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arXiv 2003.06914math.AP
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Hamilton-Jacobi方程的反问题与包络线

The inverse problem for Hamilton-Jacobi equations and semiconcave envelopes

Carlos Esteve, Enrique Zuazua

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摘要：

本文研究了一类时变Hamilton-Jacobi方程的反问题或反设计问题。更准确地说，给定目标函数$u_T$和时间范围$T>0$，我们的目标是构建所有初始条件，对于这些初始条件，黏性解在时间$T$与$u_T$一致。由于这类非线性方程的普遍性，目标可能无法达到。我们首先研究至少存在一个初始条件，导致系统的给定目标。自然的候选者，这确实允许确定的可达性$u_T$，是一个通过反转的方程中的时间方向，考虑$u_T$作为终端条件。在这种情况下，我们使用向后黏性解的概念，它提供了终值问题的存在性和唯一性。我们还给出了一个基于微分不等式的等价的可达性条件，该条件将目标的可达性与它的可达性联系起来。然后，对于$u_T$可达的情况，我们构造解在时间$T$与$u_T$重合的所有初始条件的集合。注意，一般来说，这样的初始条件不是唯一的。最后，当目标$u_T$不一定是可达的情况下，我们研究的投影$u_T$的可达的目标，通过解决问题的时间向后，然后向前。然后，该投影被识别为完全非线性障碍问题的解，并且可以被解释为$u_T$的最小可达目标包络，即由$u_T$从下面限定的最小可达目标。

英文摘要：

We study the inverse problem, or inverse design problem, for a time-evolution Hamilton-Jacobi equation. More precisely, given a target function $u_T$ and a time horizon $T>0$, we aim to construct all the initial conditions for which the viscosity solution coincides with $u_T$ at time $T$. As it is common in this kind of nonlinear equations, the target might not be reachable. We first study the existence of at least one initial condition leading the system to the given target. The natural candidate, which indeed allows determining the reachability of $u_T$, is the one obtained by reversing the direction of time in the equation, considering $u_T$ as terminal condition. In this case, we use the notion of backward viscosity solution, that provides existence and uniqueness for the terminal-value problem. We also give an equivalent reachability condition based on a differential inequality, that relates the reachability of the target with its semiconcavity properties. Then, for the case when $u_T$ is reachable, we construct the set of all initial conditions for which the solution coincides with $u_T$ at time $T$. Note that in general, such initial conditions are not unique. Finally, for the case when the target $u_T$ is not necessarily reachable, we study the projection of $u_T$ on the set of reachable targets, obtained by solving the problem backward and then forward in time. This projection is then identified with the solution of a fully nonlinear obstacle problem, and can be interpreted as the semiconcave envelope of $u_T$, i.e. the smallest reachable target bounded from below by $u_T$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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