抛物型最优控制问题的精细可度量化凸松弛
Fine metrizable convex relaxations of parabolic optimal control problems
摘要：
由无限维空间中的演化问题（例如抛物型边值问题）控制的非凸最优控制问题需要对某种（最好是凸的）紧化进行连续（并且可能也是光滑）的扩展，称为松弛，以保证其解的存在性，并方便使用相对传统的工具进行分析。当控制在勒贝格空间的某些子集中取值时，通常的扩展要么太粗糙（实际上只允许非常有限的非线性），要么太精细（不可度量化）。为了克服这些缺点，一个妥协的凸紧化在这里设计，结合经典的技术与Choquet理论的杨措施。这适用于抛物型最优控制问题的存在性和最优性条件的关注。
英文摘要：
Nonconvex optimal-control problems governed by evolution problems in infinite-dimensional spaces (as e.g. parabolic boundary-value problems) needs a continuous (and possibly also smooth) extension on some (preferably convex) compactification, called relaxation, to guarantee existence of their solutions and to facilitate analysis by relatively conventional tools. When the control is valued in some subsets of Lebesgue spaces, the usual extensions are either too coarse (allowing in fact only very restricted nonlinearities) or too fine (being nonmetrizable). To overcome these drawbacks, a compromising convex compactification is here devised, combining classical techniques for Young measures with Choquet theory. This is applied to parabolic optimal control problems as far as existence and optimality conditions concerns.
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