brief input to state stability of pd controlled robotic systems a sufficient condition for input to state stability is presented for systems admitting a lyapunov function with negative semi definite derivative an application to pd controlled robotic manipulators is also described

the discrete spectrum of the perturbed periodic schrodinger operator in the large coupling constant limit let a be a periodic schrodinger operator and let v 0 0 be a decaying potential we study the number of the eigenvalues of the operator a a v 0 inside a fixed interval 1 2 we obtain an asymptotic formula for as

singularities of energy minimizing maps from the ball to the sphere we study maps from the unit ball b in r3 to the unit sphere s2 in r3 which minimize dirichlet s energy integral varepsilon varphi int b nabla varphi 2 dv

trapping and cascading of eigenvalues in the large coupling limit we consider eigenvaluese of the hamiltonianh v w w compactly supported in the limit forw 0 we find monotonic convergence ofe to the eigenvalues of a limiting operatorh associated with an exterior dirichlet problem and we estimate the rate of convergence for 1 dimensional systems in 1 dimensional systems withw 0 or withw changing sign we do not find convergence instead we find a cascade phenomenon in which as each eigenvaluee stays near a dirichlet eigenvalue for a long interval of lengtho sqrt lambda of the scaling range quickly drops to the next lower dirichlet eigenvalue stays there for a long interval drops again and so on as a result for most large values of the discrete spectrum ofh is close to that ofe but when reaches a transition region the entire spectrum quickly shifts down by one we also explore the behavior of several explicit models as

observers to the aid of strictification of lyapunov functions abstract we present a procedure for modifying a weak lyapunov function v into a strict one for this we augment the given function with an auxiliary function v a obtained as a lyapunov function associated with the error dynamics given by an observer designed for the system with the derivative of v as an output function

a nested matrosov theorem and persistency of excitation for uniform convergence in stable nonautonomous systems a new infinitesimal sufficient condition is given for uniform global asymptotic stability ugas for time varying nonlinear systems it is used to show that a certain relaxed persistency of excitation condition called uniform spl delta persistency of excitation u spl delta pe is sufficient for uniform global asymptotic stability in certain situations u spl delta pe of the right hand side of a time varying differential equation is also shown to be necessary under a uniform lipschitz condition the infinitesimal sufficient condition for ugas involves the inner products of the flow field with the gradients of a finite number of possibly sign indefinite locally lipschitz lyapunov like functions these inner products are supposed to be bounded by functions that have a certain nested or triangular negative semidefinite structure this idea is reminiscent of a previous idea of matrosov who supplemented a lyapunov function having a negative semidefinite derivative with an additional function having a derivative that is definitely nonzero where the derivative of the lyapunov function is zero for this reason we call the main result a nested matrosov theorem the utility of our results on stability analysis is illustrated through the well known case study of the nonholonomic integrator

bound states for nano tubes with a dislocation abstract as a model for an interface in solid state physics we consider two real valued potentials v 1 and v 2 on the cylinder or tube s r r z where we assume that there exists an interval a 0 b 0 which is free of spectrum of v k for k 1 2 we are then interested in the spectrum of h t v t for t r where v t x y v 1 x y for x 0 and v t x y v 2 x t y for x 0 while the essential spectrum of h t is independent of t we show that discrete spectrum related to the interface at x 0 is created in the interval a 0 b 0 at suitable values of the parameter t provided v 2 has some essential spectrum in a 0 we do not require v 1 or v 2 to be periodic we furthermore show that the discrete eigenvalues of h t are lipschitz continuous functions of t if the potential v 2 is locally of bounded variation

partial regularity of minimizers of p x growth functionals with p x 1 abstract we prove partial regularity of minimizers u for functionals of the following type a u a i j x u d u i d u j p x 2 g x u d u d x assuming that a i j x u and p x are sufficiently smooth and that p x 1 we prove that u c 0 0 for some 0 1 and an open set 0 with h m 1 0 0 where h s denotes the s dimensional hausdorff measure and 1 inf p x

on the integrated density of states for crystals with randomly distributed impurities in the present paper we discuss spectral properties of a periodic schrodinger operator which is perturbed by randomly distributed impurities such operators occur as simple models for crystals or semi conductors with impurities while the spectrum itself is independent of the concentrationp of impurities for 0 p 1 we focus our attention on the limiting behavior of the integrated density of states p of the random schrodinger operator inside a spectral gap of the periodic operator asp 0 denoting byu0 the set of eigenvalues in the gap of the reference problem having precisely one impurity located at the origin say we show that the integrated density of states concentrates around the points ofu0 in the sense that p ue is of orderp for any fixed e neighborhoodue ofu0 while p k c p2 for any compact subsetk of the gap which does not intersectue

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

global stability results for switched systems based on weak lyapunov functions in this paper we study the stability of nonlinear and time varying switched systems under restricted switching we approach the problem by decomposing the system dynamics into a nominal like part and a perturbation like one most stability results for perturbed systems are based on the use of strong lyapunov functions i e functions of time and state whose total time derivative along the nominal system trajectories is bounded by a negative definite function of the state however switched systems under restricted switching may not admit strong lyapunov functions even when asymptotic stability is uniform over the set of switching signals considered the main contribution of the current paper consists in providing stability results that are based on the stability of the nominal like part of the system and require only a weak lyapunov function these results may have wider applicability than results based on strong lyapunov functions the results provided follow two lines first we give very general global uniform asymptotic stability results under reasonable boundedness conditions on the functions that define the dynamics of the nominal like and the perturbation like parts of the system second we provide input to state stability iss results for the case when the nominal like part is switched linear time varying we provide two types of iss results standard iss that involves the essential supremum norm of the input and a modified iss that involves a power type norm

existence proofs for harmonic mappings with the help of a maximum principle in this paper we want to use a maximum principle to give existence proofs for harmonic mappings by the direct methods in the calculus of variations in particular we discuss a rather general existence theorem for the dirichlet problem as well as a new proof of lemaire s theorem on the existence of harmonic mappings between closed surfaces which completely avoids the main difficulty of the so far known proofs namely to ensure that the limit of a minimizing sequence does not fall out of the prescribed homotopy class as stefan hildebrandt kindly pointed out to me this maximum principle is based on the same idea as the one in 6 lemma 6 although i was originally led to it by looking at the cut off function of meier in 17 i want to take this opportunity to thank stefan hildebrandt for the interest he has always shown in my work

partial regularity of p x harmonic maps let g x and hij u be uniformly elliptic symmetric matrices and assume that hij u and p x 2 are sufficiently smooth we prove partial regularity of minimizers for the functional f u z

on a theorem of deift and hempel we provide an alternative proof of the main result of deift and hempel 1 on the existence of eigenvalues ofv dimensional schrodinger operatorsh h0 w in spectral gaps ofh0

liquid crystals and the monotonicity formula abstract we give numerical evidence that the monotonicity formula holds for the liquid crystal functional and study part of the associated bifurcation diagram

remark on the dirichlet problem for harmonic maps from the disc into the 2 sphere in this paper we prove a theorem concerning existence and nonexistence of minimising harmonic maps in each connected component using the framework of brezis coron and soyeur

critical lieb thirring bounds in gaps and the generalized nevai conjecture for finite gap jacobi matrices we prove bounds of the form e i d h dist e e h 1 2 l 1 norm of a perturbation where i is a gap included are gaps in continuum one dimensional periodic schrodinger operators and finite gap jacobi matrices where we get a generalized nevai conjecture about an l 1 condition implying a szeg condition one key is a general new form of the birman schwinger bound in gaps

asymptotic stability equals exponential stability and iss equals finite energy gain if you twist your eyes in this paper we show that uniformly global asymptotic stability for a family of ordinary differential equations is equivalent to uniformly global exponential stability under a suitable nonlinear change of variables the same is shown for input to state stability and input to state exponential stability and for input to state exponential stability and a nonlinear h estimate

eigenvalue branches of the schrodinger operatorh λw in a gap of σ h the authors study the eigenvalue branches of the schrodinger operatorh w in a gap of h in particular they consider questions of asymptotic distribution of eigenvalues and bounds on the number of branches they also address the completeness problem

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

blow up and global existence for heat flows of harmonic maps in this paper it is proved that the solution to the evolution problem for harmonic maps blows up in finite time if the initial map belongs to some nontrivial homotopy class and the initial energy is sufficiently small

the discrete spectrum in the spectral gaps of semibounded operators with non sign definite perturbations abstract given two self adjoint operators a and v v v we study the motion of the eigenvalues of the operator a t a tv as t increases let 0 and let be a regular point for a we consider the quantities n v n v and n 0 v defined as the number of eigenvalues of the operator a t that pass point from the right to the left from the left to the right or change the direction of their motion exactly at point respectively as t increases from 0 to 0 we study asymptotic characteristics of these quantities as in the present paper the results obtained previously o l safronov comm math phys 193 1998 233 243 are extended and given new applications to differential operators

strict lyapunov function constructions under lasalle conditions with an application to lotka volterra systems we provide new techniques for building explicit global strict lyapunov functions for broad classes of periodic time varying nonlinear systems satisfying lasalle conditions we illustrate our work using the lotka volterra model which plays a fundamental role in bioengineering we use our strict lyapunov constructions to prove robustness of the lotka volterra tracking dynamics to uncertainty in the death rates

liouville theorems partial regularity and holder continuity of weak solutions to quasilinear elliptic systems this paper describes the connections between liouville type theorems and interior regularity results for bounded weak solutions of quasilinear elliptic systems with quadratic growth it is shown that equivalence does in general hold only in some restricted sense a complete correspondence can be established in certain cases e g for small solutions and for minima of quadratic variational integrals

invariance principles allowing of non lyapunov functions for estimating attractor of discrete dynamical systems this technical note establishes several versions of invariance principles for describing the eventual dynamical behaviors of discrete dynamical systems instead of the requirement of the so called lyapunov functions in the classical lasalle invariance principle some more relaxed conditions are imported the established invariance principles thus can be applied to a more general class of discrete dynamical systems for classifying their orbits into two categories based on the eventual dynamical behaviors and the proposed classification scheme is suitable for theoretically and numerically estimating the local or global attractors produced by the discrete dynamical systems the practical usefulness of the analytical results is verified by systematically investigating several representative discrete systems

a simplified design for strict lyapunov functions under matrosov conditions we construct strict lyapunov functions for broad classes of nonlinear systems satisfying matrosov type conditions our new constructions are simpler than the designs available in the literature we illustrate the practical interest of our designs using a globally asymptotically stable biological model

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

on the regularity of harmonic functions and spherical harmonic maps defined on lattices abstract a growth lemma for certain discrete symmetric laplacians defined on a lattice z d zd rd with spacing is proved the lemma implies a de giorgi theorem that the harmonic functions for these laplacians are equi holder continuous 0 these results are then applied to establish regularity properties for the harmonic maps defined on zd and taking values in an n dimensional sphere sn uniform in questions of the convergence 0 and the dirichlet problem for these discrete harmonic maps are also addressed

eigenvalue branches of the perturbed maxwell operator m λd in a gap of σ m the propagation of guided waves in photonic crystal fibers pcfs is studied a pcf can be regarded as a perfectly two dimensional photonic crystal with a line defect along the axial direction this problem can be treated as an eigenvalue problem for a family of noncompact self adjoint operators under the assumption that the background spectrum has a gap we prove that a line defect can create an eigenvalue of any given fixed value in the gap provided that the defect is strong enough based on a decoupling of regions in r2 by means of dirichlet and neumann boundaries then using the trace ideal estimates we study asymptotic distribution of eigenvalues and bounds on the number of eigenvalue branches in particular we show that if the defect is weak enough no eigenvalues can be created inside the gap

further remarks on strict input to state stable lyapunov functions for time varying systems we study the stability properties of a class of time varying non linear systems we assume that non strict input to state stable iss lyapunov functions for our systems are given and posit a mild persistency of excitation condition on our given lyapunov functions which guarantee the existence of strict iss lyapunov functions for our systems next we provide simple direct constructions of explicit strict iss lyapunov functions for our systems by applying an integral smoothing method we illustrate our constructions using a tracking problem for a rotating rigid body

the discrete spectrum of the periodic maxwell operator perturbed by a decreasing potential the propagation of electromagnetic waves in dielectric medium with decreasing perturbations defects is studied we use the maxwell operator m p x and the perturbed operator m p x q x to describe the background medium and perturbed medium respectively where 0 is the coupling constant under the assumption that m has a spectral gap g we prove that a defect can create an eigenvalue of any given fixed value g for some 0 provided that the support of the defect is large enough we also consider the asymptotic distribution of eigenvalues in the gap and bounds on the number of eigenvalue branches

spectral stability under tunneling we study the spectral properties of multiple well schrodinger operators on n we give in particular upper bounds on energy shifts due to tunnel effect and localization properties of wave packets our methods are based on agmon type estimates for resolvents in classically forbidden regions and geometric perturbation theory our results are valid also for an infinite number of wells arbitrary spectral type and in non semi classical regimes

on the asymptotic distribution of the eigenvalue branches of the schrodinger operator h λw in a spectral gap of h on considere des operateurs de schrodinger h v dans l espace de hilbert l 2 r avec un potentiel v satisfaisant 1 vel r on suppose que l intervalle a b est contenu dans l ensemble resolvant de h on obtient des estimations asymptotiques sur le nombre des branches de valeurs propes dans a b des familles d operateurs h w 0 quand ou w est un potentiel a courte portee non negatif

further constructions of control lyapunov functions and stabilizing feedbacks for systems satisfying the jurdjevic quinn conditions for a broad class of nonlinear systems we construct smooth control lyapunov functions we assume our systems satisfy appropriate generalizations of the jurdjevic quinn conditions we also design state feedbacks of arbitrarily small norm that render our systems integral input to state stable to actuator errors

on eigenvalues in gaps for perturbed magnetic schrodinger operators we study schrodinger operators h0 with a gap in the essential spectrum perturbed by either a decreasing electric potential or a decreasing magnetic field in both cases the strength of the perturbation is measured by a coupling constant 0 here we are mainly interested in the asymptotic behavior as of certain counting functions for the eigenvalues that are produced by the perturbation inside the spectral gap the case where we perturb by a potential can be handled using current technology even if h0 contains a fixed magnetic background for perturbations by magnetic fields however we require rather strong assumptions like exponential decay of the perturbations to obtain a lower bound on the counting function to gain some additional intuition we use separation of variables in the closely related model of a schrodinger operator with constant magnetic field in r2 perturbed by a rotationally symmetric magnetic field that decays at infinity

strong lyapunov functions for systems satisfying the conditions of la salle we present a construction of a strong lyapunov function whose derivative is negative definite along the solutions of the system using another weak lyapunov function whose derivative along the solutions of the system is negative semidefinite the construction can be carried out if a lie algebraic condition that involves the weak lyapunov function and the system vector field is satisfied our main result extends to general nonlinear systems the strong lyapunov function construction presented in a previous paper that was valid only for homogeneous systems

density results relative to the dirichlet energy of mappings into a manifold let y be a smooth compact oriented riemannian manifold without boundary weak limits of graphs of smooth maps u k b n y with an equibounded dirichlet integral give rise to elements of the space cart 2 1 b n y assume that y is 1 connected and that its 2 homology group has no torsion in any dimension n we prove that every element t in cart 2 1 b n x y with no singular vertical part can be approximated weakly in the sense of currents by a sequence of graphs of smooth maps u k b n y with dirichlet energies converging to the energy of t

existence of solutions for semilinear elliptic equations with indefinite linear part we study equations on rn lu w x u p 1 ulu f u where l is an indefinite schrodinger operator 0 w x 0 as and subcritical nonlinearities we prove existence of solutions u h1 rn via an inversion to a dual variational problem

regularity of harmonic maps with prescribed singularities in this paper we studied the regularity problem for harmonic maps into hyperbolic spaces with prescribed singularities along codimension two submanifolds this is motivated from one of hawking s conjectures on the uniqueness of kerr solutions among all axially symmetric asymptotically flat stationary solutions to the vacuum einstein equation in general relativity

boundary regularity and sufficient conditions for strong local minimizers abstract in this paper we present a new proof of the sufficiency theorem for strong local minimizers concerning c 1 extremals at which the second variation is strictly positive the results are presented in the quasiconvex setting in accordance with the original statement by grabovsky and mengesha 31 the strategy that we follow relies on a decomposition theorem that allows to split a sequence of variations into its oscillating and its concentrating parts as well as on a sufficiency result according to which smooth extremals are spatially local minimizers furthermore we prove partial regularity up to the boundary for strong local minimizers in the non homogeneous case and a full regularity result for lipschitz extremals with gradient of vanishing mean oscillation as a consequence we also establish a sufficiency result for this class of extremals the regularity results are established via a blow up argument

eigenvalues in gaps and decoupling by neumann boundary conditions abstract continuing earlier investigations we show the existence of eigenvalues of schrodinger operators h w r inside a spectral gap of h v under suitable conditions on v and w our method of proof is based on a decoupling of regions in r v by means of dirichlet or neumann boundary conditions where in particular the use of neumann boundary conditions in the decoupling process leads to a new asymptotic results for eigenvalue counting functions

partially constrained boundary conditions with energy minimizing mappings on considere la regularite a la frontiere pour une application minimisant l energie u m n pour deux types de conditions aux limites frontiere libre et condition limite partiellement contrainte

on feedback stabilization of smooth nonlinear systems this paper deals with the problem of stabilization of nonlinear systems we present a version of the well known jurdjevic quinn theorem 1978 concerning affine systems with dissipative drift for smooth but not necessarily analytic systems an example and counterexample are proposed

on the existence of eigenvalues of the schrodinger operatorh λw in a gap of σ h the authors prove a number of results asserting the existence of eigenvalues of the schrodinger operatorh w in a gap of h whereh v andv andw are bounded the existence of these eigenvalues is an important element in the theory of the colour of crystals the basic theorems are proved in v stronger results forv 1 are also presented

regularity of minimizers of relaxed problems for harmonic maps we prove that every minimizer on h1 s2 of the relaxed energy u 2 8 l u where 0 1 and l u is the length of a minimal connection connecting the singularities of u is smooth except at a finite number of points

boundary regularity in variational problems we prove that if u omega subset mathbb r n to mathbb r n is a solution to the dirichlet variational problem mathop rm min limits w int omega f x w dw rm d x quad rm subject to quad w equiv u 0 rm on partial omega involving a regular boundary datum u 0 and a regular integrand f x w dw strongly convex in dw and satisfying suitable growth conditions then mathcal h n 1 almost every boundary point is regular for u in the sense that du is holder continuous in a relative neighborhood of the point the existence of even one such regular boundary point was previously not known except for some very special cases treated by jost meier math ann 262 549 561 1983 our results are consequences of new up to the boundary higher differentiability results that we establish for minima of the functionals in question the methods also allow us to improve the known boundary regularity results for solutions to non linear elliptic systems and in some cases to improve the known interior singular sets estimates for minimizers moreover our approach allows for a treatment of systems and functionals with rough coefficients belonging to suitable sobolev spaces of fractional order

an l p regularity theory for harmonic maps motivated by the harmonic map heat flow we consider maps between riemannian manifolds such that the tension field belongs to an lp space under an appropriate smallness condition a certain degree of regularity follows for suitable solutions of the harmonic map heat flow we have a partial regularity result as a consequence

eigenvalue bounds in the gaps of schrodinger operators and jacobi matrices abstract we consider c a b where a is selfadjoint with a gap a b in its spectrum and b is relatively compact we prove a general result allowing b of indefinite sign and apply it to obtain a v d 2 bound for perturbations of suitable periodic schrodinger operators and a not quite lieb thirring bound for perturbations of algebro geometric almost periodic jacobi matrices

boundary regularity and the dirichlet problem for harmonic maps in a previous paper 10 we developed an interior regularity theory for energy minimizing harmonic maps into riemannian manifolds in the first two sections of this paper we prove boundary regularity for energy minimizing maps with prescribed dirichlet boundary condition we show that such maps are regular in a full neighborhood of the boundary assuming appropriate regularity on the manifolds the boundary and the data the reader may refer to theorem 2 7 for a statement of the precise result it is not surprising that the boundary regularity is actually stronger than the partial regularity we obtained for the interior this is due to the fact that there are no nontrivial smooth harmonic maps from hemispheres s which map the boundary s 9s to a point i j n 2 and is analogous to the fact that in certain cases we were able to obtain complete regularity in the interior many authors have worked on boundary regularity for this general type of problem we mention hildebrandt and widman 5 and hamilton 4 as having obtained important results specifically for harmonic maps morrey had obtained the boundary regularity for domain dimension n 2 in conjunction with his investigation of the plateau problem in riemannian manifolds 8 in 3 of this paper we observe that the direct method gives solvability of the dirichlet problem under reasonable hypotheses on the manifolds we give as an application an amusing proof of a theorem of sacks and uhlenbeck 9 on the existence of minimal 2 spheres representing the second homotopy group of a manifold the same method gives smooth harmonic representations for k n for a certain class of manifolds n these are characterized by the nonexistence of lower dimensional harmonic spheres whose homogeneous extensions are minimal see proposition 3 4 in the last section of the paper we discuss approximation of l maps by smooth maps we give a simple example of an l map from the three dimensional ball to the two sphere which is not an l limit of continuous maps we