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This paper deals with the stabilization and the practical stabilization of nonlinear systems described by neutral functional differential equations in Hale’s form, affine in the control input. Artstein’s methodology and Sontag’s universal formula are investigated for this class of systems, by means of invariantly differentiable control Lyapunov–Krasovskii functionals. [ABSTRACT FROM AUTHOR]

It is standard worldwide, in both industrial and academic activities, to design in continuous time a stabilizing state feedback for a nonlinear system and then implement it by digital devices using sampling and zero-order holding. It is well known, from practice, that stability is rather preserved if sampling is performed at suitable high frequency. This fact has been proved in the literature, also from a theoretical point of view, for delay-fr ee nonlinear systems. An analogous result has not been proved, from the theoretical point of view, for nonlinear retarded systems. Moreover, in the case of retarded systems, implementation by means of digital devices often requires some further approximation due to nonavailability in the buffer of the value of the system variables at some past times. In order to cope with this problem, we make use here of the well-known approximation scheme based on first order splines. We introduce the standard hypothesis that there exists a stabilizing continuous-time state feedback and that there exists a related suitable Lyapunov--Krasovskii functional (in a large class already introduced in the literature) for the continuous-time, globally asymptotically stable, closed-loop system. We show, for fully nonlinear retarded systems, that, by sampling at suitable high frequency the system (finite d imensional) variable, stabilization in the sample-and-hold sense is guaranteed, when the held input is obtained as a feedback of the (first order) spline approximation of the (infinite dimensional) system state, whose entries are available at sampling times. In case of nonuniform sampling, the current sampling period (time elapsed between the latest two samplings) has to be measured (for instance by a clock device set to zero at each sampling time). A sampled-data, event-based control law is also studied. [ABSTRACT FROM AUTHOR]

In this paper, the problem of the stabilization in the sample-and-hold sense for fully nonlinear systems with an arbitrary number of arbitrary discrete as well as of distributed time delays is studied. It is shown that steepest descent feedbacks, continuous or not, induced by Lyapunov--Krasovskii functionals in a suitable (large) class, are stabilizers in the sample-and-hold sense. The fact that discontinuities are overcome by the sampling and holding process enlarges greatly the possibility of finding successful controllers for retarded nonlinear systems, by means of control Lyapunov--Krasovskii functionals. [ABSTRACT FROM AUTHOR]

In this paper a new Lyapunov-Krasovskii methodology for both the (local) asymptotic stability and the global asymptotic stability of non-linear coupled delay differential and difference equations is proposed. This methodology is based on the concept of input-to-state stability applied to the difference equation, for which a sufficient Lyapunov criterion is given, and on previous methodologies developed in the literature for linear delay descriptor systems. [ABSTRACT FROM AUTHOR]

Abstract: In [Driver, R. D. (1962). Existence and stability of solutions of a delay-differential system. Archive for Rational Mechanics and Analysis 10, 401–426] a proper definition, not involving the solution, of the derivative of the Liapunov–Krasovskii functional for retarded functional differential equations with continuous right side is given and it is showed that this definition coincides with the non-constructive one given in Krasovskii [1956. On the application of the second method of A. M. Lyapunov to equations with time delays (in Russian). Prikladnaya Matematika i Mekhanika 20, 315–327] involving the solution, for functionals V which are locally Lipschitz (and not only continuous, as it is considered in most literature). In this paper, the result by Driver is extended to a general class of retarded functional differential equations coupled with continuous time difference equations with more general right sides, verifying the Carathéodory conditions. Such result is applied to build a new Liapunov–Krasovskii theorem for studying the input-to-state stability of time-invariant neutral functional differential equations with linear difference operator. An example taken from the literature, concerning transmission lines, is reported, showing the effectiveness of the methodology. [Copyright &y& Elsevier]

Abstract: A two-step Liapunov–Krasovskii methodology for checking the asymptotic stability of nonlinear coupled delay differential and continuous time difference equations is proposed here. The feasibility of such methodology is shown by means of Liapunov–Krasovskii functionals with nonconstant kernels in the integrals, for instance discretized Liapunov–Krasovskii ones. An illustrative example taken from the literature, showing the effectiveness of the proposed method, is reported. [Copyright &y& Elsevier]