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This paper deals with finite-time boundedness (FTB) control with input constraints for linear time-invariant systems. A design method of state feedback FTB controllers with input constraints is proposed based on reachable sets of the state in finite-time period. A design method of observer-based output feedback FTB controllers is also proposed based on reachable sets of the estimated state. Both design methods include controller designs for finite-time stabilization. All design conditions are reduced to linear matrix inequalities. Numerical examples are shown to illustrate the proposed design methods.