In this study, we employ the established Carleman estimates and propagation estimates of smallness from measurable sets for real analytic functions, along with the telescoping series method, to establish an observability inequality for the degenerate parabolic equation over measurable subsets in the time-space domain. As a direct application, we formulate a captivating Stackelberg–Nash game problem and provide a proof of the existence of its equilibrium. Additionally, we characterize the set of Stackelberg–Nash equilibria and delve into the analysis of a norm optimal control problem.

中文翻译：

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简并抛物方程可测集的可观测性不等式及其应用

在本研究中，我们采用已建立的卡尔曼估计和实解析函数可测集的小传播估计，以及伸缩级数方法，建立了时空域可测子集上简并抛物线方程的可观性不等式。作为直接应用，我们提出了一个迷人的斯塔克尔伯格-纳什博弈问题，并提供了其均衡存在的证明。此外，我们还描述了 Stackelberg-Nash 均衡集的特征，并深入研究了范数最优控制问题的分析。