracy depends on the choice of hyperplanes that define the decomposition