Evolution family ($U(t,s)$)

An evolution family consists of bounded operators $U(t,s)$ for $s\le t$ satisfying $U(s,s)=I$ and $U(t,r)U(r,s)=U(t,s)$. For sufficiently regular data it solves ${\partial }_{t}U(t,s)x=A(t)U(t,s)x$ and ${\partial }_{s}U(t,s)x=-U(t,s)A(s)x$.

For a partition $s=t_0<\cdots <t_N=t$, the frozen-generator approximation isUnder the hypotheses of the nonautonomous generation theorem, these products converge strongly and uniformly on compact time triangles to the evolution family.