This paper presents a time optimal control problem with control constraints of the rectangular type for internally controlled heat equations. An existence result of time optimal controls for such a problem is established. The rectangular type of control constraints originates from the study of time optimal control problems for ordinary differential equations. In the finite dimensional case, there is no difference between such problems with control constraints of the rectangular type and those

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... the ball type, from the perspective of the study on the existence of optimal controls. Interestingly, in the infinite dimensional case, the problem with control constraints of the rectangular type differs essentially from that with control constraints of the ball type. For infinite dimensional systems, the existence for time optimal controls with constraints of the ball type has already been discussed in the literature, while the study of the rectangular type has not been touched upon as far as we know.