Discontinuous control systems by
Discontinuous control systems by
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不连续控制系统
DOI：
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发表时间：
2008
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通讯作者：
A. Seierstad
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作者：
A. Seierstad
通过一些简单的经济学例子，我们阐明了当所研究的系统在跨越某些边界时发生重大变化时解决最优控制问题的某些解决工具。 “主要变化”可能是状态在跨越边界时出现跳跃不连续性，或者微分方程的右侧发生变化。给出了一些理论结果。在给出的结果中，至少与极值场相关的充分条件应该是新的。 。引言 在经济控制问题中，有时当国家跨越边界时，底层系统会发生重大变化。例如，当一家公司的股本变为负值时，该公司可能会破产。从数学上来说，这种变化会在微分方程中引入不连续性，从而使应用标准极大值原理通常所需的假设失效。在其他情况下，当状态到达表面时，它会受到冲击（跳跃不连续性），这种情况也需要改变最大原则。下面将介绍一些体现这些特征的简单经济例子。本文的主要目的是展示可用的解决方案工具如何在简单的情况下工作。这些求解工具由最大原理的标准方程以及涉及跨越边界时肋变量跳跃的附加条件组成。下面没有关于这个跳转条件的参考，但它之前肯定已经被直接或间接地陈述和使用过，并且不止一次，至少在更多的应用工作中。 （它确实出现在 Nævdal (2001) 中，另请参见 Nævdal (2003)。）以下结果适用于上述两种类型的不连续性。当然，一般来说，有许多与状态变量的跳跃有关的结果，包括所谓的脉冲控制问题，参见例如Seierstad 和 Sydsæter (1987) 和 Arutyunov, A. (2005) 的第 3 章以及其中引用的论文。早期的经济例子包括 Arrow 和 Kurz 以及 K.Vind 的著作，以及 Kamien 和 Schwartz 的控制理论书籍，所有这些都在 Seierstad 和 Sydsæter (1987) 中引用。然而，让我们首先陈述一个标准的“连续”控制问题以及最大值原理。 1. 标准控制问题 考虑问题
By means of some simple examples from economics, we elucidate certain solution tools for the solution of optimal control problems were the system under study undergoes major changes when certain boundaries are crossed. The "major changes" may be that the state gets a jump discontinuity when crossing a boundary, or that the right hand side of the differential equation changes. Some theoretical result are presented. Among the results presented, at least the sufficient condition related to fields of extremals should be new. . Introduction In economic control problems, sometimes the underlying system undergoes a major change when the state crosses a boundary. For example, a firm may go bust, when its equity becomes negative. Mathematically speaking, such changes introduce discontinuities in the differential equation that invalidate the assumptions ordinarily required for the standard maximum principle to apply. In other situations, when the state reaches a surface it gets a kick, (a jump discontinuity), a case also necessitating changes in the maximum principle. Below, some simple economic examples are presented in which such features appear. The main purpose of this paper it to show how the solution tools available work in simple situations. These solution tools consist of the standard equations of the maximum principle plus an additional condition involving a jump in the costate variable when boundaries are crossed. There is no reference below for this jump condition, but it must have been stated and used before, directly or indirectly, and more than once, at least in more applied work. (It does appear in Nævdal (2001), see also Nævdal (2003).) The following results pertain to both types of discontinuities described above. There are in general, of course, a number of results pertaining to jumps in the state variables, including so-called impulse control problems, see e.g. chapter 3 in Seierstad and Sydsæter (1987) and Arutyunov, A. (2005) and papers referred to there. Early economic examples include the works by Arrow and Kurz, and K.Vind, as well as the control theory book by Kamien and Schwartz, all referenced in Seierstad and Sydsæter (1987). Let us, however, first state a standard "continuous" control problem, together with the maximum principle. 1. Standard Control Problem Consider the problem