on the equivalence between dissipativity and optimality of discontinuous nonlinear regulators for filippov dynamical systems in this paper we derive guaranteed gain sector and disk margins for nonlinear optimal and inverse optimal discontinuous feedback regulators that minimize a nonlinear nonquadratic performance functional for filippov dynamical systems specifically we consider dynamical systems with lebesgue measurable and locally essentially bounded vector fields characterized by differential inclusions involving filippov set valued maps in addition we extend classical dissipativity theory to address the problem of dissipative discontinuous dynamical systems these results are then used to derive extended kalman yakubovich popov conditions for characterizing necessary and sufficient conditions for dissipativity of discontinuous systems using clarke gradients and locally lipschitz continuous storage functions furthermore using the newly developed dissipativity notions we develop a return difference inequality to provide connections between dissipativity and optimality of nonlinear discontinuous controllers for filippov dynamical systems specifically using the extended kalman yakubovich popov conditions we show that our discontinuous feedback control law satisfies a return difference inequality if and only if the controller is dissipative with respect to a quadratic supply rate

discrete time asymptotic controllability implies smooth control lyapunov function we demonstrate the existence of a smooth control lyapunov function clf for difference equations asymptotically controllable to closed sets we further show that this clf may be used to construct a robust feedback stabilizer the existence of such a clf is a consequence of a more general result on the existence of weak lyapunov function under the assumption of weak asymptotic stability of a closed not necessarily compact set for a difference inclusion

backstepping with bounded feedbacks for time varying systems a family of time varying nonlinear systems is globally uniformly asymptotically stabilized by bounded feedbacks constructed through a new extension of the backstepping approach explicit expressions of control laws and lyapunov functions are given

a unified view of transport equations abstract distribution functions of many static transport equations are found using the maximum entropy principle the equations of constraint which contain the relevant dynamical information are simply the low lying moments of the distributions systems subject to conservative forces have also been considered in this approach determining the solutions to the transport equations no longer requires solving a partial differential equation but instead experimentally determining the low lying moments and potentials

towards a generic constructive nonlinear control design tool using relaxed control in this technical note we revisit a control design approach for general non affine nonlinear systems using relaxed control using the notion of relaxed input where the ordinary real valued control input is replaced by a measure valued control input we are able to manipulate the original system such that the resulting relaxed system is amenable to control we show the applicability of using relaxed input for transforming a non affine nonlinear system into an affine one such that the relaxed system has desirable control properties finally we present a practical implementation of relaxed input via ordinary control input which is based on the sampled data of the relaxed input

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

input to state stabilization of nonlinear discrete time systems with event triggered controllers abstract this paper studies an event triggered control problem of nonlinear discrete time systems in the presence of external disturbances in particular we focus on the practically interesting situation where the controller update at step k is determined by the feedback information x k 1 measured at step k 1 the presence of external disturbance makes it hard to predict the magnitude of x k based on x k 1 and then to design a desired event triggering data transmission mechanism in this paper refined tools of input to state stability iss and the nonlinear small gain theorem are developed to estimate the influence of external disturbances before an iss induced design is proposed to solve the problem the proposed solution leads to a new event triggering data transmission mechanism which drastically differs from the existing event triggered control results

multiple limit cycles in a continuous culture vessel with variable yield abstract determining the number of limit cycles for a continuous culture vessel system is always useful in analyzing the system we prove the conditions that guarantee there exist three limit cycles for the chemostat with variable yield that was first proposed by huang limit cycles in a continuous fermentation model j math chem 5 1990 287 296 and by pilyugin and waltman multiple limit cycles in the chemostat with variable yield math boisci 182 2003 151 166

yield trajectory tracking for hyperbolic age structured population systems abstract for population systems modeled by age structured hyperbolic partial differential equations pdes that are bilinear in the input and evolve with a positive valued infinite dimensional state global stabilization of constant yield set points was achieved in prior work seasonal demands in biotechnological production processes give rise to time varying yield references for the proposed control objective aiming at a global attractivity of desired yield trajectories multiple non standard features have to be considered a non local boundary condition a pde state restricted to the positive orthant of the function space and arbitrary restrictive but physically meaningful input constraints moreover we provide control lyapunov functionals ensuring an exponentially fast attraction of adequate reference trajectories to achieve this goal we make use of the relation between first order hyperbolic pdes and integral delay equations leading to a decoupling of the input dependent dynamics and the infinite dimensional internal one furthermore the dynamic control structure does not necessitate exact knowledge of the model parameters or online measurements of the age profile with a galerkin based numerical simulation scheme using the key ideas of the karhunen loeve decomposition we demonstrate the controller s performance

input to state stabilization of linear systems with positive outputs this paper considers the problem of stabilization of linear systems for which only the magnitudes of outputs are measured it is shown that if a system is controllable and observable then one can find a stabilizing controller which is robust with respect to observation noise in the iss sense

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 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

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

on dynamical properties of general dynamical systems and differential inclusions abstract let f be a general dynamical system defined on a complete locally compact metric space x we give a slightly improved version of the lyapunov characterization of asymptotic stability in one of our previous works and provide a short self contained proof for the existence of arbitrarily small positively invariant neighborhoods of compact asymptotically stable sets in the present context based on these results we prove that the uniform attractors of f are connected and are stable with respect to perturbations under appropriate conditions we are also interested in the dynamical properties of differential inclusion x t f x t on r m first we show that if no solutions of the system blow up in finite future time then its reachable mapping f is a general dynamical system then we discuss some asymptotic stability properties of the system in particular we prove that if there exists a nonempty compact connected subset m of r m such that m attracts a neighborhood of itself then the system has a connected uniform attractor a we also prove that a is stable with respect to both internal and external perturbations more precisely we prove that when 0 is sufficiently small the perturbed system x t con f x t b 1 b 1 has a connected uniform attractor a moreover h a a 0 as 0 where h is the hausdorff distance in r m

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

backstepping with bounded feedbacks an extension of the backstepping approach is proposed it allows to globally asymptotically stabilize by bounded feedbacks families of nonlinear control systems explicit expressions of control laws and lyapunov functions are given

a general result on the stabilization of linear systems using bounded controls we present two constructions of controllers that globally stabilize linear systems subject to control saturation we allow essentially arbitrary saturation functions the only conditions imposed on the system are the obvious necessary ones namely that no eigenvalues of the uncontrolled system have positive real part and that the standard stabilizability rank condition hold one of the constructions is in terms of a neural network type one hidden layer architecture while the other one is in terms of cascades of linear maps and saturations

inverse optimality in robust stabilization the concept of a robust control lyapunov function bf rclf is introduced and it is shown that the existence of an bf rclf for a control affine system is equivalent to robust stabilizability via continuous state feedback this extends artstein s theorem on nonlinear stabilizability to systems with disturbances it is then shown that every bf rclf satisfies the steady state hamilton jacobi isaacs hji equation associated with a meaningful game and that every member of a class of pointwise min norm control laws is optimal for such a game these control laws have desirable properties of optimality and can be computed directly from the bf rclf without solving the hji equation for the upper value function

feedback stabilization and lyapunov functions given a locally defined nondifferentiable but lipschitz lyapunov function we employ it in order to construct a discontinuous feedback law which stabilizes the underlying system to any given tolerance a converse result shows that suitable lyapunov functions of this type exist under mild assumptions we also establish that the feedback in question possesses a robustness property relative to measurement error despite the fact that it may not be continuous

a hybrid robust non homogeneous finite time differentiator a variant of super twisting differentiator is proposed lyapunov function is designed and an estimate on finite time of derivatives estimation is given the differentiator is equipped with hybrid adaptation algorithm that ensures global differentiation ability independently on amplitude of the differentiated signal and measurement noise

a remark on the converging input converging state property suppose that an equilibrium is asymptotically stable when external inputs vanish then every bounded trajectory which corresponds to a control which approaches zero and which lies in the domain of attraction of the unforced system must also converge to the equilibrium this well known but hard to cite fact is proved and slightly generalized here

global asymptotic behavior of a chemostat model with discrete delays this paper studies the global asymptotic behavior of an exploitative competition model between n species in a chemostat the model incorporates discrete time delays to describe the delay in the conversion of nutrient consumed to viable biomass and hence includes delays simultaneously in variables of nutrient and species concentrations in the case where only two species are engaged in competition it is shown that competitive exclusion holds for any monotone growth response functions sufficient conditions are also obtained for the model to exhibit competitive exclusion in the n species case in regard to the delay effects on the qualitative outcome of competition it is demonstrated that when the delays are relatively small the predictions of the model are identical with the predictions given by corresponding models without time delays however including large delays in the model may alter the predicted outcome of competition the techniques used also work when different removal rates are permitted an

further results on stabilization of periodic trajectories for a chemostat with two species we discuss an important class of problems involving the tracking of prescribed trajectories in the chemostat model we provide new tracking results for chemostats with two species and one limiting substrate based on lyapunov function methods in particular we use a linear feedback control of the dilution rate and an appropriate time varying substrate input concentration to produce a locally exponentially stable oscillatory behavior this means that all trajectories for the nutrient and corresponding species concentrations in the closed loop chemostat that stay near the oscillatory reference trajectory are attracted to the reference trajectory exponentially fast we also obtain a globally stable oscillatory reference trajectory for the species concentrations using a nonlinear feedback control depending on the dilution rate and the substrate input concentration this guarantees that all trajectories for the closed loop chemostat dynamics are attracted to the reference trajectory finally we construct an explicit lyapunov function for the corresponding global error dynamics we demonstrate the efficacy of our method in a simulation

information theory explanation of the fluctuation theorem maximum entropy production and self organized criticality in non equilibrium stationary states jaynes information theory formalism of statistical mechanics is applied to the stationary states of open non equilibrium systems first it is shown that the probability distribution p of the underlying microscopic phase space trajectories over a time interval of length satisfies p exp 2kb where is the time averaged rate of entropy production of three consequences of this result are then derived 1 the fluctuation theorem which describes the exponentially declining probability of deviations from the second law of thermodynamics as 2 the selection principle of maximum entropy production for non equilibrium stationary states empirical support for which has been found in studies of phenomena as diverse as the earth s climate and crystal growth morphology and 3 the emergence of self organized criticality for flux driven systems in the slowly driven limit the explanation of these results on general information theoretic grounds underlines their relevance to a broad class of stationary non equilibrium systems in turn the accumulating empirical evidence for these results lends support to jaynes formalism as a common predictive framework for equilibrium and non equilibrium statistical mechanics

statistical mechanics of the us supreme court we build simple models for the distribution of voting patterns in a group using the supreme court of the united states as an example the maximum entropy model consistent with the observed pairwise correlations among justices votes an ising spin glass agrees quantitatively with the data while all correlations perhaps surprisingly are positive the effective pairwise interactions in the spin glass model have both signs recovering the intuition that ideologically opposite justices negatively influence each another despite the competing interactions a strong tendency toward unanimity emerges from the model organizing the voting patterns in a relatively simple energy landscape besides unanimity other energy minima in this landscape or maxima in probability correspond to prototypical voting states such as the ideological split or a tightly correlated conservative core the model correctly predicts the correlation of justices with the majority and gives us a measure of their influence on the majority decision these results suggest that simple models grounded in statistical physics can capture essential features of collective decision making quantitatively even in a complex political context

a small gain approach to robust event triggered control of nonlinear systems this paper presents a new approach to event triggered control for nonlinear uncertain systems by using the notion of input to state stability iss and the nonlinear small gain theorem the contribution of this paper is threefold first it is proved that infinitely fast sampling can be avoided if the system is input to state stabilizable with the sampling error as the external input and the corresponding iss gain is locally lipschitz no assumption on the existence of known iss lyapunov functions is made in the discussions moreover the forward completeness problem with event triggered control is studied systematically by using iss small gain arguments second the proposed approach gives rise to a new self triggered sampling strategy for a class of nonlinear systems subject to external disturbances if an upper bound of the external disturbance is known then the closed loop system can be designed to be robust to the external disturbance and moreover the system state globally asymptotically converges to the origin if the external disturbance decays to zero third a new design method is developed for event triggered control of nonlinear uncertain systems in the strict feedback form it is particularly shown that the iss gain with the sampling error as the input can be designed to satisfy the proposed condition for event triggered control and self triggered control

a multipoint continued fraction expansion for linear system reduction this note presents a multipoint continued fraction expansion mcfe to obtain reduced models for linear time invariant systems algorithms for the expansion and inversion of the mcfe are outlined a realization for the mcfe of a transfer function is suggested and the corresponding mcfe canonical state space model is established a new similarity transformation matrix is constructed to transform a state space model in the phase variable canonical form to one in the mcfe canonical form the attractive feature of the present mcfe is that it is general in form and the resulting reduced models approximate well the original system in both the frequency and time responses

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

relaxed lyapunov criteria for robust global stabilisation of non linear systems the notion of the restricted robust control lyapunov function rclf is introduced and is exploited for the design of robust feedback stabilisers for non linear systems particularly it is shown for systems with input constraints that restricted rclfs can be easily obtained while rclfs are not available moreover it is shown that the use of restricted rclfs usually results in different feedback designs from the ones obtained by the use of the standard rclf methodology using the restricted rclfs feedback design methodology a simple controller that guarantees robust global stabilisation of a perturbed chemostat model 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

smooth patchy control lyapunov functions a smooth patchy control lyapunov function for a nonlinear system consists of an ordered family of smooth local control lyapunov functions whose open domains form a locally finite cover of the state space of the system and which satisfy certain further increase or decrease conditions we prove that such a control lyapunov function exists for any asymptotically controllable nonlinear system we also show a construction based on such a control lyapunov function of a stabilizing hybrid feedback that is robust to measurement noise

isotropic spatial filters for suppression of spurious noise waves in sub gridded fdtd simulation a class of isotropic spatial filters is proposed to suppress spurious noise waves due to sub gridding in finite difference time domain simulations the proposed filters are suitable for both two dimensional and three dimensional applications a simple procedure is introduced for the a priori determination of the required filter order and the frequency with which the filter is applied based on the maximum temporal frequency bandwidth of the electromagnetic source the proposed filters are easy to implement in the context of the standard yee algorithm and are applicable for computational domains that involve quite arbitrary material and structural inhomogeneities

stabilization of a chemostat model with haldane growth functions and a delay in the measurements the stabilization of equilibria in chemostats with measurement delays is a complex and challenging problem and is of significant ongoing interest in bioengineering and population dynamics in this paper we solve an output feedback stabilization problem for chemostat models having two species one limiting substrate and either haldane or monod growth functions our feedback stabilizers depend on a given linear combination of the species concentrations which are both measured with a constant time delay the values of the delays are unknown instead one only knows an upper bound on the delays and we allow the upper bound to be arbitrarily large the stabilizing feedback depends on the known upper bound for the delays as well our work is based on a lyapunov krasovskii argument

nonlinear control synthesis by convex optimization a stability criterion for nonlinear systems recently derived by the third author can be viewed as a dual to lyapunov s second theorem the criterion is stated in terms of a function which can be interpreted as the stationary density of a substance that is generated all over the state space and flows along the system trajectories toward the equilibrium the new criterion has a remarkable convexity property which in this note is used for controller synthesis via convex optimization recent numerical methods for verification of positivity of multivariate polynomials based on sum of squares decompositions are used

deterministic reduced order macromodels for computing the broadband radiation field pattern of antenna arrays in fdtd the enhanced nodal order reduction enor technique is used to perform model order reduction mor on a macromodel of a subgridded finite difference time domain fdtd region which contains the deterministic fine structural features 1 of an aperture antenna array laying in a larger and relatively coarser fdtd grid to solve the broadband time varying electromagnetic fields emanating from the antenna array and to obtain the far field radiation patterns the far field radiation pattern computed in fdtd is correlated against results computed through physical optics approximation this work lays the foundation for a separate but related work to follow in which we further develop the above deterministic macromodel into a stochastic macromodel that addresses the problem of uncertainty in fine features of structures with disparate spatial scales

a delayed chemostat model with general nonmonotone response functions and differential removal rates a chemostat model with general nonmonotone response functions is considered the nutrient conversion process involves time delay we show that under certain conditions when n species compete in the chemostat for a single resource that is allowed to be inhibitory at high concentrations the competitive exclusion principle holds in the case of insignificant death rates the result concerning the attractivity of the single species survival equilibrium already appears in the literature several times see h m el owaidy m ismail asymptotic behavior of the chemostat model with delayed response in growth chaos solitons fractals 13 2002 787 795 h m el owaidy a a moniem asymptotic behavior of a chemostat model with delayed response growth appl math comput 147 2004 147 161 s yuan m han z ma competition in the chemostat convergence of a model with delayed response in growth chaos solitons fractals 17 2003 659 667 however the proofs are all incorrect in this paper we provide a correct proof that also applies in the case of differential death rates in addition we provide a local stability analysis that includes sufficient conditions for the bistability of the single species survival equilibrium and the washout equilibrium thus showing the outcome can be initial condition dependent moreover we show that when the species specific death rates are included damped oscillations may occur even when there is no delay thus the species specific death rates might also account for the damped oscillations in transient behavior observed in experiments

generation of fdtd subcell equations by means of reduced order modeling adapted finite difference time domain fdtd update equations exist for a number of objects that are smaller than the grid step such as wires and thin slots we provide a technique that automatically generates new fdtd update equations for small objects our presentation focusses on 2d fdtd we start from the fdtd equations in a fine grid where the time derivative is not discretised this yields a large state space model that is drastically reduced with a reduced order modeling technique the reduced state space model is then translated into new fdtd update equations that can be used in an fdtd simulation in the same way as the existing update equations for wires and thin slots this technique is applied to a number of numerical problems showing the accuracy and versatility of the proposed method

bifurcation for a functional yield chemostat when one competitor produces a toxin abstract a model with general yield functions f i s i 1 2 of competition in the chemostat of two competitors for a single nutrient when one of the competitors produces toxin against its opponent is studied in this paper the conditions in terms of the relevant parameters for the hopf bifurcation of the three dimensional system have been proved which implies the existence of limit cycles in the 3 d system

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

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

maximum entropy production and the fluctuation theorem recently the author used an information theoretical formulation of non equilibrium statistical mechanics maxent to derive the fluctuation theorem ft concerning the probability of second law violating phase space paths a less rigorous argument leading to the variational principle of maximum entropy production mep was also given here a more rigorous and general mathematical derivation of mep from maxent is presented and the relationship between mep and the ft is thereby clarified specifically it is shown that the ft allows a general orthogonality property of maximum information entropy to be extended to entropy production itself from which mep then follows the new derivation highlights mep and the ft as generic properties of maxent probability distributions involving anti symmetric constraints independently of any physical interpretation physically mep applies to the entropy production of those macroscopic fluxes that are free to vary under the imposed constraints and corresponds to selection of the most probable macroscopic flux configuration in special cases maxent also leads to various upper bound transport principles the relationship between maxent and previous theories of irreversible processes due to onsager prigogine and ziegler is also clarified in the light of these results

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

maximum entropy approach for deducing amino acid interactions in proteins we present a maximum entropy approach for inferring amino acid interactions in proteins subject to constraints pertaining to the mean numbers of various types of equilibrium contacts for a given sequence or a set of sequences we have carried out several kinds of tests for a two dimensional lattice model with just two types of amino acids with very promising results we also show that the method works very well even when the mean numbers of contacts are not known and therefore can be applied to real proteins

asymptotic stability and smooth lyapunov functions abstract we establish that differential inclusions corresponding to upper semicontinuous multifunctions are strongly asymptotically stable if and only if there exists a smooth lyapunov function since well known concepts of generalized solutions of differential equations with discontinuous right hand side can be described in terms of solutions of certain related differential inclusions involving upper semicontinuous multifunctions this result gives a lyapunov characterization of asymptotic stability of either filippov or krasovskii solutions for differential equations with discontinuous right hand side in the study of weak as opposed to strong asymptotic stability the existence of a smooth lyapunov function is rather exceptional however the methods employed in treating the strong case of asymptotic stability are applied to yield a necessary condition for the existence of a smooth lyapunov function for weakly asymptotically stable differential inclusions this is an extension to the context of lyapunov functons of brockett s celebrated covering condition from continuous feedback stabilization theory

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

smooth lyapunov functions for hybrid systems part ii pre asymptotically stable compact sets it is shown that pre asymptotic stability which generalizes asymptotic stability of a compact set for a hybrid system satisfying mild regularity assumptions is equivalent to the existence of a smooth lyapunov function this result is achieved with the intermediate result that asymptotic stability of a compact set for a hybrid system is generically robust to small state dependent perturbations as a special case we state a converse lyapunov theorem for systems with logic variables and use this result to establish input to state stabilization using hybrid feedback control the converse lyapunov theorems are also used to establish semiglobal practical robustness to slowly varying weakly jumping parameters to temporal regularization to the insertion of jumps according to an ldquoaverage dwell timerdquo rule and to the insertion of flow according to a ldquoreverse average dwell timerdquo rule

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

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

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