Filippov's and Filippov–Ważewski's Theorems on Closed Domains
Filippov's and Filippov–Ważewski's Theorems on Closed Domains
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DOI：
10.1006/jdeq.2000.3711
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发表时间：
2000-03
影响因子：
2.4
通讯作者：
H. Frankowska;F. Rampazzo
中科院分区：
文献类型：
--
作者：
H. Frankowska;F. Rampazzo
著名的Filippov定理表明，给定一个微分包含x′∈F（t, x）的轨迹x1:[0， +∞[∑R n]，其集值映射F在t中可测，k-Lipschitz在x中可测，对于任意初始条件x2(0)∈R n，存在一个从x2(0)开始的轨迹x2（·），使得|x1(t)−x2(t)|≤ekt |x1(0)−x2(0)|。Filippov - Wazewski定理建立了用从相同初始条件出发的原始包含项x′∈F（t, x）的轨迹逼近凸化微分包含项x′∈co F（t, x）的任何轨迹的可能性。在本文中，我们将这两个定理推广到状态变量x被约束于开放子集Θ∧R n的闭包的情况。后者被允许是非光滑的。我们对F和Θ施加了一个广义的sononer型条件，得到了上述经典结果在无限视界约束问题上的推广。讨论了该方法在带状态约束的最优控制问题的值函数正则性研究中的应用。
Abstract The celebrated Filippov's theorem implies that, given a trajectory x1: [0, +∞[↦ R n of a differential inclusion x′∈F(t, x) with the set-valued map F measurable in t and k-Lipschitz in x, for any initial condition x2(0)∈ R n, there exists a trajectory x2(·) starting from x2(0) such that |x1(t)−x2(t)|⩽ekt |x1(0)−x2(0)|. Filippov– Wazewski's theorem establishes the possibility of approximating any trajectory of the convexified differential inclusion x′∈ co F(t, x) by a trajectory of the original inclusion x′∈F(t, x) starting from the same initial condition. In the present paper we extend both theorems to the case when the state variable x is constrained to the closure of an open subset Θ⊂ R n. The latter is allowed to be non smooth. We impose a generalized Soner type condition on F and Θ, yielding extensions of the above classical results to infinite horizon constrained problems. Applications to the study of regularity of value functions of optimal control problems with state constraints are discussed as well.