Optimal design and hyperbolic problems

Quite often practical problems of optimal design have no solution. This situation can be alleviated by relaxation, where one needs generalised materials which can mathematically be defined by using the theory of homogenisation. First mathematical results in this direction for general (nonperiodic) materials were obtained by Murat and Tartar. We present some results in optimal design where the equation of state is hyperbolic. The control function is related to the response of vibrating material under the given external force. As the problem under consideration has no solution, we consider its relaxation to H-closure of the original set of controls