Energy methods for stability of bilinear systems with oscillatory inputs. (English) Zbl 0845.93070
The author considers the dynamical system
\[
\dot x = \left( A + \sum^m_{i = 1} u_i (\omega t) B_i \right) x
\]
where \(A, B_1, \dots, B_m\) are constant \(n \times n\) matrices, \(x (t)\in\mathbb{R}^n\), \(u_i\) is a piecewise continuous periodic function with period \(T\) and mean 0 and for \(i,j = 1, \dots, m\), \(B_i B_j = 0\). He studies the stability of such systems under high-frequency, periodic forcing. He defines “the averaged potential” for linearizations of the systems on the (reduced) configuration space, and he shows that stable motions of the forced system are associated with minimum values of this quantity. A few examples are given to illustrates the results obtained.
Reviewer: Leslaw Socha (Katowice)
MSC:
| 93D20 | Asymptotic stability in control theory |
| 70Q05 | Control of mechanical systems |
| 34C29 | Averaging method for ordinary differential equations |
| 93B18 | Linearizations |
| 93B17 | Transformations |
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