I'm wondering if there is any idea to bound the norm of the initial data by the norm of corresponding solution? To make it clear, consider the following abstract Cauchy problem: $x'(t)=Ax(t)$, in $(0,T)$, with $x(0)=x_0$, where $A$ generates a $C_0$-semigroup on a Banach space $X$. I'm looking for some ideas to get an inequality of type $$\|x_0\|_X\le C_1\|x(t)\|_X+C_2\|x(\tau)\|_X+C_3 \|x'(t)\|_X, \qquad 0<t<T,$$ for a fixed $\tau$ and positive constants $C_i$, for a bounded set of initial data (maybe in a special domain).

I found an idea which uses the Sobolev embedding $W^{s,p}(0,T;X) \subset C([0,T];X)$ under restrictions on $p$ and $s$. I guess a result from regularity theory (for analytic semigroups for example) or something similar would help. Do you know any similar idea?