(In-)Stability of Differential Inclusions by Philipp Braun & Lars Grüne & Christopher M. Kellett

Author:Philipp Braun & Lars Grüne & Christopher M. Kellett
Language: eng
Format: epub
ISBN: 9783030763176
Publisher: Springer International Publishing

(4.14)

is known as Artstein’s circles in the literature. For and the dynamics can be described in the form of a differential inclusion (2.​1). The function

(4.15)

is a control Lyapunov function in the Dini sense according to Definition 4.4 (see [5]), which implies weak -stability according to Theorem 4.5. The control Lyapunov function is shown in Fig. 4.4.

Fig. 4.4Nonsmooth control Lyapunov function (4.15) for the dynamics (4.14). The function V is nonsmooth on the -axis

The time reversal system is not weakly completely unstable since all solutions of Artstein’s circles with initial value are bounded for all .

More explicitly, all solutions of the dynamical system (4.14) are described through circles, where the radius of the circle is defined by the initial value. The input u can only change the direction (left or right) and the velocity of the solution. Fig. 4.5 shows solutions of (4.14) for different initial conditions (left) and the phase portrait (right) with respect to constant input signals and , respectively.

Fig. 4.5On the left, solutions of the dynamics (4.14) for different initial conditions and . On the right, phase portrait of (4.14) for (blue) and (red)

Download

Copyright Disclaimer:
This site does not store any files on its server. We only index and link to content provided by other sites. Please contact the content providers to delete copyright contents if any and email us, we'll remove relevant links or contents immediately.