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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]

Time optimal control problems for some non-smooth systems in general form are considered. The non-smoothness is caused by singularity. It is proved that Pontryagin's maximum principle holds for at least one optimal relaxed control. Thus, Pontryagin's maximum principle holds when the optimal classical control is a unique optimal relaxed control. By constructing an auxiliary controlled system which admits the original optimal classical control as its unique optimal relaxed control, one get a chance to get Pontryagin's maximum principle for the original optimal classical control. Existence results are also considered. [ABSTRACT FROM AUTHOR]

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]