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

sharp local well posedness for the good boussinesq equation abstract in the present article we prove the sharp local well posedness and ill posedness results for the good boussinesq equation on 1d torus the initial value problem is locally well posed in h 1 2 t and ill posed in h s t for s 1 2 well posedness result is obtained from reduction of the problem into a quadratic nonlinear schrodinger equation and the contraction argument in suitably modified x s b spaces the proof of the crucial bilinear estimates in these spaces especially in the lowest regularity rely on some bilinear estimates for periodic functions in x s b spaces which are generalization of the bilinear refinement of the l 4 strichartz estimate on r our result improves the known local well posedness in h s t with s 3 8 given by oh stefanov 2012 similar ideas also establish the sharp local well posedness in h 1 2 r and ill posedness below h 1 2 for the nonperiodic case which improves the result of tsugawa and the author 2010 in h s r with s 1 2

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

on potential wells and vacuum isolating of solutions for semilinear wave equations abstract in this paper we study the initial boundary value problem of semilinear wave equations u tt u u p 1 u x t 0 u x 0 u 0 x u t x 0 u 1 x x u x t 0 x t 0 where r n is a bounded domain 1 1 n 2 n 2 for n 3 first by using a new method we introduce a family of potential wells which include the known potential well as a special case then by using it we obtain some new existence theorems of global solutions and prove that for any e 0 d d is the depth of the known potential well all solutions with initial energy e 0 satisfying 0 h 0 1

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

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

global existence of cauchy problem for boussinesq paradigm equation in this paper we study the cauchy problem for the boussinesq paradigm equation u t email protected b email protected t email protected email protected email protected u where f u email protected u p new functionals are introduced and their sign preserving properties under the flow of the equation are studied the existence of a global weak solution with supercritical initial energy is proved the explicit value of the critical energy constant is found

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

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

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

global existence and nonexistence of the initial boundary value problem for the dissipative boussinesq equation abstract a class of multidimensional dissipative boussinesq equations is considered under three different cases of initial energy we prove the existence and uniqueness of global solutions in the energy space and provide sufficient conditions of finite time blow up solutions the sharp result is given as e 0 d particular attention is paid to initial energy at supercritical initial energy e 0 d level which is novel contribution of this paper and new methods will be introduced and some analysis techniques are needed in addition the asymptotic behavior of the global solutions is studied

the cauchy problem for the rosenau equation abstract this paper considers the existence and the uniqueness of the solution for the cauchy problem of the rosenau equation u t t u x 4 u x 4 t t u x x f u x x we prove the existence and the uniqueness of the global solution under some conditions of f u and give the condition for nonexistence of a global solution for f u u p u with 0 and p 0 the global existence and finite time blow up for the problem are proved with the aid of the potential well method

maximal unipotent subgroups of finitary linear groups finitary linear groups have been in recent years the subject of intensiveinvestigations the structure of these groups has been studied and nowmuch is known about the locally solvable or locally nilpotent finitary lineargroups or about the locally finite ones a remarkable achievement is forinstance the classification of the simple locally finite finitary linear groups recently completed by j hall

global existence and finite time blow up for a class of semilinear pseudo parabolic equations abstract in this paper we study a class of semilinear pseudo parabolic equations by introducing a family of potential wells we prove the invariance of some sets global existence nonexistence and asymptotic behavior of solutions with initial energy j u 0 d moreover we obtain finite time blow up with high initial energy j u 0 d by comparison principle

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

decay and scattering of small solutions of a generalized boussinesq equation abstract we study the long time behavior of small solutions of the initial value problem for a generalized boussinesq equation we obtain a lower bound for the degrees of nonlinearity which allows us to establish a nonlinear scattering result for small perturbations that is the small solutions of the nonlinear problem behave asymptotically like the solution of the associated linear problem under certain hypotheses we can construct a scattering operator for the boussinesq equation which carries a neighborhood of 0 in the energy space x into x

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

global existence of smooth solutions and stability of solitary waves for a generalized boussinesq equation certain generalizations of one of the classical boussinesq type equations u tt u xx u 2 u xx xx are considered it is shown that the initial value problem for this type of equation is always locally well posed it is also determined that the special solitary wave solutions of these equations are nonlinearly stable for a range of their phase speeds these two facts lead to the conclusion that initial data lying relatively close to a stable solitary wave evolves into a global solution of these equations this contrasts with the results of blow up obtained recently by kalantarov and ladyzhenskaya for the same type of equation and casts additional light upon the results for the original version of this class of equations obtained via inverse scattering theory by deift tomei and trubowitz

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