Flexible manipulators have received wide attention because they are more realistic than their rigid counterparts in many practical conditions. The control of the motion and vibration of an elastic model has inspired many studies in the past decades. In this paper, a finite element model is developed and used for the modeling of the single-link flexible arm. A new adaptive genetic algorithm (AGA) is proposed to optimize the feedback gains of a proportional derivative (PD) controller for the control of the motion and vibration of the flexible arm. The proposed technique is based on a new adaptive mutation operator to adjust the probability of mutation of each chromosome based on the average fitness value at each generation. The results are compared with those obtained by applying simple GA and a direct evaluation of the objective function in the domain. It has been shown that the new developed adaptive mechanism is faster in convergence and the obtained solutions are of higher fitness values. Compared analysis shows the effectiveness of the proposed adaptive GA over traditional genetic algorithms that have the defects of premature convergence and stagnation when applied in optimization problems. Three different objective functions based on minimizing the error function are tested and compared. Multi-objective error fitness function combined with AGA has given the best system response. Simulation results demonstrate the effectiveness of the proposed technique which encourages for further research into the application of the technique in controlling multiple link flexible robots for further applications.