Abstract
We consider forced Lur’e systems in which the linear dynamic component is an infinite-dimensional well-posed system. Numerous physically motivated delay and partial differential equations are known to belong to this class of infinite-dimensional systems. We present refinements of recent incremental input-to-state stability results (Guiver in SIAM J Control Optim 57:334–365, 2019) and use them to derive convergence results for trajectories generated by Stepanov almost periodic inputs. In particular, we show that the incremental stability conditions guarantee that for every Stepanov almost periodic input there exists a unique pair of state and output signals which are almost periodic and Stepanov almost periodic, respectively. The almost periods of the state and output signals are shown to be closely related to the almost periods of the input, and a natural module containment result is established. All state and output signals generated by the same Stepanov almost periodic input approach the almost periodic state and the Stepanov almost periodic output in a suitable sense, respectively, as time goes to infinity. The sufficient conditions guaranteeing incremental input-to-state stability and the existence of almost periodic state and Stepanov almost periodic output signals are reminiscent of the conditions featuring in well-known absolute stability criteria such as the complex Aizerman conjecture and the circle criterion.
Highlights
The analysis of solutions of differential equations, the right-hand side of which exhibits almost periodic time dependence, has a long history and the relevant literature is vast, see, for example, [1,8,9,14]
Based on the incremental input-to-state stability (ISS) result in Theorem 3.4, we show that incremental versions of certain classical sufficient conditions for absolute stability such as the complex Aizerman conjecture [19,20], the small-gain theorem [13,44] and the circle criterion [22,44] guarantee that, for a given Stepanov almost periodic input v∗, there exists a corresponding unique state/output trajectory (x∗, y∗) with x∗ almost periodic and y∗ Stepanov almost periodic, and, for any input/state/output trajectory (v, x, y) such that v(t) approaches v∗(t) as t → ∞ in a natural sense, the behaviour of (x, y) is asymptotically identical to that of (x∗, y∗)
Before we come to the main result of this paper, we provide some relevant background on almost periodic functions
Summary
The analysis of solutions of differential equations, the right-hand side of which exhibits almost periodic time dependence, has a long history and the relevant literature is vast, see, for example, [1,8,9,14]. We provide a refinement of the incremental ISS results in [16] and use them to analyse the asymptotic behaviour of the Lur’e system shown in Fig. 1 in response to Stepanov almost periodic inputs. Based on the incremental ISS result in Theorem 3.4, we show that incremental versions of certain classical sufficient conditions for absolute stability such as the complex Aizerman conjecture [19,20], the small-gain theorem [13,44] and the circle criterion [22,44] (or variations thereof) guarantee that, for a given Stepanov almost periodic input v∗, there exists a corresponding unique state/output trajectory (x∗, y∗) with x∗ almost periodic and y∗ Stepanov almost periodic, and, for any input/state/output trajectory (v, x, y) such that v(t) approaches v∗(t) as t → ∞ in a natural sense, the behaviour of (x, y) is asymptotically identical to that of (x∗, y∗).
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