completion of a set of rules modulo a set of equations abstract church rosser properties are first presented depending on an arbitrary relation r an equivalence relation e and a reduction relation r e used to compute normal forms of r modulo e terminating rewriting systems operating on equational congruence classes of terms of a free algebra are then considered in this framework the church rosser property is proved decidable for a very general reduction relation which may take into account the left linearity of rules for efficiency reasons under the only assumption of existence of a complete and finite unification algorithm for the underlying equational theory whose congruence classes are assumed to be finite this extends previous results by lankford and ballantyne peterson and stickel huet jouannaud a general completion procedure for mixed sets of rules and equations is then presented that generalizes and improves peterson and stickel s one in addition to computing a church rosser set of rules when it terminates it yields a semi decision pro

leanest quasi orderings a convenient method for defining a quasi ordering such as those used for proving termination of rewriting is to choose the minimum of a set of quasi orderings satisfying some desired traits unfortunately a minimum in terms of set inclusion can be non existent even when an intuitive minimum exists we suggest an alternative to set inclusion called leanness show that leanness is a partial order on quasi orderings and provide sufficient conditions for the existence of a leanest member of a set of total well founded quasi orderings

unification algorithms cannot be combined in polynomial time we establish that there is no polynomial time general combination algorithm for unification in finitary equational theories unless the complexity class p of counting problems is contained in the class fp of function problems solvable in polynomial time the prevalent view in complexity theory is that such a collapse is extremely unlikely for a number of reasons including the fact that the containment of p in fp implies that p np our main result is obtained by establishing the intractability of the counting problem for general ag unification where ag is the equational theory of abelian groups more specifically we show that computing the cardinality of a minimal complete set of unifiers for general ag unification is a p hard problem in contrast ag unification with constants is known to be solvable in polynomial time since an algorithm for general ag unification can be obtained as a combination of a polynomial time algorithm for ag unification with constants and a polynomial time algorithm for syntactic unification it follows that no polynomial time general combination algorithm exists unless p is contained in fp this implication of our main results holds not only for the combination of unification algorithms but also for the combination of constraint solvers as well we also show that the counting problem for boolean ring unification is p hard this gives a lower bound on the performance of all algorithms for general br unification

control effort considerations in the stabilization of uncertain dynamical systems abstract this paper investigates the problem of designing a state feedback control to stabilize an uncertain nonlinear system we focus attention on the amplitude norm of the controller which is used to achieve this end the uncertain system is described by a state equation which contains uncertain parameters which are unknown but bounded a lyapunov function is used to establish the stability of the closed loop system the paper gives necessary and sufficient concitions for the uncertain system to be stabilizable with a given lyapunov function furthermore a procedure is indicated for the construction of the desired feedback control law

moment lyapunov exponent and stability index for linear conservative system with small random perturbation asymptotic expansion series for the moment lyapunov exponent and stability index are constructed and justified for the two dimensional linear stochastic system close to a harmonic oscillator as an example a one degree of freedom mechanical system parametrically excited in stiffness and damping is considered and several terms of the expansion are obtained

lyapunov exponents and rotation number of two dimensional systems with telegraphic noise exact expressions for the exponential growth rates lyapunov exponents and rotation numbers of two dimensional systems perturbed by telegraphic noise are derived asymptotic expansions for small large slow and fast noise and the white noise limit are constructed then results are applied to the random oscillator stochastic stability in particular stabilization by noise and stochastic bifurcation

exact results for the lyapunov exponents of certain linear ito systems the lyapunov exponents of certain linear ito systems are considered the approach exploits the relations between the lyapunov exponents and the p moment lyapunov exponents as has been observed and studied by arnold siam j appl math 44 1984 pp 793 802 using the method that was developed in a leizarowitz explicit expressions for the lyapunov exponents of certain markov processes preprint the differential equations that are satisfied by the function t x 0 to e x t x 0 p and its laplace transform are derived in the two dimensional case this study yields explicit expressions for the lyapunov exponents

further facts about input to state stabilization previously published results about input to state stabilizability are shown to hold even for systems which are not linear in controls provided that a more general type of feedback is allowed applications to certain stabilization problems and coprime factorizations as well as comparisons to other results on input to state stability are briefly discussed

on solving equations and disequations we are interested in the problem of solving a system sl tl 1 i n pj qj 1 j m of equations and disequations also known as disunification solutions to disunification problems are substitutions for the variables of the problem that make the two terms of each equation equal but leave those of the disequations different we investigate this in both algebraic and logical contexts where equality is defined by an equational theory and more generally by a definitive clause equality theory e we show how e disunification can be reduced to e unification that is solving equations only and give a disunification algorithm for theories given a unification algorithm in fact this result shows that for theories in which the solutions of all unification problems can also be represented finitely we sketch how disunification can be applied to handle negation in logic programming with equality in a similar style to colmerauer s logic programming with rational trees and to represent many solutions to ac unification problems by a few solutions to aci disunification problems

almost sure instability of the random harmonic oscillator the problem of almost sure stability of the random harmonic oscillator with a jump process coefficient is studied it is shown that the sample paths are almost surely unstable and an asymptotic series expansion in the powers of the noise strength for the asymptotic exponential growth rate is obtained

a geometrical approach to multiset orderings we survey different ways of ordering multisets and give a classification of multiset orderings based on the notion of a cone in rn this enables us to derive new results about the dominance ordering and the standard multiset ordering

on the complexity of recognizing the hilbert basis of a linear diophantine system the problem of computing the hilbert basis of a homogeneous linear diophantine system over nonnegative integers is often considered in automated deduction and integer programming in automated deduction the hilbert basis of a corresponding system serves to compute the minimal complete set of associative commutative unifiers whereas in integer programming the hilbert bases are tightly connected to integer polyhedra and to the notion of total dual integrality in this paper we sharpen the previously known result that the problem asking whether a given solution belongs to the hilbert basis of a given system is conp complete we show that the problem has a pseudopolynomial algorithm if the number of equations in the system is fixed but it is conp complete in the strong sense if the given system is unbounded this result is important in the scope of automated deduction where the input is given in unary and therefore the previously known conp completeness result was unusable moreover we show that from the complexity standpoint it is not important to know the underlying homogeneous linear diophantine system when we ask whether a given set of vectors constitutes a hilbert basis

l_p stabilization of integrator chains subject to input saturation using lyapunov based homogeneous design consider the n th integrator dot x j nx sigma u e n where x in mathbb r n u in mathbb r j n is the n th jordan block and e n 0 cdots 0 1 t in mathbb r n we provide easily implementable state feedback laws u k x which not only render the closed loop system globally asymptotically stable but also are finite gain l p stabilizing with arbitrarily small gain as in a saberi p hou and a stoorvogel ieee trans automat control 45 2000 pp 1042 1052 these l p stabilizing state feedbacks are built from homogeneous feedbacks appearing in finite time stabilization of linear systems we also provide additional l infty stabilization results for the case of both internal and external disturbances of the n th integrator namely for the perturbed system dot x j nx e n sigma k x d d where d in mathbb r and d in mathbb r n

a lyapunov like characterization of asymptotic controllability it is shown that a control system in bf r n is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can be made negative by appropriate choices of controls

a formula connecting sample and moment stability of linear stochastic systems given the linear stochastic system dot x a xi t x x in mathbb r d xi t some nice diffusion sample and moment stability of x equiv 0 are studied via the sample lyapunov exponent lambda and the lyapunov exponent of the pth mean g p of a solution generalizing a result of molchanov for the undamped linear oscillator it is shown that g p is a smooth convex function of p in mathbb r with g 0 0 and g 0 lambda for trace a 0 a second zero of g at p d is detected this allows one to draw simple but quite complete conclusions about the relation between sample and moment stability

stabilization with relaxed controls cannot in general be stabilized using a continuous closed loop control u x even if each state separately can be driven asymptotically to the origin an example is analyzed in section 2 in this paper we examine the possibility of stabilizing such systems with a continuous closed loop relaxed control we find indeed that the family of systems stabilizable with relaxed controls is larger than the family of those stabilizable with ordinary controls an even larger class is obtained if the continuity of the closed loop at the origin is not required the latter class includes all one dimensional systems for which states can be driven asymptotically to the origin this result does not hold in two dimensional systems and we provide a counter example it should be pointed that relaxed control type stabilization is used both in theory and in practice the method is known as dither we shall comment on the similarities lyapunov functions for the system 1 help us in the construction of the continuous closed loop stabilizers in fact we find that the existence of a smooth lyapunov function is equivalent to the existence of a stabilizing closed loop which is continuous except possibly at the origin an additional condition on the lyapunov function implies the continuity at the origin as well we present these results in section 4 after a brief introduction of closed loop relaxed controls notations and terminology in section 3 prior to that in section 2 we discuss an example illustrating the power of relaxed controls in the particular case of systems linear in the controls relaxed controls can be replaced by ordinary controls this is discussed in section 5 the role of lyapunov functions in the stability and stabilization theories is of course well known examples of systems with lyapunov functions are available in the literature we display some in section 6 along with general comments on the construction applications and counterexamples including one which cannot be continuously stabilized yet possesses a nonsmooth lyapunov function closed loop stabilization with ordinary controls is analyzed extensively in the literature see sontag 8 sussmann ll and references therein lyapunov functions techniques in stabil

inverse optimality in robust stabilization the concept of a robust control lyapunov function bf rclf is introduced and it is shown that the existence of an bf rclf for a control affine system is equivalent to robust stabilizability via continuous state feedback this extends artstein s theorem on nonlinear stabilizability to systems with disturbances it is then shown that every bf rclf satisfies the steady state hamilton jacobi isaacs hji equation associated with a meaningful game and that every member of a class of pointwise min norm control laws is optimal for such a game these control laws have desirable properties of optimality and can be computed directly from the bf rclf without solving the hji equation for the upper value function

combination of constraint solvers for free and quasi free structures abstract when combining languages for symbolic constraints one is typically faced with the problem of how to treat mixed constraints the two main problems are 1 how to define a combined solution structure over which these constraints are to be solved and 2 how to combine the constraint solving methods for pure constraints into one for mixed constraints the paper introduces the notion of a free amalgamated product as a possible solution to the first problem we define so called quasi free structures called strong simply combinable structures in a previous publication as a generalization of free structures for quasi free structures over disjoint signatures we describe a canonical amalgamation construction that yields the free amalgamated product the combination techniques known from unification theory can be used to combine constraint solvers for quasi free structures over disjoint signatures into a solver for their free amalgamated product in addition to term algebras modulo equational theories i e free algebras the class of quasi free structures contains many solution structures that are of interest in constraint logic programming such as the algebra of rational trees feature structures and domains consisting of hereditarily finite well founded or non well founded nested sets and lists

brief paper input to state stability of switched systems and switching adaptive control in this paper we prove that a switched nonlinear system has several useful input to state stable iss type properties under average dwell time switching signals if each constituent dynamical system is iss this extends available results for switched linear systems we apply our result to stabilization of uncertain nonlinear systems via switching supervisory control and show that the plant states can be kept bounded in the presence of bounded disturbances when the candidate controllers provide iss properties with respect to the estimation errors detailed illustrative examples are included

input output to state stability this work explores lyapunov characterizations of the input output to state stability ioss property for nonlinear systems the notion of ioss is a natural generalization of the standard zero detectability property used in the linear case the main contribution of this work is to establish a complete equivalence between the ioss property and the existence of a certain type of smooth lyapunov function as corollaries one shows the existence of norm estimators and obtains characterizations of nonlinear detectability in terms of relative stability and of finite energy estimates

strong iiss for a class of systems under saturated feedback this paper proposes sufficient conditions under which nonlinear input affine systems can be made strongly iiss in the presence of actuator saturation strong iiss was recently proposed as a compromise between the strength of input to state stability iss and the generality of integral input to state stability iiss it ensures in particular that solutions are bounded provided that the disturbance magnitude is below a certain threshold and that they tend to the origin in response to any vanishing disturbance we propose a growth rate condition under which the bounded feedback law proposed by lin and sontag for disturbance free nonlinear systems ensures strong iiss in the presence of perturbations we illustrate our findings on the angular velocity control of a spacecraft with limited power thrusters in the specific case of linear time invariant systems with neutrally stable internal dynamics we provide a simple static state feedback that ensures strong iiss in presence of actuator saturations this second result is illustrated by the robust stabilization of the harmonic oscillator in both cases we provide an estimate of the maximum disturbance amplitude that can be tolerated without compromising solutions boundedness

simultaneous nonlinear model predictive control and state estimation an output feedback approach to model predictive control that combines state estimation and control into a single minmax optimization is introduced for discrete time nonlinear systems specifically a criterion that involves finite forward and backward horizons is minimized with respect to control input variables and is maximized with respect to the unknown initial state as well as disturbance and measurement noise variables under appropriate assumptions that encode controllability and observability we show that the state of the closed loop remains bounded and that a bound on tracking error can be found for trajectory tracking problems we also introduce a primaldual interior point method that can be used to efficiently solve the minmax optimization problem and show in simulation examples that the method succeeds even for severely nonlinear and non convex problems

unification in commutative idempotent monoids abstract embedding sets as a datastructure into resolution based deduction requires a unification algorithm respecting associativity commutativity and idempotence of the constructor function union we shall present such an algorithm and prove its correctness and termination although this algorithm is very simple the enormous proliferation of unifiers restricts its value to theoretical considerations for practical purposes the more special unitary theory of boolean expressions seem to be better suited

a unifying integral iss framework for stability of nonlinear cascades we analyze nonlinear cascades in which the driven subsystem is integral input to state stable iss and we characterize the admissible integral iss gains for stability this characterization makes use of the convergence speed of the driving subsystem and allows a larger class of gain functions when the convergence is faster we show that our integral iss gain characterization unifies different approaches in the literature which restrict the nonlinear growth of the driven subsystem and the convergence speed of the driving subsystem the result is used to develop a new observer based backstepping design in which the growth of the nonlinear damping terms is reduced

viability control for a class of underactuated systems this paper addresses the feedback control design for a class of nonholonomic systems which are subject to inequality state constraints defining a constrained viability set k based on concepts from viability theory the necessary conditions for selecting viable controls for a nonholonomic system are given so that system trajectories starting in k always remain in k furthermore a class of state feedback control solutions for nonholonomic systems are redesigned by means of switching control so that system trajectories starting in k converge to a goal set g in k without ever leaving k the proposed approach can be applied in various problems whose objective can be recast as controlling a nonholonomic system so that the resulting trajectories remain forever in a subset k of the state space until they converge into a goal target set g in k the motion control for an underactuated marine vehicle in a constrained configuration set k is treated as a case study the set k essentially describes the limited sensing area of a vision based sensor system and viable control laws which establish convergence to a goal set g in k are constructed the robustness of the proposed control approach under a class of bounded external perturbations is also considered the efficacy of the methodology is demonstrated through simulation results

complexity of nilpotent unification and matching problems we consider the complexity of equational unification and matching problems where the equational theory contains a nilpotent function i e a function f satisfying f x x 0 where 0 is a constant we show that nilpotent unification and matching are np complete in the presence of associativity and commutativity the problems still remain np complete however when 0 is also assumed to be the unity for the function f the problems are solvable in polynomial time we also show that the problem remains in p even when a homomorphism is added an application of this result to a subclass of set constraints is illustrated second order matching modulo nilpotence is shown to be undecidable

inverse optimal design of input to state stabilizing nonlinear controllers we show that input to state stabilizability as defined by sontag 1989 1995 is both necessary and sufficient for the solvability of a hamilton jacobi isaacs equation associated with a meaningful differential game problem similar to but more general than the nonlinear h sub spl infin problem the significance of the result stems from the fact that constructive solutions to the input to state stabilization problem are available presented in the paper and that as shown here inverse optimal controllers possess margins on input to state stability against a certain class of input unmodeled dynamics rather than completion of squares the main tools in our analysis are legendre fenchel transformations and the general form of young s inequality

a characterization of integral input to state stability the notion of input to state stability iss is now recognized as a central concept in nonlinear systems analysis it provides a nonlinear generalization of finite gains with respect to supremum norms and also of finite l sup 2 gains it plays a central role in recursive design coprime factorizations controllers for nonminimum phase systems and many other areas in this paper a newer notion that of integral input to state stability iiss is studied the notion of iiss generalizes the concept of finite gain when using an integral norm on inputs but supremum norms of states in that sense generalizing the linear h sup 2 theory it allows one to quantify sensitivity even in the presence of certain forms of nonlinear resonance we obtain several necessary and sufficient characterizations of the iiss property expressed in terms of dissipation inequalities and other alternative and nontrivial characterizations

nonlinear scaling of i iss lyapunov functions while nonlinear scalings of lyapunov functions are also lyapunov functions we provide examples that the same statement does not necessarily hold for input to state stable iss lyapunov functions or for integral iss iiss lyapunov functions we provide sufficient conditions under which a nonlinear scaling of an iss or iiss lyapunov function is also an iss or iiss lyapunov function we also introduce a generalization of the iiss lyapunov function which we term a dissipative iiss lyapunov function that allows for a larger class of nonlinear scalings so that nonlinear scalings of dissipative iiss lyapunov functions are again dissipative iiss lyapunov functions

a universal formula for stabilization with bounded controls abstract we provide a formula for stabilizing feedback law using a bounded control under the assumption that an appropriate control lyapunov function is known such a feedback smooth away from the origin and continuous everywhere is known to exist via artstein s theorem as in the unbounded control case treated in a previous note we provide an explicit and universal formula given by an algebraic function of lie derivatives in particular we extended to the bounded case the result that the feedback can be chosen analytic if the lyapunov function and the vector fields defining the system are analytic

completion for unification abstract syntactic theories have the nice property that a unification algorithm may be computed directly from the form of the axioms of a specific presentation called resolvent of the theory in this work we present and prove a completion algorithm that for a given presentation returns a resolvent set of axioms whenever it terminates

a universal construction of artstein s theorem on nonlinear stabilization abstract this note presents an explicit proof of the theorem due to artstein which states that the existence of a smooth control lyapunov function implies smooth stabilizability moreover the result is extended to the real analytic and rational cases as well the proof uses a universal formula given by an algebraic function of lie derivatives this formula originates in the solution of a simple riccati equation

refutational theorem proving using term rewriting systems abstract in this paper we propose a new approach to theorem proving in first order logic based on the term rewriting method first for propositional calculus we introduce a canonical term rewriting system for boolean algebra this system enables us to transform the first order predicate calculus into a form of equational logic and to develop several complete strategies both clausal and nonclausal for first order theories based on the knuth bendix completion procedure more importantly our strategies can deal with predicate logic and built in equational theories in a uniform and effective way we also describe an implementation and comparisons with some other first order theorem proving methods

towards a foundation of completion procedures as semidecision procedures completion procedures originated from the seminal work of knuth and bendix are well known as procedures for generating confluent rewrite systems i e decision procedures for equational theories in this paper we present a new abstract framework for the utilization of completion procedures as semidecision procedures for theorem proving the key idea in our approach in that a semidecision process should be target oriented i e keep into account the target theorem to be proved for the inference rules of a completion procedure we present target oriented schemes of contraction inference rules i e inference rules that delete sentences which are redundant for proving the target for the search plan we give a target oriented definition offairness according to which not all critical pairs need to be considered we prove that our notion of fairness together with the refutational completeness of the inference rules is sufficient for a completion procedure to be a semidecision procedure by relaxing the requirement of considering all critical pairs our target oriented framework should be more suitable for designing efficient procedures for theorem proving the generation of decision procedures is included as a special side effect and all the results of the classical approach to completion are re obtained in our framework the application of completion to disprove inductive conjectures i e the so called inductionless induction method is also covered as a semidecision process finally we present according to our framework some equational completion procedures based on unfailing knuth bendix completion

computing with rewrite systems term rewriting systems that is sets of directed equations provide a paradigm of computation with particularly simple syntax and semantics rewrite systems may be used for straightforward computation by simplifying terms we show how in addition restricted forms of the knuth bendix completion procedure may be used to interpret logic programs written as a set of equivalence preserving rewrite rules we discuss verification issues and also illustrate the use of the full completion procedure to synthesize rewrite programs from specifications

orderings for term rewriting systems abstract methods of proving that a term rewriting system terminates are presented they are based on the intuitive notion of simplification orderings orderings in which any term that is syntactically simpler than another is smaller than the other as a consequence of kruskal s tree theorem any nonterminating system must be self embedding in the sense that it allows for the derivation of some term from a simpler one thus termination is guaranteed if every rule in the system is a reduction in some simplification ordering most of the orderings that have been used for proving termination are indeed simplication orderings using this notion often allows for much easier proofs a particularly useful class of simplification orderings the recursive path orderings is defined examples of the use of simplication orderings in termination proofs are given

dedekind multisets and function shells abstract a multiset is a set containing repeated elements we define the unique multiset determined by a function based on an idea formulated by dedekind in 1888 the class of all functions which determine the same multiset is called a function shell the formal theory of multisets is revised to allow for infinite repetitions of elements the revised theory of multisets is then interpreted as a theory of function shells with this interpretation a new algebra of functions is defined and interesting properties of function shells are investigated

reliability of elastic structures driven by random loads we study the stability of an elastic column driven by a fluctuating load we consider a sequence of approximations to the continuous column by first a simple pendulum with an impulsive random force applied at the hinge then by a double pendulum and finally by an n fold pendulum with a vertical or tangential random load we find a stability criterion which depends on the noise to damping ratio even if the fluctuating load is sub critical on the average we show that the structure may not be stable and its energy may reach any level in this case we calculate the mean time to reach a given energy level and use this time as an index of reliability of the structure we consider both the vertically loaded column the euler column and the tangentially loaded column follower load two types of noise are considered a random impulsive force and gaussian white noise in either case the qualitative behavior of the pendulum or the column is the same

approximate bisimulation relations for constrained linear systems in this paper we define the notion of approximate bisimulation relation between two continuous systems while exact bisimulation requires that the observations of two systems are and remain identical approximate bisimulation allows the observations to be different provided the distance between them remains bounded by some parameter called precision approximate bisimulation relations are conveniently defined as level sets of a so called bisimulation function which can be characterized using lyapunov like differential inequalities for a class of constrained linear systems we develop computationally effective characterizations of bisimulation functions that can be interpreted in terms of linear matrix inequalities and optimal values of static games we derive a method to evaluate the precision of the approximate bisimulation relation between a constrained linear system and its projection this method has been implemented in a matlab toolbox matisse an example of use of the toolbox in the context of safety verification is shown

first order dynamics driven by rapid markovian jumps the effect of rapid random jumps on the behavior of deterministically stable dynamics is considered the random driving force noise is taken to be a state dependent markov jump process with either finite or infinite state space while the jump rate is high o 1 varepsilon the jump size is assumed to be o 1 for a particle that is deterministically confined to a potential well the stationary joint probability density function of the state and noise variables the probability distribution of the exit points and the mean first passage time from the well are computed in contrast to the case of white noise where these quantities depend on a barrier height determined solely by the potential here they depend on an effective barrier height determined by both the potential and the noise process the method used here is to introduce the small parameter varepsilon where 1 varepsilon is a measure of the rapid jump rate and to employ singular perturbation methods to solve the forward and

rings with lexicographic straightening law the concept of a straightening law as a means of analyzing the structure of particular commutative algebras has appeared independently in the work of a number of authors the purpose of this paper is to introduce a systematic theory of algebras endowed with a structure of a lexicographic straightening law based on a partially ordered set or lexicographic ring for short much of the work in the literature dealing with such rings fits quite naturally in the context of our theory and we sketch some of this work moreover some of our later results in particular the betti number bound in section 6 may actually be new results even for the special cases of the known examples of lexicographic rings the concept of a lexicographic ring as an interesting area of study was suggested to this author by deconcini who conjectured that a lexicographic ring is cohen macaulay if the underlying partially ordered set is so we prove this result in two different ways it has recently come to our attention that this conjecture has also been proved by deconcini et al 131 using deformation theory methods the results of this paper are arranged in two parts the first part consisting of sections 2 through 4 is more elementary and uses the technique of combinatorial decompositions as developed by garsia and this author the second part sections 6 and 7 requires some knowledge of homological algebra methods in ring theory the intermediate section 5 discusses some of the known examples of lexicographic rings in this section we also sketch some of the ways that our theory may be extended to more general contexts the main result of the first part is the transfer theorem 4 3 which enables one to transfer combinatorial decompositions from the underlying partially ordered set or more precisely from the corresponding stanley reisner ring to the lexicographic ring as a result those concepts that are expressible in terms of combinatorial decompositions are also

lyapunov exponent and rotation number of two dimensional linear stochastic systems with small diffusion by means of a singular perturbation scheme and large deviation estimates we expand the lyapunov exponent and rotation number in terms of the small noise intensity varepsilon for all 2 times 2 systems dx t varepsilon ax t varepsilon dt sqrt varepsilon sum nolimits k 1 r b k x t varepsilon circ dw t k for which a is complex diagonalizable examples are given

certainty equivalence implies detectability abstract it is shown that any stabilizing certainty equivalence control used within an adaptive control system causes the familiar interconnection of a controlled process and associated output estimator to be detectable through the estimator s output error e p for every frozen value of the index or parameter vector p upon which both the estimator and controller dynamics depend the fact that certainty equivalence implies detectability has been known for some time this has been shown to be so whenever the process model is linear and the controller and estimator models are also linear for every frozen value of p in this paper use is made of recently introduced concepts of input to state stability and detectability for nonlinear systems to prove that the same implication is valid in a more general nonlinear setting

the gain scheduling and the robust state feedback stabilization problems we consider dynamical uncertain systems of the form x spl dot a x w b x w u where w t spl isin w is an unknown but bounded time varying parameter we consider the robust state feedback stabilization problem in which we consider a control of the form u spl phi x and the gain scheduling stabilization problem in which a control of the form u spl phi x w full information control is admitted we show that for convex processes those systems in which for fixed x the set of all a x w b x w is convex the two problems are equivalent i e if there exists a locally lipschitz gain scheduling stabilizing control then there exists a robustly stabilizing control continuous everywhere except possibly at the origin thus for convex process stabilization knowledge of w t is not an advantage for the compensator then we consider the special class of polytopic lpv systems and show that there is no loss of regularity as in the general case if we pass from a gain scheduling controller to a state feedback controller in particular no discontinuity at the origin may occur we show that the existence of a dynamic controller always implies the existence of a static one finally we show that we can always find a linear gain scheduling controller for a stabilizable system this means that a possible advantage of the online measurement of the parameter w t is that this always allows for linear compensators whose implementation can be easier than that of nonlinear ones

universal construction of feedback laws achieving iss and integral iss disturbance attenuation we study nonlinear systems with both control and disturbance inputs the main problem addressed in the paper is design of state feedback control laws that render the closed loop system integral input to state stable iiss with respect to the disturbances we introduce an appropriate concept of control lyapunov function iiss clf whose existence leads to an explicit construction of such a control law the same method applies to the problem of input to state stabilization converse results and techniques for generating iiss clfs are also discussed

symbolic protocol analysis for monoidal equational theories we are interested in the design of automated procedures for analyzing the in security of cryptographic protocols in the dolev yao model for a bounded number of sessions when we take into account some algebraic properties satisfied by the operators involved in the protocol this leads to a more realistic model in comparison to what we get under the perfect cryptography assumption but it implies that protocol analysis deals with terms modulo some equational theory instead of terms in a free algebra the main goal of this paper is to setup a general approach that works for a whole class of monoidal theories which contains many of the specific cases that have been considered so far in an ad hoc way e g exclusive or abelian groups exclusive or in combination with the homomorphism axiom we follow a classical schema for cryptographic protocol analysis which proves first a locality result and then reduces the insecurity problem to a symbolic constraint solving problem this approach strongly relies on the correspondence between a monoidal theory e and a semiring s e which we use to deal with the symbolic constraints we show that the well defined symbolic constraints that are generated by reasonable protocols can be solved provided that unification in the monoidal theory satisfies some additional properties the resolution process boils down to solving particular quadratic diophantine equations that are reduced to linear diophantine equations thanks to linear algebra results and the well definedness of the problem examples of theories that do not satisfy our additional properties appear to be undecidable which suggests that our characterization is reasonably tight

simultaneous stabilization for a collection of single input nonlinear systems this paper presents a novel method for designing a controller that simultaneously stabilizes a collection of single input nonlinear systems the control lyapunov function approach is used to derive necessary and sufficient conditions for the existence of time invariant simultaneously stabilizing state feedback controllers additionally a universal formula for constructing a continuous simultaneously stabilizing controller when the provided sufficient condition is satisfied is presented for any collection of second order and third order feedback linearizable systems in canonical form global simultaneous stabilization via a single state feedback controller is shown to be always possible two examples are included for illustration

lyapunov exponent and rotation number of two dimensional linear stochastic systems with telegraphic noise in this paper the lyapunov exponent and rotation number of a family of two dimensional systems with general telegraphic noise where the unperturbed system has oscillatory behavior are studied using a martingale approach an iterative computational procedure is derived and analytic expansions in the noise strength are obtained for the lyapunov exponent and the rotation number these expansions are convergent in a nonmarginal way the positivity of the exponent is also established for the random harmonic oscillator with zero mean telegraphic noise and lower and upper bounds are derived for the exponent when the noise is the random telegraph process

instability of the harmonic oscillator with small noise we construct asymptotic expansions for the exponential growth rate lyapunov exponent and rotation number of the random oscillator when the noise is small and is defined by a temporally homogeneous markov process with a finite number of states in the case of two states the telegraph process we obtain additional terms in the expansions affording comparison between the exact values and the asymptotic formulas

modulational instability induced by randomly varying coefficients for the nonlinear schrodinger equation abstract we introduce the theory of modulational instability mi of electromagnetic waves in optical fibers the model at hand is the one dimensional nonlinear schrodinger equation with random group velocity dispersion and random nonlinear coefficient we compute the mi gain which reads as the lyapunov exponent of a random linear system we show that the distribution of the mi gain can be expressed in terms of the log normal statistics the heavy tail of this probability distribution function involves very different behaviors for the sample and moment mi gains in the anomalous dispersion regime random fluctuations of the nonlinear coefficient reduces the sample mi gain peak although the moment mi peak is enhanced and the unstable bandwidth is widened still in the anomalous dispersion regime random fluctuations of the group velocity dispersion reduces both the sample mi gain peak and the moment mi peak finally in the normal dispersion regime randomness extends the mi domain to the whole spectrum of modulations and increases the mi gain peak the linear stability analysis is confirmed by numerical simulations of the full stochastic nonlinear schrodinger equation

a unification algorithm for associative commutative functions an important component of automated theorem proving systems are unification algorithms which fred most general substitutions which when apphed to two expressions make them equivalent functions which are associative and commutative such as the arithmetic addition and multiphcatton functions are often the subject of automated theorem proving an algorithm which unifies terms whose function is associative and commutauve is presented here termmaaon soundness and completeness of the algorithm have been proved for a subclass of the general case the algorithm is an efficient alternative to other methods of handling associative commutative functions

unification in commutative semigroups abstract unification is one of the basic concepts of automated theorem proving it concerns such questions as finding solutions of finite sets of equations determining if every solution comes from a most general solution and if so determining how many most general solutions are needed to generate all solutions these solutions given in terms of substitutions are called more formally unifiers the unification type of a variery equational class of algebras is defined according to the cardinality or existence of minimal complete sets of most general unifiers of particular interest from a computational point of view are varieties of groups and semigroups so far the problem has been considered mainly for particular varieties in this paper we determine unification types for all varieties of commutative semigroups in particular we prove that for commutative semigroups the unification problem is solvable in the very strong sense that there is an algorithm which for any two finite sets 1and 2of semigroup equations produces the minimal complete set of the most general unifiers of 1over the variety of commutative semigroups generated by 2 it seems that this is the first so general decidability result in the area

global asymptotic controllability implies input to state stabilization the main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable gac control affine systems that render the closed loop systems input to state stable iss with respect to actuator errors extensions for fully nonlinear gac systems with actuator errors are also discussed our controllers have the property that they tolerate small observation noise as well

an undecidability result for agh we present an undecidability result for the verification of security protocols since the perfect cryptography assumption is unrealistic for cryptographic primitives with visible algebraic properties several recent works relax this assumption allowing the intruder to exploit these properties we are interested in the abelian groups theory in combination with the homomorphism axiom we show that the security problem for a bounded number of sessions expressed by satisfiability of symbolic deductibility constraints is undecidable obtaining in this way the first undecidability result concerning a theory for which unification is known to be decidable f baader unification in commutative theories hilbert s basis theorem and grobner bases j acm 40 3 1993 477 503

permutative rewriting and unification permutative rewriting provides a way of analyzing deduction modulo a theory defined by leaf permutative equations our analysis naturally leads to the definition of the class of unify stable axiom sets in order to enforce a simple reduction strategy we then give a uniform unification algorithm modulo theories e axiomatized this way we prove that it computes complete sets of unifiers of simply exponential cardinality and that the e unification decision problem belongs to np