The Circle Criterion and Input-to-State Stability for Infinite-Dimensional Systems∗

In this paper, the focus is on absolute stability and input-to-state stability of the feedback interconnection of an infinite-dimensional linear system Σ and a nonlinearity Φ : dom(Φ) ⊂ Lloc(R+, Y ) → L 2 loc(R+, U), where dom(Φ) denotes the domain of Φ and U and Y (Hilbert spaces) denote the input and output spaces of Σ, respectively (see Figure 1, wherein v is an essentially bounded input signal). The system Σ is assumed to belong to the rather general class of well-posed systems (see, for example, [11, 13] and the references therein) and the nonlinearity is assumed to satisfy a (generalized) sector condition. In the literature on the circle criterion for infinite-dimensional systems (see, for example, [3, 4, 5, 7, 9, 12], and the references therein), the emphasis is usually on Lor L-stability and global asymptotic or global exponential stability (or some variants thereof) of feedback systems of the type shown in Figure 1, with a static sector-bounded nonlinearity Φ in the feedback path. The new contribution of this paper as compared to the previous literature is twofold. (i) In addition to static nonlinearities, we include a class of dynamic nonlinearities which may exhibit bias, but still satisfy a generalized pointwise sector condition.