We consider the first order autonomous differential equation (ODE) $\mathbf{x}'=\mathbf{f}(\mathbf{x})$ where $\mathbf{f}: \mathbb{R}^n\to\mathbb{R}^n$ is locally Lipschitz. For a box $B_0\subseteq\mathbb{R}^n$ and $h>0$, the set of solutions $\mathbf{x}:[0,h]\to\mathbb{R}^n$ that satisfies $\mathbf{x}'(t)=\mathbf{f}(\mathbf{x}(t))$ and $\mathbf{x}(0)\in B_0$ is denoted $\mathrm{IVP}_{\mathbf{f}}(B_0,h)$. We provide a complete validated algorithm for the following \emph{End Cover Problem}: given $(\mathbf{f},B_0,\varepsilon,h)$, to compute a finite set $\mathcal{C}$ of boxes such that $$ \operatorname{End}_{\mathbf{f}}(B_0,h) \subseteq \bigcup_{B\in\mathcal{C}} B \subseteq \Bigl(\operatorname{End}_{\mathbf{f}}(B_0,h) \oplus [-\varepsilon,\varepsilon]^n\Bigr) $$ where $\operatorname{End}_{\mathbf{f}}(B_0,h)=\{\mathbf{x}(h): \mathbf{x}\in\mathrm{IVP}(B_0,h)\}$. We will also give a complexity analysis of our algorithm, and introduce an alternative technique to compute End Cover $\mathcal{C}$ based on covering the boundary of $\operatorname{End}_{\mathbf{f}}(B_0,h)$. We also introduce the \emph{Boundary Cover Problem} to compute a $\varepsilon$-cover for the boundary of $\operatorname{End}_{\mathbf{f}}(B_0,h)$, leading to a faster algorithm the above End Cover Problem. Finally, we give experimental results indicating the practicality of our techniques.

翻译：暂无翻译