Calculation of the Gradient in Quenching the Vibrations of Short Beams

The initial boundary value problem for a pseudohyperbolic equation of sixth order in terms of a spatial variable is considered. In this problem, the function to be determined includes a mixed derivative with respect to the spatial and time variables. By means of the Galerkin method, the solution may be expressed as a Fourier series in terms of a special system of basis functions. Energy estimates yield a law of energy conservation, from which a uniqueness theorem for the solution may be derived. In the case of a quadratic functional, a control problem for vibrations is considered. Specifically, the possibility of optimization by the gradient method is assessed. That entails explicitly stating the gradient of the functional to be optimized. A rule is proposed for calculating the gradient in terms of the solution of a conjugate problem.