Optimal regularity of robustness for parameterized perturbations
The authors consider a linear differential equation of the form \[ v'= [A(t)+ B(t,\lambda)]v\tag{1} \] in a Banach space, where \(A\) and \(B\) are continuous functions and \(\lambda\) is a parameter in some open subset of a Banach space. By means of a function \(\rho(t)\) a new dichotomy for (1) is introduced. It has the form \[ \begin{aligned} \| T(t,s)P(s)\|&\leq De^{a(\rho(t)- \rho(s))+ \varepsilon|\rho(s)|},\\ \| T(t,s)^{-1}Q(t)\|&\leq De^{-b(\rho(s)- \rho(t))+\varepsilon|\rho(t)|}, \end{aligned} \] for \(t\geq s\). Here, \(T(t,s)\) is the evolution operator, \(T(t,s)P(s)= P(t) T(t,s)\) for \(t,s\in J\), \(Q(t)= \text{id}- P(t)\). A notion of robustness is introduced.
scientific article; zbMATH DE number 5802607
scientific article