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This chapter presents a review of a recently developed methodology for the (adaptive) stabilization of nonlinear systems via state and output feedback. The method relies upon the notion of system immersion and is well-suited in applications where a controller for a reduced-order model is known and we would like to robustify it with respect to higher-order dynamics. This is achieved by the dynamics of the controlled plant into the desired dynamics of the reduced-order model. The method is illustrated with several practical and academic examples.

The general topic of study is Lyapunov stability of nonlinear time-varying cascaded systems. Roughly speaking these are systems in “open loop” as illustrated in the figure below.

The objective of this chapter is to describe geometric methods applied to aerospace systems: attitude control of a rigid spacecraft, orbital transfer, shuttle re-entry. We characterize the controllability of a rigid spacecraft controlled by one gas jet and the set of orbits reached in orbital transfer, depending upon the orientation of the thrust. Then we construct controls in orbital transfer and attitude control, using stabilization techniques and path planning. The optimal control is analyzed in orbital transfer, the cost being the time and in the shuttle re-entry, where the cost is the total amount of thermal flux, taking into account the state constraints. We present an analysis of extremals, solutions of the maximum principle and second-order sufficient optimality conditions.

The Hamiltonian formulation of distributed-parameter systems has been a challenging reserach area for quite some time. (A nice introduction, especially with respect to systems stemming from fluid dynamics, can be found in [26], where also a historical account is provided.) The identification of the underlying Hamiltonian structure of sets of p.d.e.s has been instrumental in proving all sorts of results on integrability, the existence of soliton solutions, stability, reduction, etc., and in existing results, see e.g. [11], [24], [18], [17], [25], [14].

The present chapter contains the material taught within the module P2 of FAP 2004. The purpose of this intensive course is first to provide an introduction to . This fashionable though quite difficult domain of pure mathematics today has been pioneered by V.P. Palamodov, M. Kashiwara and B. Malgrange around 1970, after the work of D.C. Spencer on the formal theory of systems of partial differential equations. We shall then focus on its application to control theory in order to study linear control systems defined by partial differential equations with constant or variable coefficients, also called multidimensional control systems, by means of new methods from module theory and homological algebra. We shall revisit a few basic concepts and prove, in particular, that controllability, contrary to a well established engineering tradition or intuition, is an intrinsic structural property of a control system, not depending on the choice of inputs and outputs among the control variables or even on the presentation of the control system. Our exposition will be rather elementary as we shall insist on the main ideas and methods while illustrating them through explicit examples. Meanwhile, we want to stress out the fact that these new techniques bring striking results even on classical control systems of Kalman type!

The aim of this chapter is the study of structural properties of linear systems. Systems with time-varying coefficients are considered, both in the continuousand discrete-time cases. These two cases are merged into a general framework. This study is based on . Let us briefly explain what a module is, in connection with the theory of differential and difference equations.