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In this work, we study both minimal and maximal blowup time controls for some ordinary differential equations. The existence and Pontryagin's maximum principle for these problems are derived. As a key preliminary to prove our main results, due to certain monotonicity of the controlled systems, "the initial period optimality" for an optimal triplet is built up. This property reduces our blowup time optimal control problems (where the target set is outside of the state space) to the classical ones (where the target sets are in state spaces). [ABSTRACT FROM AUTHOR]

Sharp constants for an inequality of Poincaré type are studied. The problem is solved by using optimal control theory. [ABSTRACT FROM AUTHOR]

The article offers information on proof of Stirling's Formula given by authors P. Diaconis and D. Freedman. Topics include use of convergence methods that provide short proof of Stirling's formula and its refinement, explanation of terms in the formula including Euler's Gamma function and the MacLaurin's series, and calculations involved for the proof of the formula.

Relaxed controls have proved to be very useful in studying the existence of optimal controls in optimal control theory. Many positive results have been obtained in the literature. However, negative results have also made their rare appearances. The optimal control problem considered in this paper looks quite simple. Yet, by treating such a problem, we can get interesting results, substantiating our idea as to whether an optimal control exists or not. In our opinion, the method used in the paper can be applied to more generalized cases. [ABSTRACT FROM AUTHOR]

An optimal control problem governed by quasi-linear variational inequality is considered. The cost functional to be minimized contains the solution of a quasi-linear variational inequality. To get the existence, regularity, and necessary condition for the optimal pair, a new related control problem is introduced. By proving the existence of an optimal pair to such a new problem, the existence and regularity of the optimal pair to the original problem are obtained. It turns out that the regularity obtained is sharp in general. Some necessary conditions of the optimal pair are also obtained. [ABSTRACT FROM AUTHOR]