The problem considered is that of maximizing a linear combination of the natural frequencies of vibration of a turbine disc idealization of variable thickness. The problem is formulated as a general problem in optimal control theory with the addition of inequality constraints on the state variables. Significant progress has been made in solving the problem by using purely analytical techniques based on the maximum principle of Pontryagin. These transform the problem into a nonlinear programming problem which is solved numerically by using the Heaviside penalty function transformation in conjunction with Rosenbrock's hill-climbing techniques. Available computational experience indicates that these procedures provide powerful tools for handling complex structural optimization problems.