Abstract:
This paper studies switching stabilization problems for general switched nonlinear systems. A piecewise smooth control-Lyapunov function (PSCLF) approach is proposed and a constructive way to design a stabilizing switching law is developed. The switching law is constructed via the directional derivatives of the PSCLF with a careful discussion on various technical issues that may occur on the nonsmooth surfaces. Sufficient conditions are derived to ensure stability of the closed-loop Filippov solutions including possible sliding motions. The proposed PSCLF approach contains many existing results as special cases and provides a unified framework to study nonlinear switching stabilization problems with a systematic consideration of sliding motions. Applications of the framework to switched linear systems with quadratic and piecewise quadratic control-Lyapunov functions are discussed and results stronger than the existing methods in the literature are obtained. Application to stabilization of switched nonlinear systems is illustrated through an numerical example.

Abstract:
The necessary and sufficient conditions for accurate construction of a Lyapunov function and the necessary and sufficient conditions for a set to be the asymptotic stability domain are algorithmically solved for a nonlinear dynamical system with continuous motions. The conditions are established by utilizing properties of o-uniquely bounded sets, which are explained in the paper. They allow arbitrary selection of an o-uniquely bounded set to generate a Lyapunov function.

Abstract:
In the sliding-mode control of nonlinear systems with uncertainties and disturbances, we prove that the partial derivative of the Gaussian fuzzy basic function vector with respect to the state vector is bounded under any condition, thus resolving the key problem in combining a second-order dynamic sliding-mode control with the fuzzy identification. In addition, we design a second-order dynamic terminal-sliding-mode control which converges in a finite period of time without chattering. The output of the fuzzy disturbance-observer is employed as the compensation signal for the adaptive robust control. The stability of the system is proved by using Lyapunov theorem. The proposed control scheme is applied to the attitude-angles tracking of a near space vehicle; the increment of the convergence time in this application of higherorder sliding-mode control has been analyzed. Results show that this control scheme effectively suppresses the chattering and is with strong robustness, fast tracking speed, and high precision. Compared with the conventional terminal-slidingmode control, the second-order dynamic terminal-sliding-mode control causes limited increment of the convergence time, demonstrating the efficacy of this control scheme in engineering application.

Abstract:
Time-invariant nonlinear systems with differentiable motions are considered. The algorithmic necessary and sufficient conditions are established in various forms for one-shot construction of a Lyapunov function, for asymptotic stability of a compact invariant set and for the exact determination of the asymptotic stability domain of the invariant set.

Abstract:
A new scheme of an adaptive fuzzy sliding mode controller for a class of uncertain nonlinear systems is proposed in this paper. The design is based on a modified Lyapunov function and the approximation capability of the second type fuzzy systems. In addition, the approach is able to avoid the requirement of the upper bound of the first time derivative of the control gain, which is assumed to be known a priori in some of the existing adaptive fuzzy/neural network control schemes. By theoretical analysis, the closed loop fuzzy control system is proven to be globally stable in the sense that all signals involved are bounded, with tracking errors converging to zero.

Abstract:
This paper presents a proof that existence of a polynomial Lyapunov function is necessary and sufficient for exponential stability of sufficiently smooth nonlinear ordinary differential equations on bounded sets. The main result states that if there exists an n-times continuously differentiable Lyapunov function which proves exponential stability on a bounded subset of R^n, then there exists a polynomial Lyapunov function which proves exponential stability on the same region. Such a continuous Lyapunov function will exist if, for example, the right-hand side of the differential equation is polynomial or at least n-times continuously differentiable. The proof is based on a generalization of the Weierstrass approximation theorem to differentiable functions in several variables. Specifically, we show how to use polynomials to approximate a differentiable function in the Sobolev norm W^{1,\infty} to any desired accuracy. We combine this approximation result with the second-order Taylor series expansion to find that polynomial Lyapunov functions can approximate continuous Lyapunov functions arbitrarily well on bounded sets. Our investigation is motivated by the use of polynomial optimization algorithms to construct polynomial Lyapunov functions.

Abstract:
The boundedness of the motions of the dynamical system described by a differential inclusion with control vector is studied. It is assumed that the right-hand side of the differential inclusion is upper semicontinuous. Using positionally weakly invariant sets, sufficient conditions for boundedness of the motions of a dynamical system are given. These conditions have infinitesimal form and are expressed by the Hamiltonian of the dynamical system.

Abstract:
Two years ago, Blanco and Fournier (Blanco S. and Fournier R., Europhys. Lett. 2003) calculated the mean first exit time of a domain of a particle undergoing a randomly reoriented ballistic motion which starts from the boundary. They showed that it is simply related to the ratio of the volume's domain over its surface. This work was extended by Mazzolo (Mazzolo A., Europhys. Lett. 2004) who studied the case of trajectories which start inside the volume. In this letter, we propose an alternative formulation of the problem which allows us to calculate not only the mean exit time, but also the mean residence time inside a sub-domain. The cases of any combinations of reflecting and absorbing boundary conditions are considered. Lastly, we generalize our results for a wide class of stochastic motions.

Abstract:
The study of chaos in relativistic systems has been hampered by the observer dependence of Lyapunov exponents (LEs) and of conditions, such as orbit boundedness, invoked in the interpretation of LEs as indicators of chaos. Here we establish a general framework that overcomes both difficulties and apply the resulting approach to address three fundamental questions: how LEs transform under Lorentz and Rindler transformations and under transformations to uniformly rotating frames. The answers to the first and third questions show that inertial and uniformly rotating observers agree on a characterization of chaos based on LEs. The second question, on the other hand, is an ill-posed problem due to the event horizons inherent to uniformly accelerated observers.

Abstract:
This paper proposes a robust control method based on sliding mode design for two-level quantum systems with bounded uncertainties. An eigenstate of the two-level quantum system is identified as a sliding mode. The objective is to design a control law to steer the system's state into the sliding mode domain and then maintain it in that domain when bounded uncertainties exist in the system Hamiltonian. We propose a controller design method using the Lyapunov methodology and periodic projective measurements. In particular, we give conditions for designing such a control law, which can guarantee the desired robustness in the presence of the uncertainties. The sliding mode control method has potential applications to quantum information processing with uncertainties.