sharp constant for a 2d anisotropic sobolev inequality with critical nonlinearity abstract for the 2 dimensional anisotropic sobolev inequality of the form r 2 u 6 d x d y r 2 u x 2 d x d y 2 r 2 d x 1 u y 2 d x d y it is proved that the sharp smallest positive constant is exactly as 3 r 2 x 2 d x d y 2 where is a minimal action solution of u x x u 4 u x d x 1 u y y

blow up and instability of solitary wave solutions to a generalized kadomtsev petviashvili equation in this paper we consider a generalized kadomtsev petviashvili equation in the form ut uxxx upux uyy x y e r2 t 0 it is shown that the solutions blow up in finite time for the supercritical power of nonlinearity p 4 3 with p the ratio of an even to an odd integer moreover it is shown that the solitary waves are strongly unstable if 2 p 4 that is the solutions blow up in finite time provided they start near an unstable solitary wave

discrete time asymptotic controllability implies smooth control lyapunov function we demonstrate the existence of a smooth control lyapunov function clf for difference equations asymptotically controllable to closed sets we further show that this clf may be used to construct a robust feedback stabilizer the existence of such a clf is a consequence of a more general result on the existence of weak lyapunov function under the assumption of weak asymptotic stability of a closed not necessarily compact set for a difference inclusion

multirate sampling and input to state stable receding horizon control for nonlinear systems abstract in this paper a robust receding horizon control for multirate sampled data nonlinear systems with bounded disturbances is presented the proposed receding horizon control is based on the solution of bolza type optimal control problems for the approximate discrete time model of the nominal system low measurement rate is assumed it is shown that the multistep receding horizon controller that stabilizes the nominal approximate discrete time model also practically input to state stabilizes the exact discrete time system with disturbances

subharmonic oscillations of forced pendulum type equations where f is periodic with minimal period t and mean value zero we have in mind as a particular case the pendulum equation where g x a sin x first results on the existence of subharmonic orbits in a neighborhood of a given periodic motion were obtained by birkhoff and lewis cf 3 and 143 by perturbation type techniques rabinowitz 151 was able to prove the existence of subharmonic solutions for hamiltonian systems by the use of variational methods his approach is not of local type like the one in 3 and enables one to obtain a sequence of solutions whose minimal period tends toward infinity in the case when the hamiltonian function has subquadratic or superquadratic growth these results have been extended in various directions cf 2 5 6 8 13 16 181 local results on subharmonies for the forced pendulum equation can be found in 19 hamiltonian systems with periodic nonlinearity were studied by conley and zehnder 6 they proved the existence of subharmonic solutions under some assumptions on the nondegenerateness of the solutions by the use of morse conley theory in this paper we will prove the existence of subharmonic oscillations of a pendulum type equation by the use of classical morse theory together with an iteration formula for the index due to bott 4 and developed in 7 and l

periodic solutions for nonlinear second order difference equations abstract this paper is devoted to introduce a new approach to investigate the existence of periodic solutions for nonlinear difference equations transforming a second order nonlinear difference equation on a finite discrete segment with periodic boundary conditions into a continuous system we obtain the existence conditions of periodic solutions of the equation the proof relies on coincidence degree theory and matrix theory without using green s function

on uniform asymptotic stability of time varying parameterized discrete time cascades recently a framework for controller design of sampled data nonlinear systems via their approximate discrete time models has been proposed in the literature in this paper we develop novel tools that can be used within this framework and that are useful for tracking problems in particular results for stability analysis of parameterized time varying discrete time cascaded systems are given this class of models arises naturally when one uses an approximate discrete time model to design a stabilizing or tracking controller for a sampled data plant while some of our results parallel their continuous time counterparts the stability properties that are considered the conditions that are imposed and the the proof techniques that are used are tailored for approximate discrete time systems and are technically different from those in the continuous time context a result on constructing strict lyapunov functions from nonstrict ones that is of independent interest is also presented we illustrate the utility of our results in the case study of the tracking control of a mobile robot this application is fairly illustrative of the technical differences and obstacles encountered in the analysis of discrete time parameterized systems

optimization based stabilization of sampled data nonlinear systems via their approximate discrete time models we present results on numerical regulator design for sampled data nonlinear plants via their approximate discrete time plant models the regulator design is based on an approximate discrete time plant model and is carried out either via an infinite horizon optimization problem or via a finite horizon with terminal cost optimization problem in both cases we discuss situations when the sampling period t and the integration period h used in obtaining the approximate discrete time plant model are the same or they are independent of each other we show that using this approach practical and or semiglobal stability of the exact discrete time model is achieved under appropriate conditions

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

survey constrained model predictive control stability and optimality model predictive control is a form of control in which the current control action is obtained by solving at each sampling instant a finite horizon open loop optimal control problem using the current state of the plant as the initial state the optimization yields an optimal control sequence and the first control in this sequence is applied to the plant an important advantage of this type of control is its ability to cope with hard constraints on controls and states it has therefore been widely applied in petro chemical and related industries where satisfaction of constraints is particularly important because efficiency demands operating points on or close to the boundary of the set of admissible states and controls in this review we focus on model predictive control of constrained systems both linear and nonlinear and discuss only briefly model predictive control of unconstrained nonlinear and or time varying systems we concentrate our attention on research dealing with stability and optimality in these areas the subject has developed in our opinion to a stage where it has achieved sufficient maturity to warrant the active interest of researchers in nonlinear control we distill from an extensive literature essential principles that ensure stability and use these to present a concise characterization of most of the model predictive controllers that have been proposed in the literature in some cases the finite horizon optimal control problem solved on line is exactly equivalent to the same problem with an infinite horizon in other cases it is equivalent to a modified infinite horizon optimal control problem in both situations known advantages of infinite horizon optimal control accrue

multirate versions of sampled data stabilization of nonlinear systems the problem of state feedback sampled data stabilization of nonlinear systems is considered under the low measurement rate constraint and in the presence of not necessarily small time delay in the measurement channel a multirate control scheme is proposed that utilizes a numerical integration scheme to approximately predict the current state from the delayed measurements for both the controller emulation approach and the approach based on approximate discrete time model of the system we show that under standard assumptions the closed loop multirate sampled data system is asymptotically stable in the semiglobal practical sense an illustrative example of sampled data control of vertical take off and landing aircraft is discussed

brief paper continuous time controller redesign for digital implementation a trajectory based approach given a continuous time nonlinear closed loop system we investigate sampled data feedback laws for which the trajectories of the sampled data closed loop system converge to the continuous time trajectories with a prescribed rate of convergence as the length of the sampling interval tends to zero we derive necessary and sufficient conditions for the existence of such sampled data feedback laws and in case of existence provide explicit redesign formulas and algorithms for these controllers

periodic and subharmonic solutions for a class of superquadratic hamiltonian systems abstract some existence theorems are obtained for periodic and subharmonic solutions of a class of superquadratic hamiltonian systems by the minimax methods in critical point theory specially a local link theorem due to li and willem and the generalized mountain pass theorem respectively

robust global stabilisability by means of sampled data control with positive sampling rate this work proposes a notion of robust reachability of one set from another set under constant control this notion is used to construct a control strategy involving sequential set to set reachability which guarantees robust global stabilisation of non linear sampled data systems with positive sampling rate sufficient conditions for robust reachability of one set from another under constant control are also provided the proposed method is illustrated through a number of examples including the study of the sampled data stabilisation problem of the chemostat

stabilization of sampled data nonlinear systems by receding horizon control via discrete time approximations abstract results on stabilizing receding horizon control of sampled data nonlinear systems via their approximate discrete time models are presented the proposed receding horizon control is based on the solution of bolza type optimal control problems for the parametrized family of approximate discrete time models this paper investigates both situations when the sampling period t is fixed and the integration parameter h used in obtaining approximate model can be chosen arbitrarily small and when these two parameters coincide but they can be adjusted arbitrary sufficient conditions are established which guarantee that the controller that renders the origin to be asymptotically stable for the approximate model also stabilizes the exact discrete time model for sufficiently small integration and or sampling parameters

periodic oscillations for a nonlinear suspension bridge model abstract we look for time periodic solutions of the suspension bridge equation lazer and mckenna showed that for a certain configuration of the parameters one may expect the existence of large amplitude periodic solutions having the same period as the forcing term we prove the existence of large scale subharmonic solutions

stabilizing receding horizon control of nonlinear time varying systems a receding horizon control scheme for nonlinear time varying systems is proposed which is based on a finite horizon optimization problem with a terminal state penalty the penalty is equal to the cost that would be incurred over an infinite horizon by applying a locally stabilizing linear control law to the nonlinear system assuming only stabilizability of the linearized system around the desired equilibrium the new scheme ensures exponential stability of the equilibrium as the length of the optimization horizon goes from zero to infinity the domain of attraction moves from the basin of attraction of the linear controller toward the basin of attraction of the infinite horizon nonlinear controller stability robustness in the face of system perturbations is also established

subharmonics for convex nonautonomous hamiltonian systems on etudie des solutions periodiques du systeme hamiltonien jx h t x ou j est la matrice symplectique standard et h c 2 r r 2n r est t periodique en la premiere variable pour tout t x h t x note le gradient partiel par rapport a la seconde variable

on stability of sets for sampled data nonlinear inclusions via their approximate discrete time models and summability criteria this paper consists of two main parts in the first part we provide a framework for stabilization of arbitrary not necessarily compact closed sets for sampled data nonlinear differential inclusions via their approximate discrete time models we generalize d nesic a r teel and p v kokotovic systems control lett 38 1999 pp 259 270 theorem 1 in several different directions we consider stabilization of arbitrary closed sets plants described as sampled data differential inclusions and arbitrary dynamic controllers in the form of difference inclusions our result does not require the knowledge of a lyapunov function for the approximate model which is a standing assumption in d nesic and a r teel ieee trans automat control 49 2004 pp 1103 1122 and d nesic a r teel and p v kokotovic systems control lett 38 1999 pp 259 270 theorem 2 we present checkable conditions that one can use to conclude semiglobal asymptotic stability or global exponential stability ges of the sampled data system via appropriate properties of its approximate discrete time model in the second part we present sufficient conditions for stability of parameterized difference inclusions that involve various summability criteria on trajectories of the system to conclude global asymptotic stability gas or ges and they represent discrete time counterparts of results given in a r teel e panteley and a loria math control signals systems 15 2002 pp 177 201 these summability criteria are not lyapunov based and they are tailored to be used within our above mentioned framework for stabilization of sampled data differential inclusions via their approximate discrete time models we believe that these tools will be a useful addition to the toolbox for controller design for sampled data nonlinear systems via their approximate discrete time models

sufficient conditions for stabilization of sampled data nonlinear systems via discrete time approximations given a parameterized by sampling period t family of approximate discrete time models of a sampled nonlinear plant and given a family of controllers stabilizing the family of plant models for all t sufficiently small we present conditions which guarantee that the same family of controllers semi globally practically stabilizes the exact discrete time model of the plant for sufficiently small sampling periods when the family of controllers is locally bounded uniformly in the sampling period the inter sample behavior can also be uniformly bounded so that the time varying sampled data model of the plant is uniformly semi globally practically stabilized the result justifies controller design for sampled data nonlinear systems based on the approximate discrete time model of the system when sampling is sufficiently fast and the conditions we propose are satisfied our analysis is applicable to a wide range of time discretization schemes and general plant models

an existence theorem for discrete time infinite horizon optimal control problems an existence theorem is given for a general class of deterministic infinite horizon discrete time optimal control problems the hypotheses of the theorem are weak and can be easily verified the objective function may be either a summation or supremum over time

a general framework to design stabilizing nonlinear model predictive controllers abstract we propose a new model predictive control mpc framework to generate feedback controls for time varying nonlinear systems with input constraints we provide a set of conditions on the design parameters that permits to verify a priori the stabilizing properties of the control strategies considered the supplied sufficient conditions for stability can also be used to analyse the stability of most previous mpc schemes the class of nonlinear systems addressed is significantly enlarged by removing the traditional assumptions on the continuity of the optimal controls and on the stabilizability of the linearized system some important classes of nonlinear systems including some nonholonomic systems can now be stabilized by mpc in addition we can exploit increased flexibility in the choice of design parameters to reduce the constraints of the optimal control problem and thereby reduce the computational effort in the optimization algorithms used to implement mpc

a framework for stabilization of nonlinear sampled data systems based on their approximate discrete time models a unified framework for design of stabilizing controllers for sampled data differential inclusions via their approximate discrete time models is presented both fixed and fast sampling are considered in each case sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete time plant models also stabilizes the exact discrete time plant model for sufficiently small integration and or sampling periods previous results in the literature are extended to cover 1 continuous time plants modeled as differential inclusions 2 general approximate discrete time plant models 3 dynamical discontinuous controllers modeled as difference inclusions and 4 stability with respect to closed arbitrary not necessarily compact sets

stabilization of strictly dissipative discrete time systems with discounted optimal control abstract we consider stabilization of an equilibrium point via infinite horizon discounted optimal control in discrete time in addition to applications in economics and social sciences discounted optimal control is a commonly used numerical technique guaranteeing solvability of certain classes of optimal control problems in this paper we present conditions based on strict dissipativity that ensure that the optimally controlled system is asymptotically stable or practically asymptotically stable these conditions are shown to be complementary to recently proposed conditions based on a detectability property illustrative examples are provided

applications of a zp index theory to periodic solutions for a class of functional differential equations abstract by means of a new geometrical index with z p group actions multiplicity results for a certain class of nonautonomous time periodic functional differential systems are obtained

solutions of minimal period for a class of nonconvex hamiltonian systems and applications to the fixed energy problem on considere le systeme hamiltonien jz h z ou j x y y x x y r n r n h est non convexe a un bon comportement superquadratique et satisfait une condition simple reliant ses coefficients de superquadraticite et l exposant dans ce cas on demontre que pour t 0 l existence de solutions pour le systeme hamiltonien de periode minimale t

multiplicity of positive periodic solutions in the superlinear indefinite case via coincidence degree abstract we study the periodic boundary value problem associated with the second order nonlinear differential equation u c u a t a t g u 0 where g u has superlinear growth at zero and at infinity a t is a periodic sign changing weight c r and 0 is a real parameter our model includes for c 0 the so called nonlinear hill s equation we prove the existence of 2 m 1 positive solutions when a t has m positive humps separated by m negative ones in a periodicity interval and is sufficiently large thus giving a complete solution to a problem raised by g j butler in 1976 the proof is based on mawhin s coincidence degree defined in open possibly unbounded sets and applies also to neumann boundary conditions our method also provides a topological approach to detect subharmonic solutions

subharmonics for not uniformly coercive hamiltonian systems abstract in this paper we prove the existence of subharmonic solutions with prescribed minimal period for the hamiltonian system u t j h t u t when the hamiltonian h is subquadratic with unbounded gradient and not uniformly coercive

subharmonic solutions with prescribed minimal period for nonautonomous hamiltonian systems on etablit un resultat de multiplicite pour une certaine classe de systemes hamiltoniens periodiques en temps

a set oriented approach to optimal feedback stabilization we present a numerical construction of an optimal control based feedback law for the stabilization of discrete time nonlinear control systems the feedback is based on a recently developed numerical solution method for optimal control problems using set oriented and graph theoretic algorithms we show how this method can be used to construct approximately optimal and stabilizing feedback laws and present an a posteriori error estimation technique for the adaptive generation of the underlying set oriented space discretization

homogeneous state feedback stabilization of homogenous systems we show that for any asymptotically controllable homogeneous system in euclidean space not necessarily lipschitz at the origin there exists a homogeneous control lyapunov function and a homogeneous possibly discontinuous state feedback law stabilizing the corresponding sampled closed loop system if the system satisfies the usual local lipschitz condition on the whole space we obtain semiglobal stability of the sampled closed loop system for each sufficiently small fixed sampling rate if the system satisfies a global lipschitz condition we obtain global exponential stability for each sufficiently small fixed sampling rate the control lyapunov function and the feedback are based on the lyapunov exponents of a suitable auxiliary system and admit a numerical approximation

formulas relating kl stability estimates of discrete time and sampled data nonlinear systems abstract we provide an explicit kl stability or input to state stability iss estimate for a sampled data nonlinear system in terms of the kl estimate for the corresponding discrete time system and a k function describing inter sample growth it is quite obvious that a uniform inter sample growth condition plus an iss property for the exact discrete time model of a closed loop system implies uniform iss of the sampled data nonlinear system our results serve to quantify these facts by means of comparison functions our results can be used as an alternative to prove and extend results in 1 or extend some results in 4 to a class of nonlinear systems finally the formulas we establish can be used as a tool for some other problems which we indicate

periodic and subharmonic solutions of a class of superquadratic hamiltonian systems abstract periodic solutions and infinitely distinct subharmonic solutions are obtained for a class of nonconvex and nonautonomous superquadratic hamiltonian systems z j h z z t by using the minimax methods in critical point theory

command governors for constrained nonlinear systems a method is described for set point tracking in nonlinear systems when pointwise in time input and or state inequality constraints are to be enforced it consists of adding to a primal compensated system a nonlinear device called command governor cg whose action is based on the current state set point and prescribed constraints the cg selects at any time the system input via a receding horizon strategy from a virtual sequence amongst all possible command sequences by solving a constrained quadratic optimization problem provided that the initial state is admissible the overall system is proved to fulfil the constraints and have desirable performance stability properties

stabilization of nonlinear delay systems using approximate predictors and high gain observers we provide a solution to the heretofore open problem of stabilization of systems with arbitrarily long delays at the input and output of a nonlinear system using output feedback only the solution is global employs the predictor approach over the period that combines the input and output delays addresses nonlinear systems with sampled measurements and with control applied using a zero order hold and requires that the sampling holding periods be sufficiently short though not necessarily constant our approach considers a class of globally lipschitz strict feedback systems with disturbances and employs an appropriately constructed successive approximation of the predictor map a high gain sampled data observer and a linear stabilizing feedback for the delay free system the obtained results guarantee robustness to perturbations of the sampling schedule and different sampling and holding periods are considered the approach is specialized to linear systems where the predictor is available explicitly

multiple periodic solutions for superquadratic second order discrete hamiltonian systems abstract some multiplicity results are obtained for periodic solutions of the nonautonomous superquadratic second order discrete hamiltonian systems 2 u t 1 f t u t 0 t z by using critical point theory especially a three critical points theorem proposed by brezis and nirenberg

small gain type sufficient conditions for sampled data feedback stabilization for autonomous composite systems small gain type sufficient conditions are established guaranteeing weak global asymptotic stabilization for composite autonomous systems by means of a sampled data feedback our main result generalizes the well known small gain theorem for global asymptotic stability for composite systems with no controls under the presence of the input to state stability the sufficient conditions we propose enable us to derive a set of algebraic sufficient conditions for the existence of a sampled data feedback stabilizer with zero order hold the work is a continuation of author s work published in same journal

a z p borsuk ulam theorem and index theory with a multiplicity result in partial differential equations on demontre un theoreme de borsuk ulam z p generalise c est le resultat topologique essentiel pour developper un theoreme d indice z p on demontre un resultat de multiplicite pour un systeme d equations aux derivees partielles elliptiques non lineaires

nonlinear oscillations and boundary value problems for hamiltonian systems we prove the existence of solutions of various boundary value problems for nonautonomous hamiltonian systems with forcing terms begin gathered dot x t h p t x t p t g t hfill dot p t h x t x t p t f t hfill end gathered