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Basic Properties of Control Systems

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Book cover Invariance Entropy for Deterministic Control Systems

Part of the book series: Lecture Notes in Mathematics ((LNM,volume 2089))

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Abstract

This introductory chapter provides the necessary background on control systems that we need for the development of the entropy theory. In Sect. 1.1, we give the definition of a control system which is basically the one from Sontag’s book (Mathematical Control Theory. Deterministic Finite-Dimensional Systems, 2nd edn., vol. 6, 1998). We also introduce particular classes of (time-invariant) systems, namely topological, linear, and smooth systems. Section 1.2 introduces the subclass of smooth systems which our main focus is on, the class of smooth systems given by differential equations. The third section serves for the introduction of several useful control-theoretic notions. In Sect.1.4, we prove elementary properties of the control flow associated with a control-affine system. Moreover, we establish the notions of control and chain control sets, and we give the proofs for some elementary properties of these objects. Most of this material stems from the book of Colonius and Kliemann (The Dynamics of Control, 2000). Finally, in Sect. 1.5, the linearization of a smooth system given by differential equations along a fixed trajectory and related notions are studied.

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Notes

  1. 1.

    Usually, by a flow one understands a continuous-time dynamical system. Essentially, we also use the terminology in this way, but we make an exception here.

  2. 2.

    Throughout the book, the abbreviation “a.e.” stands for “(Lebesgue) almost everywhere”.

  3. 3.

    Recall: A subset \(\mathcal{F}\) of the function space \({\mathcal{C}}^{0}(X,Y )\) of continuous maps from a compact topological space X into a metric space Y is relatively compact if and only if \(\mathcal{F}\) is equicontinuous and each of the sets \(\{f(x)\}_{f\in \mathcal{F}}\), x ∈ X, has a compact closure.

  4. 4.

    Here we mean that z is locally absolutely continuous as a curve in T M, cf. Sect. A.3.

  5. 5.

    By \(\|\cdot \|_{[0,\tau ]}\) we denote the L ∞-norm on \({L}^{\infty }([0,\tau ], {\mathbb{R}}^{m})\).

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    Christoph Kawan

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Kawan, C. (2013). Basic Properties of Control Systems. In: Invariance Entropy for Deterministic Control Systems. Lecture Notes in Mathematics, vol 2089. Springer, Cham. https://doi.org/10.1007/978-3-319-01288-9_1

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