Relaxed Controls and Variational Problems
Relaxed Controls and Variational Problems
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DOI：
10.1137/0305027
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发表时间：
1967-08
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影响因子：
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通讯作者：
E. J. McShane
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作者：
E. J. McShane
导论.在研究某类曲线上的函数M的极小化或极大化问题时，控制理论的符号学已显示出其在理论和应用上的价值。一个很大的优势，制定的是，它大大方便了调查ofminimizing曲线中的控制，在某些时候设置在边界的区域，他们被限制。所有经典的变分法单积分问题都可以很容易地转化为最优控制问题，最坏的情况下只需对拉格朗日和Bolza问题的边方程作一个适当的限制[1]，[2]。然而，大多数最优控制理论，虽然不是全部（el。[5]，[21]），是在控制是有界的假设下发展起来的。当然，这对于许多有用的应用程序来说已经足够了。然而，即使对于应用来说，最好还是放弃这种限制;当经典变分法中简单而熟悉的问题被重新表述为最优控制问题时，它往往不成立。当状态空间中的每一点的可达时间导数形成一个有界闭凸集时，最优路径就存在，正如我们已经知道的那样。[17][18][19][19]当不满足凸性要求时，可以用可达集的凸船体代替可达集，从而产生”松弛控制”问题。但是，在这种情况下，特别是当控制是无界的时候，要找到最优路径所满足的必要条件是很不容易的。（Warga为有界控制做了这件事[19]，[20]。另一种设备成为可在1937年，当年轻人设计的对象称为”广义曲线”，这些包括普通曲线，以及其他对象，可被视为一个正确定义的替代品的unrealizable想法的曲线，其衍生物不断变化，并以无限的速度从一个到另一个给定的一组值。这些方法在证明可能非线性微分方程问题解的存在性定理中具有很大的实用价值。Young在[22]-[24]中证明了曲线被推广且没有边条件的问题解的存在性。这已经被扩展到具有边条件的问题，无论是传统形式[11]。最优控制形式[6]。
Introduction. In the study of the problems of minimizing or maximizing a functionM of a curve on some assigned class of curves, the symbolism of control theory has shown itself to be valuable both in the theory and in the applications. One great advantage of the formulation is that it greatly facilitates the investigation ofminimizing curves in which the controls are at some times set at the boundary of the region to which they are confined. All the classical single-integral problems of the calculus of variations can be easily rephrased as optimal control problems, with at worst a mild re-striction on the side-equations in Lagrange and Bolza problems [1],[2]. However, most of the theory of optimal control, though not all (el.[5],[21]), has been developed under the hypothesis that the controls are bounded. Certainly this is enough for a great number of useful applications. Nevertheless, even for applications it would be better to dispense with this re-striction; and when simple andfamiliar problems of the classical calculus of variations are reformulated as optimal control problems, it often fails to hold.When for each point in state-space the attainable time-derivativesof the state form a bounded closed convex set, an optimal path exists, as has been kn. own fora long time [17],[8]. When the convexity requirement failswe can replace the attainable set by its convex hull, producing the" relaxed control" problem. But it is far from easyto find the necessary conditions satisfied by optimal paths in thiscase, especially if controls are unbounded.(Warga did this for bounded controls [19],[20].) Another device became available in 1937, when Young devised objects called" generalized curves"; these include ordinary curves, andalso other objects which may be regarded as a correctly-defined substitute for the un-realizable idea of a curve whose derivative is changing incessantly and with infinite rapidity from one to another of a given set of values. These have proved to be of great utility in proving theorems on he existence of solutions of problems with possibly nonlinearside differential equations. Young showed in [22]-[24] the existence of solutions of problems in which the curves are generalized and there are no side conditions. This has been ex-tended to problems with side conditions, both in traditional form [11.] and in optimal-control form [6].