ISSN:
1573-2878
Keywords:
Control theory
;
synchronization theory
;
periodic solutions
;
maximum amplitude
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract We consider differential equations of the form $$\ddot x + \in f(x,\dot x) + x = \in u$$ , where ε 〉0 is supposed to be small. For piecewise continuous controlsu(t), satisfying |u(t)| ≤ 1, we present sufficient conditions for the existence of 2π-periodic solutions with a given amplitude. We present a method for determining the limiting behavior of controlsū ε for which the equation has a 2π-periodic solution with a maximum amplitude and for determining the limit of this maximum amplitude as ε tends to zero. The results are applied to the linear system $$\ddot x + \in \dot x + x = \in u$$ , the Duffing equation $$\ddot x + \in (x - 1)\dot x + x = \in u$$ , and the Van der Pol equation $$\ddot x + \in (x^2 - 1)\dot x + x = \in u$$ .
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00939134
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