on a common quadratic lyapunov function for widely distant systems a sufficient condition concerning the construction of a common quadratic lyapunov function for widely distant linear systems is presented the condition is set on a mutual relation matrix whose size is just equal to the number of the coefficient matrices

probabilistic analysis of two heuristics for the 3 satisfiability problem an algorithm for the 3 satisfiability problem is presented and a probabilistic analysis is performed the analysis is based on an instance distribution which is parameterized to simulate a variety of sample characteristics the algorithm assigns values to variables appearing in a given instance of 3 satisfiability one at a time using the unit clause heuristic and a maximum occurring literal selection heuristic at each step a variable is chosen randomly from a subset of variables which is usually large the algorithm runs in polynomial time and it is shown that the algorithm finds a solution to a random instance of 3 satisfiability with probability bounded from below by a constant greater than zero for a range of parameter values the heuristics studied here can be used to select variables in a backtrack algorithm for 3 satisfiability experiments have shown that for about the same range of parameters as above the backtrack algorithm using the heuristics finds a solution in polynomial average time

probabilistic analysis of a generalization of the unit clause literal selection heuristics for the k satisfiability problem abstract two algorithms for the k satisfiability problem are presented and a probabilistic analysis is performed the analysis is based on an instance distribution which is parametrized to simulate a variety of sample characteristics the algorithms assign values to literals appearing in a given instance of k satisfiability one at a time until a solution is found or it is discovered that further assignments cannot lead to finding a solution one algorithm chooses the next literal from a unit clause if one exists and randomly from the set of remaining literals otherwise the other algorithm uses a generalization of the unit clause rule as a heuristic for selecting the next literal at each step a literal is chosen randomly from a clause containing the least number of literals the algorithms run in polynomial time and it is shown that they find a solution to a random instance of k satisfiability with probability bounded below by a constant greater than zero for two different ranges of parameter values it is also shown that the second algorithm mentioned finds a solution with probability approaching one for a wide range of parameter values

hukuhara differentiability of interval valued functions and interval differential equations on time scales abstract using the concept of the generalized hukuhara difference in this paper we introduce and study the differentiability and the integrability for the interval valued functions on time scales some illustrative examples to interval differential equations on time scales are presented

geometry of the limit sets of linear switched systems this paper is concerned with asymptotic stability properties of linear switched systems under the hypothesis that all the subsystems share a nonstrict quadratic lyapunov function we provide a large class of switching signals for which a large class of switched systems are asymptotically stable for this purpose we define what we call nonchaotic inputs which generalize the different notions of inputs with dwell time next we turn our attention to the behavior for possibly chaotic inputs finally we give a sufficient condition for a system composed of a pair of hurwitz matrices to be asymptotically stable for all inputs

stability of discrete linear inclusion let m ai be a set of linear operators on rn the discrete linear inclusion dli m is the set of possible trajectories xi i 0 such that xn ainain 1 ai1x0 where x0 rn and aij m we study several notions of stability for dli m including absolute asymptotic stability aas which is that all products ain ai1 0 as n we mainly study the case that m is a finite set we give criteria for the various forms of stability two new approaches are taken one relates the question of aas of dli m to formal language theory and finite automata while the second connects the aas property to the structure of a lie algebra associated to the elements of m more generally the discrete linear inclusion dli m makes sense for m contained in a banach algebra b we prove some results for aas in this case and give counterexamples showing that some results valid for finite sets of operators on rn are not true for finite sets m in a general banach algebras b

on nirenberg type conditions for higher order systems on time scales we study the existence of periodic solutions for a nonlinear system of nth order differential equations on time scales assuming a suitable nirenberg type condition we prove the existence of at least one solution of the problem using mawhin s coincidence degree

homoclinic orbits and periodic solutions for a class of hamiltonian systems on time scales abstract in this paper we are concerned with a second order non autonomous hamiltonian system on time scales t u t v u t u t f t t t under certain conditions the existence and multiplicity of periodic solutions are obtained for this hamiltonian system on time scales by using the saddle point theory the least action principle as well as the three critical point theorem in addition the existence of homoclinic orbit is obtained as a limit of 2 k t periodic solutions of a given sequence of hamiltonian system on time scales by means of the mountain pass theorem and the standard minimizing argument

a dynamic matrix exponential via a matrix cylinder transformation abstract in this work we develop a new matrix exponential on time scales via a cylinder transformation with a component wise locally integrable square matrix subscript our resulting matrix function can be written in terms of the matrix exponential of a lebesgue integral added to a logarithmic sum in terms of the gaps of a general time scale under strict commutativity conditions we show our dynamic matrix exponential is equivalent to the one in the standard literature finally we demonstrate that our matrix exponential satisfies a nonlinear dynamic integral equation

integrable equations on time scales integrable systems are usually given in terms of functions of continuous variables on r in terms of functions of discrete variables on z and recently in terms of functions of q variables on kq we formulate the gel fand dikii gd formalism on time scales by using the delta differentiation operator and find more general integrable nonlinear evolutionary equations in particular they yield integrable equations over integers difference equations and over q numbers q difference equations we formulate the gd formalism also in terms of shift operators for all regular discrete time scales we give a method allowing to construct the recursion operators for integrable systems on time scales finally we give a trace formula on time scales and then construct infinitely many conserved quantities casimirs of the integrable systems on time scales

many hard examples for resolution for every choice of positive integers c and k such that k 3 and c 2 k 0 7 there is a positive number e such that with probability tending to 1 as n tends to a randomly chosen family of cn clauses of size k over n variables is unsatisfiable but every resolution proof of its unsatisfiability must generate at least 1 e n clauses

exponential stabilization of switched time varying systems with delays and disturbances abstract this paper deals with exponential stabilization for a class of switched time varying systems by taking time varying delays and nonlinear disturbances into consideration time dependent switching signals have been characterized in terms of metzler matrices such that the resulting system is globally exponentially stable compared with preceding works we introduce a model transformation and an approach without involving the lyapunov krasovskii functional to derive new exponential stability criteria for switched time varying systems under the average dwell time switching numerical examples show that the obtained theoretical results can be applied to some cases not covered by some existing results

a sufficient condition for the stability of discrete time systems with state dependent coefficient matrices there is no systematic method of checking the stability of general nonlinear systems in this study we focus on discrete time autonomous systems with state dependent coefficient matrices sufficient conditions for local and global asymptotic stabilities are given for upper triangularizable systems the transformation matrix for triangularization is not required explicitly for checking such conditions moreover the system may have nondifferentiability at the origin which is an advantage of our approach over lyapunov s indirect method

invertibility of switched nonlinear systems this article addresses the invertibility problem for switched nonlinear systems affine in controls the problem is concerned with reconstructing the input and switching signal uniquely from given output and initial state we extend the concept of switch singular pairs introduced recently to nonlinear systems and develop a formula for checking if the given state and output form a switch singular pair a necessary and sufficient condition for the invertibility of switched nonlinear systems is given which requires the invertibility of individual subsystems and the nonexistence of switch singular pairs when all the subsystems are invertible we present an algorithm for finding switching signals and inputs that generate a given output in a finite interval when there is a finite number of such switching signals and inputs detailed examples are included to illustrate these newly developed concepts

on robust lie algebraic stability conditions for switched linear systems abstract this paper presents new sufficient conditions for exponential stability of switched linear systems under arbitrary switching which involve the commutators lie brackets among the given matrices generating the switched system the main novel feature of these stability criteria is that unlike their earlier counterparts they are robust with respect to small perturbations of the system parameters two distinct approaches are investigated for discrete time switched linear systems we formulate a stability condition in terms of an explicit upper bound on the norms of the lie brackets for continuous time switched linear systems we develop two stability criteria which capture proximity of the associated matrix lie algebra to a solvable or a solvable plus compact lie algebra respectively

on the 45 region and the uniform asymptotic stability of classes of second order parameter varying and switched systems in this paper we present sufficient conditions for the stability of a number of classes of time varying systems these results are of interest for two reasons firstly the conditions presented take the form of matrix pencil eigenvalue criteria and are therefore co ordinate independent and easily verifiable secondly we show a direct relationship between the form of our criteria and the choice of lyapunov function used to demonstrate stability quadratic or unic these results indicate the importance of the region of the complex plane known as the 45 region

on equilibria stability and instability of hopfield neural networks existence and uniqueness of equilibrium as well as its stability and instability of a continuous time hopfield neural network are studied a set of new and simple sufficient conditions are derived

uniform exponential stability of first order dynamic equations with several delays abstract this paper is concerned with the uniform exponential stability of ordinary and delay dynamic equations after revealing the equivalence between various types of uniform exponential stability definitions on time scales with bounded graininess and demonstrating their relation when the graininess is arbitrary we confine our attention to the uniform exponential stability of ordinary dynamic equations we introduce and prove the bohl perron criterion for delay dynamic equations if for any bounded right hand side the solution of the delay dynamic equation with bounded coefficients and delays is bounded then the trivial solution of the equation is uniformly exponentially stable we also obtain some corollaries of this criterion based on these results explicit exponential stability tests are derived for delay dynamic equations with nonnegative coefficients which are illustrated with an example on a nonstandard time scale

stability analysis of switched systems using variational principles an introduction many natural and artificial systems and processes encompass several modes of operation with a different dynamical behavior in each mode switched systems provide a suitable mathematical model for such processes and their stability analysis is important for both theoretical and practical reasons we review a specific approach for stability analysis based on using variational principles to characterize the most unstable solution of the switched system we also discuss a link between the variational approach and the stability analysis of switched systems using lie algebraic considerations both approaches require the use of sophisticated tools from many different fields of applied mathematics the purpose of this paper is to provide an accessible and self contained review of these topics emphasizing the intuitive and geometric underlying ideas

higher order self adjoint boundary value problems on time scales in this study higher order self adjoint differential expressions on time scales and their associated self adjoint boundary conditions are discussed the symmetry property of the corresponding green s functions is shown together with specific formulas of green s functions for select time scales

hamiltonian systems on time scales linear and nonlinear hamiltonian systems are studied on time scales t we unify symplectic flow properties of discrete and continuous hamiltonian systems a chain rule which unifies discrete and continuous settings is presented for our so called alpha derivatives on generalized time scales this chain rule allows transformation of linear hamiltonian systems on time scales under simultaneous change of independent and dependent variables thus extending the change of dependent variables recently obtained by dosl and hilscher we also give the legendre transformation for nonlinear euler lagrange equations on time scales to hamiltonian systems on time scales

dynamical realizations of finite volterra series in this paper realizations of finite volterra series are viewed as nonlinear analytic input output systems with state space described by an analytic manifold for a minimal realization guaranteed by h j sussmann the state space which is unique up to diffeomorphism is shown to have the homogeneous space structure of a nilmanifold the quotient of two nilpotent lie groups the structure of nilmanifolds as described by a malcev is used to show that for these systems the state space has a vector space structure as a consequence of this result it is shown that a minimal realization of a finite volterra series can be described as a cascade of linear subsystems with polynomial link maps in which the dimension o f each linear subsystem is independent of the realization considered

the laplace transform on time scales revisited abstract in this work we reexamine the time scale laplace transform as defined by bohner and peterson m bohner a peterson dynamic equations on time scales an introduction with applications birkhauser boston 2001 m bohner a peterson laplace transform and z transform unification and extension methods appl anal 9 1 2002 155 162 in particular we give conditions on the class of functions which have a transform develop an inversion formula for the transform and further we provide a convolution for the transform the notion of convolution leads to considering its algebraic structure in particular the existence of an identity element motivating the development of the dirac delta functional on time scales applications and examples of these concepts are given

experimental results on the crossover point in random 3 sat abstract determining whether a propositional theory is satisfiable is a prototypical example of an np complete problem further a large number of problems that occur in knowledge representation learning planning and other areas of ai are essentially satisfiability problems this paper reports on the most extensive set of experiments to date on the location and nature of the crossover point in satisfiability problems these experiments generally confirm previous results with two notable exceptions first we have found that neither of the functions previously proposed accurately models the location of the crossover point second we have found no evidence of any hard problems in the under constrained region in fact the hardest problems found in the under constrained region were many times easier than the easiest unsatisfiable problems found in the neighborhood of the crossover point we offer explanations for these apparent contradictions of previous results

singular value conditions for stability of dynamic switched systems abstract in this paper the stability of a certain class of time varying systems evolving over nonuniformly spaced discrete domains is analyzed switched systems used here in the context of dynamic equations over time scale domains arise naturally in applications when a continuous time system is discretized via a sample and hold method with multiple sample rates the stability of switched systems is typically deduced by appealing to certain interrelated properties of the subsystems such as pairwise commutativity 24 simultaneous diagonalization 7 simultaneous triangularizability 8 or other lie algebraic conditions 1 which imply the existence of a common quadratic lyapunov function a novel approach is used here to determine the existence of quadratic lyapunov functions which does not rely on how the subsystems interact with each other this new method instead examines the role they each play in the aggregate system by way of singular value conditions results implying switched system stability and instability are developed under the two primary methods of examining such systems how the system behaves during arbitrary switching and how it behaves under the influence of a particular switching signal several examples illuminating the results are provided throughout the text as well as important insight discussing certain bounds on these results as they apply to stability theory in general

uniqueness and existence results for initial value problems on time scales through a reciprocal problem and applications this paper is devoted to deriving the existence uniqueness and approximation of solutions to a first order initial value problem on an interval of an arbitrary time scale taking values in a different time scale we obtain the results considering dynamic equations whose nonlinear part is nonnegative and satisfies a type of inverse caratheodory s conditions and discontinuity several examples are given to illustrate the applicability of these results

sobolev s spaces on time scales and its applications to a class of second order hamiltonian systems on time scales abstract in this paper we study the sobolev s spaces on time scales and their properties as applications we present a recent approach via variational methods and the critical point theory to obtain the existence of solutions for a class of second order hamiltonian systems on time scales by establishing a proper variational setting three existence results for systems under consideration are obtained finally three examples are presented to illustrate the feasibility and effectiveness of the existence results

common polynomial lyapunov functions for linear switched systems in this paper we consider linear switched systems dot x t a u t x t x in r n u in u a u u in u compact and the problem of asymptotic stability for arbitrary switching functions uniform with respect to switching uas given a uas system it is always possible to build a common polynomial lyapunov function our main result is that the degree of that common polynomial lyapunov function is not uniformly bounded over all the uas systems this result answers a question raised by dayawansa and martin a generalization to a class of piecewise polynomial lyapunov functions is given

line integrals and green s formula on time scales in this paper we study curves parametrized by a time scale parameter introduce line delta and nabla integrals along time scale curves and obtain an analog of green s formula in the time scale setting

factorization method on time scales abstract we present an approach to the factorization method for second order difference equations on time scales we construct hilbert spaces of functions on the time scale and show how to construct a chain of intertwined first order difference operators a k in the sense that a k and a k act as ladder operators generating new solutions of eigenproblem of a k a k

stability and stabilization of a class of switched nonlinear systems with time varying delay abstract this paper investigates the problems of exponential stability and robust stabilization for a class of switched nonlinear systems with time varying delay using the average dwell time method a new exponential stability criterion is derived in terms of linear matrix inequalities lmis it is shown that the obtained criteria can provide less conservative results than some existing ones then we propose a state feedback control law which ensures the exponential stability of the closed loop system finally numerical examples are given to illustrate the effectiveness of the proposed results

results related to threshold phenomena research in satisfiability lower bounds we present a history of results related to the threhold phenomena which arise in the study of random conjunctive normal form cnf formulas in a companion paper d achlioptas theoret comput sci this volume the major ideas used to achieve many of the lower bounds results on the location of the threshold are described in an informal intuitive manner

upper bounds on the satisfiability threshold we present a survey of upper bounds which has been established up to now on the satisfiability threshold of random k sat formulae the ideas which led to these bounds and the techniques used to obtain them are explained a companion paper in this volume present a survey of the lower bounds

solving convex programming problems with equality constraints by neural networks abstract this paper presents a neural network approach for solving convex programming problems with equality constraints after defining the energy function and neural dynamics of the proposed neural network we show the existence of an equilibrium point at which the neural dynamics becomes asymptotically stable it is shown that under proper conditions an optimal solution of the underlying convex programming problems is an equilibrium point of the neural dynamics and vise versa the configuration of the proposed neural network with an exact layout is provided for solving linear programming problems the operational characteristics of the neural network are demonstrated by numerical examples

on stabilisability of nonlinear systems on time scales in this article stabilisability of nonlinear finite dimensional control systems on arbitrary time scales is studied the classical results on stabilisation of nonlinear continuous time and discrete time systems are extended to systems on arbitrary time scales with bounded graininess function it is shown that uniform exponential stability of the linear approximation of a nonlinear system implies uniform exponential stability of the nonlinear system then this result is used to show a similar implication for uniform exponential stabilisability

foundations of nabla fractional calculus on time scales and inequalities here we develop the nabla fractional calculus on time scales then we produce related integral inequalities of types poincare sobolev opial ostrowski and hilbert pachpatte finally we give inequality applications on the time scales r z

mimetic methods on measure chains we introduce the divergence and the gradient for functions defined on a measure chain and this includes as special cases both continuous derivatives and discrete forward differences it is shown that in one dimension subject to dirichlet boundary conditions the divergence and the gradient are negative adjoints of each other and that the divergence of the gradient is negative semidefinite these are well known results in tile continuous theory and hence mimic those properties also for the case of a general measure chain 2001 elsevier science ltd all rights reserved

stability of switched systems a lie algebraic condition we present a sucient condition for asymptotic stability of a switched linear system in terms of the lie algebra generated by the individual matrices namely if this lie algebra is solvable then the switched system is exponentially stable for arbitrary switching in fact we show that any family of linear systems satisfying this condition possesses a quadratic common lyapunov function we also discuss the implications of this result for switched nonlinear systems c 1999 elsevier science b v all rights reserved

configurations of steady states for hopfield type neural networks the dependence of the steady states on the external input vector i for the continuous time and discrete time hopfield type neural networks of n neurons is discussed conditions for the existence of one or several paths of steady states are derived it is shown that in some conditions for an external input i there may exist at least 2 n exponentially stable steady states called configuration of steady states and their regions of attraction are estimated this means that there exist 2 n paths of exponentially stable steady states defined on a certain set of input values conditions assuring the transfer of a configuration of exponentially stable steady states to another configuration of exponentially stable steady states by successive changes of the external input are obtained these results may be important for the design and maneuvering of hopfield type neural networks used to analyze associative memories

a non autonomous hamiltonian system on time scales abstract in this paper we are concerned with a second order non autonomous hamiltonian system on time scales t of the form u 2 t f t u t a e t 0 t t u 0 u t 0 u 0 u t 0 some results on the existence and multiplicity of periodic solutions are obtained for the hamiltonian system by means of the saddle point theorem the least action principle as well as the three critical point theorem

on the stabilizability of discrete time switched linear systems novel conditions and comparisons in this paper we deal with the stabilizability property for discrete time switched linear systems a recent necessary and sufficient characterization of stabilizability based on set theory is considered as the reference for comparing the computation oriented sufficient conditions the classical bmi conditions based on lyapunov metzler inequalities are considered and extended novel lmi conditions for stabilizability derived from the geometric ones are presented that permit to combine generality with computational affordability for the different conditions the geometrical interpretations are provided and the induced stabilizing switching laws are given the relations and the implications between the stabilizability conditions are analyzed to infer and compare their conservatism and their complexity

l2 spaces and boundary value problems on time scales abstract in this paper we consider a second order sturm liouville type boundary value operator of the form l u p u q u on an arbitrary bounded time scale t for suitable functions p q together with suitable boundary conditions operators of this type on time scales have normally been considered in a setting involving banach spaces of continuous functions on t in this paper we introduce a space l 2 t of square integrable functions on t and sobolev type spaces h n t n 1 consisting of l 2 t functions with nth order generalised l 2 t type derivatives we prove some basic functional analytic results for these spaces and then formulate the operator l in this setting in particular we allow p h 1 t while q l 2 t this generalises the usual conditions that p c rd 1 t q c rd 0 t 2 we give some immediate applications of the functional analytic results to l such as positivity injectivity invertibility and compactness of the inverse we also construct a green s function for l the analogues of these results on real intervals are well known and are fundamental to the usual sturm liouville theory on such intervals

periodic solutions for a two neuron network with delays in this paper we use the coincidence degree theory to establish new results on the existence of w periodic solutions for a two neuron network model

length of prime implicants and number of solutions of random cnf formulae consider a uniform distribution of r cnf formulae in conjunctive normal form with cn clauses each with r distinct literals over a set of n variables a prime implicant j of a formula is a consistent conjunction of literals which implies but ceases to imply when deprived of any one literal the normalized length of j is the ratio of the number of its literals to the number of variables occurring in we show that for any e 0 and for some range of values of c depending on r almost every r cnf formula 1 either is satisfiable and any one of its prime implicants has a normalized length at least equal to mr c 1 e rc e and at most equal to mr c 1 e rc e mr c and mr c being well defined as functions of c 2 or is unsatisfiable a first practical consequence is when testing the satisfiability of r cnf formulae by procedures such as the well known davis putnam and loveland procedure for almost every r cnf formula when it is satisfiable the proportion of variables which must be assigned a value by such procedures in order to find a solution is at least equal to mr c 1 e rc e a second consequence is that almost every r cnf formula when it is satisfiable has an exponential actual number of solutions i e the number of solutions defined on the variables occurring in the formula at least equal to 2 1 e rc mr c e n moreover for r 2 3 we show th for any c it is at least equal to 20 03n 20 012n respectively

on exponential almost sure stability of random jump systems this paper is concerned with a class of random jump systems represented by transition operators which includes switched linear systems with strictly stationary switching signals in infinite modes space as its special case a series of necessary and sufficient conditions are established for almost sure stability of this class of random jump systems under different scenarios the stability criteria obtained are further extended to markov jump linear systems with infinite states and hence a unified approach to describing the almost sure stability of mjlss is addressed under this context all the results in the work are developed for both the continuous and discrete time systems

sharp thresholds of graph properties and the sat problem consider g n p to be the probability space of random graphs on n vertices with edge probability p we will be considering subsets of this space defined by monotone graph properties a monotone graph property p is a property of graphs such that a p is invariant under graph automorphisims b if graph h has property p then so does any graph g having h as a subgraph a monotone symmetric family of graphs is a family defined by such a property one of the first observations made about random graphs by erdos and renyi in their seminal work on random graph theory 12 was the existence of threshold phenomena the fact that for many interesting properties p the probability of p appearing in g n p exhibits a sharp increase at a certain critical value of the parameter p bollobas and thomason proved the existence of threshold functions for all monotone set properties 6 and in 14 it is shown that this behavior is quite general and that all monotone graph properties exhibit threshold behavior i e the probability of their appearance increases from values very close to 0 to values close to 1 in a very small interval more precise analysis of the size of the threshold interval is done in 7 this threshold behavior which occurs in various settings which arise in combinatorics and computer science is an instance of the phenomenon of phase transitions which is the subject of much interest in statistical physics one of the main questions that arises in studying phase transitions is how sharp is the transition for example one of the motivations for this paper arose from the question of the sharpness of the phase transition for the property of satisfiability of a random kcnf boolean formula nati linial who introduced me to this problem suggested that although much concrete analysis was being performed on this problem the best approach would be to find general conditions for sharpness of the phase transition answering the question posed in 14 as to the relation between the length of the threshold interval and the value of the critical probability in this paper we indeed introduce a simple condition and prove it is sufficient stated roughly in the setting of random graphs the main theorem states that if a property has a coarse threshold then it can be approximated by the property of having certain given graphs as a subgraph this condition can be applied in a more

lower bounds for random 3 sat via differential equations it is widely believed that the probability of satisfiability for random k sat formulae exhibits a sharp threshold as a function of their clauses to variables ratio for the most studied case k 3 there have been a number of results during the last decade providing upper and lower bounds for the threshold s potential location all lower bounds in this vein have been algorithmic i e in each case a particular algorithm was shown to satisfy random instances of 3 sat with probability 1 o 1 if the clauses to variables ratio is below a certain value we show how differential equations can serve as a generic tool for analyzing such algorithms by rederiving most of the known lower bounds for random 3 sat in a simple uniform manner

representations and ostrowski type inequalities on time scales here we give univariate and multivariate representations of the montgomery type for hybrid functions on time scales based on these we establish univariate and multivariate ostrowski type inequalities on time scales domains these compare the average of a function to its values the estimates involve the higher order delta and nabla derivatives and partial derivatives we finish with applications on the time scales r and z

on green s functions and positive solutions for boundary value problems on time scales in this paper we offer a form of self adjoint differential equations on time scales so that the associated green s function is found symmetric in the usual sense for this purpose together with the delta derivative we employ the nabla derivative as well we introduce the concepts of lebesgue delta and nabla integrals on time scales next sign properties of the green s function are investigated and existence results for positive solutions of nonlinear boundary value problems are established upper and lower bounds for these positive solutions also are given

nonlinear integro differential equations and optimal control problems on time scales this paper is mainly concerned with a class of optimal control problems of a system governed by the nonlinear integro differential equation on time scales using the contraction mapping principle the existence and uniqueness of a weak solution of the nonlinear integro differential equation on time scales are proved discussing l t 1 strong weak lower semicontinuity of integral functional on time scales we give sufficient conditions for the existence of optimal control by the virtue of the integration by parts formula on time scales the necessary conditions of optimality are derived

integration on time scales in this paper we study the process of riemann and lebesgue integration on time scales the relationship of the riemann and lebesgue integrals is considered and a criterion for riemann integrability is established

criteria of asymptotic stability of differential and difference inclusions encountered in control theory abstract for a class of differential inclusions to which many of the practically important control systems can be reduced necessary and sufficient conditions for asymptotic stability of the zero solution are established by the method of lyapunov functions similar results are obtained for the corresponding class of difference inclusions

necessary conditions of optimality for a class of optimal control problems on time scales a class of optimal control problems of a system governed by linear dynamic equations on time scales with quadratic cost functional is considered by the lebesgue d integral theory and the sobolev type space h 1 on time scales the weak solution of linear dynamic equations on time scales for both initial value problem and backward problem are introduced therefore the necessary conditions of optimality are presented some typical examples are given for demonstration

brief paper componentwise ultimate bound and invariant set computation for switched linear systems we present a novel ultimate bound and invariant set computation method for continuous time switched linear systems with disturbances and arbitrary switching the proposed method relies on the existence of a transformation that takes all matrices of the switched linear system into a convenient form satisfying certain properties the method provides ultimate bounds and invariant sets in the form of polyhedral and or mixed ellipsoidal polyhedral sets is completely systematic once the aforementioned transformation is obtained and provides a new sufficient condition for practical stability we show that the transformation required by our method can easily be found in the well known case where the subsystem matrices generate a solvable lie algebra and we provide an algorithm to seek such transformation in the general case an example comparing the bounds obtained by the proposed method with those obtained from a common quadratic lyapunov function computed via linear matrix inequalities shows a clear advantage of the proposed method in some cases

lie algebraic stability criteria for switched systems it was recently shown that a family of exponentially stable linear systems whose matrices generate a solvable lie algebra possesses a quadratic common lyapunov function which implies that the corresponding switched linear system is exponentially stable for arbitrary switching in this paper we prove that the same properties hold under the weaker condition that the lie algebra generated by given matrices can be decomposed into a sum of a solvable ideal and a subalgebra with a compact lie group the corresponding local stability result for nonlinear switched systems is also established moreover we demonstrate that if a lie algebra fails to satisfy the above condition then it can be generated by a family of stable matrices such that the corresponding switched linear system is not stable relevant facts from the theory of lie algebras are collected at the end of the paper for easy reference

statistical mechanics of the random k satisfiability model the random k satisfiability problem consisting in verifying the existence of an assignment of n boolean variables that satisfy a set of m ensuremath alpha n random logical clauses containing k variables each is studied using the replica symmetric framework of diluted disordered systems we present an exact iterative scheme for the replica symmetric functional order parameter together for the different cases of interest k 2 kg 3 and k ensuremath gg 1 the calculation of the number of solutions which allowed us phys rev lett 76 3881 1996 to predict a first order jump at the threshold where the boolean expressions become unsatisfiable with probability one is thoroughly displayed in the case k 2 the rigorously known critical value ensuremath alpha 1 of the number of clauses per boolean variable is recovered while for kg 3 we show that the system exhibits a replica symmetry breaking transition the annealed approximation is proven to be exact for large k

on the freezing of variables in random constraint satisfaction problems the set of solutions of random constraint satisfaction problems zero energy groundstates of mean field diluted spin glasses undergoes several structural phase transitions as the amount of constraints is increased this set first breaks down into a large number of well separated clusters at the freezing transition which is in general distinct from the clustering one some variables spins take the same value in all solutions of a given cluster in this paper we introduce and study a message passing procedure that allows to compute for generic constraint satisfaction problems the sizes of the rearrangements induced in response to the modification of a variable these sizes diverge at the freezing transition with a critical behavior which is also investigated in details we apply the generic formalism in particular to the random satisfiability of boolean formulas and to the coloring of random graphs the computation is first performed in random tree ensembles for which we underline a connection with percolation models and with the reconstruction problem of information theory the validity of these results for the original random ensembles is then discussed in the framework of the cavity method

on a disconjugacy criterion for second order dynamic equations on time scales in this paper a sufficient condition for disconjugacy of second order differential equations on time scales is given for this purpose a mean value result on time scales is employed

a converse lyapunov theorem for a class of dynamical systems which undergo switching the authors investigate the stability of a system in which the dynamics at any instant in time will follow one of a fixed set of vector fields they allow switching between members of the family of vector fields to be completely random the focus of this paper is to prove a converse lyapunov theorem for this class of systems

weak solutions for the dynamic cauchy problem in banach spaces abstract this paper is devoted to unify and extend the results of the existence of the weak solutions of continuous and discrete cauchy problem in banach spaces we offer the existence of the weak solution of dynamic cauchy problem on an infinite time scale the measure of weak noncompactness and the fixed point theorem of kubiaczyk are used to prove the main result

stability and stabilizability of switched linear systems a survey of recent results during the past several years there have been increasing research activities in the field of stability analysis and switching stabilization for switched systems this paper aims to briefly survey recent results in this field first the stability analysis for switched systems is reviewed we focus on the stability analysis for switched linear systems under arbitrary switching and we highlight necessary and sufficient conditions for asymptotic stability after a brief review of the stability analysis under restricted switching and the multiple lyapunov function theory the switching stabilization problem is studied and a variety of switching stabilization methods found in the literature are outlined then the switching stabilizability problem is investigated that is under what condition it is possible to stabilize a switched system by properly designing switching control laws note that the switching stabilizability problem has been one of the most elusive problems in the switched systems literature a necessary and sufficient condition for asymptotic stabilizability of switched linear systems is described here

solvability of sturm liouville problems on time scales at resonance under the appropriate growth conditions on the inhomogeneous term we present the existence of the nontrivial solution for sturm liouville boundary value problems on time scales at resonance by making use of coincidence degree theory of mawhin the interesting point is the nonlinearity f which is involved with first order d derivative explicitly

on the satisfiability threshold of formulas with three literals per clause in this paper we present a new upper bound for randomly chosen 3 cnf formulas in particular we show that any random formula over n variables with a clauses to variables ratio of at least 4 4898 is as n grows large asymptotically almost surely unsatisfiable the previous best such bound due to dubois in 1999 was 4 506 the first such bound independently discovered by many groups of researchers since 1983 was 5 19 several decreasing values between 5 19 and 4 506 were published in the years between we believe that the probabilistic techniques we use for the proof are of independent interest

on the connection between the hilger and radon nikodym derivatives abstract we show that the hilger derivative on time scales is a special case of the radon nikodym derivative with respect to the natural measure associated with every time scale moreover we show that the concept of delta absolute continuity agrees with the one from measure theory in this context

existence and stability of equilibria of the continuous time hopfield neural network the qualitative properties of the continuous time hopfield neural network are investigated by using the cauchy existence theorem of ordinary differential equations a set of new and simple sufficient conditions guaranteeing the existence and uniqueness of equilibrium is derived furthermore global and asymptotical stability conditions are established by means of lyapunov method

stability and controllability of planar conewise linear systems abstract this paper presents a fairly complete treatment of stability and controllability of piecewise linear systems defined on a conic partition of r 2 this includes necessary and sufficient conditions for stability and controllability as well as establishing that controllability implies stabilizability by piecewise linear state feedback a key tool in the approach is the study of the poincare map

stability analysis of a class of switched linear systems on non uniform time domains abstract this paper deals with the stability analysis of a class of switched linear systems on non uniform time domains the considered class consists of a set of linear continuous time and linear discrete time subsystems first some conditions are derived to guarantee the exponential stability of this class of systems on time scales with bounded graininess function when the subsystems are exponentially stable these results are extended when considering an unstable discrete time subsystem or an unstable continuous time subsystem some examples illustrate these results

on stability bohl exponent and bohl perron theorem for implicit dynamic equations in this paper we develop a stability theory for implicit dynamic equations which is a general form of differential algebraic equations and implicit difference equations we derive some results abo

on existence of optimal control governed by a class of the first order linear dynamic systems on time scales in this paper by using the approximation of classical solution we introduce the definition of solution and prove the existence and uniqueness of solutions of the first order linear dynamic systems on time scales existence of lagrange optimal control problem governed by the first order linear dynamic systems on time scales is also presented for illustration some examples of optimal control problems on time scales are also discussed

stability by linear approximation for time scale dynamical systems abstract we study systems on time scales that are generalizations of classical differential or difference equations and appear in numerical methods in this paper we consider linear systems and their small nonlinear perturbations in terms of time scales and of eigenvalues of matrices we formulate conditions sufficient for stability by linear approximation for non periodic time scales we use techniques of central upper lyapunov exponents a common tool of the theory of linear odes to study stability of solutions also time scale versions of the famous chetaev s theorem on conditional instability are proved in a nutshell we have developed a completely new technique in order to demonstrate that methods of non autonomous linear ode theory may work for time scale dynamics

absolute continuity and existence of solutions to dynamic inclusions in time scales we provide some properties for absolutely continuous functions in time scales then we consider a class of dynamical inclusions in time scales and extend to this class a convergence result of a sequence of almost inclusion trajectories to a limit which is actually a trajectory of the inclusion in question we also introduce the so called euler solution to dynamical systems in time scales and prove its existence a combination of the existence of euler solutions with the compactness type result described above ensures the existence of an actual trajectory for the dynamical inclusion when the setvalued vector field is nonempty compact convex and has closed graph

multiple positive solutions of singular dirichlet problems on time scales via variational methods abstract this paper is devoted to using variational techniques and critical point theory to derive some sufficient conditions for the existence of multiple positive solutions to a singular second order dynamic equation with homogeneous dirichlet boundary conditions