On limitations of the sampled-data feedback for nonparametric dynamical systems
The authors investigate the stability of scalar closed-loop systems of the form \[ dx_t = f(x_t)\,dt + u_t \, dt + \sigma \, dW_t\quad (\sigma =0 \text{ or }\sigma > 0)\tag{1} \] with sampled-data feedback control \[ u_t = u_{kh} = g(x_0, x_h,\dots , x_{kh}) \quad\text{for }t \in [kh, (k+1)h)\tag{2} \] in dependence of the sampling rate \(h>0\). The results are the following. In the case of a function \(f\) that is locally Lipschitz and grows linearly in the sense that there are positive constants \(L\) and \(c\) with \( | f(x)| \leq L| x| + c\) (for all \(x \in {\mathbb R}\)), two constants \(b_1\) and \(b_2\) with the following properties are computed. If the sampling rate \(h < b_1 \frac{1}{L}\), then there is a feedback control (2) (given explicitly) such that for all functions \(f\) of the kind considered the closed-loop system (1), (2) is globally stable (in the deterministic case for \(\sigma = 0\)) and globally mean-square stable (in the presence of noise, \(\sigma> 0\)). However, if \(h> b_2 \frac{1}{L}\), then there is always a function \(f^*\) in the class considered such that for any control (2) \(| x_{kh}| \to \infty\). In the case that the function \(f\) grows faster than linearly and noise is present (\(\sigma>0\)), there is no sampling rate \(h>0 \), however fast, which stabilizes the system (1).
scientific article; zbMATH DE number 1895563