on the synthesis of a stabilizing feedback control via lie algebraic methods let the n dimensional system of differential equations dx dt x x t have p in mathbb r n as a rest solution i e x p 0 even in cases when this rest solution is unstable one can often induce a strong stability asymptotic stability by the inclusion of one or more controls e g via a controlled system a dx dt x x uy x where say u leqq 1 lie theory gives a computable sufficient condition to determine when by use of the control u one can steer a full n dimensional neighborhood of p to p by solutions of a this condition is assumed to hold one prefers a feedback control i e that u u x the main result in this paper is an algorithm which determines a modified stabilizing feedback control specifically for given varepsilon 0 one measures the current state q and the algorithm determines u t q 0 leqq t leqq varepsilon such that the solution x cdot u of a initiating from q and corresponding to this control u satisfies distance x varepsilon u p q p in fact iterates are theoretically shown to converge to p numerical examples computed via a simple fortran program are included these substantiate the strong stability achieved via such a modified feedback control

on switching stabilizability for continuous time switched linear systems this technical note studies switching stabilization problems for continuous time switched linear systems we consider four types of switching stabilizability defined under different assumptions on the switching control input the most general switching stabilizability is defined as the existence of a measurable switching signal under which the resulting time varying system is asymptotically stable discrete switching stabilizability is defined similarly but requires the switching signal to be piecewise constant on intervals of uniform length in addition we define feedback stabilizability in filippov sense respectively sample and hold sense as the existence of a feedback law under which closed loop filippov solution respectively sample and hold solution is asymptotically stable it is proved that the four switching stabilizability notions are equivalent and their sufficient and necessary condition is the existence of a piecewise quadratic control lyapunov function that can be expressed as the pointwise minimum of a finite number of quadratic functions

generalized sliding mode control for multi input nonlinear systems the purpose of this paper is to present a framework for the design of sliding mode based controllers for multi input affine nonlinear systems in this framework the control effort goes to zero as the state approaches the sliding surface a key feature for practical implementation the sliding surface is a submanifold with codimension equal to the dimension of the involutive distribution g generated by the controlled vector fields rather than the dimension of their linear span we also allow nonzero vectors in g to be tangent to the sliding surface the singular case in sliding mode control

on brockett s condition for smooth stabilizability and its necessity in a context of nonsmooth feedback the necessity of brockett s condition for stabilizability of nonlinear systems by smooth feedback is shown by an argument based on properties of a degree for set valued maps to persist when the class of controls is enlarged to include discontinuous feedback

on the existence of local smooth repulsive stabilizing feedbacks in dimension three abstract given an affine control system in r 3 subject to the hormander s condition at the origin we prove the existence of a local smooth repulsive stabilizing feedback at the origin our construction is based on the classical homogenization procedure on the existence of a semiconcave control lyapunov function and on the classification of singularities of semiconcave functions in dimension two

the synthesis of universal feedback pursuit strategies in differential games we show how any generalized supersolution of the hamilton jacobi equation can be used to construct a feedback pursuit strategy which guarantees to any given tolerance a capture time not exceeding the solution s value if the supersolution is the value function then a near optimal pursuit strategy is obtained in this way an important feature of the construction is its universal nature i e the fact that the feedback law is uniformly effective on compact sets of initial conditions this implies in particular that the feedback construction is one that exploits nonoptimal behavior on the part of the evader

a receding horizon generalization of pointwise min norm controllers control lyapunov functions clfs are used in conjunction with receding horizon control to develop a new class of receding horizon control schemes in the process strong connections between the seemingly disparate approaches are revealed leading to a unified picture that ties together the notions of pointwise min norm receding horizon and optimal control this framework is used to develop a clf based receding horizon scheme of which a special case provides an appropriate extension of sontag s formula the scheme is first presented as an idealized continuous time receding horizon control law the issue of implementation under discrete time sampling is then discussed as a modification these schemes are shown to possess a number of desirable theoretical and implementation properties an example is provided demonstrating their application to a nonlinear control problem finally stronger connections to both optimal and pointwise min norm control are proved

a rigorous reduction of the l2 stability of the solutions to a nonlinear binary reaction diffusion system of pde s to the stability of the solutions to a linear binary system of ode s abstract a basic peculiar lyapunov functional v is introduced for the dynamical systems generated by a pair of nonlinear reaction diffusion pde s with nonconstant coefficients the sign of v and of its derivative along the solutions is linked through an immediate simple relation to the eigenvalues by using v and the l 2 norm the non linear l 2 stability instability is rigorously reduced to the stability instability of the solutions to a linear binary system of ode s

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

a lyapunov like characterization of asymptotic controllability it is shown that a control system in bf r n is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can be made negative by appropriate choices of controls

a general notion of global asymptotic controllability for time varying systems and its lyapunov characterization a general notion of global asymptotic controllability to a given equilibrium of a time varying system is introduced and is shown that this property is equivalent to the existence of a lower semicontinuous control lyapunov function the work constitutes an extension of albertini and sontag 1999 paper in same journal dealing with same subject under stricter hypotheses

solutions to hybrid inclusions via set and graphical convergence with stability theory applications motivated by questions in stability theory for hybrid dynamical systems we establish some fundamental properties of the set of solutions to such systems using the notion of a hybrid time domain and general results on set and graphical convergence we establish under weak regularity and local boundedness assumptions that the set of solutions is sequentially compact and upper semicontinuous with respect to initial conditions and system perturbations the general facts are then used to establish several results for the behavior of hybrid systems that have asymptotically stable compact sets these results parallel what is already known for differential inclusions and difference inclusions for example the basin of attraction for a compact attractor is relatively open the attractivity is uniform from compact subsets of the basin of attraction and asymptotic stability is robust with respect to small perturbations

sufficient conditions for asymptotic controllability and hybrid feedback stabilization for a class of time varying systems for a class of time varying systems including those whose dynamics have triangular structure we provide sufficient conditions for global asymptotic controllability and global asymptotic stabilization by means of a hybrid time varying feedback

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

complete global analysis of a two scale network sirs epidemic dynamic model with distributed delay and random perturbations the model validation results are established the basic reproduction numbers at three human vector contact levels are computed the stochastic stability of equilibria is proven interplay between stability and human mobility are exhibited in real life scenarios a stochastic sirs epidemic dynamic model with distributed time delay for vector borne diseases in two scale network structured populations is presented the distributed delay accounts for the varying incubation period of the disease furthermore the disease dynamics is influenced by random environmental perturbations in the disease transmission process as well as the two scale human mobility process the basic reproduction numbers at three human vector contact levels in the two scale population are computed and the results for the stochastic asymptotic stability of the equilibria are presented moreover the asymptotic stability results are exhibited in several real life scenarios and the significance of the results are presented numerical simulation results are presented

extended invariance principle for nonautonomous switched systems stability analysis is developed for nonlinear nonautonomous switched systems trajectories of which admit generally speaking a nonunique continuation on the right for these systems krasovskii lasalle s invariance principle is extended in such a manner to remain true even in the nonautonomous case in addition to a nonsmooth lyapunov function with negative semidefinite time derivatives along the system trajectories the extended invariance principle involves a coupled indefinite function to guarantee asymptotic stability of the system in question as an illustration of the capabilities of this principle a switched regulator of a fully actuated manipulator with frictional joints is constructed

norm controllability of nonlinear systems in this paper we introduce and study the notion of norm controllability for nonlinear systems this property captures the responsiveness of a system with respect to applied inputs which is quantified via the norm of an output as a main contribution we obtain several lyapunov like sufficient conditions for norm controllability some of which are based on higher order derivatives of a lyapunov like function various aspects of the proposed concept and the sufficient conditions are illustrated by several examples including a chemical reactor application furthermore for the special case of linear systems we establish connections between norm controllability and standard controllability

suboptimal receding horizon control for continuous time systems solving a continuous time optimal control problem under state and control constraints is known to be a very hard task in this note we propose a suboptimal solution based on the euler auxiliary system eas we show that we can determine a continuous time stabilizing control whose cost not only converges to the optimal as the eas time parameter vanishes but it is also upper bounded by the discrete time cost no matter how such a parameter is chosen in particular continuous time linear problems with convex cost can be solved by considering a fictitious receding horizon scheme both stability and constraints satisfaction are guaranteed for the continuous time system this scheme turns out to be very useful when due to unstable or poorly damped dynamics the digital implementation of the control requires a very small virtually zero sampling time since the time parameter of the eas can be much greater than the sampling time without compromising stability with a strongly reduced computational burden

links between asymptotic controllability and persistence of excitation for a class of time varying systems abstract for a class of nonlinear time varying parameterized systems we prove that persistence of excitation implies global asymptotic controllability to the equilibrium with respect to a set of parameters the result is used to derive sufficient conditions for global asymptotic controllability for composite systems

robust global stabilisability by means of sampled data control with positive sampling rate this work proposes a notion of robust reachability of one set from another set under constant control this notion is used to construct a control strategy involving sequential set to set reachability which guarantees robust global stabilisation of non linear sampled data systems with positive sampling rate sufficient conditions for robust reachability of one set from another under constant control are also provided the proposed method is illustrated through a number of examples including the study of the sampled data stabilisation problem of the chemostat

stabilisation of nonlinear systems by discontinuous dynamic state feedback in this paper a dynamic state feedback that stabilizes general nonlinear affine in the control systems is proposed it is deduced from the solution of an hamilton jacobi like equality with two boundary conditions the asymptotic stability of the closed loop system is proved and some examples are given an interesting feature of this method is that the so obtained feedback takes the form of a generalised sliding mode control in the sense that it is a dynamic feedback on one half part of the space and a static feedback on the other half with eventually a discontinuity on the surface dividing the space

vaccination policies and nonlinear force of infection generalization of an observation by alexander and moghadas 2004 it has been recently showed in m e alexander s m moghadas periodicity in an epidemic model with a generalized non linear incidence math biosci 189 2004 75 96 that one of the possible causes of unexpected failures of some vaccination campaigns may be the nonlinearity of the force of infection the aim of this paper is to give a simple and biologically meaningful sufficient condition for the globally stable eradication of diseases whose spreading is described via a generic class of sir and seir epidemic models ruled by nonlinear force of infection we consider a scenario in which there are both traditional and pulse vaccination strategies

the mathematics of infectious diseases many models for the spread of infectious diseases in populations have been analyzed mathematically and applied to specific diseases threshold theorems involving the basic reproduction number r 0 the contact number sigma and the replacement number r are reviewed for the classic sir epidemic and endemic models similar results with new expressions for r 0 are obtained for mseir and seir endemic models with either continuous age or age groups values of r 0 and sigma are estimated for various diseases including measles in niger and pertussis in the united states previous models with age structure heterogeneity and spatial structure are surveyed

universal formulas for feedback stabilization with respect to minkowski balls this note provides explicit algebraic stabilizing formulas for clf s when controls are restricted to certain minkowski balls in euclidean space feedbacks of this kind are known to exist by a theorem of artstein but the proof of artstein s theorem is nonconstructive the formulas are obtained from a general feedback stabilization technique and are used to construct approximation solutions to some stabilization problems

control of systems without drift via generic loops this paper proposes a simple numerical technique for the steering of arbitrary analytic systems with no drift it is based on the generation of nonsingular loops which allow linearized controllability along suitable trajectories once such loops are available it is possible to employ standard newton or steepest descent methods as classically done in numerical control the theoretical justification of the approach relies on results establishing the genericity of nonsingular controls as well as a simple convergence lemma

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

on the lyapunov stability for sirs epidemic models with general nonlinear incidence rate abstract we study an epidemic model for infections with non permanent acquired immunity sirs the incidence rate is assumed to be a general nonlinear function of the susceptibles and the infectious classes by using a peculiar lyapunov function we obtain necessary and sufficient conditions for the local nonlinear stability of equilibria conditions ensuring the global stability are also obtained unlike the recent literature on this subject here no restrictions are required about the monotonicity and concavity of the incidence rate with respect to the infectious class among the applications the noteworthy case of a convex incidence rate is provided

sampled data feedback practical semi global controllability and stabilization for time varying systems we deal with the problems of the closed loop controllability and stabilization for time varying systems whose dynamics are in general unbounded with respect to time it is shown that a general notion of global asymptotic controllability being in general non uniform with respect to initial values of time implies existence of a family of continuous control lyapunov functions and this enables the establishment of practical semi global closed loop controllability by means of a sampled data time varying feedback moreover the usual concept of global asymptotic controllability being uniform with respect to the initial values of time implies practical semi global stabilization by sampled data feedback the work is a continuation of a recent paper of the author in the same journal and extends krichman s 2000 result relying practical stabilization of time varying systems with bounded in time dynamics

feedback stabilization of a class of nonsmooth systems in this paper we investigate global stabilization of nonsmooth systems the aim is to provide sufficient condition based on standard derivative for feedback stabilization of nonsmooth systems by means of a feedback of class cr r 1 the main result is a generalization of a result concerning smooth systems see for instance j control optimization 32 1994 358 to systems that are only continuous and locally lipschitz

a framework for stabilization of nonlinear sampled data systems based on their approximate discrete time models a unified framework for design of stabilizing controllers for sampled data differential inclusions via their approximate discrete time models is presented both fixed and fast sampling are considered in each case sufficient conditions are presented which guarantee that the controller that stabilizes a family of approximate discrete time plant models also stabilizes the exact discrete time plant model for sufficiently small integration and or sampling periods previous results in the literature are extended to cover 1 continuous time plants modeled as differential inclusions 2 general approximate discrete time plant models 3 dynamical discontinuous controllers modeled as difference inclusions and 4 stability with respect to closed arbitrary not necessarily compact sets

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

on the stabilization in finite time of locally controllable systems by means of continuous time varying feedback law it is proven that if for any positive time t there exists an open loop control u a t depending continuously on the initial data a vanishing for a 0 and steering a small neighborhood of 0 into 0 in time t then the control system can be locally stabilized in small time by means of continuous time varying feedback law provided that the dimension of the state space is at least 4 and the strong accessibility rank condition holds

remarks on asymptotic controllability and sampled data feedback stabilization for autonomous systems a weak concept of sampled data feedback stabilization for the case of autonomous systems is presented and its lyapunov characterization is established sufficient conditions are derived for the existence of a time varying sampled data feedback exhibiting global stabilization for certain class of nonlinear systems

homogeneous state feedback stabilization of homogenous systems we show that for any asymptotically controllable homogeneous system in euclidean space not necessarily lipschitz at the origin there exists a homogeneous control lyapunov function and a homogeneous possibly discontinuous state feedback law stabilizing the corresponding sampled closed loop system if the system satisfies the usual local lipschitz condition on the whole space we obtain semiglobal stability of the sampled closed loop system for each sufficiently small fixed sampling rate if the system satisfies a global lipschitz condition we obtain global exponential stability for each sufficiently small fixed sampling rate the control lyapunov function and the feedback are based on the lyapunov exponents of a suitable auxiliary system and admit a numerical approximation

propagating asymptotic controllability through integrators for a class of systems with triangular structure we derive sufficient conditions for global asymptotic controllability which according to clarke ledyaev sontag subbotin theorem guarantees asymptotic stabilization by discontinuous static feedback

nonlinear regulation the piecewise linear approach this paper approaches nonlinear control problems through the use of discrete time piecewise linear systems these are systems whose next state and output maps are both described by pl maps i e by maps which are affine on each of the components of a finite polyhedral partition various results on state and output feedback observers and inverses standard for linear systems are proved for pl systems many of these results are then used in the study of more general both discrete and continuous time systems using suitable approximations

on the topological structure of attraction basins for differential inclusions abstract we show that when a compact set is globally asymptotically stable under the action of a differential inclusion satisfying certain regularity properties there exists a smooth differential equation rendering the same compact set globally asymptotically stable the regularity properties assumed in this work stem from the consideration of krasovskii filippov solutions to discontinuous differential equations and the robustness of asymptotic stability under perturbation in particular the results in this work show that when a compact set cannot be globally asymptotically stabilized by continuous feedback due to topological obstructions it cannot be robustly globally asymptotically stabilized by discontinuous feedback either the results follow from converse lyapunov theory and parallel what is known for the local stabilization problem

shape from patterns regularization we present a theory for the recovery of the shape of a surface covered with small elements texels the theory is based on the apparent surface pattern distortion in the image and fits the regularization paradigm recently introduced in computer vision by poggio et al 1985 a mapping is defined on the basis of measurement of the local distortions of a repeated unknown texture pattern due to the image projection from this a constraint on the gradient orientation is determined from the apparent area of a texture element the analysis is done under an approximation of the perspective projection called paraperspective the resulting algorithm is applied to several synthetic and real images to demonstrate its performance

control lyapunov functions and zubov s method for finite dimensional nonlinear control systems we study the relation between asymptotic null controllability and control lyapunov functions it is shown that control lyapunov functions clfs may be constructed on the domain of asymptotic null controllability as viscosity solutions of a first order pde that generalizes zubov s equation the solution is also given as the value function of an optimal control problem from which several regularity results may be obtained

continuous control lyapunov functions for asymptotically controllable time varying systems this paper shows that for time varying systems global asymptotic controllability to a given closed subset of the state space is equivalent to the existence of a continuous control lyapunov function with respect to the set

asymptotic stabilisation of a class of nonlinear systems via sampled data output feedback control this article considers sampled data output feedback control of a class of nonlinear systems in output feedback form the underlying continuous time controller is designed based on backstepping technique employing a linear dynamic filter and globally asymptotically stabilises the system rigorous analysis shows that when implemented with a sampling and zero order hold device the sampled data version of the continuous time controller semi globally asymptotically stabilises the original system given that the sampling period is smaller than a specific value which depends on the initial values and nonlinearity of the system simulation results of a physical system are included in the end

hybrid feedback control and robust stabilization of nonlinear systems in this paper we show for any nonlinear system that is asymptotically controllable to a compact set that a logic based hybrid feedback can achieve asymptotic stabilization that is robust to small measurement noise actuator error and external disturbance the construction of such a feedback hinges upon recasting a stabilizing patchy feedback in a hybrid framework by making it dynamic with a discrete state while insisting on semicontinuity and closedness properties of the hybrid feedback and of the resulting closed loop hybrid system the robustness of stability is then shown as a generic property of hybrid systems having the said regularity properties auxiliary results give uniformity of convergence and of overshoots for hybrid systems and give a characterization of asymptotic stability of compact sets

stability and controllability of planar conewise linear systems abstract this paper presents a fairly complete treatment of stability and controllability of piecewise linear systems defined on a conic partition of r 2 this includes necessary and sufficient conditions for stability and controllability as well as establishing that controllability implies stabilizability by piecewise linear state feedback a key tool in the approach is the study of the poincare map

global asymptotic controllability implies input to state stabilization the main problem addressed in this paper is the design of feedbacks for globally asymptotically controllable gac control affine systems that render the closed loop systems input to state stable iss with respect to actuator errors extensions for fully nonlinear gac systems with actuator errors are also discussed our controllers have the property that they tolerate small observation noise as well

remarks regarding the gap between continuous lipschitz and differentiable storage functions for dissipation inequalities appearing in h control abstract this paper deals with the regularity of solutions of the hamilton jacobi inequality which arises in h control it shows by explicit counterexamples that there are gaps between existence of continuous and locally lipschitz positive definite and proper solutions and between lipschitz and continuously differentiable ones on the other hand it is shown that it is always possible to smooth out solutions provided that an infinitesimal increase in gain is allowed

some remarks on stabilization by means of discontinuous feedbacks by means of ryan s necessary condition and of some examples we see how the interpretation of solutions of systems with discontinuous right hand side influences stabilizability results for systems which do not admit continuous stabilizing feedback laws