on the existence of weak solutions of perturbated systems with critical growth sur un domaine borne de r n on considere le probleme de dirichlet pour le systeme elliptique a u g u f sur et u 0 ou a u est un operateur quasi lineaire et la perturbation g u depend de u et de d u on etudie des conditions suffisantes sur a u et sur g u pour l existence de solutions faibles

strict lyapunov functions for time varying systems with persistency of excitation we study the stability of the origin for a class of linear time varying systems with a drift that may be divided in two parts under the action of the first a function of the trajectories is guaranteed to converge to zero under the action of the second the solutions are restricted to a periodic orbit hence by assumption the systems trajectories are bounded our main results focus on two generic case studies that are motivated by common nonlinear control problems model reference adaptive control control of nonholonomic systems tracking control problems to name a few then based on the standing assumption that the systems dynamics is persistently excited we construct a time dependent lyapunov function that has a negative definite derivative our main statements may be regarded as off the shelf tools of analysis for linear and nonlinear time varying systems

quantum mechanical scattering by impenetrable periodic surfaces in this paper we investigate the existence and completeness of the wave operators w s limt exp ith p exp ith0 corresponding to the quantum mechanical scattering of nonrelativistic particles by certain classes of impenetrable noncompact surfaces bounding domains r 2 which contain a half space and are contained in another half space here h0 is the usual negative distributional laplacian in h0 l2 r h is the negative dirichlet laplacian in h l2 and p is an appropriate identification operator under these conditions we prove by elementary methods that w exist as partially isometric operators whose initial sets have a transparent physical meaning suppose now that the domain r also has the periodicity property x x x l x when l ranges over a bravais lattice in r 1 where we write x r as x x with x r 1 and x r then a ranw hscatt h and b w are asymptotically complete in the sense that h hscatt h hsurf h here hscatt h and hsur

quasilinear elliptic equations with quadratic growth in unbounded domains on donne des resultats d existence et de regularite pour des equations elliptiques quasilineaires a croissance quadratique par rapport au gradient du type au f x u grad u 0 dans u 0 sur

singular quasilinear equations with quadratic growth in the gradient without sign condition given a bounded domain in r n a nd af unctiona l q with q n 2 we study the existence of a positive solution for the quasilinear problem

quasilinear elliptic problems with singular and homogeneous lower order terms abstract we deal with singular quasilinear elliptic equations namely u u x u q u q 1 f x in u 0 in u 0 on where is a bounded smooth domain of r n n 3 r 1 q 2 0 l and 0 f l p for some p n 2 we completely describe the set of values of the parameter for which the problem admits solution thus we study existence nonexistence and uniqueness of bounded weak solutions in both cases f 0 and f 0

uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions in this paper we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions

multiplicity results in the non coercive case for an elliptic problem with critical growth in the gradient abstract we consider the boundary value problem p u c x u x u 2 h x u h 0 1 l where r n n 3 is a bounded domain with smooth boundary it is assumed that c h belong to l p for some p n with c 0 as well as l and 1 0 for some 1 r it is known that when 0 problem p has at most one solution in this paper we study under various assumptions the structure of the set of solutions of p assuming that 0 our study unveils the rich structure of this problem we show in particular that what happen for 0 influences the set of solutions in all the half space 0 h 0 1 l most of our results are valid without assuming that h has a sign if we require h to have a sign we observe that the set of solutions differs completely for h 0 and h 0 we also show when h has a sign that solutions not having this sign may exists some uniqueness results of signed solutions are also derived the paper ends with a list of open problems

nonlinear elliptic equations with singular boundary conditions and stochastic control with state constraints hamiltoniens sousquadratiques conditions aux limites infinies et donnees d explosion hamiltoniens superquadratiques formulation viscosite des conditions aux limites le probleme ergodique commande stochastique optimale avec contraintes d etat

non linear elliptic and parabolic equations involving measure data in this paper we prove the existence of solutions for equations of the type div a du f in a bounded open set u 0 on where a is a possibly non linear function satisfying some coerciveness and monotonicity assumptions and f is a bounded measure we also consider the equation div a du g u f in u 0 on with f l1 or f m u u 0 and the parabolic equivalent of the first elliptic equation

lyapunov conditions for finite time stability of time varying time delay systems in this paper we develop the lyapunov razumikhin method to finite time stability fts and finite time contractive stability ftcs of time delay systems several lyapunov based sufficient conditions for establishing these fts properties are obtained then the theoretical results are applied to fts and ftcs for a class of linear time varying ltv time delay system the efficiency of the proposed criteria is illustrated by three numerical examples where a stabilizing memoryless controller for ftcs of a second order ltv system with time delay is proposed

solvability of perturbed elliptic equations with critical growth exponent for the gradient sur un domaine borne cr n on considere le probleme de dirichlet a u g u f u 0 ou a u t 1 n 1 d i a i x u d u definit un operateur pseudomonotone d un espace de sobolev h 0 1 p dans son espace dual et la perturbation g u g x u d u satisfait une condition de croissance

razumikhin type theorems for time delay systems with persistent impulses abstract the main results of the paper are generalizations of the razumikhin classical theorems for stability analysis of impulsive time delay systems by employing lyapunov razumikhin method and impulsive control theory several razumikhin type theorems for uniform stability and global exponential stability are obtained the significance and novelty of the results lie in that the criteria for stability of impulsive time delay systems admit the existence of persistent impulses and unbounded time varying delays two examples are provided to show the effectiveness of the proposed results

existence results for some quasilinear parabolic equations on considere le probleme de cauchy pour une equation dont la partie principale est un operateur parabolique quasilineaire le terme non lineaire additionnel a au plus une croissance quadratique par rapport au gradient de la solution on demontre l existence d une solution bornee faible sous des hypotheses de regularite minimales en supposant l existence d une sous solution et d une supersolution pour le probleme

continuum of solutions for an elliptic problem with critical growth in the gradient abstract we consider the boundary value problem p u h 0 1 l u c x u x u 2 h x where r n n 3 is a bounded domain with smooth boundary it is assumed that c 0 c h belong to l p for some p n 2 and that l we explicitly describe a condition which guarantees the existence of a unique solution of p when 0 and we show that these solutions belong to a continuum the behaviour of the continuum depends in an essential way on the existence of a solution of p 0 it crosses the axis 0 if p 0 has a solution otherwise it bifurcates from infinity at the left of the axis 0 assuming that p 0 has a solution and strengthening our assumptions to x 1 0 and h 0 we show that the continuum bifurcates from infinity on the right of the axis 0 and this implies in particular the existence of two solutions for any 0 sufficiently small

elliptic problems with nonlinear terms depending on the gradient and singular on the boundary abstract in this paper we consider the problem p u u q u q f x in u 0 on where r n is a bounded domain 1 q 2 r and f 0 we prove that 1 if q 1 then problem p has a distributional solution for all f l 1 and all 0 2 if 1 q 0 then problem p has a solution for all f l s where s n q if n 2 and without any restriction on 3 if q 2 and 1 q 0 then problem p has infinitely many solutions under suitable hypotheses on f 4 if 0 q and f l 1 satisfies 1 f inf w 0 1 2 2 d x f 2 d x 0 then problem p has a positive solution if 0 1 f and no positive solution for large

lower semicontinuity of functionals via the concentration compactness principle in this paper we show the weak lower semicontinuity of some classes of functionals using the concentration compactness principle of p l lions these functionals involve an integral term and we do not know whether it can be handled by the de giorgi theorem the semicontinuity result allows us to prove several existence results for quasilinear elliptic equations

nonlinear elliptic systems in stochastic game theory on etudie des systemes d equations aux derivees partielles non lineaires provenant de la theorie des jeux differentiels stochastiques au sens de nash

scattering by periodic surfaces if r 2 is an exterior domain containing a half space and contained in a half space we prove that the wave operators w s limt exp ith jexp ith0 are partial isometries and that the invariance principle w w holds for suitable real functions on r admissible functions here h0 is the negative distributional laplacian in l2 r 2 h is hd or hn the negative dirichlet or neumann laplacians in h l2 respectively j is an appropriate identification operator and w are defined as were w but with h0 and h replaced by h0 and h respectively suppose in addition that has a suitable periodicity property and that when h hn it has a certain mild local compactness property then we prove 1 that w are asymptotically complete in the sense that ran w hscatt h h hsurf h 2 that hscatt h hscatt h and hsurf h hsurf h for each admissible function and hence that w are asymptotically complete in a similar sense for each such

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

some new criteria on p th moment stability of stochastic functional differential equations with markovian switching this note gives some new razumikhin type theorems on th moment stability of stochastic functional differential equations with markovian switching sfdewms by using auxiliary ordinary differential equation the main results of this note allow the diffusion operator associated with the underlying sfdewms of the lyapunov function along a solution of the system to be not always negative an example is provided to illustrate the effectiveness of the proposed results

stabilizability of systems with exponential dichotomy in this paper we introduce the concept of controllability into a closed subspace for time varying linear systems various characterizations are given and the dual relation is discussed this concept is used to present a necessary and sufficient condition for the stabilizability of systems with exponential dichotomy

robustness of stability of time varying linear systems abstract this paper introduces the concept of stability radius for time varying linear systems invariance properties of the stability radius are analysed for the group of bohl transformations we also explore the relationship between the stability radius the norm of a certain perturbation operator and the solvability of a nonstandard differential riccati equation as an application we construct robust lyapunov functions and show how they can be used to analyze robustness with respect to nonlinear perturbations

existence and non existence of positive solutions for nonlinear elliptic singular equations with natural growth abstract in this paper we analyze the existence non existence and uniqueness of positive solutions of some nonlinear elliptic equations containing singular terms and natural growth in the gradient term we use an adequate sub super solution method to prove the existence of solutions the characterization of eigenvalues and the integrability of the term u 2 u 2 for the non existence and results from arcoya segura de leon for the uniqueness

on asymptotic stability of linear time varying systems this paper is concerned with asymptotic stability analysis of linear time varying ltv systems with the help of the notion of stable functions some differential lyapunov inequalities dlis based necessary and sufficient conditions are derived for testing asymptotic stability exponential stability and uniformly exponential stability of general ltv systems with the help of the concept of non uniformly exponential stability a class of upper triangular ltv systems is carefully investigated based on the developed stability analysis approaches a couple of numerical examples with some of them borrowed from the literature is provided to illustrate the effectiveness of the proposed theoretical results

a priori estimates and bifurcation of solutions for an elliptic equation with semidefinite critical growth in the gradient abstract we study nonnegative solutions of the boundary value problem p u c x u x u 2 h x u h 0 1 l where is a smooth bounded domain of r n c l h l r for some r n 2 and c h 0 our main motivation is to study the semidefinite case namely unlike in previous work on the subject we do not assume to be uniformly positive in nor even positive everywhere in space dimensions up to n 5 we establish uniform a priori estimates for weak solutions of p when 0 is bounded away from 0 this is proved under the assumption that the supports of and c intersect a condition that we show to be actually necessary and in some cases we further assume that is uniformly positive on the support of c and or some other conditions as a consequence of our a priori estimates assuming that p 0 has a solution we deduce the existence of a continuum c of solutions such that the projection of c onto the axis is an interval of the form 0 a for some a 0 and that the continuum c bifurcates from infinity to the right of the axis 0 in particular for each 0 small enough problem p has at least two distinct solutions

lyapunov functions for persistently excited cascaded time varying systems application to consensus we present some results on stability of linear time varying systems with particular structures such systems appear in diverse problems which include the analysis of adaptive systems persistently excited observers and consensus of systems interconnected through time varying links the originality of our statements rely in the fact that we provide smooth strict lyapunov functions hence our proofs are constructive and direct moreover we establish uniform global exponential stability with explicit stability and decay estimates for illustration we address a brief but representative case study of consensus of lagrangian systems interconnected through unreliable links

regularizing effect for some p laplacian systems abstract we study existence and regularity of weak solutions for the following p laplacian system p u a 1 u r 2 u f u w 0 1 p p u r w 0 1 p where is an open bounded subset of r n n 2 p v div v p 2 v is the p laplacian operator for 1 p n a 0 r 1 0 p 1 and f belongs to a suitable lebesgue space in particular we show how the coupling between the equations in the system gives rise to a regularizing effect producing the existence of finite energy solutions

stability analysis of linear time varying time delay systems by non quadratic lyapunov functions with indefinite derivatives abstract this paper presents a stability analysis of linear time varying ltv time delay systems by using non quadratic lyapunov functions and functionals two types of sufficient conditions are proposed for testing several different kinds of stability such as asymptotic stability exponential stability and uniformly exponential stability for ltv systems without delay then by constructing suitable non quadratic lyapunov functions functionals that are respectively the time varying weighted l 1 and l norms of the state variables and by using properties of the uniformly asymptotically stable uas function and the recently established improved razumikhin and krasovskii stability theorems both delay dependent and delay independent conditions are proposed for testing uniformly exponential stability of a class of ltv time delay systems the time derivatives of the non quadratic lyapunov functions functionals along the solutions are allowed to be indefinite namely to take both negative and positive values numerical examples show that the non quadratic lyapunov functions functionals based methods are more efficient than the existing ones that are based on quadratic functions functionals

uniform global asymptotic stability of a class of adaptively controlled nonlinear systems we give a new explicit global strict lyapunov function construction for the error dynamics for adaptive tracking control problems under an appropriate persistency of excitation condition we then allow time varying uncertainty in the unknown parameters in this case we construct input to state stable lyapunov functions under suitable bounds on the uncertainty provided the regressor also satisfies an affine growth condition this lets us quantify the effects of uncertainties on both the tracking and the parameter estimation we illustrate our results using rossler systems

scattering theory for high order operators in domains with infinite boundary abstract a general scattering theory is studied for high order elliptic operators in domains with infinite boundary

uniqueness results for nonlinear elliptic equations with a lower order term abstract we prove the uniqueness of the weak solution of noncoercive nonlinear elliptic problems whose model is u w 0 1 p div a x 1 u 2 p 2 2 u b x 1 u 2 1 2 f in d where is a bounded open subset of r n n 2 p satisfies p 2 n n 1 a is a function belonging to l such that a x 0 f belongs to the dual space w 1 p b belongs to some lebesgue space l r with r r n p and belongs to the interval 0 n p r with n p r and r n p functions which are specified below

solvability of uniformly elliptic fully nonlinear pde we get existence uniqueness and non uniqueness of viscosity solutions of uniformly elliptic fully nonlinear equations of the hamilton jacobi bellman isaacs type with unbounded ingredients and quadratic growth in the gradient without hypotheses of convexity or properness some of our results are new even for equations in divergence form

existence and nonexistence of solutions for singular quadratic quasilinear equations we study both existence and nonexistence of nonnegative solutions for nonlinear elliptic problems with singular lower order terms that have natural growth with respect to the gradient whose model is 8

construction of strict lyapunov krasovskii functionals for time varying time delay systems abstract for stability analysis of time varying time delay systems it is known that time derivatives time shifts in the discrete time setting of the lyapunov krasovskii functionals lkfs for both internal stability and input to state stability iss can be indefinite which can improve the resulting stability conditions in some cases however the non strict lkfs may be insufficient and inconvenient to use in practice in this paper based on the non strict iss lkfs for time varying time delay systems and with the help of uniformly asymptotically stable uas and uniformly exponentially bounded ueb scalar functions three classes of strict iss lkfs are constructed such that their time derivatives are strictly negative definite along the trajectories of the considered system a general construction of strict iss lkfs by using the positive definite and uniformly bounded solution to a scalar lyapunov differential equation is established which includes the proposed three classes of strict iss lkfs as special cases the approaches are also extended to deal with a lyapunov inequality whose right hand side is indefinite and nonlinear both continuous time and discrete time systems are considered the effectiveness of the proposed methods is illustrated by some examples borrowed from the literature

input to state stability of nonlinear systems based on an indefinite lyapunov function abstract this study investigated the input to state stability iss and integral iss iiss of nonlinear systems and discovered new sufficient conditions for them these conditions are more relaxed than others in that they employ an indefinite lyapunov function rather than a negative definite one thus a nonlinear time varying system satisfying them has uniform asymptotic stability two numerical examples show their effectiveness and advantages

on construction of lyapunov functions for scalar linear time varying systems abstract it is known that the construction of lyapunov functions for scalar linear time varying systems is related with solutions to the scalar lyapunov differential equation whose solution involves both improper integrals and double integrals and thus are not easy to compute in general this paper establishes a systematic method for constructing lyapunov functions for scalar linear time varying systems the constructed lyapunov functions involve an integral of the system parameter with a weighting function over a finite interval explicit conditions are imposed on the weighting function and the integral interval such that the lyapunov function is both positive definite and uniformly bounded and its time derivative is negative definite as a result constructive solutions to the associated scalar lyapunov differential equations are also obtained the established method includes some existing ones as special cases examples demonstrate the effectiveness of the proposed methods

radial solutions of the dirichlet problem for a class of quasilinear elliptic equations arising in optometry abstract this paper deals with the quasilinear elliptic problem div u 1 u 2 a x u b x 1 u 2 in b u 0 on b where b is an open ball in r n with n 2 and a b c 1 b are given radially symmetric functions with a x 0 in b this class of anisotropic prescribed mean curvature equations appears in the description of the geometry of the human cornea as well as in the modeling theory of capillarity phenomena for compressible fluids unlike all previous works published on these subjects existence and uniqueness of solutions of the above problem are here analyzed in the case where the coefficients a b are not necessarily constant and no sign condition is assumed on b

existence of solutions for some doubly degenerate parabolic equations with natural growth terms abstract this paper deals with a class of doubly degenerate parabolic equations of the type b u t div a x t u u f x t u u f where b is a strictly increasing c 1 function under some suitable hypothesis we prove that there exists at least a bounded solution to this problem in standard weak sense

optimal results for parabolic problems arising in some physical models with critical growth in the gradient respect to a hardy potential abstract we deal with the following parabolic problem u t u u p u x 2 f u 0 in 0 t u x t 0 on 0 t u x 0 u 0 x x where r n n 3 is a bounded regular domain such that 0 or r n p 1 0 and f 0 u 0 0 are in a suitable class of functions there are deep differences with respect to the heat equation 0 the main features in the paper are the following if 0 there exists a critical exponent p such that for p p there is no nontrivial local solution p is optimal in the sense that if p p there exists solution for suitable data if we consider the cauchy problem i e r n we find the same phenomenon about the critical power p as above moreover there exists a fujita type exponent f p in the sense that independently of the initial datum for 1 p f any solution blows up in a finite time respect to an integral norm this is a major difference with respect to the heat equation 0

on the estimation of the consensus rate of convergence in graphs with persistent interconnections abstractthe aim of the current article is to establish myriad convergence rate estimates to consensus for time varying graphs with persistent interaction several novel analysis methodologies for consensus protocols employing the notions of persistence of excitation and lyapunov functions are provided the estimates are compared with each other and existing literature numerical simulations on test examples are illustrated to support the theoretical findings

a unified razumikhin type criterion on input to state stability of time varying impulsive delayed systems abstract this paper discusses the problem of input to state stability iss of time varying impulsive delayed systems by introducing a switching parameter and using the notion of average impulsive interval a unified razumikhin type criterion on iss which is simultaneously effective for stabilizing impulses and destabilizing impulses is derived the condition which requires the coefficient of the estimated upper bound of the derivative of a lyapunov function to be constant in the existing results on iss of impulsive systems is weakened the results in this paper allow the coefficient of the derivative of a lyapunov function to be time varying function which can be both positive and negative and may even be unbounded furthermore the impulsive intervals of an impulsive sequence are allowed to have arbitrarily small lower bound and large enough upper bound simultaneously as a by product a unified criterion on iss for time varying impulsive delay free systems is also presented two examples are presented to illustrate the effectiveness of our results

the p dirichlet to neumann operator with applications to elliptic and parabolic problems abstract in this paper we investigate the dirichlet to neumann operator associated with second order quasi linear operators of p laplace type for 1 p which acts on the boundary of a bounded lipschitz domain in r d for d 2 we establish well posedness and holder continuity with uniform estimates of weak solutions of some elliptic boundary value problems involving the dirichlet to neumann operator by employing these regularity results of weak solutions of elliptic problems we show that the semigroup generated by the negative dirichlet to neumann operator on l q enjoys an l q c 0 smoothing effect and the negative dirichlet to neumann operator on the set of continuous functions on the boundary of the domain generates a strongly continuous and order preserving semigroup moreover we establish convergence in large time with decay rates of all trajectories of the semigroup and in the singular case 1 e 2 d d 2 p 2 for some e 0 we give upper estimates of the finite time of extinction

quantum mechanical scattering in exterior domains with impenetrable periodic boundaries and short range potentials we study the scattering of a nonrelativistic particle in an exterior domain open connected subset r 2 containing a half space and contained in another half space and having an impenetrable periodic boundary impenetrable means that generalized homogeneous dirichlet conditions are imposed on we prove the existence and completeness of the wave operators w limt exp ith1 p exp ith0 corresponding to the scattering of a nonrelativistic particle in by the combined effect of the boundary and a short range potential present in here h0 is the negative distributional laplacian in the hilbert space h0 l2 r h1 d v d being the dirichlet laplacian in the hilbert space h l2 v an operator of multiplication in h by a bounded measurable function v x on having the periodicity of the boundary and p h0 h an identification operator the operators w model the quantum mechanical scattering of low energy atoms by crystal surfaces with v modeling the interaction between the

brief strict lyapunov functions for time varying systems uniformly asymptotically stable periodic time varying systems for which is known a lyapunov function with a derivative along the trajectories non positive and negative definite in the state variable on non empty open intervals of the time are considered for these systems strict lyapunov functions are constructed

remarks on the uniqueness for quasilinear elliptic equations with quadratic growth conditions abstract in this note we present some uniqueness and comparison results for a class of problem of the form 0 1 l u h x u u h x u h 0 1 l where r n n 2 is a bounded domain l is a general elliptic second order linear operator with bounded coefficients and h is allowed to have a critical growth in the gradient in some cases our assumptions prove to be sharp

some elliptic problems with singular natural growth lower order terms abstract in this paper we deal with a nonlinear elliptic problem whose model is u b d u 2 u f in u 0 on where b 0 and f 0 belongs to a lebesgue space we prove the existence of positive solutions in suitable sobolev spaces depending on f and b