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For a special class of systems, it is shown that the existence of a common quadratic Lyapunov function (CQLF) is necessary and sufficient for the stability of an associated switched system under arbitrary switching. Furthermore, it is shown that the existence of a CQLF for N (N > 2) subsystems is equivalent to the existence of a CQLF for every pair of subsystems. An algorithm is proposed to compute a CQLF for the subsystems, when it exists, using the left and right eigenvectors of a critical matrix obtained from a matrix pencil. Index Terms--Common quadratic Lyapunov function (CQLF), M-matrix, stability, switched systems.

A new Lyapunov-Krasovskii functional is constructed for delayed cellular neural networks, and the S-procedure is employed to handle the nonlinearities. An improved global asymptotic stability criterion is also derived that is a generalization of, and an improvement over, previous results. Numerical examples demonstrate the effectiveness of the criterion. Index Terms--Delayed cellular neural networks, global asymptotic stability, linear matrix inequality (LMI), S-procedure.

The linear stability properties of Euler's method and quaternion-based algorithm are examined. The Lyapunov exponent is found to be positive in the Euler method and hence this method is unstable at all wave vectors whereas the quaternion algorithm is significantly more stable.

An algorithm and its intended application to large-scale system decentralized controller designs are presented in this paper. Index Terms - Decentralized control, large-scale systems.

A new linear robust state-feedback controller that guarantees uniform boundedness for a class of discrete-time uncertain systems with multi-inputs is presented, based on the second method of Lyapunov. Furthermore, an upper bound of the uncertainties has been derived in order to make the closed-loop system stable. Index Terms - Discrete-time systems, Lyapunov stability, robust control, uncertain systems.

Sufficient conditions for the global asymptotic stability of the equilibrium [x.sub.e] = 0 of discrete-time dynamical systems which have saturation nonlinearities on part of the states are established. We utilize a class of positive definite and radially unbounded Lyapunov functions in establishing our results. When using quadratic form Lyapunov functions, our results involve necessary and sufficient conditions under which positive definite matrices can be used to generate Lyapunov functions for the systems considered herein.

Several delay-dependent criteria to test the stability of time-delay systems that were proposed require solving the Lyapunov matrix equation. This can be a troublesome task and often non-trivial. In this note, a delay-dependent sufficient condition that guarantees the robust stability of linear uncertain time-delay systems is presented. The stability test criterion derived in this paper is based on induced norms and matrix measures. The salient feature of the result obtained is its simplicity and ease in testing the robust stability of uncertain time-delay linear systems.

In this letter, we develop a new adjustment rule for a perceptron with a saturating nonlinearity that ensures perfect classification when the input patterns are linearly separable. The proof is based on the Lyapunov stability formalism, is widely used in deterministic process identification, and is rather straightforward. It should therefore be of pedagogical interest.

Stability in the sense of Lyapunov for the adaptive robot controller proposed by Slotine and Li is proved in this note. The result is a generalization of previous work, where the feedback gain matrix was assumed to be constant and diagonal, while in this paper the feedback gain matrix is only assumed to be uniformly positive definite.

Block-diagonalization of a singularly perturbed system requires the solution of the Riccati equation and the Lyapunov equation. A new approach is suggested for both equations, using Taylor expansions. The convergence is studied in detail; it is shown that it is ensured under restrictions which are less restrictive than the case of previously suggested methods.

Lower eigenvalue summation (including trace) bounds for the solution of the discrete algebraic Riccati and Lyapunov matrix equations are presented. These are tighter than or supplement existing results.

Upper bounds for summations including the trace, and for products including the determinant, of the eigenvalues of the solution of the continuous algebraic Lyapunov matrix equation are presented. The majority of the bounds are tighter than those in the literature, and some are new.

Byline: Toshio Yoshimura This paper is concerned with an adaptive output tracking control (AOTC) for a class of uncertain discrete-time systems via sliding mode theory. It is assumed that the uncertain discrete-time systems are described by a state equation with mismatched uncertainties found in unmodelled dynamics and external disturbances, and the uncertainties are expressed in a parameterised form. The estimates for the uncertainties are taken by using the proposed weighted least squares estimator (WLSE), and then the proposed AOTC is designed on the basis of the output from the estimator via sliding mode theory. It is verified by using the Lyapunov method that the estimation error equation is asymptotically stable and the uncertain discrete-time systems under the action of the proposed AOTC are ultimately stable. The effectiveness of the proposed method is indicated by the simulation experiment in a simple numerical example.

Byline: Salem Zerkaoui, Fabrice Druaux, Edouard Leclercq, Dimitri Lefebvre Adaptive control by means of neural networks for non-linear dynamical systems is an open issue. For real world applications, practitioners have to pay attention to external disturbances, parameters uncertainty and measurement noise, as long as these factors will influence the stability of the closed loop system. As a consequence, robust stability of the closed loop controlled by neural network is an important issue that must be considered. Our contribution concerns the robustness analysis and synthesis of adaptive indirect control scheme. This scheme is based on fully connected neural networks, and inspired from the standard real time recurrent learning. This analysis is concerned by combining Lyapunov approach and linearisation around the nominal parameters to establish analytical sufficient conditions for the global robust stability of adaptive neural network controller. Advantages of the proposed algorithm are suggested according to simulation examples.

The strength of microscopic and macroscopic sensitivity to initial conditions due to chaotic dynamics is characterized by the Lyapunov exponents. They give the rates of exponential growth of the vector differences between two nearby trajectories along different directions in phase space.

The early proofs of Lyapunov's theorem on the convexity of vector measures are long and elaborate, thus a new proof for is discussed. It is described that the proof for Lyapunov theorem can be obtained as a consequence of the intermediate value theorem.

The classical works of Steinhaus, Dubins and Spanier highlighting the problem of division of the cake so that everyone is satisfied with the share received, are discussed. It is observed that the Lyapunov Convexity Theorem cannot be generalized to guarantee a convex range for a vector of normalized, strongly subadditive, and strongly continuous set functions.

A system with a negative Lyapunov exponent (LE) in the absence of noise has a positive stochastic Lyapunov exponent (SLE) on the introduction of noise. The strong influence of underlying deterministic chaotic invariant sets help in the revelation of the chaotic dynamics in stochastic systems.

This note considers the robust stability problem for a class of time-delay systems with norm-bounded, and possibly time-varying, uncertainty. Based on the discretized Lyapunov functional approach, a stability criterion is derived. The time-delay is assumed constant and known. Numerical examples show that the results obtained by this new criterion significantly improve the estimate of the stability limit over some existing results in the literature. Index Terms--Linear matrix inequality (LMI), Lyapunov functional, stability, time-delay, uncertainty.

The transients induced in power systems by a large disturbance can initiate unexpected events such as protection operation or converter misfiring. This paper proposes a direct technique for assessing the likelihood of such events. The approach is based on Lyapunov (energy) function methods, but with the critical energy redefined to take account of protection operating characteristics and viability constraints. Power systems have many protection devices and many constraints that must be monitored. The paper provides a method of identifying the subset of protection devices and/or viability constraints that are most vulnerable for any particular disturbance. Controlling UEP (unstable equilibrium point) ideas underlie the algorithm. Index Terms--Lyapunov (energy) functions, power system protection, power system security assessment, viability constraints.

A stability criterion for uncertain systems with time-varying delay is derived via the Lyapunov functional approach. By checking the Hamiltonian matrix and solving an algebraic Riccati equation, a new bound on allowable stability preserving nonlinear perturbations is presented. The result obtained here is shown to be less conservative than others reported in the literature. Index Terms--Robust stability, time-delay systems, uncertain systems.

A one-dimensional iterative chaotic map with infinite collapses within symmetrical region [-1, 0) [union] (0, +1] is proposed. The stability of fixed points and that around the singular point are analyzed. Higher Lyapunov exponents of proposed map show stronger chaotic characteristics than some iterative and continuous chaotic models usually used. There exist inverse bifurcation phenomena and special main periodic windows at certain positions shown in the bifurcation diagram, which can explain the generation mechanism of chaos. The chaotic model with good properties can be generated if choosing the parameter of the map properly. Stronger inner pseudorandom characteristics can also be observed through [chi square] test on the supposition of even distribution. This chaotic model may have many advantages in practical use. Index Terms--Bifurcation, iterative chaotic map with infinite collapses, Lyapunov exponents, main periodic windows, stability, [chi square] test.

In this note, we consider production-distribution systems with buffer and capacity constraints. For such systems, we assume that the model is not known exactly. More precisely, the entries of the matrix representing the system structure may be affine functions of some uncertain time-varying parameters that take values within assigned bounds. We give stabilizability conditions that can be checked, in principle, by solving a min-max problem on the surface of the state-space (buffer level space) unit ball. Then, we consider a special case in which each uncertain parameter affects a single column of the system matrix and is independent of all the other ones. In this case, we propose a mixed integer linear programming algorithm to check the stabilizability conditions and we provide a stabilizing control in an explicit form. Index Terms--Constrained control, inventory systems, Lyapunov functions, min-max control, mixed integer linear programming, uncertain systems.

In this note, we establish new Lyapunov and Lagrange stability results in the pth mean for a class of discontinuous stochastic dynamical systems. We apply these results in the qualitative analysis of a class of digital feedback control systems that are subjected to multiplicative and additive disturbances in the plants. The present results constitute natural extensions of our earlier results for discontinuous deterministic dynamical systems. Index Terms--Boundedness in the pth mean, discontinuous, moment stability, stability in the pth mean, stochastic dynamical systems.

In this note, we analyze the discrete behavior to identify all kinds of cycles of hybrid nonlinear systems and then study the continuous behavior along each kind of cycle. Based on these analysis, we construct some continuous functions to bound Lyapunov functions along all subsystems and identify a subsequence of time points where the Lyapunov functions are nonincreasing. We use these results to derive some new sufficient conditions for the robust stability of a class of hybrid nonlinear systems with polytopic uncertainties. These conditions do not require the Lyapunov functions to be nonincreasing along each subsystem nor the whole sequence of the switchings. Furthermore, they do not require the knowledge of continuous trajectory. Index Terms--Hybrid nonlinear systems, robust stability.

This note points out several technical errors in the above paper. It is shown that the choice of Lyapunov function is inadequate for the system stability analysis. The asymptotic tracking of the unactuated joints is not achievable under certain conditions.

It is well known that unconstrained infinite-horizon optimal control may be used to construct a stabilizing controller for a nonlinear system. In this note, we show that similar stabilization results may be achieved using unconstrained finite horizon optimal control. The key idea is to approximate the tail of the infinite horizon cost-to-go using, as terminal cost, an appropriate control Lyapunov function. Roughly speaking, the terminal control Lyapunov function (CLF) should provide an (incremental) upper bound on the cost. In this fashion, important stability characteristics may be retained without the use of terminal constraints such as those employed by a number of other researchers. The absence of constraints allows a significant speedup in computation. Furthermore, it is shown that in order to guarantee stability, it suffices to satisfy an improvement property, thereby relaxing the requirement that truly optimal trajectories be found. We provide a complete analysis of the stability and region of attraction/operation properties of receding horizon control strategies that utilize finite horizon approximations in the proposed class. It is shown that the guaranteed region of operation contains that of the CLF controller and may be made as large as desired by increasing the optimization horizon (restricted, of course, to the infinite horizon domain). Moreover, it is easily seen that both CLF and infinite-horizon optimal control approaches are limiting cases of our receding horizon strategy. The key results are illustrated using a familiar example, the inverted pendulum, where significant improvements in guaranteed region of operation and cost are noted. Index Terms--Control Lyapunov functions (CLFs), model predictive control, nonlinear control design, optimal control, receding horizon control.

Nonlinear response of a flexible curved panel exhibiting bifurcation to fully developed chaos is demonstrated along with the sensitivity to small perturbations from the initial forcing. The response is determined from the measured time series at two fixed points. The panel is forced by an external nonharmonic multifrequency and monofrequency sound field. Using a low-power time-continuous control, carefully tuned to each initial forcing condition, produces large long-term effects on the dynamics toward taming chaos. Without knowledge of the initial forcing, control may be achieved by destructive interference. In this case, the control power is proportional to the loading power. Calculation of the correlation dimension and the estimation of positive Lyapunov exponents, in practice, are the proof of chaotic response.

This paper investigates the existence, uniqueness, and global exponential stability (GES) of the equilibrium point for a large class of neural networks with globally Lipschitz continuous activations including the widely used sigmoidal activations and the piecewise linear activations. The provided sufficient condition for GES is mild and some conditions easily examined in practice are also presented. The GES of neural networks in the case of locally Lipschitz continuous activations is also obtained under an appropriate condition. The analysis results given in the paper extend substantially the existing relevant stability results in the literature, and therefore expand significantly the application range of neural networks in solving optimization problems. As a demonstration, we apply the obtained analysis results to the design of a recurrent neural network (RNN) for solving the linear variational inequality problem (VIP) defined on any nonempty and closed box set, which includes the box constrained quadratic programming and the linear complementarity problem as the special cases. It can be inferred that the linear VIP has a unique solution for the class of Lyapunov diagonally stable matrices, and that the synthesized RNN is globally exponentially convergent to the unique solution. Some illustrative simulation examples are also given. Index Terms--Global exponential stability, global Lipschitz continuous activations, linear variational inequality problems, recurrent neural networks.

Since McCulloch and Pitts presented a simplified neuron model in 1943, several neuron models have been proposed. Among them, the binary maximum neuron model was introduced by Takefuji et al. and successfully applied to some combinatorial optimization problems. Takefuji et al. also presented a proof for the local minimum convergence of the maximum neural network. In this paper we discuss this convergence analysis and show that this model does not guarantee the descent of a large class of energy functions. We also propose a new maximum neuron model, the optimal competitive hopfield model (OCHOM), that always guarantees and maximizes the decrease of any Lyapunov energy function. Funabiki et al. applied the maximum neural network for the n-queens problem and showed that this model presented the best overall performance among the existing neural networks for this problem. Lee et al. applied the maximum neural network for the bipartite subgraph problem showing that the solution quality was superior to that of the best existing algorithm. However, simulation results in the n-queens problem and in the bipartite subgraph problem show that the OCHOM is much superior to the maximum neural network in terms of the solution quality and the computation time. Index Terms--Bipartite subgraph problem, combinatorial optimization, competitive Hopfield model, maximum neural network, n-queens problem, winner-take-all.

It is demonstrated that many previously reported Lyapunov-based stability conditions for time-delay systems are equivalent to the robust stability analysis of an uncertain comparison system free of delays via the use of the scaled small-gain lemma with constant scales. The novelty of this note stems from the fact that it unifies several existing stability results under the same framework, in addition, it offers insights on how new, less conservative results can be developed. Index Terms--Stability, time-delay systems.

The design of sliding mode control schemes for quantum resonant converters is introduced by means of two different approaches. First, an easy-to-use procedure for devising nonlinear control structures is established, using Lyapunov's well-known stability criteria. Second, an alternative method that provides linear sliding surfaces is also developed, considering reaching, existence, and stability conditions. The application of both control design techniques is illustrated in detail by means of three selected examples. The advantages and drawbacks of the resulting control circuits are examined. Simulation and experimental results corroborate the expected features of the close-loop quantum converters. Index Terms--Linear sliding surfaces, Lyapunov-based control, quantum resonant converters.

In this brief, an adaptive chaos control method is developed for stabilizing chaotic systems at their unknown equilibrium(s) using the invariant manifold theory. The developed method overcomes the problem that the equilibrium(s) of the chaotic systems are dependent on the unknown system parameters, which makes direct application of the conventional adaptive control difficult. Further development of the adaptive chaos control is undertaken for the situation where the parameter estimates are only allowed to vary within a bounded set due to the sensitivity of chaotic systems to parameter variations. A sufficient condition for convergence of system states and parameter estimates is obtained. The design method developed then is applied to stabilizing the Lorenz chaotic system at an unknown equilibrium. Both mathematical and computational results have demonstrated the effectiveness of this method. Index Terms--Adaptive control, chaos control, invariant manifolds, Lorenz system, Lyapunov method.

In this paper stochastic dynamic systems are studied, modeled by a countable state space Markov cost/reward chain, satisfying a Lyapunov-type stability condition. For an infinite planning horizon, risk-sensitive (exponential) discounted and average cost criteria are considered. The main contribution is the development of a vanishing discount approach to relate the discounted criterion problem with the average criterion one, as the discount factor increases to one, i.e., no discounting. In comparison to the well-established risk-neutral case, our results are novel and reveal several fundamental and surprising differences. For example, the limit of the (normalized) discounted costs, as the discount vanishes, is not equal to the average cost, but rather equal to an arithmetic mean of the average cost over a range of values for the risk sensitivity coefficient. Other significant contributions made in this paper are the use of convex analytic arguments to obtain appropriately convergent sequences and a verification theorem for the case of unbounded solutions to the average cost Poisson equation arising in the risk-sensitive case. Also of importance is the fact that our developments are very much self-contained and employ only basic probabilistic and analysis principles. Index Terms--Exponential Lyapunov stability condition, exponential utility function, Markov cost/reward chains, risk-sensitive average Poisson equation, risk-sensitive discounted and average criteria, vanishing discount approach.

This paper presents a new speed and position sensorless control method of permanent magnet synchronous motors based on the sliding mode observer. The sliding mode observer structure and its design method are described. Also, Lyapunov functions are chosen for determining the adaptive law for the speed and the stator resistance estimator. The effectiveness of the proposed observer is confirmed by the experimental results. Index Terms--PMSM (Permanent magnet synchronous motor), sensorless control, sliding mode control, stator resistance estimation.

This paper presents an adaptive sliding-mode observer for speed-sensorless field-oriented control of induction motors. The observer detects the rotor flux components in the two-phase stationary reference frame by the motor electrical equations. The motor speed is estimated by an additional relation obtained by a Lyapunov function. The analytical development of the sliding observer and the speed estimation algorithm is fully explained. Experimental results are presented, based on a TMS320F240 digital signal processor controller implementation. The system performance with different observer gains and the influence of the motor parameters deviations are shown and discussed. Index Terms--Adaptive observer, induction motor, sensorless control, sliding mode.

In this paper, an adaptive control method is presented for a class of first-order systems with nonlinear parameterization. The main features of the scheme are that a novel integral-type Lyapunov function is developed for constructing an asymptotically stable adaptive controller, and output tracking error bounds are provided to evaluate the control performance of the adaptive system. The design procedure and the effectiveness of the proposed controller are illustrated through an example study. Index Terms--Adaptive control, Lyapunov stability, nonlinear parameterization, transient performance.

This work evolved from an endeavor to derive stability criteria and Poincare normal forms for nonlinear systems associated with a nonsemisimple zero (in Part I) or a pair of imaginary eigenvalues (in Part II). The stability criteria are given in terms of the noninteracting vector restoring and restraining forces, which are motivated from the Lienard equation for nonlinear mass-damper-spring system models. Lyapunov functions are constructed explicitly to fulfill the La Salle invariant principle for local or global stability assertion. It turned out that the Lyapunov functions thus constructed apply to a wide variety of linear stability scenarios. By introducing the notions of restoring and restraining forces, how the Lyapunov functions, the stability criteria and the system dynamics interplay are also exhibited. Two distinct classes of nonlinearities which we refered to as being arithmetical and transcendental, emerged. In some sense, such systems carry nonlinear lags coexisting with the linear lead. In particular, a characteristic of the nonlinear dynamics, a staircase structure, is discovered. Further extension is also made to incorporate nondestabilizing perturbation, which bears important bifurcational implications. Index Terms--Lyapunov function/stability, nonsemisimple zero mode, Poincare normal forms.

This paper studies nonlinear systems associated with or linked to a nonsemisimple (NSS) pair of imaginary eigenvalues. Stability criteria and Poincare normal forms (PNF's) are derived via explicit construction of Lyapunov functions. It is a counterpart and continuation of Part I, which considered the case with an NSS zero eigenvalue. While originating from stability analysis for a nonsemisimple critical mode, the development extends to co-existing (semi-) simple critical modes and to stable modes, as in Part I. The present NSS imaginary pairs (NSSIP) case is shown to possess certain characteristics common to NSS zero (NSSZ), such as the nonlinear lag and the staircase structure, but retains its own identities as well. These include the less surprising, the submodular variable-pair circles which emerge from the units of the potential generator and the more remarkable stricter specification in the dynamics, which is possibly attributable to the inherited conjugacy. An elementary partial differential equation traversing the development shows potential for nontrivial generalization. Index Terms--Lyapunov function/stability, nonsemisimple imaginary pair mode, Poincare normal forms.

A disturbance observer based tracking control algorithm is presented in this paper. The key idea of the proposed method is that the plant nonlinearities and parameter variations can be lumped into a disturbance term. The lumped disturbance signal is estimated based on a plant dynamic observer. A state observer then corrects the disturbance estimation in a two-step design. First, a Lyapunov-based feedback estimation law is used. The estimation is then improved by using a feedforward correction term. The control of a telescopic robot arm is used as an example system for the proposed algorithm. Simulation results comparing the proposed algorithm against a standard adaptive control scheme and a sliding mode control algorithm show that the proposed scheme achieves superior performance, especially when large external disturbances are present. [S0022-0434(00)00802-9]

This article presents an innovative time-domain nonlinear mapping-based identification method. The method reported is applied to identify the unknown parameters of multivariable dynamic systems which are mapped by nonlinear differential equations. A systematic identification method is introduced, and a novel algorithm is developed using nonlinear error maps. An analysis of parameter convergence is provided and the regions of convergence can be found using the second method of Lyapunov. Innovative nonquadratic Lyapunov functions are designed and used. Analytical and numerical studies are performed to illustrate and validate the identification concept. The unsteady flight of a high-alpha aircraft in the longitudinal axis is chosen as a nonlinear case study. The unknown parameters are identified. Simulation results show that the model dynamics match the experimental data. The reported example demonstrates that the time-domain nonlinear mapping-based identification method ensures robustness and reduces major shortcomings in stability, convergence, and computational efficiency compared with other algorithms available.

This paper is concerned with the speed tracking control problem for a permanent-magnet synchronous motor (PMSM) in the presence of an unknown load torque disturbance. After a brief review of the mathematical model of the PMSM, a speed tracking control law using the exact linearization methodology is introduced. The tracking control algorithm is completed by adding an extended observer which provides, on the one hand, the motor speed and acceleration and, on the other hand, estimates the unknown load torque. The stability of the closed-loop system composed of a nonlinear speed tracking controller and an observer is studied by the way of Lyapunov theory. Furthermore, the decoupling of the state observer and the load torque observer is discussed. Finally, a real-time implementation and the experimental results of the proposed control strategy are presented. Index Terms--Digital signal processors, Lyapunov methods, nonlinear systems, observers, permanent-magnet synchronous motors.

In this paper, a Lyapunov-based control algorithm is developed for the force/pressure tracking control of an electro-hydraulic actuator. The controller relies on an accurate model of the system. To compensate for the parameter uncertainties, a standard parameter adaptation based on Lyapunov analysis is applied. The control law is coupled with the adaptation scheme and applied to an experimental system. Friction modeling and compensation for pressure tracking are discussed and experimental results shown. The results show that the nonlinear control algorithm together with the adaptation scheme gives a good performance for the specified task. [S0022-0434(00)00501-3]

Efficient design of high performance automatic control systems is extremely important for high technology systems. To get the best hardware cost-to-performance ratio, it is desirable to design a controller that takes full advantage of actuator capabilities, but this can lead to nonlinear behavior due to actuator saturation. The saturation nonlinearities in the system may have severe effects on system performance due to, for example, integrator windup. In this paper, a new design method is presented based on Lyapunov stability theory. By incorporating the actuator constraints directly in the design method, better utilization of the available control effort can be ensured in achieving desired system behavior. [S0022-0434(00)01801-3]

In this paper, we investigate the problem of asymptotics of Lyapunov exponents for a class of singularly perturbed nonlinear systems. We define the maximal and minimal Lyapunov exponents for the underlying systems and show, via an averaging technique, that under certain conditions, the extremal Lyapunov exponents of the original system converge to the extremal Lyapunov exponents of the averaged slow subsystem when the singular perturbation parameter tends to zero. For low-dimensional systems, the existence of Lipschitz, continuous composite state feedbacks, which asymptotically provide the minimal Lyapunov exponents, can be shown. An example is given to illustrate the potential of the proposed technique and show that the designed controller is robust for sufficiently small perturbations. Index Terms--Averaging, composite feedback, Lyapunov exponent, nonautonomous linear system, singular perturbation.

This paper presents dynamic output feedback controller design for fuzzy dynamic systems. Three kinds of controller design methods are proposed based on a smooth Lyapunov function or a piecewise smooth Lyapunov function. The controller design involves solving a set of linear matrix inequalities (LMI's) and the control laws are numerically tractable via LMI techniques. The global stability of the closed-loop fuzzy control system is also established. Index Terms--Dynamic output feedback controller, fuzzy systems, linear matrix inequality.

We propose an alternative to gain scheduling for stabilization of nonlinear systems. For a useful class of nonlinear systems, the characterization of a region of stability based on a control Lyapunov function is computationally tractable, in the sense that computation times vary polynomially with the state dimension for a fixed number of scheduling variables. Using this fact, we develop a procedure to expand the region of stability by constructing control Lyapunov functions to various trim points of the system. A Lyapunov-based control synthesis algorithm is used to construct a control law that guarantees closed-loop stability for initial conditions in the expanded region of state space. This control asymptotically recovers the optimal stability margin in the sense of a Lyapunov derivative, which in turn can be seen as a performance measure. Robustness to bounded disturbances and stabilization under bounded control are easily incorporated into this framework. In the worst case, the computational complexity of the analysis problem that develops in the new method is increased by an exponential in the disturbance dimension. Similarly, we can handle control constraints with an increase in computational complexity of no more than an exponential in the control dimension. We demonstrate the new control design procedure on an example. Index Terms--Computation time, control systems, Lyapunov methods, nonlinear systems, robustness.

We consider stochastic nonaffine nonlinear control systems [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (written in the sense of Ito), [Omega] being a standard Wiener process, for which we give a sufficient condition for global stabilization by a bounded smooth state feedback that is explicitly given. This condition generalizes the well-known Jurdjevic-Quinn result for deterministic affine control systems. Index Terms--Lyapunov functions, state feedback law, stochastic nonlinear control systems, stochastic stability.