A Razumikhin-type lemma for functional differential equations with application to adaptive control
The authors consider a class of \(m\)-input, \(m\)-output controlled functional differential equations of the form \[ \dot z(t)=g(t,z_{t}^{a})+B(t,u(t)), \quad y(t)=Cz(t), \] where \(z(t),u(t),y(t)\in\mathbb{R}^{m}\), \(z_{t}^{a}(\theta)=z(t+\theta)\), \(-a\leq\theta\leq 0\), \(g\) and \(B\) are of Carathéodory class, \(\det C\neq 0\), \(a\geq 0\). Under some restrictions on \(g\) and \(B\), an adaptive strategy -- universal for the class -- is developed for approximate tracking of any reference signal \(r(t)\) of Sobolev class \(W^{1,\infty}\), i.e., the tracking error \(e(t)=y(t)-r(t)\) should tend to the closed ball of radius \(\lambda\) centered at the origin in \(\mathbb{R}^{n}\) for any prescribed \(\lambda>0\). The result is based on a Razumikhin-type lemma on boundedness of solutions of retarded functional differential equations. The approach is also extended to provide approximate tracking for a class of dynamically perturbed functional differential equations. An application in the control of a thermal process is presented. The paper is very well written and all statements are proved.
scientific article; zbMATH DE number 1325751