By Christoph Kawan
This monograph offers an advent to the concept that of invariance entropy, the primary motivation of which lies within the have to care for verbal exchange constraints in networked regulate platforms. For the best attainable community topology, which include one controller and one dynamical method attached through a electronic channel, invariance entropy presents a degree for the smallest facts fee above which it truly is attainable to render a given subset of the kingdom house invariant via a symbolic coder-controller pair. this idea is largely corresponding to the proposal of topological suggestions entropy brought through Nair, Evans, Mareels and Moran (Topological suggestions entropy and nonlinear stabilization. IEEE Trans. Automat. keep an eye on forty nine (2004), 1585–1597). The booklet provides the rules of a concept which goals at discovering expressions for invariance entropy by way of dynamical amounts akin to Lyapunov exponents. whereas either discrete-time and continuous-time structures are taken care of, the emphasis lies on structures given by means of differential equations.
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Extra info for Invariance Entropy for Deterministic Control Systems: An Introduction
Indeed, if there were two control sets D1 and D2 with D1 \ D2 ¤ ;, then D1 [ D2 would satisfy properties (1) and (2) of control sets and hence would be contained in a control set D D1 [ D2 , which contradicts maximality of both D1 and D2 . In the following, we assume that ˙ additionally satisfies the following three properties: (A) Continuity in time: T D R; (B) The maps 't;! x/ V . These three properties are satisfied in particular for smooth systems given by differential equations with bounded control range.
19. x/ and suppose that for an element x 2 D there are > 0 and ! 2 U with '. / 2 D. / 2 D for all t 2 Œ0; . Proof. y/. '. //. /. /, there is a neighborhood V of x with 't;! V / W . 1 2 U and 1 > 0 with '. 1 / 2 V , because x 2 D. t 1 / for t 2 Œ 1 ; 1 C t/: This control function satisfies '. t; '. / 2 't;! y/. W; t u The only control sets which are interesting for us are those with nonempty interior, which can be characterized as follows. 20. x/ and assume that the interior of D is nonempty. Then D is a control set.
0 . R; Rm / such that 1 ! e. p 2T -periodically to R. p 2 W as desired. t u Before we return to the continuity proof of ˚, we provide some additional information about the dynamical properties of the shift flow. Since we do not use these properties, we omit their proofs. 16. The shift flow W R U ! 4 Control Flow and Control Sets 21 (iii) If ˝ contains more than one point, then has sensitive dependence on initial conditions, that is, there is ı > 0 such that for every ! 2 U and every neighborhood N of !