radial solutions of δu f u 0 with prescribed numbers of zeros u 0 0 u r f 0 as r 03 1 2 here n 1 is a real parameter and the function f satisfies the following hypotheses f 1 f r r is locally lipschitz continuous f2 uf u 0 and j 0 such that f u 0 on 0 pi f u 0 on uc 0 f u 0 on b 01 f u 0 on a pi where f u f r f s ds f4 f u k u jujp u g u where

asymptotic controllability implies feedback stabilization it is shown that every asymptotically controllable system can be globally stabilized by means of some discontinuous feedback law the stabilizing strategy is based on pointwise optimization of a smoothed version of a control lyapunov function iteratively sending trajectories into smaller and smaller neighborhoods of a desired equilibrium a major technical problem and one of the contributions of the present paper concerns the precise meaning of solution when using a discontinuous controller

stabilization with relaxed controls cannot in general be stabilized using a continuous closed loop control u x even if each state separately can be driven asymptotically to the origin an example is analyzed in section 2 in this paper we examine the possibility of stabilizing such systems with a continuous closed loop relaxed control we find indeed that the family of systems stabilizable with relaxed controls is larger than the family of those stabilizable with ordinary controls an even larger class is obtained if the continuity of the closed loop at the origin is not required the latter class includes all one dimensional systems for which states can be driven asymptotically to the origin this result does not hold in two dimensional systems and we provide a counter example it should be pointed that relaxed control type stabilization is used both in theory and in practice the method is known as dither we shall comment on the similarities lyapunov functions for the system 1 help us in the construction of the continuous closed loop stabilizers in fact we find that the existence of a smooth lyapunov function is equivalent to the existence of a stabilizing closed loop which is continuous except possibly at the origin an additional condition on the lyapunov function implies the continuity at the origin as well we present these results in section 4 after a brief introduction of closed loop relaxed controls notations and terminology in section 3 prior to that in section 2 we discuss an example illustrating the power of relaxed controls in the particular case of systems linear in the controls relaxed controls can be replaced by ordinary controls this is discussed in section 5 the role of lyapunov functions in the stability and stabilization theories is of course well known examples of systems with lyapunov functions are available in the literature we display some in section 6 along with general comments on the construction applications and counterexamples including one which cannot be continuously stabilized yet possesses a nonsmooth lyapunov function closed loop stabilization with ordinary controls is analyzed extensively in the literature see sontag 8 sussmann ll and references therein lyapunov functions techniques in stabil

invariance principles for hybrid systems with connections to detectability and asymptotic stability this paper shows several versions of the lasalle s invariance principle for general hybrid systems the broad framework allows for nonuniqueness of solutions zeno behaviors and does not insist on continuous dependence of solutions on initial conditions instead only a mild structural property involving graphical convergence of solutions is posed the general invariance results are then specified to hybrid systems given by set valued data further results involving invariance as well as observability detectability and asymptotic stability are given

structure of positive radial solutions of semilinear elliptic equations abstract we study the positive radial solutions of a semilinear elliptic equation u f u 0 wheref u has a supercritical growth order for smallu 0 and a subcritical growth order for largeu by showing the uniqueness of positive solutions behaving likeo x 2 n at infinity we give an almost complete description for the structure of positive radial solutions as a consequence we also prove the uniqueness of positive solutions of the nonlinear dirichlet problem for the equation in a finite ball

stationary solutions for the 2d critical dirac equation with kerr nonlinearity abstract in this paper we prove the existence of an exponentially localized stationary solution for a two dimensional cubic dirac equation it appears as an effective equation in the description of nonlinear waves for some condensed matter bose einstein condensates and nonlinear optics optical fibers systems the nonlinearity is of kerr type that is of the form 2 and thus not lorenz invariant we solve compactness issues related to the critical sobolev embedding h 1 2 r 2 c 2 l 4 r 2 c 4 thanks to a particular radial ansatz our proof is then based on elementary dynamical systems arguments

nonpathological lyapunov functions and discontinuous caratheodory systems differential equations with discontinuous right hand side and solutions intended in caratheodory sense are considered for these equations sufficient conditions which guarantee both lyapunov stability and asymptotic stability in terms of nonsmooth lyapunov functions are given an invariance principle is also proven

existence of carathe spl acute odory solutions in nonlinear systems with discontinuous switching feedback controllers in this note we consider the existence of a carathe spl acute odory solution in a single input single output nonlinear system with a discontinuous switching feedback controller the main contribution is to show that if the nonlinear system can be transformed into a global normal form then we can specify the value of the discontinuous switching feedback controller on the switching hypersurface so that the closed loop system has a carathe spl acute odory solution

a nonsmooth lyapunov function and stability for ode s of caratheodory type abstract the problem of lyapunov stability for systems of ode s of caratheodory type is considered it is proved that without the necessity of calculating dini or clarke generalized gradients the locally lipschitz lyapunov function can follow different kinds of stability

stationary solitons of a three wave model generated by type ii second harmonic generation in quadratic media abstract in this paper we prove the uniqueness of stationary standing wave solutions of an optical model generated by type ii second harmonic generation shg with behaviors tending to zero at infinity under certain conditions on parameters in addition we provide the same issues for the dirichlet boundary value problems on the ball centered at the origin a classification of solutions for radial case is also established

an integral invariance principle for differential inclusions with applications in adaptive control the byrnes martin integral invariance principle for ordinary differential equations is extended to differential inclusions on bbb r n the extended result is applied in demonstrating the existence of adaptive stabilizers and servomechanisms for a variety of nonlinear system classes

asymptotic convergence of constrained primal dual dynamics this paper studies the asymptotic convergence properties of the primal dual dynamics designed for solving constrained concave optimization problems using classical notions from stability analysis we motivate the need for this study by providing an example that rules out the possibility of employing the invariance principle for hybrid automata to study asymptotic convergence we understand the solutions of the primal dual dynamics in the caratheodory sense and characterize their existence uniqueness and continuity with respect to the initial condition we use the invariance principle for discontinuous caratheodory systems to establish that the primal dual optimizers are globally asymptotically stable under the primal dual dynamics and that each solution of the dynamics converges to an optimizer

low energy nodal solutions to the yamabe equation abstract given an isoparametric function f on the n dimensional sphere we consider the space of functions w f to reduce the yamabe equation on the round sphere into a singular ode on w in the interval 0 of the form w h r sin r w w 4 n 2 w w 0 and boundary conditions w 0 0 w where h is a monotone function with exactly one zero on 0 and 0 is a constant for any positive integer k we obtain a solution with exactly k zeroes yielding solutions to the yamabe equation with exactly k connected isoparametric hypersurfaces as nodal set the idea of the proof is to consider the initial value problems on the singularities and then to solve the corresponding double shooting problem matching the values of w and w at the unique zero of h in particular we obtain solutions with exactly one zero providing solutions of the yamabe equation with low energy

the converse lyapunov s theorem for odes of caratheodory type abstract the converse problem of lyapunov stability for systems of odes of caratheodory type is considered it is proved that if the right hand side of an ode satisfies only the osgood condition the uniform stability of the origin is sufficient to the existence of a locally lipschitz continuous lyapunov function actually the uniform stability is equivalent to the robust stability in this case moreover as an auxiliary result the generalization of the famous gronwall bellman bihari inequality is also proved

lasalle yoshizawa corollaries for nonsmooth systems in this technical note two generalized corollaries to the lasalle yoshizawa theorem are presented for nonautonomous systems described by nonlinear differential equations with discontinuous right hand sides lyapunov based analysis methods that achieve asymptotic convergence when the candidate lyapunov derivative is upper bounded by a negative semi definite function in the presence of differential inclusions are presented a design example illustrates the utility of the corollaries

generalized lyapunov method for discontinuous systems abstract nonlinear dynamical systems described by differential equations with discontinuous right hand side and solutions intended in filippov sense are considered based on filippov differential inclusion and a new chain rule for differentiating regular functions along filippov solution trajectories different kinds of stability and convergence results are presented moreover we investigate the stability and convergence for the corresponding perturbation system of the discontinuous system and some new criteria are addressed

on random frequent universality abstract we give a probabilistic construction of frequently hypercyclic or universal entire functions using birkhoff s ergodic theorem we apply the same ideas to construct random frequently universal vectors for the polynomials of the weighted backward shift on the classical real or complex l p space

flow stability of patchy vector fields and robust feedback stabilization the paper is concerned with patchy vector fields a class of discontinuous piecewise smooth vector fields that were introduced by the authors to study feedback stabilization problems we prove the stability of the corresponding solution set w r t a wide class of impulsive perturbations these results yield the robustness of patchy feedback controls in the presence of measurement errors and external disturbances

multiplicity results for sign changing bound state solutions of a semilinear equation abstract in this paper we give conditions on f so that the problem u f u 0 x r n n 2 has at least two radial bound state solutions with any prescribed number of zeros and such that u 0 belongs to a specific subinterval of 0 this property will allow us to give conditions on f so that this problem has at least any given number of radial solutions having a prescribed number of zeros

uniqueness of non negative solutions of semilinear equations in rn on etudie l unicite des solutions positives classiques u x du probleme u f u 0 dans r n u x 0 quand x avec n 1 et x x 1 x n

the role of convexity in saddle point dynamics lyapunov function and robustness this paper studies the projected saddle point dynamics associated to a convex concave function which we term saddle function the dynamics consists of gradient descent of the saddle function in variables corresponding to convexity and projected gradient ascent in variables corresponding to concavity we examine the role that the local and or global nature of the convexity concavity properties of the saddle function plays in guaranteeing convergence and robustness of the dynamics under the assumption that the saddle function is twice continuously differentiable we provide a novel characterization of the omega limit set of the trajectories of this dynamics in terms of the diagonal blocks of the hessian using this characterization we establish global asymptotic convergence of the dynamics under local strong convexity concavity of the saddle function when strong convexity concavity holds globally we establish three results first we identify a lyapunov function that decreases strictly along the trajectory for the projected saddle point dynamics when the saddle function corresponds to the lagrangian of a general constrained convex optimization problem second for the particular case when the saddle function is the lagrangian of an equality constrained optimization problem we show input to state stability iss of the saddle point dynamics by providing an iss lyapunov function third we use the latter result to design an opportunistic state triggered implementation of the dynamics various examples illustrate our results

some remarks on abstract universal series abstract we introduce a new class of universal series in the abstract setting this class lies between two already known classes of universal series by two examples we differentiate these classes these examples are the first examples in the case where all classes are non void we also prove that the class of universal series where the approximating indexes have upper density 1 is dense and g and that the class of lower frequently universal series is of the first category

frequently hypercyclic operators we investigate the subject of linear dynamics by studying the notion of frequent hypercyclicity for bounded operators t on separable complex f spaces t is frequently hypercyclic if there exists a vector x such that for every nonempty open subset u of x the set of integers n such that t n x belongs to u has positive lower density we give several criteria for frequent hypercyclicity and this leads us in particular to study linear transformations from the point of view of ergodic theory several other topics which are classical in hypercyclicity theory are also investigated in the frequent hypercyclicity setting

dual approach to lyapunov stability abstract we investigate the problem of lyapunov stability for nonautonomous odes for which the right hand side is only lebesgue measurable with respect to time t x f t x we describe how to use the dual lyapunov function in dual space and the hamilton jacobi inequality to formulate sufficient conditions for the stability of the zero solution in primary and dual space

the existence and uniqueness of bound state solutions of a semi linear equation we investigate the existence and uniqueness of bounded radially symmetric solutions in r3 of uf u 0 where f u is continuous piecewise linear and has three distinct real zeros we give a detail

sign changing stationary solutions and blowup for the two power nonlinear heat equation in a ball abstract consider the nonlinear heat equation 0 1 u t u u p 1 u u q 1 u where t 0 and x the unit ball of r n n 3 with dirichlet boundary conditions let h be a radially symmetric sign changing stationary solution of 0 1 we prove that the solution of 0 1 with initial value h blows up in finite time if 1 0 is sufficiently small and if 1 q p p s n 2 n 2 and p sufficiently close to p s this proves that the set of initial data for which the solution is global is not star shaped around 0

semiconcave control lyapunov functions and stabilizing feedbacks we study the general problem of stabilization of globally asymptotically controllable systems we construct discontinuous feedback laws and particularly we make it possible to choose these continuous outside a small set closed with measure zero of discontinuity in the case of control systems which are affine in the control moreover this set of singularities is shown to be repulsive for the caratheodory solutions of the closed loop system under an additional assumption

finite time consensus for multi agent networks with unknown inherent nonlinear dynamics the objective of this paper is to analyze the finite time convergence of a nonlinear but continuous consensus algorithm for multi agent networks with unknown inherent nonlinear dynamics due to the existence of the unknown inherent nonlinear dynamics the stability analysis and the finite time convergence analysis are more challenging than those under the well studied consensus algorithms for known linear systems for this purpose we propose a novel comparison based tool by using this tool it is shown that the proposed nonlinear consensus algorithm can guarantee finite time convergence if the directed switching interaction graph has a directed spanning tree at each time interval specifically the finite time convergence is shown by comparing the closed loop system under the proposed consensus algorithm with some well designed closed loop system whose stability properties are easier to obtain moreover the stability and the finite time convergence of the closed loop system using the proposed consensus algorithm under a general directed switching interaction graph can even be guaranteed by the stability and the finite time convergence of some well designed nonlinear closed loop system under some special directed switching interaction graph this provides a stimulating example for the potential applications of the proposed comparison based tool in the stability analysis of linear nonlinear closed loop systems by making use of known results in linear nonlinear systems

radial solutions of superlinear equations on ℝn part i global variational approach we present a global variational approach to the search for multiple nodal solutions u h 1 n to a class of elliptic equations of type u x f x u x x n where n 2 f is superlinear and subcritical and f x 0

lyapunov stability theory of nonsmooth systems this paper develops nonsmooth lyapunov stability theory and lasalle s invariance principle for a class of nonsmooth lipschitz continuous lyapunov functions and absolutely continuous state trajectories computable tests based on filipov s differential inclusion and clarke s generalized gradient are derived the primary use of these results is in analyzing the stability of equilibria of differential equations with discontinuous right hand side such as in nonsmooth dynamic systems or variable structure control