Nonhomogeneous Linear Systems

Abstract

In this section, we consider the linear system

$$\left\{ {\begin{array}{*{20}c} {{\rm{\dot x}}\left( {\rm{t}} \right) = {\rm{L}}\left( {{\rm{t}},{\rm{x}}_{\rm{t}} } \right) + {\rm{h}}\left( {\rm{t}} \right),{\rm{t}} \ge \sigma } \\ {{\rm{x}}_\sigma = {\rm{\phi }}} \\ \end{array}} \right.$$

((16.1))

or equivalently

$$\left\{ {\begin{array}{*{20}c} {{\rm{\dot x}}\left( {\rm{t}} \right) = {\rm{\phi }}\left( 0 \right) + \int_\sigma ^{\rm{t}} {{\rm{L}}\left( {{\rm{s,x}}_{\rm{s}} } \right){\rm{ds}} + \int_\sigma ^{\rm{t}} {{\rm{h}}\left( {\rm{s}} \right){\rm{ds}}} } {\rm{, t}} \ge \sigma } \\ {{\rm{x}}_\sigma = {\rm{\phi }}} \\ \end{array}} \right.$$

((16.2))

where $h \in L_1^{{\rm{loc}}} \left( {\left[ {\sigma,\infty } \right],{\rm{R}}^{\rm{n}} } \right)$ the space of functions mapping [σ,∞) →Rⁿ which are Lebesgue integrable on each compact set of [σ,∞). Also, we assume L(t,ϕ) is linear in ϕ and, in addition, there is an n × n matrix function η(t,θ) measurable in t,θ, of bounded variation in θ on [−r,o] for each t, and there is an $\ell \in L_1^{{\rm{loc}}} \left( {\left[ { - \infty,\infty } \right],{\rm{R}}} \right)$ such that

$${\rm{L}}\left( {{\rm{t,\phi }}} \right) = \int_{ - {\rm{r}}}^0 {\left[ {{\rm{d}}_\theta {\rm{\eta }}\left( {{\rm{t}},\theta } \right)} \right]{\rm{\phi }}\left( \theta \right)} $$

((16.3))

$$\left| {{\rm{L}}\left( {{\rm{t}},{\rm{\phi }}} \right)} \right| \le \ell \left( {\rm{t}} \right)\left| {\rm{\phi }} \right|$$

((16.4))

for all t ∈ (−∞,∞), ϕ ∈ C.