$s$-almost cross-$t$-intersecting families for vector spaces

Let $V$ be an $n$-dimensional vector space over the finite field ${\mathbb{F}}_q$, and $\left[\genfrac{}{}{0px}{}{V}{k}\right]$ denote the family of all $k$-dimensional subspaces of $V$. The families $\mathcal{F},\mathcal{G}\subseteq \left[\genfrac{}{}{0px}{}{V}{k}\right]$ are said to be cross-$t$-intersecting if $\dim (F\cap G)\ge t$ for all $F\in \mathcal{F},G\in \mathcal{G}$. Two families $\mathcal{F}$ and $\mathcal{G}$ are called $s$-almost cross-$t$-intersecting if each member of $\mathcal{F}$ (resp. $\mathcal{G}$) is $t$-disjoint with at most $s$ members of $\mathcal{G}$ (resp. $\mathcal{F}$). In this paper, we discribe the structure of $s$-almost cross-$t$-intersecting families with maximum product of their sizes. In addition, we prove a stability result.