differentiability of palmer s linearization theorem and converse result for density functions we study differentiability properties in a particular case of the palmer s linearization theorem which states the existence of a homeomorphism h between the solutions of a linear ode system having exponential dichotomy and a quasilinear system indeed if the linear system is uniformly asymptotically stable sufficient conditions ensuring that h is a c2 preserving orientation diffeomorphism are given as an application we generalize a converse result of density functions for a nonlinear system in the nonautonomous case

large time behavior of energy in exponentially decreasing solutions of the navier stokes equations abstract we study large time behavior of exponentially decreasing global strong solutions of the navier stokes equations we show as the main result that these solutions are characterized by the asymptotic disappearance of low and high frequencies and concentration of energy in middle frequencies we also describe a relationship between the spectrum of the stokes operator and the class of exponentially decreasing global strong solutions

stability criteria with nonlinear eigenvalues for diagonalizable nonlinear systems stability of the linear system especially the diagonalizable system can be verified by computing the signs of the real parts of its eigenvalues in this paper we generalize this result to the diagonalizable nonlinear system in terms of nonlinear eigenvalues according to the main results the unique equilibrium point of the diagonalizable system is globally asymptotically stable if a specific set of nonlinear eigenvalues is not positive at each point in the state space

a dual to lyapunov s stability theorem lyapunov s second theorem is a standard tool for stability analysis of ordinary differential equations here we introduce a theorem which can be viewed as a dual to lyapunov s result from existence of a scalar function satisfying certain inequalities it follows that almost all trajectories of the system tend to zero the scalar function has a physical interpretation as the stationary density of a substance that is generated in all points of the state space and flows along the system trajectories if the stationary density is bounded everywhere except at a singularity in the origin then almost all trajectories tend towards the origin the weaker notion of stability allows for applications also in situations where lyapunov s theorem cannot be used moreover the new criterion has a striking convexity property related to control synthesis

a lyapunov density criterion for almost everywhere stability of a class of lipschitz continuous and almost everywhere c 1 nonlinear systems this paper is concerned with the analysis of almost everywhere stability aes of a class of nonlinear systems via lyapunov densities while many of the previous results on lyapunov densities assume that the vector field is continuously differentiable this paper deals with nonlinear systems that can have points of non differentiability a key to prove aes is monotonicity of a measure defined with a lyapunov density to deduce that the set of non convergent initial values has zero lebesgue measure the monotonicity condition of a measure involves a change of variables in integrals of the lyapunov density over a set such a change of variables along trajectories is not available for vector fields that are not globally continuously differentiable even though they are smooth at almost all points of the state space in this paper an alternative inequality is presented that proves monotonicity of a measure along trajectories of a class of nonlinear systems that are lipschitz continuous and almost everywhere c

almost global stochastic stability we develop a method to prove almost global stability of stochastic differential equations in the sense that almost every initial point with respect to the lebesgue measure is asymptotically attracted to the origin with unit probability the method can be viewed as a dual to lyapunov s second method for stochastic differential equations and extends the deterministic result of a rantzer syst control lett 42 2001 pp 161 168 the result can also be used in certain cases to find stabilizing controllers for stochastic nonlinear systems using convex optimization the main technical tool is the theory of stochastic flows of diffeomorphisms

lower bound of l2 decay of the navier stokes flow in the half space r n and its asymptotic behavior in the frequency space abstract we consider the asymptotic behavior of weak solutions of the navier stokes equations in the half space r n we obtain the lower bound of the energy decay of the navier stokes flow by means of the profile of the initial data indeed we construct a class of the initial data which causes the slow decay of the navier stokes flow with an explicit rate furthermore we investigate the asymptotic behavior of concentration in the frequency space

attractors of nonlinear dynamical systems with a weakly monotone measure abstract this paper is devoted to the study of attractive sets for dynamical systems in a metric space with a measure it is assumed that the measure of a set of points in the phase space is increasing along the flow we prove that an invariant set is an attractor for almost all initial conditions under some extra assumptions for a system of autonomous ordinary differential equations we present attractivity conditions in terms of the divergence with a density function unlike previous results in the literature our approach allows the use of a wider class of density functions if the divergence vanishes on a set of positive measure as an example the attitude stabilization problem of a rigid body is solved by using an affine feedback control for the kinematic equations in terms of quaternions

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

conditions for asymptotic energy and enstrophy concentration in solutions to the navier stokes equations abstract let 0 a be the power of the stokes operator a and r a be the range of a we show as the main result of the paper that if w is a nonzero global weak solution to the navier stokes equations satisfying the strong energy inequality and w 0 r a then the energy of the solution w concentrates asymptotically in frequencies smaller than or equal to the finite number c 1 2 lim sup t a 1 2 w t 2 w t 2 in the sense that lim t e w t w t 1 for every c 1 2 where e 0 is the resolution of the identity of a we also obtain an explicit convergence rate in the limit above and similar results for the enstrophy of w defined as a 1 2 w

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

density functions for navigation function based systems in this paper we present a scheme for constructing density functions for systems that are almost globally asymptotically stable i e systems for which all trajectories converge to an equilibrium except for a set of measure zero using navigation functions nfs although recently proven converse theorems guarantee the existence of density functions for such systems such results are only existential and the construction of a density function for almost globally asymptotically stable systems remains a challenging task we show that for a specific class of dynamical systems that are defined based on an nf a density function can be easily derived from the system s underlying nf

stability in the almost everywhere sense a linear transfer operator approach abstract the problem of almost everywhere stability of a nonlinear autonomous ordinary differential equation is studied using a linear transfer operator framework the infinitesimal generator of a linear transfer operator perron frobenius is used to provide stability conditions of an autonomous ordinary differential equation it is shown that almost everywhere uniform stability of a nonlinear differential equation is equivalent to the existence of a non negative solution for a steady state advection type linear partial differential equation we refer to this non negative solution verifying almost everywhere global stability as lyapunov density a numerical method using finite element techniques is used for the computation of lyapunov density

combination of lyapunov and density functions for stability of rotational motion lyapunov methods and density functions provide dual characterizations of the solutions of a nonlinear dynamic system this work exploits the idea of combining both techniques to yield stability results that are valid for almost all the solutions of the system based on the combination of lyapunov and density functions analysis methods are proposed for the derivation of almost input to state stability and of almost global stability in nonlinear systems the techniques are illustrated for an inertial attitude observer where angular velocity readings are corrupted by non idealities

analysis of positive invariance and almost regional attraction via density functions with converse results this note is concerned with analysis of positive invariance of nonlinear systems and convergence of trajectories in a region of the state space via density functions if there exists a density function that is positive inside a set containing an equilibrium point and tends to zero as approaches to the boundary the set is positively invariant and almost all of the trajectories starting from there converge to the equilibrium converse results are also provided to prove the existence of such density functions

asymptotic energy concentration in the phase space of the weak solutions to the navier stokes equations abstract we study the asymptotic behavior of the energy of weak solutions of navier stokes equations as t we characterize the space of the initial data which causes a concentration of the kinetic energy in the phase space moreover an explicit convergence rate is obtained

on necessary conditions for almost global stability in the year 2000 the sufficiency of the existence of density functions for the almost global stability of a system was proved in this note we prove the necessity of the existence of density functions for the particular case of global asymptotical stability

noncommutative finite dimensional manifolds i spherical manifolds and related examples we exhibit large classes of examples of noncommutative finite dimensional manifolds which are non formal deformations of classical manifolds the main result of this paper is a complete description of noncommutative three dimensional spherical manifolds a noncommutative version of the sphere s 3 defined by basic k theoretic equations we find a 3 parameter family of deformations of the standard 3 sphere s 3 and a corresponding 3 parameter deformation of the 4 dimensional euclidean space 4 for generic values of the deformation parameters we show that the obtained algebras of polynomials on the deformed 4 u only depend on two parameters and are isomorphic to the algebras introduced by sklyanin in connection with the yang baxter equation it follows that different can span the same this equivalence generates a foliation of the parameter space this foliation admits singular leaves reduced to a point these critical points are either isolated or fall in two 1 parameter families up to the simple operation of taking the fixed algebra by an involution these two families are identical and we concentrate here on c for the above isomorphism with the sklyanin algebra breaks down and the corresponding algebras are special cases of deformations a notion which we generalize in any dimension and various contexts and study in some detail here and this point is crucial the dimension is not an artifact i e the dimension of the classical model but is the hochschild dimension of the corresponding algebra which remains constant during the deformation besides the standard noncommutative tori examples of deformations include the recently defined noncommutative 4 sphere as well as m dimensional generalizations noncommutative versions of spaces and quantum groups which are deformations of various classical groups we develop general tools such as the twisting of the clifford algebras in order to exhibit the spherical property of the hermitian projections corresponding to the noncommutative dimensional spherical manifolds a key result is the differential self duality properties of these projections which generalize the self duality of the round instanton

dynamical noncommutative spheres we introduce a family of noncommutative 4 spheres such that the instanton projector has its first chern class trivial ch 1 e b b we construct for them a 4 dimensional cycle and calculate explicitly the chern connes pairing for the instanton projector

almost global attraction in planar systems abstract in this work we present a result relating the recent ideas of almost global stability and density functions with the classical poincare bendixson theory for planar systems