论文标题
受控Moreau的最小时间功能
The minimum time function for the controlled Moreau's Sweeping Process
论文作者
论文摘要
令$ c（t）$，$ t \ geq0 $为Lipschitz设置值地图，带有闭合和（轻度非）凸值，$ f（t，x，x，u）$是地图，LipsChitz连续W.R.T. $ x $。我们考虑在$ c $的图中达到目标$ s $的问题。 $ t（t_ {0}，x_ {0}）$。 Dynamics $（\ star）$称为干扰（或Moreau）过程。我们为$ t $提供了足够的条件，可以是有限的，连续的，并通过汉密尔顿 - 雅各比的不平等来表征$ t $。我们方法的关键工具是$（\ star）$的$ s $的弱和强流不变性的特征。由于存在普通的锥体$ n_ {c（t）}（x）$，因此$（\ star）$的右侧侧面隐含在c（t）$中隐含状态约束$ x（t）\，并且与$ x $相对于$ x $而不是Lipschitz。
Let $C(t)$, $t\geq0$ be a Lipschitz set-valued map with closed and (mildly non-)convex values and $f(t, x,u)$ be a map, Lipschitz continuous w.r.t. $x$. We consider the problem of reaching a target $S$ within the graph of $C$ subject to the differential inclusion \[ (\star)\qquad \dot{x} \in -N_{C(t)}(x) + G(t,x) \] starting from $x_{0}\in C(t_{0})$ in the minimum time $T(t_{0},x_{0})$. The dynamics $(\star)$ is called a perturbed sweeping (or Moreau) process. We give sufficient conditions for $T$ to be finite and continuous and characterize $T$ through Hamilton-Jacobi inequalities. Crucial tools for our approach are characterizations of weak and strong flow invariance of a set $S$ subject to $(\star)$. Due to the presence of the normal cone $N_{C(t)}(x)$, the right hand side of $(\star)$ contains implicitly the state constraint $x(t)\in C(t)$ and is not Lipschitz continuous with respect to $x$.
