symbolic model checking for real time systems abstract we describe finite state programs over real numbered time in a guarded command language with real valued clocks or equivalently as finite automata with real valued clocks model checking answers the question which states of a real time program satisfy a branching time specification given in an extension of ctl with clock variables we develop an algorithm that computes this set of states symbolically as a fixpoint of a functional on state predicates without constructing the state space for this purpose we introduce a calculus on computation trees over real numbered time unfortunately many standard program properties such as response for all nonzeno execution sequences during which time diverges cannot be characterized by fixpoints we show that the expressiveness of the timed calculus is incomparable to the expressiveness of timed ctl fortunately this result does not impair the symbolic verification of implementable real time programs those whose safety constraints are machine closed with respect to diverging time and whose fairness constraints are restricted to finite upper bounds on clock values all timed ctl properties of such programs are shown to be computable as finitely approximable fixpoints in a simple decidable theory

optimization of lyapunov invariants in verification of software systems the paper proposes a control theoretic framework for verification of numerical software systems and puts forward software verification as an important application of control and systems theory the idea is to transfer lyapunov functions and the associated computational techniques from control systems analysis and convex optimization to verification of various software safety and performance specifications these include but are not limited to absence of overflow absence of division by zero termination in finite time absence of dead code and certain user specified assertions central to this framework are lyapunov invariants these are properly constructed functions of the program variables and satisfy certain properties analogous to those of lyapunov functions along the execution trace the search for the invariants can be formulated as a convex optimization problem if the associated optimization problem is feasible the result is a certificate for the specification

bifurcations in a system of interacting fronts we show that the bifurcation scenario in a high dimensional system with interacting moving fronts can be related to the universal u sequence which is known from the symbolic analysis of iterated one dimensional maps this connection is corroborated for a model of a semiconductor superlattice which describes the complex dynamics of electron accumulation and depletion fronts by a suitable poincare section we reduce the dynamics to a low dimensional iterated map for which in the most elementary case the bifurcation points can be determined analytically

nontrivial solutions of kirchhoff type problems via the yang index abstract we obtain nontrivial solutions of a class of nonlocal quasilinear elliptic boundary value problems using the yang index and critical groups

existence of a positive solution to kirchhoff type problems without compactness conditions abstract the existence of a positive solution to a kirchhoff type problem on r n is proved by using variational methods and the new result does not require usual compactness conditions a cut off functional is utilized to obtain the bounded palais smale sequences

reachability analysis of dynamical systems having piecewise constant derivatives abstract in this paper we consider a class of hybrid systems namely dynamical systems with piecewise constant derivatives pcd systems such systems consist of a partition of the euclidean space into a finite set of polyhedral sets regions within each region the dynamics is defined by a constant vector field hence discrete transitions occur only on the boundaries between regions where the trajectories change their direction with respect to such systems we investigate the reachability question given an effective description of the systems and of two polyhedral subsets p and q of the state space is there a trajectory starting at some x p and reaching some point in q our main results are a decision procedure for two dimensional systems and an undecidability result for three or more dimensions

verification of hybrid automata diagnosability by abstraction a notion of diagnosability for hybrid systems is defined which generalizes the common notion of observability we propose an abstraction procedure to translate a hybrid automaton into a timed automaton in order to verify observability and diagnosability properties we introduce a procedure to check diagnosability and show that for the system class of our abstraction namely for a subclass of timed automata the durational graphs the verification problem belongs to the complexity class p we apply our procedure to an electromagnetic valve system for camless engines

verification of bounded discrete horizon hybrid automata we consider the class of o minimally definable hybrid automata with a bounded discrete transition horizon we show that for every hybrid automata in this class there exists a bisimulation of finite index and that the bisimulation quotient can be effectively constructed when the underlying o minimal theory is decidable more importantly we give natural specifications for hybrid automata which ensure the boundedness of discrete transition horizons in addition we show that these specifications are reasonably tight with respect to the decidability of the models and that they can model modern day real time and embedded systems as a result the analysis of several problems for these systems admit effective algorithms we provide a representative example of a hybrid automaton in this class unlike previously examined subclasses of o minimally defined hybrid automata with decidable verification properties and extended o minimal hybrid automata we do not impose re initialization of the continuous variables in a memoryless fashion when a discrete transition is taken our class of hybrid systems has both rich continuous dynamics and strong discrete continuous coupling showing that it is not necessary to either simplify the continuous dynamics or restrict the discrete dynamics to achieve decidability

the lie algebraic structure of symmetries generated by hereditary symmetries based on the works of li zhu and co workers 1985 a general lie algebraic structure is given for symmetries of integrable systems which are generated by hereditary symmetries

lie algebraic structures of 1 1 dimensional lax integrable systems an approach of constructing isospectral flows kl nonisospectral flows k and their implicit representations of a general lax integrable system is proposed by introducing product function matrices it is shown that the two sets of flows and of related symmetries both constitute infinite dimensional lie algebras with respect to the commutator given in this paper algebraic properties for some well known integrable systems such as the akns system the generalized harry dym system and the n wave interaction system are obtained as particular examples

existence of positive solutions for the nonlinear kirchhoff type equations in rn abstract in this paper we consider the following nonlinear problem of kirchhoff type a r n u 2 d x u v x u u p 1 u x r n u h 1 r n u 0 where n 3 a are positive constants 1 p max 3 n 2 n 2 and v x is a positive continuous potential satisfying v 0 v x lim inf x v x v using variational methods and a cutoff technique we prove the existence of positive solution to the above equation for all 0 small

lie algebra contractions and symmetries of the toda hierarchy the lie algebra l of generalized and point symmetries of the equations in the toda hierarchy is shown to be a semidirect sum of two infinite dimensional lie algebras one perfect the other abelian in the continuous limit the structure of the lie algebra changes a contraction occurs with the lattice spacing as the contraction parameter in particular for the toda equation itself a set of five elements involving both point symmetries and generalized ones contracts to the point symmetry algebra of the potential kdv equation

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

computing reachable states for nonlinear biological models in this paper we describe reachability computation for continuous and hybrid systems and its potential contribution to the process of building and debugging biological models we summarize the state of the art for linear systems and then develop a novel algorithm for computing reachable states for nonlinear systems we report experimental results obtained using a prototype implementation applied to several biological models we believe these results constitute a promising contribution to the analysis of complex models of biological systems

integrable systems associated with the schrodinger spectral problem in the plane associated with the schrodinger spectral problem in the plane the nonlinear integrable systems are investigated the commutative properties of equations of the flow and their symmetries and mastersymmetries are derived in spite of the noncanonicality of the related extended recursion operator by applying the inverse scattering transform the systems can be solved and then the correspondences between commuting resp noncommuting flows and isospectral resp nonisospectral deformations of the spectral problem are found

process algebra for hybrid systems we propose a process algebra obtained by extending a combination of the process algebra with continuous relative timing from baeten and middelburg process algebra with timing springer berlin 2002 chapter 4 and the process algebra with propositional signals from baeten and bergstra theoret comput sci 177 1977 381 405 the proposed process algebra makes it possible to deal with the behaviour of hybrid systems i e systems in which the instantaneous state transitions caused by performing actions are alternated with continuous state evolutions this process algebra has in addition to equational axioms rules to derive equations with the help of real analysis

computational techniques for reachability analysis of max plus linear systems this work discusses a computational approach to reachability analysis of max plus linear mpl systems a class of discrete event systems widely used in synchronization and scheduling applications given a set of initial states we characterize and compute its reach tube namely the collection of set of reachable states regarded step wise as reach sets by an alternative characterization of the mpl dynamics we show that the exact computation of the reach sets can be performed quickly and compactly by manipulations of difference bound matrices and further derive worst case bounds on the complexity of these operations the approach is also extended to backward reachability analysis the concepts and results are elucidated by a running example and we further illustrate the performance of the approach by a numerical benchmark the technique comfortably handles twenty dimensional mpl systems i e with twenty continuous state variables and as such it outperforms the state of the art alternative approaches in the literature

the algorithmic analysis of hybrid systems we present a general framework for the formal specification and algorithmic analysis of hybrid systems a hybrid system consists of a discrete program with an analog environment we model hybrid systems as finite automata equipped with variables that evolve continuously with time according to dynamical laws for verification purposes we restrict ourselves to linear hybrid systems where all variables follow piecewise linear trajectories we provide decidability and undecidability results for classes of linear hybrid systems and we show that standard program analysis techniques can be adapted to linear hybrid systems in particular we consider symbolic model checking and minimization procedures that are based on the reachability analysis of an infinite state space the procedures iteratively compute state sets that are definable as unions of convex polyhedra in multidimensional real space we also present approximation techniques for dealing with systems for which the iterative procedures do not converge

infinitely many positive solutions for kirchhoff type problems abstract this paper is concerned with the existence of infinitely many positive solutions to a class of kirchhoff type problem a b u 2 d x u f x u in and u 0 on where is a smooth bounded domain of r n a b 0 0 and f r r is a caratheodory function satisfying some further conditions we obtain a sequence of a e positive weak solutions to the above problem tending to zero in l with f being more general than that of k perera z zhang nontrivial solutions of kirchhoff type problems via the yang index j differential equations 221 2006 246 255 z zhang k perera sign changing solutions of kirchhoff type problems via invariant sets of descent flow j math anal appl 317 2006 456 463

decidable integration graphs abstract integration graphsare a computational model developed in the attempt to identify simple hybrid systems with decidable analysis problems we start with the class ofconstant slope hybrid systems cshs in which the right hand side of all differential equations is an integer constant we refer to continuous variables whose right hand side constants are always 1 astimers all other continuous variables are calledintegrators the first result shown in the paper is that simple questions such as reachability of a given state are undecidable for even this simple class of systems to restrict the model even further we impose the requirement that no test that refers to integrators may appear within a loop in the graph this restricted class of cshs is calledintegration graphs the main results of the paper are that the reachability problem of integration graphs is decidable for two special cases the case of a single timer and the case of a single test involving integrators the expressive power of the integration graphs formalism is demonstrated by showing that some typical problems studied within the context of the calculus of durations and timed statecharts can be formulated as reachability problems for restricted integration graphs and a high fraction of these fall into the subclasses of a single timer or a single test involving integrators

symbolic reachability analysis of fifo channel systems with nonregular sets of configurations abstract we address the verification problem of fifo channel systems we apply the symbolic analysis principle to these systems we represent their sets of configurations using structures called constrained queue content decision diagrams cqdds combining finite state automata with linear arithmetical constraints on number of occurrences of symbols we show that cqdds allow forward and backward reachability analysis of systems with nonregular sets of configurations moreover we prove that cqdds allow to compute the exact effect of the repeated execution of any fixed cycle in the transition graph of a system we use this fact to define a generic reachability analysis semi algorithm parametrized by a set of cycles given a set of configurations this semi algorithm performs a least fixpoint calculation to construct the set of its successors or predecessors at each step this calculation is accelerated by considering the cycles in as additional meta transitions in the transition graph generalizing the approach of boigelot and godefroid cav 96 lecture notes in computer science vol 1102 springer berlin 1996

a game theoretic approach to fault diagnosis and identification of hybrid systems physical systems can fail for this reason the problem of identifying and reacting to faults has received a lot of attention in the control and computer science communities in this paper we study the fault diagnosis problem for hybrid systems from a game theoretical point of view a hybrid system is a system mixing continuous and discrete behaviours that cannot be faithfully modelled neither by using a formalism with continuous dynamics only nor by a formalism including only discrete dynamics we model hybrid systems as hybrid automata and add distinguished actions to describe faults we define a fault identification game on them using two players the environment and the identifier the environment controls the evolution of the system and chooses whether and when a fault occurs the identifier observes the external behaviour of the system and announces whether a fault has occurred or not existence of a winning strategy for the identifier implies that faults can be detected correctly while computing such a winning strategy corresponds to implementing an identifier for the system we will show how to determine the existence of a winning strategy and how to compute it for all decidable classes of hybrid automata that admit a finite bisimulation quotient

toward unified analysis and controller synthesis for a class of hybrid systems abstract this work defines a new class of hybrid systems called state based switched sbs systems that have numerous important engineering applications the characterizing feature of these systems is that the discrete event dynamics are associated with the continuous time state making a specific function be equal to zero the choice of this function is application specific and for the closed loop sbs systems defined in this paper it is related to the execution of a desired set of tasks from a pre specified mission plan for this broad class of sbs systems the paper presents a unified analysis and controller synthesis methodology based on lyapunov theory depending on the details of the mission plan the closed loop hybrid system will be divided into two subclasses sequential and non sequential the controller design procedure for both subclasses consists of the same two steps finding a control law and finding a stabilizing switching rule for static state and output feedback of sequential hybrid systems the paper proposes a new hybrid sequential sliding mode controller it is proven that the control mission can be accomplished for sequential hybrid systems under static state and output feedback using this new controller a similar framework is investigated for the more complex class of nonsequential hybrid systems and a systematic procedure for designing the switching rule is presented for some specific instances of these systems

hierarchies of nonlinear integrable equations and their symmetries in 2 1 dimensions abstract for a given nonlinear integrable equation in 2 1 dimensions an approach is described to construct the hierarchies of equations and relevant lie algebraic properties the commutability and noncommutability of equations of the flow their symmetries and mastersymmetries are then derived as direct results of these algebraic properties the details for the modified kadomtsev petviashvilli equation are shown as an example and the main results for the 2 1 dimensional caudrey dodd gibbon katera sawada equation are given

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

hybrid control synthesis for eventuality specifications using level set methods this paper is concerned with the extraction of controllers for hybrid systems with respect to eventuality specifications given a hybrid system modelled by a hybrid automaton and a target set of states the objective is to compute the maximal set of initial states together with the hybrid control policy such that all the trajectories of the controlled system reach the target in finite time due to the existence of set valued disturbance inputs the problem is studied in a game theoretic framework having shown that a least restrictive solution does not exist we propose a dynamic programming algorithm that computes the maximal initial set and a controller with the desired property to implement the algorithm reachable sets of pursuit evasion differential games need to be computed for that reason level set methods are employed where the boundary of the reachable set is characterized as the zero level set of a hamilton jacobi equation the procedure for the numerical extraction of the controller is presented in detail and examples illustrate the methodology finally to demonstrate the practical character of our results a control design problem in the benchmark system of the batch evaporator is considered as an eventuality synthesis problem and solved using the proposed methodology

on the decidability and complexity of problems for restricted hierarchical hybrid systems we study variants of a recently introduced hybrid system model called a hierarchical piecewise constant derivative hpcd these variants loosely called restricted hpcds form a class of natural models with similarities to many other well known hybrid system models in the literature such as stopwatch automata rectangular automata and pcds we study the complexity of reachability and mortality problems for variants of rhpcds and show a variety of results depending upon the allowed powers these models form a useful tool for the study of the complexity of such problems for hybrid systems due to their connections with existing models we show that the reachability problem and the mortality problem are co np hard for bounded 3 dimensional rhpcds 3 rhpcds reachability is shown to be in pspace even for n dimensional rhpcds we show that for an unbounded 3 rhpcd the reachability and mortality problems become undecidable for a nondeterministic variant of 2 rhpcds the reachability problem is shown to be pspace complete

discrete time control for rectangular hybrid automata rectangular hybrid automata model digital control programs of analog plant environments we study rectangular hybrid automata where the plant state evolves continuously in real numbered time and the controller samples the plant state and changes the control state discretely only at the integer points in time we prove that rectangular hybrid automata have finite bisimilarity quotients when all control transitions happen at integer times even if the constraints on the derivatives of the variables vary between control states this is in contrast with the conventional model where control transitions may happen at any real time and already the reachability problem is undecidable based on the finite bisimilarity quotients we give an exponential algorithm for the symbolic sampling controller synthesis of rectangular automata we show our algorithm to be optimal by proving the problem to be exptime hard we also show that rectangular automata form a maximal class of systems for which the sampling controller synthesis problem can be solved algorithmically

a framework for worst case and stochastic safety verification using barrier certificates this paper presents a methodology for safety verification of continuous and hybrid systems in the worst case and stochastic settings in the worst case setting a function of state termed barrier certificate is used to certify that all trajectories of the system starting from a given initial set do not enter an unsafe region no explicit computation of reachable sets is required in the construction of barrier certificates which makes it possible to handle nonlinearity uncertainty and constraints directly within this framework in the stochastic setting our method computes an upper bound on the probability that a trajectory of the system reaches the unsafe set a bound whose validity is proven by the existence of a barrier certificate for polynomial systems barrier certificates can be constructed using convex optimization and hence the method is computationally tractable some examples are provided to illustrate the use of the method

algorithmic analysis of polygonal hybrid systems part i reachability in this work we are concerned with the formal verification of two dimensional non deterministic hybrid systems namely polygonal differential inclusion systems spdis spdis are a class of non deterministic systems that correspond to piecewise constant differential inclusions on the plane for which we study the reachability problem our contribution is the development of an algorithm for solving exactly the reachability problem of spdis we extend the geometric approach due to maler and pnueli o maler a pnueli reachability analysis of planar multi linear systems in c courcoubetis ed cav 93 in lncs vol 697 springer verlag 1993 pp 194 209 to non deterministic systems based on the combination of three techniques the representation of the two dimensional continuous time dynamics as a one dimensional discrete time system using poincare maps the characterization of the set of qualitative behaviors of the latter as a finite set of types of signatures and acceleration used to explore reachability according to each of these types

lax representations and lax operator algebras of isospectral and nonisospectral hierarchies of evolution equations a method of constructing lax representations for isospectral and nonisospectral hierarchies of evolution equations is proposed it is shown that under some simple but essential conditions the corresponding lax operators may constitute infinite dimensional lie algebras with respect to the binary operation given in this paper furthermore the detailed analyses for the kdv couple the akns couple and a new couple of isospectral and nonisospectral hierarchies of integrable equations are presented as examples

lie algebraic structure for the akns system a new approach is proposed to construct the master symmetry and to prove its lie algebraic structure for the akns system

algebraic structure of discrete zero curvature equations and master symmetries of discrete evolution equations an algebraic structure related to discrete zero curvature equations is established it is used to give an approach for generating master symmetries of the first degree for systems of discrete evolution equations and an answer to why there exist such master symmetries the key of the theory is to generate nonisospectral flows t l l 0 from the discrete spectral problem associated with a given system of discrete evolution equations three examples are given

a decidable class of planar linear hybrid systems abstract in this paper we show the decidability of a new subclass of linear hybrid automata these automata are planar that is consist of two state variables monotonic along some direction in the plane and have identity resets that is the values of the continuous variables are not reset to a different value on a discrete transition our proof uses a combination of a tree construction capturing the edge to edge reachability and a finite bisimulation construction our class strictly contains the class of two dimensional piecewise constant derivative systems for which decidability of the reachability problem is known

the cg coefficient for the non special gel fand basis of the group su m n for the five particle system using the outer product reduction coefficient of the ordinary permutation group the authors have calculated the clebsch gordan coefficient for the non special gel fand basis of the graded unitary group su m n for the five particle system

algorithmic analysis of polygonal hybrid systems part ii phase portrait and tools polygonal differential inclusion systems spdi are a subclass of planar hybrid automata which can be represented by piecewise constant differential inclusions the reachability problem as well as the computation of certain objects of the phase portrait is decidable in this paper we show how to compute the viability controllability and invariance kernels as well as semi separatrix curves for spdis we also present the tool speedi which implements a reachability algorithm and computes phase portraits of spdis

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

new set of symmetries and lie algebraic structures of the toda lattice hierarchy by introducing the new time dependence of the spectral parameter we construct two sets of symmetries which are different from the centerless kac moody virasoro algebras for the isospectral toda lattice hierarchy

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

sign changing solutions of kirchhoff type problems via invariant sets of descent flow we obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow

complementarity modeling of hybrid systems a complementarity framework is described for the modeling of certain classes of mixed continuous discrete dynamical systems the use of such a framework is well known for mechanical systems with inequality constraints but we give a more general formulation which also applies for instance to switching control systems the main theoretical results in the paper are concerned with uniqueness of smooth continuations the solution of this problem requires the construction of a map from the continuous state to the discrete state a crucial technical tool is the so called linear complementarity problem from mathematical programming and we introduce various generalizations of this problem

positive solutions for the kirchhoff type problem involving general critical growth part i existence theorem involving general critical growth abstract in this paper we consider the following kirchhoff type problem a r 3 u 2 d x b r 3 u 2 d x u b u f u in r 3 u h 1 r 3 u 0 in r 3 where 0 is a parameter a b are positive constants and f reaches the critical growth without the ambrosetti rabinowitz condition we prove the existence of positive solutions for the kirchhoff type problem with a general critical nonlinearity we also study the asymptotics of solutions as 0 numerical solutions for related problems will be discussed in the second part

synthesis and infeasibility analysis for stochastic models of biochemical systems using statistical model checking and abstraction refinement the stochastic dynamics of biochemical reaction networks can be modeled using a number of succinct formalisms all of whose semantics are expressed as continuous time markov chains ctmc while some kinetic parameters for such models can be measured experimentally most are estimated by either fitting to experimental data or by performing ad hoc and often manual search procedures we consider an alternative strategy to the problem and introduce algorithms for automatically synthesizing the set of all kinetic parameters such that the model satisfies a given high level behavioral specification our algorithms which integrate statistical model checking and abstraction refinement can also report the infeasibility of the model if no such combination of parameters exists behavioral specifications can be given in any finitely monitorable logic for stochastic systems including the probabilistic and bounded fragments of linear and metric temporal logics the correctness of our algorithms is established using a novel combination of arguments based on survey sampling and uniform continuity we prove that the probability of a measurable set of paths is uniformly and jointly continuous with respect to the kinetic parameters under a suitable technical condition we also show that the unbiased statistical estimator for the probability of a measurable set of paths is monotonic in the parameter space we apply our algorithms to two benchmark models of biochemical signaling and demonstrate that they can efficiently find parameter regimes satisfying a given high level behavioral specification in particular we show that our algorithms can synthesize up to 6 parameters simultaneously which is more than that reported by any other synthesis algorithm for stochastic systems moreover when parameter estimation is desired as opposed to synthesis we show that our approach can scale to even higher dimensional spaces by identifying the single parameter combination that maximizes the probability of the behavior being true in an 11 dimensional system

recursion operators and bi hamiltonian structures in multidimensions ii the algebraic properties of exactly solvable evolution equations in one spatial and one temporal dimensions have been well studied in particular the factorization of certain operators called recursion operators establishes the bi hamiltonian nature of all these equations recently we have presented the recursion operator and the bi hamiltonian formulation of the kadomtsev petviashvili equation a two spatial dimensional analogue of the korteweg devries equation here we present the general theory associated with recursion operators for bi hamiltonian equations in two spatial and one temporal dimensions as an application we show that general classes of equations which include the kadomtsev petviashvili and the davey stewartson equations possess infinitely many commuting symmetries and infinitely many constants of motion in involution under two distinct poisson brackets furthermore we show that the relevant recursion operators naturally follow from the underlying isospectral eigenvalue problems

impulse differential inclusions a viability approach to hybrid systems impulse differential inclusions are introduced as a framework for modeling hybrid phenomena connections to standard problems in the area of hybrid systems are discussed conditions are derived that allow one to determine whether a set of states is viable or invariant under the action of an impulse differential inclusion for sets that violate these conditions methods are developed for approximating their viability and invariance kernels that is the largest subset that is viable or invariant under the action of the impulse differential inclusion the results are demonstrated on examples

k symmetries and tau symmetries of evolution equations and their lie algebras on the basis of tu s work 1951 the author establishes a more general skeleton on k symmetries and tau symmetries of evolution equations and their lie algebraic structures and discuss carefully the corresponding symmetries and lie algebras for kdv and jaulent miodek hierarchies

hamiltonian structure of discrete soliton systems we describe an approach for investigating the hamiltonian structures of the lattice isospectral evolution equations associated with a general discrete spectral problem by using the so called implicit representations of the isospectral flows we demonstrate the existence of the recursion operator l which is a strong and hereditary symmetry of the flows it is then proven that every equation in the isospectral hierarchy possesses the hamiltonian structure if l has a skew symmetric factorization and the first equation ut k 0 in the hierarchy satisfies some simple condition we obtain related properties such as the implectic symplectic factorization of l the liouville complete integrability and the multi hamiltonian structures of the isospectral hierarchy four examples are given

robustness of temporal logic specifications for continuous time signals in this paper we consider the robust interpretation of metric temporal logic mtl formulas over signals that take values in metric spaces for such signals which are generated by systems whose states are equipped with non trivial metrics for example continuous or hybrid robustness is not only natural but also a critical measure of system performance thus we propose multi valued semantics for mtl formulas which capture not only the usual boolean satisfiability of the formula but also topological information regarding the distance e from unsatisfiability we prove that any other signal that remains e close to the initial one also satisfies the same mtl specification under the usual boolean semantics finally our framework is applied to the problem of testing formulas of two fragments of mtl namely metric interval temporal logic mitl and closed metric temporal logic clmtl over continuous time signals using only discrete time analysis the motivating idea behind our approach is that if the continuous time signal fulfills certain conditions and the discrete time signal robustly satisfies the temporal logic specification then the corresponding continuous time signal should also satisfy the same temporal logic specification

low dimensional hybrid systems decidable undecidable don t know even though many attempts have been made to define the boundary between decidable and undecidable hybrid systems the affair is far from being resolved more and more low dimensional systems are being shown to be undecidable with respect to reachability and many open problems in between are being discovered in this paper we present various two dimensional hybrid systems for which the reachability problem is undecidable we show their undecidability by simulating minsky machines their proximity to the decidability frontier is understood by inspecting the most parsimonious constraints necessary to make reachability over these automata decidable we also show that for other two dimensional systems the reachability question remains unanswered by proving that it is as hard as the reachability problem for piecewise affine maps on the real line which is a well known open problem

inclusion dynamics hybrid automata hybrid systems are dynamical systems with the ability to describe mixed discrete continuous evolution of a wide range of systems consequently at first glance hybrid systems appear powerful but recalcitrant neither yielding to analysis and reasoning through a purely continuous time modeling as with systems of differential equations nor open to inferential processes commonly used for discrete state transition systems such as finite state automata a convenient and popular model called hybrid automata was introduced to model them and has spurred much interest on its tractability as a tool for inference and model checking in a general setting intuitively a hybrid automaton is simply a finite state automaton with each state augmented by continuous variables which evolve according to a set of well defined continuous laws each specified separately for each state this article investigates both the notion of hybrid automaton and the model checking problem over such a structure in particular it relates first order theories and analysis results on multivalued maps and reduces the bounded reachability problem for hybrid automata whose continuous laws are expressed by inclusions x f x t to a decidability problem for first order formulaeover the reals furthermore the paper introduces a class of hybrid automata for which the reachability problem can be decided and shows that the problem of deciding whether a hybrid automaton belongs to this class can be again decided using first order formulaeover the reals despite the fact that the bisimulation quotient for this class of hybrid automata can be infinite we show that our techniques permit effective model checking for a nontrivial fragment of ctl