positive invariance of constrained affine dynamics and its applications to hybrid systems and safety verification motivated by long time dynamic analysis of hybrid systems and safety verification problems this paper addresses fundamental positive invariance issues of an affine dynamical system on a general polyhedron and their applications necessary and sufficient algebraic conditions are established for the existence of a positively invariant set of an affine system on a polyhedron using the tools of lexicographic relation and long time oscillatory dynamic analysis a linear program based algorithm is proposed to verify these conditions and its computational complexity is analyzed the positive invariance results are applied to obtain an explicit characterization of global switching behaviors of piecewise affine systems further the positive invariance techniques developed in this paper are exploited to show the decidability of safety verification of a class of affine dynamics on semialgebraic sets

decidability for a temporal logic used in discrete event system analysis the type of plant considered is one that can be modelled by a non deterministic finite state machine p the regulator is a deterministic finite state machine r the closed loop system is formed by connecting p and r in a regulator configuration formulae in a propositional temporal language are used to describe the behaviour of the closed loop system it is shown that there is a mechanical procedure which for a given p and r and a temporal formula will determine in a finite number of steps whether or not must be true this decidability result could be proven using other known results on temporal logic the proof given here shows that the behaviour of the closed loop system may safely be assumed to be ultimately periodic formulae of a given complexity say n will be true in all possible runs of the system just in case they are true in all ultimately periodic runs with the period and the onset of periodicity bounded by a certain function of n a synthesis result follows immediately from th

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

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

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

monotonic reach control on polytopes the technical note studies the problem of making the trajectories of an affine system defined on a polytopic state space reach a prescribed facet of the polytope in finite time without first leaving the polytope the focus is on solvability by continuous piecewise affine feedback and we formulate a variant of the problem in which trajectories exit in a monotonic sense this allows to obtain necessary and sufficient conditions for solvability in certain geometric situations

controlling a class of nonlinear systems on rectangles in this paper we focus on a particular class of nonlinear affine control systems of the form xdot f x bu where the drift f is a multi affine vector field i e affine in each state component the control distribution b is constant and the control u is constrained to a convex set for such a system we first derive necessary and sufficient conditions for the existence of a multiaffine feedback control law keeping the system in a rectangular invariant we then derive sufficient conditions for driving all initial states in a rectangle through a desired facet in finite time if the control constraints are polyhedral we show that all these conditions translate to checking the feasibility of systems of linear inequalities to be satisfied by the control at the vertices of the state rectangle this work is motivated by the need to construct discrete abstractions for continuous and hybrid systems in which analysis and control tasks specified in terms of reachability of sets of states can be reduced to searches on finite graphs we show the application of our results to the problem of controlling the angular velocity of an aircraft with gas jet actuators

a control problem for affine dynamical systems on a full dimensional polytope given an affine system on a full dimensional polytope the problem of reaching a particular facet of the polytope using continuous piecewise affine state feedback is studied necessary conditions and sufficient conditions for the existence of a solution are derived in terms of linear inequalities on the input vectors at the vertices of the polytope special attention is paid to affine systems on full dimensional simplices in this case the necessary and sufficient conditions are equivalent and a constructive procedure yields an affine feedback control law that solves the reachability problem under consideration

guaranteed cost control of affine nonlinear systems via partition of unity method we consider the problem of guaranteed cost control gcc of affine nonlinear systems in this paper firstly the general affine nonlinear system with the origin being its equilibrium point is represented as a linear like structure with state dependent coefficient matrices secondly partition of unity method is used to approximate the coefficient matrices as a result of which the original affine nonlinear system is equivalently converted into a linear like system with modeling error a gcc law is then synthesized based on the equivalent model in the presence of modeling error under certain error condition the control law ensures that the system under control is asymptotically stable as well as that a given cost function is upper bounded a suboptimal gcc law can be obtained via solving an optimization problem in terms of linear matrix inequality lmi in stead of state dependent riccati equation sdre or hamilton jacobi equations that are usually required in solving nonlinear optimal control problems finally a numerical example is provided to illustrate the validity of the proposed method

symbolic control of stochastic systems via approximately bisimilar finite abstractions symbolic approaches for control design construct finite state abstract models that are related to the original systems then use techniques from finite state synthesis to compute controllers satisfying specifications given in a temporal logic and finally translate the synthesized schemes back as controllers for the original systems such approaches have been successfully developed and implemented for the synthesis of controllers over non probabilistic control systems in this paper we extend the technique to probabilistic control systems modelled by controlled stochastic differential equations we show that for every stochastic control system satisfying a probabilistic variant of incremental input to state stability and for every given precision e 0 a finite state transition system can be constructed which is e approximately bisimilar to the original stochastic control system moreover we provide results relating stochastic control systems to their corresponding finite state transition systems in terms of probabilistic bisimulation relations known in the literature we demonstrate the effectiveness of the construction by synthesizing controllers for stochastic control systems over rich specifications expressed in linear temporal logic our technique enables automated correct by construction controller synthesis for stochastic control systems which are common mathematical models employed in many safety critical systems subject to structured uncertainty

controller synthesis for robust invariance of polynomial dynamical systems using linear programming in this paper we consider a control synthesis problem for a class of polynomial dynamical systems subject to bounded disturbances and with input constraints more precisely we aim at synthesizing at the same time a controller and an invariant set for the controlled system under all admissible disturbances we propose a computational method to solve this problem given a candidate polyhedral invariant we show that controller synthesis can be formulated as an optimization problem involving polynomial cost functions over bounded polytopes for which effective linear programming relaxations can be obtained then we propose an iterative approach to compute the controller and the polyhedral invariant jointly each iteration of the approach mainly consists in solving two linear programs one for the controller and one for the invariant and is thus computationally tractable finally we show with several examples the usefulness of our method in applications

global dynamics for switching systems and their extensions by linear differential equations abstract switching systems use piecewise constant nonlinearities to model gene regulatory networks this choice provides advantages in the analysis of behavior and allows the global description of dynamics in terms of morse graphs associated to nodes of a parameter graph the parameter graph captures spatial characteristics of a decomposition of parameter space into domains with identical morse graphs however there are many cellular processes that do not exhibit threshold like behavior and thus are not well described by a switching system we consider a class of extensions of switching systems formed by a mixture of switching interactions and chains of variables governed by linear differential equations we show that the parameter graphs associated to the switching system and any of its extensions are identical for each parameter graph node there is an order preserving map from the morse graph of the switching system to the morse graph of any of its extensions we provide counterexamples that show why possible stronger relationships between the morse graphs are not valid

temporal logic motion planning for dynamic robots in this paper we address the temporal logic motion planning problem for mobile robots that are modeled by second order dynamics temporal logic specifications can capture the usual control specifications such as reachability and invariance as well as more complex specifications like sequencing and obstacle avoidance our approach consists of three basic steps first we design a control law that enables the dynamic model to track a simpler kinematic model with a globally bounded error second we built a robust temporal logic specification that takes into account the tracking errors of the first step finally we solve the new robust temporal logic path planning problem for the kinematic model using automata theory and simple local vector fields the resulting continuous time trajectory is provably guaranteed to satisfy the initial user specification

an input output simulation approach to controlling multi affine systems for linear temporal logic specifications this article presents an input output simulation approach to controlling multi affine systems for linear temporal logic ltl specifications which consists of the following steps first the state space is partitioned into rectangles each of which satisfies atomic ltl propositions then we study the control of multi affine systems on rectangles including the control based on the exit sub region to drive all trajectories starting from a rectangle to exit through a facet and the control to stabilise the multi affine system towards a desired point with the proposed controllers a finitely abstracted transition system is constructed which is shown to be input output simulated by the rectangular transition system of the multi affine system since the input output simulation preserves ltl properties the controller synthesis of the multi affine system for ltl specifications is achieved by designing a nonblocking supervisor for the abstracted transition system and by implementing the resulting supervisor to

determinant formulas for class numbers in function fields in this paper by extending kucera s idea to the function field case we obtain several determinant formulas involving the real class number and the relative class number of any subfield of cyclotomic function fields we also provide several examples using these determinant formulas

safety controller synthesis using human generated trajectories this paper focuses on the task of safety controller synthesis that is designing a controller that will take a system from any point within a compact set of initial states to a point inside a set of acceptable goal states while never entering any state that is deemed unsafe to do this we use a human generated trajectory based approach we introduce the control autobisimulation function which is the analog of the control lyapunov function for approximate bisimulation we consider a class of hybrid systems and use this function to determine a set of admissible feedback control laws that guarantee trajectory robustness for underlying dynamics that are linear affine feedback linearizable and differentially flat this property ensures that any trajectory of the closed loop system that is initialized within some neighborhood of a nominal trajectory will stay within some tube of the nominal trajectory when given the same input this feedback control and input can be used as the controller for some subset of the initial states we demonstrate how to combine multiple trajectories into a synthesized controller that satisfies the safety problem

symbolic models for networks of control systems in this note we propose symbolic models for networks of discrete time nonlinear control systems if each subsystem composing the network admits an incremental input to state stable lyapunov function and if some small gain theorem type conditions are satisfied a network of symbolic models each one associated with each subsystem composing the network is proposed and shown to be approximately bisimilar to the original network with any desired accuracy

supervisory control of discrete event systems with ctl temporal logic specifications the supervisory control problem of discrete event systems with temporal logic specifications is studied the full branching time logic of ctl is used for expressing specifications of discrete event systems the control problem of ctl is reduced to the decision problem of ctl a small model theorem for the control of ctl is obtained it is shown that the control problem of ctl resp ctl is complete for deterministic double resp single exponential time a sound and complete supervisor synthesis algorithm for the control of ctl is provided special cases of the control of computation tree logic ctl and linear time temporal logic are also studied

on algorithmic analysis of transcriptional regulation by ltl model checking studies of cells in silico can greatly reduce the need for expensive and prolonged laboratory experimentation the use of model checking for the analysis of biological networks has attracted much attention recently the practical limitations are still the size of the model and the time needed to generate the state space this paper is focused on the model checking approach for analysis of piecewise linear deterministic models of genetic regulatory networks firstly the qualitative simulation algorithm of de jong et al that builds the heart of genetic network analyzer gna is revisited and its time complexity is studied in detail secondly a novel algorithm that reduces the state space generation time is introduced the new algorithm is developed as an abstraction of the original gna algorithm finally a fragment of linear time temporal logic for which the provided abstraction is conservative is identified efficiency of the new algorithm when implemented in the parallel model checking environment is demonstrated on a set of experiments performed on randomly modified biological models in general the achieved results bring a new insight into the field of qualitative simulation emerging in the context of systems biology

symbolic models for nonlinear control systems without stability assumptions finite state models of control systems were proposed by several researchers as a convenient mechanism to synthesize controllers enforcing complex specifications most techniques for the construction of such symbolic models have two main drawbacks either they can only be applied to restrictive classes of systems or they require the exact computation of reachable sets in this paper we propose a new abstraction technique that is applicable to any nonlinear sampled data control system as long as we are only interested in its behavior in a compact set moreover the exact computation of reachable sets is not required the effectiveness of the proposed results is illustrated by synthesizing a controller to steer a vehicle

dynamical properties of hybrid automata hybrid automata provide a language for modeling and analyzing digital and analogue computations in real time systems hybrid automata are studied here from a dynamical systems perspective necessary and sufficient conditions for existence and uniqueness of solutions are derived and a class of hybrid automata whose solutions depend continuously on the initial state is characterized the results on existence uniqueness and continuity serve as a starting point for stability analysis lyapunov s theorem on stability via linearization and lasalle s invariance principle are generalized to hybrid automata

cyclotomic units and stickelberger ideals of global function fields in this paper we define the group of cyclotomic units and stickelberger ideals in any subfield of the cyclotomic function field we also calculate the index of the group of cyclotomic units in the total unit group in some special cases and the index of stickelberger ideals in the integral group ring

equivalence of dynamical systems by bisimulation a general notion of bisimulation is defined for linear input state output systems using analogies with the theory of concurrent processes a characterization of bisimulation and an algorithm for computing the maximal bisimulation relation is derived using geometric control theory bisimulation is shown to be a notion which unifies the concepts of state space equivalence and state space reduction and which allows to study equivalence of systems with nonminimal state space dimension the notion of bisimulation is especially powerful for nondeterministic dynamical systems and leads in this case to a notion of equivalence which is finer than equality of external behavior for abstractions of systems it is shown how the results specialize to previously obtained results by other authors extensions of the main results to the nonlinear case are provided

brief paper reachability and stabilization of discrete time affine systems with disturbances this paper studies the fundamental problems whether an affine system affected by additive disturbances is robustly transferable from a source set simplex to a target set polytope and whether it is robustly stabilizable with its state constrained in a simplex first a necessary and sufficient condition is derived for the existence of affine feedback control that solves the robust reachability problem further investigation is provided for two situations relying on whether the union of the source set and the target set is convex or non convex for the former one a necessary and sufficient condition is obtained in the form of linear inequalities while for the latter several computationally feasible sufficient conditions are found second we show that robust stabilization subject to a state constraint is equivalent to find a feasible solution to a linear equation once it is known that either of the problems has a solution by checking the derived conditions design of control laws is then straightforward

k3 surfaces with non symplectic automorphisms of prime order in this paper we present the classification of non symplectic automorphisms of prime order on k3 surfaces i e we describe the topological structure of their fixed locus and determine their invariant lattice in cohomology we provide new results for automorphisms of order 5 and 7 and alternative proofs for higher orders moreover for any prime p we identify the irreducible components of the moduli space of k3 surfaces with a non symplectic automorphism of order p

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

failure diagnosis of discrete event systems with linear time temporal logic specifications the paper studies failure diagnosis of discrete event systems dess with linear time temporal logic ltl specifications the ltl formulas are used for specifying failures in the system the ltl based specifications make the specification specifying process easier and more user friendly than the formal language automata based specifications and they can capture the failures representing the violation of both liveness and safety properties whereas the prior formal language automaton based specifications can capture the failures representing the violation of only the safety properties such as the occurrence of a faulty event or the arrival at a failed state prediagnosability and diagnosability of dess in the temporal logic setting are defined the problem of testing prediagnosability and diagnosability is reduced to the problem of model checking an algorithm for the test of prediagnosability and diagnosability and the synthesis of a diagnoser is obtained the complexity of the algorithm is exponential in the length of each specification ltl formula and polynomial in the number of system states and the number of specifications the requirement of nonexistence of unobservable cycles in the system which is needed for the diagnosis algorithms in prior methods to work is relaxed finally a simple example is given for illustration

symbolic models for nonlinear control systems affected by disturbances in the last few years there has been a growing interest in the use of symbolic models for the formal verification and control design of purely continuous or hybrid systems symbolic models are abstract descriptions of continuous systems where one symbol corresponds to an aggregate of continuous states in this article we face the problem of deriving symbolic models for nonlinear control systems affected by disturbances the main contribution of this article is in proposing symbolic models that can be effectively constructed and that approximate nonlinear control systems affected by disturbances in the sense of alternating approximate bisimulation

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

an algorithm to find maximal state constraint sets for discrete time linear dynamical systems with bounded controls and states an algorithm to find a polyhedral approximant to the maximal state constraint set given state and input bounds is suggested for linear discrete time dynamical systems

control of systems integrating logic dynamics and constraints this paper proposes a framework for modeling and controlling systems described by interdependent physical laws logic rules and operating constraints denoted as mixed logical dynamical mld systems these are described by linear dynamic equations subject to linear inequalities involving real and integer variables mld systems include linear hybrid systems finite state machines some classes of discrete event systems constrained linear systems and nonlinear systems which can be approximated by piecewise linear functions a predictive control scheme is proposed which is able to stabilize mld systems on desired reference trajectories while fulfilling operating constraints and possibly take into account previous qualitative knowledge in the form of heuristic rules due to the presence of integer variables the resulting on line optimization procedures are solved through mixed integer quadratic programming miqp for which efficient solvers have been recently developed some examples and a simulation case study on a complex gas supply system are reported

hierarchically consistent control systems large scale control systems typically possess a hierarchical architecture in order to manage complexity higher levels of the hierarchy utilize coarser models of the system resulting from aggregating the detailed lower level models in this layered control paradigm the notion of hierarchical consistency is important as it ensures the implementation of high level objectives by the lower level system in this paper we define a notion of modeling hierarchy for continuous control systems and obtain characterizations for hierarchically consistent linear systems with respect to controllability objectives as an interesting byproduct we obtain a hierarchical controllability criterion for linear systems from which we recover the best of the known controllability algorithms from numerical linear algebra

deadness and how to disprove liveness in hybrid dynamical systems what if we designed a tool to automatically prove the dynamical properties of systems for which analytic proof is difficult or impossible to obtain such a tool would represent a significant advance in the understanding of complex dynamical systems with nonlinearities this is precisely what this paper offers a solution to the problem of automatically proving some dynamic stability properties of complex systems with multiple discontinuities and modes of operation modelled as hybrid dynamical systems for this purpose we propose a reinterpretation of some stability properties from a computational viewpoint chiefly by using the computer science concepts of safety and liveness however these concepts need to be redefined within the framework of hybrid dynamical systems in computer science terms here we consider the problem of automatically disproving the liveness properties of nonlinear hybrid dynamical systems for this purpose we define a new property which we call deadness this is a dynamically aware property of a hybrid system which if true disproves the liveness property by means of a finite execution we formally define this property and give an algorithm which can derive deadness properties automatically for a type of liveness property called inevitability we show how this algorithm works for three different examples that represent three classes of hybrid systems with complex behaviours

class numbers of some abelian extensions of rational function fields let p be a monic irreducible polynomial in this paper we generalize the determinant formula for h k pn of bae and kang and the formula for h kpn of jung and ahn to any subfields k of the cyclotomic function field kpn by using these formulas we calculate the class numbers h k h k of all subfields k of kp when q and deg p are small

classification of non symplectic automorphisms on k3 surfaces which act trivially on the neron severi lattice abstract we treat non symplectic automorphisms on k 3 surfaces which act trivially on the neron severi lattice in this paper we classify non symplectic automorphisms of prime power order especially 2 power order on k 3 surfaces i e we describe their fixed locus

improved delay dependent stability criteria for linear systems with multiple time varying delays this paper concentrates on the stability analysis of continuous time linear systems with multiple time varying delays the time delays are assumed to be bounded and have or not lower and upper boun

on invariant polyhedra of continuous time linear systems this note presents some conditions of existence of positively invariant polyhedra for linear continuous time systems these conditions are first described algebraically then interpreted on the basis of the system eigenstructure then a simple state feedback placement method is described for solving some linear regulation problems under constraints

synthesis for constrained nonlinear systems using hybridization and robust controllers on simplices in this technical note we propose an approach to controller synthesis for a class of constrained nonlinear systems it is based on the use of a hybridization that is a hybrid abstraction of the nonlinear dynamics this abstraction is defined on a triangulation of the state space where on each simplex of the triangulation the nonlinear dynamics is conservatively approximated by an affine system subject to disturbances except for the disturbances this hybridization can be seen as a piecewise affine hybrid system on simplices for which appealing control synthesis techniques have been developed in the past decade we extend these techniques to handle systems subject to disturbances by synthesizing and coordinating local robust affine controllers defined on the simplices of the triangulation we show that the resulting hybrid controller can be used to control successfully the original constrained nonlinear system our approach though conservative can be fully automated and is computationally tractable to show its effectiveness in practical applications we apply our method to control a pendulum mounted on a cart

observability analysis of conewise linear systems via directional derivative and positive invariance techniques belonging to the broad framework of hybrid systems conewise linear systems clss form a class of lipschitz piecewise linear systems subject to state triggered mode switchings motivated by state estimation of nonsmooth switched systems this paper exploits directional derivative and positive invariance techniques to characterize finite time and long time local observability of a general cls for the former observability notion directional derivative results are developed from the simple switching property and they yield improved observability conditions for the latter notion we focus on the case where a nominal trajectory has finitely many switchings in order to characterize long time behaviors of the cls necessary and sufficient conditions are obtained for the interior of a positively invariant cone by employing these conditions we establish connections between finite time and long time local observability underlying positive invariance properties are unveiled

nonlinear regulation the piecewise linear approach this paper approaches nonlinear control problems through the use of discrete time piecewise linear systems these are systems whose next state and output maps are both described by pl maps i e by maps which are affine on each of the components of a finite polyhedral partition various results on state and output feedback observers and inverses standard for linear systems are proved for pl systems many of these results are then used in the study of more general both discrete and continuous time systems using suitable approximations

admissible sets and feedback control for discrete time linear dynamical systems with bounded controls and states for linear discrete time dynamical systems necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set a variable structure linear state feedback controller given in terms of the controls at the vertices of the polyhedral state constraint set is presented

on reachability and controllability of switched boolean control networks this paper investigates the reachability and controllability of switched boolean control networks bcns by using the semi tensor product method first a kind of switching input state transfer graph is defined based on which and the algebraic expression of switched bcns we thus propose a new kind of switching input state incidence matrix that contains the complete information of the switching input state transfer graph second by the switching input state incidence matrix several necessary and sufficient conditions are obtained for the reachability and controllability of switched bcns finally an algorithm is established to find a switching sequence and a control scheme to realize the reachability of switched bcns in the shortest time the study of an illustrative example shows that the new results obtained in this paper are very effective in investigating the reachability and controllability of switched bcns

circular units of function fields a unit index class number formula is proved for subfields of cyclotomic function fields in analogy with similar results for subfields of cyclotomic number fields let m be a positive integer and let cm exp 27ri m let km q 4m denote the mth cyclotomic field and k its maximal real subfield the ring of integers in km resp k is z cm resp z 4m c 1 in 2 sinnott showed that the index of circular units in the full group of units of q cm equals the class number of z 4m cm multiplied by a power of 2 which depends exclusively on the number of prime factors of m sinnott 3 subsequently generalized this result to an arbitrary abelian field there is a parallel setup for function fields of characteristic p let fq be the finite field with q elements let rt fq t be the ring of polynomials over fq with t transcendental over fq and let fq t be the field of rational functions over fq to each polynomial m e rt one can associate an extension km called the mth cyclotomicfunctionfield which enjoys properties analogous to those of the cyclotomic number field km in particular galovich and rosen 1 proved the analogue of sinnott s theorem in this setting the purpose of this paper is to extend this unit index class number formula to an arbitrary subfield of km let k be any subfield of the mth cyclotomic function field m monic g the galois group of k over ifq t k the maximal subfield of k in which oo splits ok ok the integral closure of fq t in k k and ok the unit group of ok in 3 we define a subgroup c of ok which we call the circular units of k our main result is that c has finite index in ok and that this index may be written in the form 0k c h ok c k where h ok is the class number of ok and ck is a rational number whose definition does not involve h ok we now briefly describe the contents of the rest of this paper in 1 we present the relevant definitions and facts in the function field setting we also state the analytic class number formula in 2 we review ordinary distributions on lfq t rt discuss an index notation and obtain a preliminary result on the structure of a certain module the circular units are introduced in 3 and received by the editors february 1 1990 and in revised form november 27 1991 1991 mathematics subject classification primary 1 lr58 d 1994 american mathematical society 0002 9947 94 1 00 25 per page

an obstruction to solvability of the reach control problem using affine feedback this paper studies the reach control problem rcp using affine feedback on simplices the contributions of this paper are threefold first we identify a new obstruction to solvability of the rcp using affine feedback and provide necessary and sufficient conditions for occurrence of such an obstruction second for two input systems these conditions are formulated in terms of scalar linear inequalities third computationally efficient necessary conditions are proposed for checking the obstruction for multi input systems as feasibility programs in terms of linear inequalities in contrast to the previous work in the literature no assumption is imposed on the set of possible equilibria so the results are applicable to the general rcp

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

flow functions control flow functions and the reach control problem the paper studies the reach control problem rcp to make trajectories of an affine system defined on a polytopic state space reach and exit a prescribed facet of the polytope in finite time without first leaving the polytope we introduce the notion of a flow function which provides the analog of a lyapunov function for the equilibrium stability problem a flow function comprises a scalar function that decreases along closed loop trajectories and its existence is a necessary and sufficient condition for closed loop trajectories to exit the polytope it provides an analysis tool for determining if a specific instance of rcp is solved without the need for calculating the state trajectories of the closed loop system results include a variant of the lasalle principle tailored to rcp an open problem is to identify suitable classes of flow functions we explore functions of the form v x max v i x and we give evidence that these functions arise naturally when rcp is solved using continuous piecewise affine feedbacks next we introduce the notion of a control flow function it is shown that the artstein sontag theorem of control lyapunov functions has direct analogies to rcp via control flow functions

k3 surfaces with non symplectic automorphisms of 2 power order abstract this paper concerns complex algebraic k3 surfaces with an automorphism which acts trivially on the neron severi group complementing a result by vorontsov and kond we determine those k3 surfaces where the order of the automorphism is a 2 power and equals the rank of the transcendental lattice we also study the arithmetic of these k3 surfaces and comment on relations with mirror symmetry

constrained regulation of linear continuous time dynamical systems abstract in this paper the linear constrained regulation problem lcrp for continuous time dynamical systems is studied the first part of the paper deals with the existence of linear state feedback control laws that transfer asymptotically to the origin all initial states belonging to a given polyhedral subset of the state space while respecting linear constraints on both state and control vectors then the lcrp is formulated as an optimization problem which can be solved by applying linear programming techniques

lyapunov stability of a class of discrete event systems discrete event systems des are dynamical systems which evolve in time by the occurrence of events at possibly irregular time intervals logical des are a class of discrete time des with equations of motion that are most often nonlinear and discontinuous with respect to event occurrences recently there has been much interest in studying the stability properties of logical des and several definitions for stability and methods for stability analysis have been proposed here the authors introduce a logical des model and define stability in the sense of lyapunov and asymptotic stability for logical des then the authors show that more conventional analysis of stability which employs appropriate lyapunov functions can be used for logical des the authors provide a general characterization of the stability properties of automata theoretic des models petri nets and finite state systems furthermore the lyapunov stability analysis approach is illustrated on a manufacturing system that processes batches of n different types of parts according to a priority scheme to prove properties related to the machine s ability to reorient itself to achieve safe operation and a load balancing problem in computer networks to study the ability of the system to achieve a balanced load to minimize underutilization

control to facet by piecewise affine output feedback the control to facet problem plays an important role in the design of feedback controllers for piecewise affine hybrid systems on polytopes in the literature necessary conditions and sufficient conditions for solvability by static state feedback exist in this paper we extend these results to the case of continuous piecewise affine static output feedback for the construction of a controller a triangulation of the output polytope is made which satisfies additional conditions to guarantee compatibility with the induced subdivision of the state polytope in the state feedback case the use of this special type of triangulation is not required

survey paper set invariance in control the properties of positively invariant sets are involved in many different problems in control theory such as constrained control robustness analysis synthesis and optimization in this paper we provide an overview of the literature concerning positively invariant sets and their application to the analysis and synthesis of control systems

hybrid control of a bioreactor with quantized measurements we consider the problem of global stabilization of an unstable bioreactor model e g for anaerobic digestion when the measurements are discrete and in finite number quantized with control of the dilution rate the measurements define regions in the state space and they can be perfect or uncertain i e without or with overlaps we show that under appropriate assumptions a quantized control may lead to global stabilization trajectories have to follow some transitions between the regions until the final region where they converge toward the reference equilibrium on the boundary between regions the solutions are defined as a filippov differential inclusion if the assumptions are not fulfilled sliding modes may appear and the transition graphs are not deterministic

temporal logic analysis of gene networks under parameter uncertainty the lack of precise numerical information for the values of biological parameters severely limits the development and analysis of models of genetic regulatory networks to deal with this problem we propose a method for the analysis of genetic regulatory networks under parameter uncertainty we consider models based on piecewise multiaffine differential equations dynamical properties expressed in temporal logic and intervals for the values of uncertain parameters the problem is then either to guarantee that the system satisfies the expected properties for every possible parameter value the corresponding parameter set is then called valid or to find valid subsets of a given parameter set the proposed method uses discrete abstractions and model checking and allows for efficient search of the parameter space however the abstraction process creates spurious behaviors in the abstract systems along which time does not progress consequently the verification of liveness properties expressing that something will eventually happen and implicitly assuming progress of time often fails a solution to this second problem is proposed using the notion of transient regions this approach has been implemented in a tool for robust verification of gene networks and applied to the tuning of a synthetic network built in e coli

non symplectic automorphisms of order 3 on k3 surfaces in this paper we study k3 surfaces with a non symplectic automorphism of order 3 in particular we classify the topological structure of the fixed locus of such automorphisms and we show that it determines the action on cohomology this allows us to describe the structure of the moduli space and to show that it has three irreducible components