on boundedness of solutions of state periodic systems a multivariable cell structure approach many dynamical systems are periodic with respect to several state variables this periodicity typically leads to the coexistence of multiple invariant solutions equilibria or limit cycles as a consequence while there are many classical techniques for analysis of boundedness and stability of such systems most of these only permit to establish local properties motivated by this a new sufficient criterion for global boundedness of solutions of such a class of nonlinear systems is presented the proposed method is inspired by the cell structure approach developed by leonov and noldus and characterized by two main advances first the conventional cell structure framework is extended to the case of dynamics which are periodic with respect to multiple states second by introducing the notion of a leonov function the usual definiteness requirements of standard lyapunov functions are relaxed to sign indefinite functions furthermore it is shown that under mild additional conditions the existence of a leonov function also ensures input to state stability i e robustness with respect to exogenous perturbations the performance of the proposed approach is demonstrated via the global analysis of boundedness of trajectories for a nonlinear system

on a jumping problem for quasilinear elliptic equations let us denote by k the eigenvalues of the operator with homogeneous dirichlet condition first studied in 1 when 1 2 this problem has been widely investigated in the case that some eigenvalue k belongs to the interval see e g 14 13 10 and references therein the methods used are often a combination of topological and variational techniques in this paper we will be concerned with a jumping problem associated to a quasilinear elliptic equation of the form n

relative entropy exponential utility and robust dynamic pricing in the area of dynamic revenue management optimal pricing policies are typically computed on the basis of an underlying demand rate model from the perspective of applications this approach implicitly assumes that the model is an accurate representation of the real world demand process and that the parameters characterizing this model can be accurately calibrated using data in many situations neither of these conditions are satisfied indeed models are usually simplified for the purpose of tractability and may be difficult to calibrate because of a lack of data moreover pricing policies that are computed under the assumption that the model is correct may perform badly when this is not the case this paper presents an approach to single product dynamic revenue management that accounts for errors in the underlying model at the optimization stage uncertainty in the demand rate model is represented using the notion of relative entropy and a tractable reformulation of the robust pricing problem is obtained using results concerning the change of probability measure for point processes the optimal pricing policy is obtained through a version of the so called isaacs equation for stochastic differential games and the structural properties of the optimal solution are obtained through an analysis of this equation in particular i closed form solutions for the special case of an exponential nominal demand rate model ii general conditions for the exchange of the max and the min in the differential game and iii the equivalence between the robust pricing problem and that of single product revenue management with an exponential utility function without model uncertainty are established through the analysis of this equation

stabilization of a class of slow fast control systems at non hyperbolic points in this document we deal with the local asymptotic stabilization problem of a class of slow fastsystems or singularly perturbed ordinary differential equations the systems studied here have the following properties 1 they have one fast and an arbitrary number of slow variables and 2 they have a non hyperbolic singularity at the origin of arbitrary degeneracy our goal is to stabilize such a point the presence of the aforementioned singularity complicates the analysis and the controller design in particular the classical theory of singular perturbations cannot be used we propose a novel design based on geometric desingularization which allows the stabilization of a non hyperbolic point of singularly perturbed control systems our results are exemplified on a didactic example and on an electric circuit

solutions of asymptotically linear operator equations via morse theory abstract in this paper we use the classical morse theory of critical points to study the existence and multiplicity of solutions of a class of asymptotically linear operator equations as special cases we include multiplicity results for semilinear elliptic bvp s as well as existence and multiplicity of periodic solutions of semilinear wave equation and of hamiltonian systems these results simplify and complement recent work of amann zehnder and castro lazer author

a non quadratic criterion for stability of forced oscillations abstract a new test for stability of forced oscillations in nonlinear systems is proposed the result is illustrated by an example of a saturated pi control of an integrator with a harmonic reference signal the range of frequencies and amplitudes that guarantee stability of the corresponding oscillations is found

chasing demand learning and earning in a changing environment we consider a dynamic pricing problem in which a seller faces an unknown demand model that can change over time the amount of change over a time horizon of t periods is measured using a variation metric that allows for a broad spectrum of temporal behavior given a finite variation budget we first derive a lower bound on the expected performance gap between any pricing policy and a clairvoyant who knows a priori the temporal evolution of the underlying demand model and then we design families of near optimal pricing policies the revenue performance of which asymptotically matches said lower bound we also show that the seller can achieve a substantially better revenue performance in demand environments that change in bursts than in demand environments that change smoothly among other things quantifying the net effect of the volatility in the demand environment on the seller s revenue performance

morse theory for continuous functionals abstract using the notion of weak slope introduced by m degiovanni and m marzocchi 1994 ann mat pura appl 167 73 100 and also independently by c katriel 1994 ann inst h poincare anal non lineaire11 189 209 we obtain some results in morse theory for continuous functionals having isolated critical points the main tools are deformation properties generalizing classical ones for smooth functionals and refining those of j n corvellec m degiovanni and m marzocchi 1993 topol methods nonlinear anal 1 151 171 we obtain for example generalized morse relations through the notion of critical group which has a natural link with the weak slope these results are developed in view of applications to non smooth variational problems

technical communique uniquely connecting frequency domain representations of given order polynomial wiener hammerstein systems the notion of frequency response functions has been generalized to nonlinear systems in several ways however a relation between different approaches has not yet been established in this paper frequency domain representations for nonlinear systems are uniquely connected for a class of nonlinear systems specifically by means of novel analytical results the generalized frequency response function gfrf and the higher order sinusoidal input describing function hosidf for polynomial wiener hammerstein systems are explicitly related assuming the linear dynamics are known necessary and sufficient conditions for this relation to exist and results on the uniqueness and equivalence of the hosidf and gfrf are provided finally this yields an efficient computational procedure for computing the gfrf from the hosidf and vice versa

contraction analysis of nonlinear distributed systems contraction theory is a comparatively recent dynamic analysis and non linear control system design tool based on an exact differential analysis of convergence this paper extends contraction theory to local and global stability analysis of important classes of non linear distributed dynamics such as convection diffusion reaction processes lagrangian and hamilton jacobi dynamics and optimal controllers and observers by contrast with stability proofs based on energy dissipation stability and convergence can be determined for energy based systems excited by time varying inputs the hamilton jacobi bellman controller and a similar optimal non linear observer design are studied based on explicitly computable conditions on the convexity of the cost function these stability conditions extend the well known conditions on controllability and observability grammians for linear time varying systems without requiring the unknown transition matrix of the underlying differential dynamics

dynamic pricing without knowing the demand function risk bounds and near optimal algorithms we consider a single product revenue management problem where given an initial inventory the objective is to dynamically adjust prices over a finite sales horizon to maximize expected revenues realized demand is observed over time but the underlying functional relationship between price and mean demand rate that governs these observations otherwise known as the demand function or demand curve is not known we consider two instances of this problem i a setting where the demand function is assumed to belong to a known parametric family with unknown parameter values and ii a setting where the demand function is assumed to belong to a broad class of functions that need not admit any parametric representation in each case we develop policies that learn the demand function on the fly and optimize prices based on that the performance of these algorithms is measured in terms of the regret the revenue loss relative to the maximal revenues that can be extracted when the demand function is known prior to the start of the selling season we derive lower bounds on the regret that hold for any admissible pricing policy and then show that our proposed algorithms achieve a regret that is close to this lower bound the magnitude of the regret can be interpreted as the economic value of prior knowledge on the demand function manifested as the revenue loss due to model uncertainty

a negative bendixson like criterion for a class of hybrid systems a condition which ensures the absence of periodic orbits for nonsmooth dynamical systems is presented the condition is a higher dimensional generalization of bendixson s criterion applicable to differential inclusions that are useful in the description of hybrid systems the main argument is based on contraction analysis of the d measured volume along the system trajectories a connection to methods for estimating the hausdorff dimension is emphasized for a class of hybrid systems described by a linear system with relay feedback the conditions are presented in the form of linear matrix inequalities a simple but illustrative example is analyzed

a contraction approach to the hierarchical analysis and design of networked systems this brief is concerned with the stability of continuous time networked systems using contraction theory a result is established on the network structure and the properties of the individual component subsystems and their couplings to ensure the overall contractivity of the entire network specifically it is shown that a contraction property on a reduced order matrix that quantifies the interconnection structure coupled with contractivity expansion estimates on the individual component subsystems suffices to ensure that trajectories of the overall system converge towards each other

contraction based stabilisation of nonlinear singularly perturbed systems and application to high gain feedback abstractrecent development of contraction theory based analysis has opened the door for inspecting differential behaviour of singularly perturbed systems in this paper a contraction theory based framework is proposed for stabilisation of singularly perturbed systems the primary objective is to design a feedback controller to achieve bounded tracking error for both standard and non standard singularly perturbed systems this framework provides relaxation over traditional quadratic lyapunov based method as there is no need to satisfy interconnection conditions during controller design algorithm moreover the stability bound does not depend on smallness of singularly perturbed parameter and robust to additive bounded uncertainties combined with high gain scaling the proposed technique is shown to assure contraction of approximate feedback linearisable systems these findings extend the class of nonlinear systems which can be made contracting

the nonstochastic multiarmed bandit problem in the multiarmed bandit problem a gambler must decide which arm of k nonidentical slot machines to play in a sequence of trials so as to maximize his reward this classical problem has received much attention because of the simple model it provides of the trade off between exploration trying out each arm to find the best one and exploitation playing the arm believed to give the best payoff past solutions for the bandit problem have almost always relied on assumptions about the statistics of the slot machines in this work we make no statistical assumptions whatsoever about the nature of the process generating the payoffs of the slot machines we give a solution to the bandit problem in which an adversary rather than a well behaved stochastic process has complete control over the payoffs in a sequence of t plays we prove that the per round payoff of our algorithm approaches that of the best arm at the rate o t 1 2 we show by a matching lower bound that this is the best possible we also prove that our algorithm approaches the per round payoff of any set of strategies at a similar rate if the best strategy is chosen from a pool of n strategies then our algorithm approaches the per round payoff of the strategy at the rate o log n1 2 t 1 2 finally we apply our results to the problem of playing an unknown repeated matrix game we show that our algorithm approaches the minimax payoff of the unknown game at the rate o t 1 2

critical points for multiple integrals of the calculus of variations in this paper we deal with the existence of critical points of functional defined on the sobolev space w01 p p 1 by j u int limits omega vartheta x u du dx text where is a bounded open subset of n even for very simple examples in n the differentiability of j u can fail to overcome this difficulty we prove a suitable version of the ambrosetti rabinowitz mountain pass theorem applicable to functionals which are not differentiable in all directions existence and multiplicity of nonnegative critical points are also studied through the use of this theorem

three critical points theorem and its application to quasilinear elliptic equations abstract in this paper we prove a pucci serrin type three critical points theorem for continuous functionals and study its application to quasilinear elliptic equations with natural growth

a link between input output stability and lyapunov stability it is shown that incremental boundedness of an input output operator ensures global asymptotic stability of any motion a reciprocal statement is set

a contraction theory based analysis of the stability of the deterministic extended kalman filter the contraction properties of the extended kalman filter viewed as a deterministic observer for nonlinear systems are analyzed the approach relies on the study of an auxiliary virtual dynamical system some conditions under which exponential convergence of the state error can be guaranteed are derived moreover contraction provides a simple formalism to study some robustness properties of the filter especially with respect to measurement errors as illustrated by a simplified inertial navigation example this technical note sheds another light on the theoretical properties of this popular observer

convergence properties for discrete time nonlinear systems three similar convergence notions are considered two of them are the long established notions of convergent dynamics and incremental stability the other is the more recent notion of contraction analysis all three convergence notions require that all solutions of a system converge to each other in this note we investigate the differences between these convergence properties for discrete time and time varying nonlinear systems by comparing the properties in pairs and using examples we also demonstrate a time varying smooth lyapunov function characterization for each of these convergence notions and with appropriate assumptions we provide several sufficient conditions to establish relationships between these properties in terms of lyapunov functions

a sliding mode approach to stabilization of nonlinear markovian jump singularly perturbed systems this paper presents a sliding mode control smc method for a class of nonlinear markovian jump singularly perturbed systems mjspss the system is subject to parameter uncertainties and partly unknown transition probabilities to fully employ the model characteristics of such a hybrid system a novel integral type switching function is firstly designed by adopting the e dependent stochastic lyapunov function method sufficient conditions are presented to ensure the mean square asymptotic stability of the sliding mode dynamics a mode dependent fuzzy smc law is then synthesized to induce and maintain the sliding motion despite partly unknown transition probabilities and parameter uncertainties finally the developed method is applied to stabilize a modified series dc motor system

convergent dynamics a tribute to boris pavlovich demidovich we review and pay tribute to a result on convergent systems by the russian mathematician boris pavlovich demidovich in a sense demidovich s approach forms a prelude to a field which is now called incremental stability of dynamical systems developments on incremental stability are reviewed from a historical perspective

doubly resonant semilinear elliptic problems via nonsmooth critical point theory abstract we consider the existence of weak solutions for classical doubly resonant semilinear elliptic problems we show how the main technical assumptions can be used to define appropriate metrics on the underlying function space so that extensions of the results already known in the literature can be obtained using only basic facts from critical point theory for continuous functionals on complete metric spaces

subdifferential calculus and nonsmooth critical point theory a general critical point theory for continuous functions defined on metric spaces has been recently developed in this paper a new subdifferential related to that theory is introduced in particular results on the subdifferential of a sum are proved an example of application to pdes is sketched detailed applications to pdes are developed in separate papers

a contraction theory approach to singularly perturbed systems in this technical note we revisit standard results on singular perturbations and multiple time scales using convergence analysis tools based on nonlinear contraction theory specifically assuming that the fast and slow subsystems are each partially contracting we obtain explicit bounds on the convergence rate of the trajectories to the slow manifold and on the asymptotic error between the trajectories of the singularly perturbed system and those of the reduced system as an application example we illustrate the design of a biomolecular insulation device

brief paper global nonlinear output regulation convergence based controller design in this paper we present output feedback controllers solving the global output regulation problem for a class of nonlinear systems the proposed controllers are based on the notion of convergent systems the presented solution extends well established results on the linear output regulation problem and the local nonlinear output regulation problem to the global case for lur e systems which are not necessarily in the output feedback form the proposed controllers can be found by solving the regulator equations and certain linear matrix inequalities for systems in the output feedback form with uncertain parameters and uncertain nonlinearities we provide a robust regulator that does not rely on the minimum phaseness assumption on the system which is crucial in the previous regulator designs for output feedback systems the results are illustrated by examples

a differential lyapunov framework for contraction analysis lyapunov s second theorem is an essential tool for stability analysis of differential equations the paper provides an analog theorem for incremental stability analysis by lifting the lyapunov function to the tangent bundle the lyapunov function endows the state space with a finsler structure incremental stability is inferred from infinitesimal contraction of the finsler metrics through integration along solutions curves

nontrivial solutions of operator equations and morse indices of critical points of min max type on montre que si les hypotheses du theoreme de point selle sont satisfaites si dim y 2 et si f 1 c contient seulement des points critiques non degeneres alors f 1 c contient au moins un point critique u 0 tel que 2 indice de morse u 0 dim y

a mathematical model for the dynamics of clustering abstract the formation of several clusters arising from attracting forces between nonidentical entities or agents is a phenomenon observed in diverse fields think of people gathered through a mutual interest swarm behaviour of animals or clustering of oscillators in brain cells we introduce a dynamic model of mutually attracting agents for which we prove that the long term behaviour consists of agents organized into several groups or clusters we have completely characterized the cluster structure i e the number of clusters and their composition by means of a set of inequalities in the parameters of the model and have identified the intensity of the attraction as a key parameter governing the transition between different cluster structures the versatility of the model will be illustrated by discussing its relation to the kuramoto model and by describing how it applies to a system of interconnected water basins

acceleration of stochastic approximation by averaging a new recursive algorithm of stochastic approximation type with the averaging of trajectories is investigated convergence with probability one is proved for a variety of classical optimization and identification problems it is also demonstrated for these problems that the proposed algorithm achieves the highest possible rate of convergence

dynamic pricing with multiple products and partially specified demand distribution we study a dynamic pricing problem with multiple products and infinite inventories the demand for these products depends on the selling prices and on parameters unknown to the seller their value can be learned from accumulating sales data using statistical estimation techniques the quality of the parameter estimates is influenced by the amount of price dispersion however a large amount of variation in the selling prices can be costly since it means that suboptimal prices are used the seller thus needs to balance optimizing the quality of the parameter estimates and optimizing instant revenue i e exploitation and exploration in this study we propose a pricing policy for this dynamic pricing problem the key idea is to use at each time period the price that is optimal with respect to current parameter estimates with an additional constraint that ensures sufficient price dispersion we measure the price dispersion by the smallest eigenvalue of the design matrix and show how a desired growth rate of this eigenvalue can be achieved by a simple quadratic constraint in the price optimization problem we study the performance of our pricing policy by providing bounds on the regret which measures the expected revenue loss caused by using suboptimal prices

design of asymptotically convergent frequency estimator using contraction theory the problem of estimating unknown frequencies of a sinusoidal signal simultaneously is a classical problem in signal and system theory many approaches and algorithms are proposed in literature to develop estimators for a measurable sinusoidal signal having multiple sinusoids with unknown amplitudes frequencies and phases in this work an asymptotically convergent frequency estimator is given for estimation of n unknown frequencies of a measurable sinusoidal signal the contraction theory approach is adopted to show the asymptotic convergence of the proposed estimator in quite a simplified manner approach given here exploits the results of contraction theory related to semi contracting systems the nonlinear estimator based on dynamic system approach guarantees global boundedness and convergence of the state and frequency estimation for all initial conditions and frequency values it further ensures simultaneous globally convergent estimation of states and frequencies of a sinusoid involving multiple frequencies numerical simulations are presented for different combinations of frequencies to justify the claim

perishability of data dynamic pricing under varying coefficient models we consider a firm that sells a large number of products to its customers in an online fashion each product is described by a high dimensional feature vector and the market value of a product is assumed to be linear in the values of its features parameters of the valuation model are unknown and can change over time the firm sequentially observes a product s features and can use the historical sales data binary sale no sale feedbacks to set the price of current product with the objective of maximizing the collected revenue we measure the performance of a dynamic pricing policy via regret which is the expected revenue loss compared to a clairvoyant that knows the sequence of model parameters in advance we propose a pricing policy based on projected stochastic gradient descent psgd and characterize its regret in terms of time t features dimension d and the temporal variability in the model parameters t we consider two settings in the first one feature vectors are chosen antagonistically by nature and we prove that the regret of psgd pricing policy is of order o t t 1t t t in the second setting referred to as stochastic features model the feature vectors are drawn independently from an unknown distribution we show that in this case the regret of psgd pricing policy is of order o d2 log t t 1t t t d