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The maximum principle is presented in the weak and general forms. The standard proofs are detailed, and the connection with the shooting method for numerical resolution is made. A brief introduction to the micro-local analysis of extremals is also provided. Regarding second-order conditions, small timeoptimality is addressed by means of high order generalized variations. As for local optimality of extremals, the conjugate point theory is introduced both for regular problems and for minimum time singular single input affine control systems. The analysis is applied to the minimum time control of the Kepler equation, and the numerical simulations for the corresponding orbit transfer problems are given. In the case of state constrained optimal control problems, necessary conditions are stated for boundary arcs. The junction and reflection conditions are derived in the Riemannian case.

In this chapter, an overview of main observability problems and possible observer designs for nonlinear systems is proposed. In particular the observer problem and related observability conditions are first given. Then, some designs are reviewed, divided into Luenberger-like designs for socalled uniformly observable systems, and Kalman-like designs for non uniformly observable ones. Finally, two directions for more designs – in the sense that they are based on the previously listed ones – are proposed: designs based on observer interconnections on the one hand, and designs based on system transformations on the other hand.

This chapter provides some of the main ideas resulting from recent developments in sampled-data control of nonlinear systems. We have tried to bring the basic parts of the new developments within the comfortable grasp of graduate students. Instead of presenting the more general results that are available in the literature, we opted to present their less general versions that are easier to understand and whose proofs are easier to follow. We note that some of the proofs we present have not appeared in the literature in this simplified form. Hence, we believe that this chapter will serve as an important reference for students and researchers that are willing to learn about this area of research.

This chapter is devoted to the stability problem of time-delay systems using time-domain approach. Some basic concepts of time-delay systems are introduced. Then, some simple Lyapunov-Krasovskii funtionals, complete Quadratic Lyapunov-Krasovskii functional and discretization scheme are introduced, with connections and extent of conservatism compared. The issue of time-varying delays are also discussed. The concept of Razumikhin Theorem is introduced. An alternative model of coupled difference-differential equations and its stability problem are also introduced.

In this section, we recall some basic elements of semigroup theory (see [25]). In particular, all the arguments of the results mentioned below can be found in [25].

Lyapunov stability theory is probably the most useful qualitative method to study the behaviour of dynamical systems; it benefits from at least 75 years of sustained development. It started with the memoir of A. M. Lyapunov [32], published in a Western2 language in [33], and, starting with the 1930s, many refinements to this stability theory have been established.

This chapter is the sequel of [7]. Its topic is the structure at infinity of discrete and continuous linear time-varying systems in a unified approach. In the time-invariant case, the linear systems in [7] are implicitly assumed to be perpetually existing and the smoothness of their behavior is not studied. In practice, however, that behavior must be sufficiently smooth (to avoid undesirable saturations of the variables, or even the destruction of the system), and the system has a limited useful life. These constraints can be taken into account by studying the structure at infinity of the system under consideration. As this system is existing during a limited period, it is called a [8]. A list of and for [7] is given at the end of the chapter.

In this appendix we discuss two different deffnitions of uniform global asymptotic stability (UGAS), both used in the literature. In the first one, UGAS is defined to be uniform local stability (ULS) plus uniform global attractivity (UGA). In the second one, it is defined to be ULS+UGA plus uniform global boundedness (UGB). We reemphasize, by means of an explicit example, that UGB is not necessarily implied by ULS and UGA, even for smooth timevarying nonlinear systems where the right-hand side’s derivative with respect to the state is bounded uniformly in time. Thus, the two definitions are truly different for nonlinear time-varying systems.