This paper applies a Lyapunov-based switched extremum seeking (LSES) control algorithm to the application of altitude optimization of airborne wind energy systems. We evaluate the stability and convergence of this control algorithm for the application at hand. Achievement of the optimal altitude to maximize energy production requires use of energy in order to search for the optimal altitude, which is continually and randomly varying. The simultaneous desires to optimize altitude and minimize control energy consumption are balanced in this paper through a variant of extremum seeking control, where the periodic perturbation signal is reduced when convergence upon an optimal altitude is detected. The signal is reinstated when the wind speed begins to deviate from its instantaneous optimal value. Because the wind shear profile (wind speed vs. altitude) is subject to continual variations, this application represents a challenging case study in LSES control. Using real wind shear data, the results presented in this paper show that the LSES controller is successful in significantly increasing the net energy production over fixed-altitude and standard extremum seeking strategies.