Forced Linear Systems

Abstract

Consider the system

$${\rm{\dot x}}\left( {\rm{t}} \right) = {\rm{L}}\left( {{\rm{x}}_{\rm{t}} } \right)\mathop = \limits^{{\rm{def}}} \int_{ - {\rm{r}}}^0 {\left[ {{\rm{d\eta }}\left( \theta \right)} \right]{\rm{x}}\left( {{\rm{t}} + \theta } \right)} $$

((25.1))

and the perturbed linear system

$${\rm{\dot x}}\left( {\rm{t}} \right) = {\rm{L}}\left( {{\rm{x}}_{\rm{t}} } \right) + {\rm{f}}\left( {\rm{t}} \right)$$

((25.2))

where f belongs to ℬ, the class of bounded continuous functions mapping (−∞,∞) into Rⁿ with the topology of uniform convergence. For any σ in (−∞,∞), we know from our variation of constants formula that the solution x of (25.2) with initial value x_σ at σ must satisfy

$$\begin{array}{*{20}c} {{\rm{x}}_{\rm{t}} = {\rm{T}}\left( {{\rm{t}} - \sigma } \right){\rm{x}}_{\rm{\sigma }} + \int_{\rm{\sigma }}^{\rm{t}} {{\rm{T}}\left( {{\rm{t}} - {\rm{s}}} \right){\rm{X}}_0 {\rm{f}}\left( {\rm{s}} \right)ds,} } & {{\rm{t}} \ge \sigma } \\\end{array}$$

((25.3))

.