This technical note considers the problem of computing extremal values of the trajectories over a given set of initial conditions as well as finding output controllers minimizing these extremal values under time-domain constraints for nonlinear systems. It is shown that upper bounds of the sought extremal values as well as candidates of the sought controllers can be computed by solving a one-parameter sequence of bilinear matrix inequality (BMI) optimizations obtained through the square matricial representation (SMR) of polynomials. Moreover, a necessary and sufficient condition is proposed to establish the tightness of the found upper bound in spite of the conservatism introduced by the nonconvexity of BMI optimizations and the chosen degree of the Lyapunov function and relaxing polynomials.