Dynamics at a Switching Intersection: Hierarchy, Isonomy, and Multiple Sliding
Dynamics at a Switching Intersection: Hierarchy, Isonomy, and Multiple Sliding
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DOI：
10.1137/13093368x
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发表时间：
2014-01-01
影响因子：
2.1
通讯作者：
Jeffrey, Mike R.
中科院分区：
文献类型：
--
作者：
Jeffrey, Mike R.
如果一组常微分方程沿沿着某个阈值是不连续的，则可以找到连续的解，即使有时是多值的。我们展示了在何种程度上可以找到独特的解决方案，在一般情况下，当阈值的形式，许多相交的流形。如果交点是横向的，那么可以找到沿着阈值滑动的许多解。它们是通过凸组合的分层应用形成一个微分包含而得到的。系统通过瞬时虚拟系统在这些解决方案之间进行选择。不需要对吸引性进行任何假设，并且所有开关都被平等对待，因此标准的“Filippov”方法以最自然的方式扩展到不连续流形的交叉点。在等效控制的设置中也给出了相应的结果，使得系统比典型的线性控制形式更一般。
If a set of ordinary differential equations is discontinuous along some threshold, solutions can be found that are continuous, if sometimes multivalued. We show the extent to which unique solutions can be found in general cases when the threshold takes the form of finitely many intersecting manifolds. If the intersections are transversal, finitely many solutions can be found that slide along the threshold. They are obtained by a hierarchical application of convex combinations to form a differential inclusion. The system chooses between these solutions by means of an instantaneous dummy system. No assumptions on attractivity are required, and all switches are treated equally, so the standard "Filippov" method is extended to intersections of discontinuity manifolds in the most natural way possible. The corresponding result in the setting of equivalent control is also given, allowing systems more general than typical linear control forms to be solved.