论文标题
汉密尔顿 - 雅各比方程和半循环信封的逆问题
The inverse problem for Hamilton-Jacobi equations and semiconcave envelopes
论文作者
论文摘要
我们研究汉密尔顿 - 雅各比方程的逆问题或反设计问题。更确切地说，给定目标函数$ u_t $和时间范围$ t> 0 $，我们旨在构建粘度解决方案与时$ t $相合的所有初始条件。由于在这种非线性方程中很常见，因此目标可能无法实现。我们首先研究了至少一种初始条件的存在，导致系统达到给定目标。自然候选人确实允许确定$ u_t $的可达到性，是通过将$ u_t $作为终端条件扭转时间方向而获得的。在这种情况下，我们使用向后粘度解决方案的概念，该解决方案为终端值问题提供了存在和唯一性。我们还基于差分不平等的差异性提供了等效的可及性条件，该条件将目标的可达性与其半腔属性相关联。然后，对于达到$ u_t $的情况，我们构建了所有初始条件的集合，该解决方案与$ t $ t $相合的所有初始条件。请注意，通常情况下，这种初始条件不是唯一的。最后，对于目标$ u_t $不一定可以达到的情况，我们研究了一组可达目标的$ u_t $的预测，该预测是通过向后求后，然后在时间上向前求解的问题。然后用完全非线性障碍物问题的解决方案来标识此投影，并可以解释为$ u_t $的半循环信封，即，从下面界定的最小可触及目标用$ u_t $。
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 $u_T$.
