Admissibility via induced delay equations (Q6572428)

The authors associate to any linear time-dependent finite-memory differential delay system\N\[\N\dot{x}(t) = L(t) x_t \quad \text{with} \quad x_{\theta}(t) := x(t+\theta)\N\]\Nan \textit{induced} system which shares many of its structural properties but is time-invariant: given the original evolution operator \(U\), they define a strongly continuous semigroup \(S\) that operates on ``stacking'' functions of two variables \(\psi\) such that \(\psi(\cdot, p)\) is the state of the system at time \(p\) and \(S(t)\) characterizes the evolution of the system during \(t\) seconds, for all initial times \(p\) simultaneously:\N\[\N(S(t) \psi)(\theta, p) := (U(p, p-t) \psi(\cdot, p-t))(\theta).\N\]\NFrom the infinitesimal generator \(A\) of \(S\), they derive the induced system as\N\[\N\frac{\partial u}{\partial t}(t, p) = (T u_t)(p) \quad \text{with} \quad T \psi := (A \psi)(0, \cdot), \; u_{\theta}(t, p) := x(t+\theta, p+\theta).\N\]\NIt is known that the existence of an exponential dichotomy for the original system is equivalent to the existence of an exponential dichotomy for its induced system. In this paper, the authors additionally prove that the admissibility of a pair of Banach spaces satisfies a similar equivalence and that for the induced system, existence of an exponential dichotomy and admissibility of the pair are equivalent, establishing the same result for the original, time-dependent system.