根据预备知识，利用紧性定理和先验估计，证明了系统最优控制的存在性。
The existence of the optimal control for the system is demonstrated via compactness theorem and prior estimates.
应用此不动点定理，在非紧的一般拓扑空间中给出了几个关于拟平衡问题的解的存在性定理。
By applying the fixed point theorem, several new existence theorems of solutions for quasi-equilibrium problems are given under noncompact setting of topological Spaces.
利用中介逻辑的完备性，本文证明了紧致性定理，即一理论有模型当且仅当其任一有穷子集有模型。
The compact theorem is proved, which states that theory T has model if and only if any finite subset of T has model.
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