Material Mixing in Space-Time and Dynamic Control in the Coefficients of Linear Hyperbolic Equations

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This work will be focused on the concept of material mixing in dynamics.This concept introduces spatio-temporal composites, i.e. material microstructuresassembled and maintained in space-time. Such composites will be investigated in the framework of a Maxwell system for moving dielectrics as well as for the relativistic equations of elastodynamics and acoustics. The relevant set of effective parameters (G-closures) will be explicitly specified for a number of practical examples.The knowledge of G-closures is crucial for the analysis of material optimizationproblems that arise in dynamics of moving media, particularly in acoustics and hydroacoustics, as well as in high frequency electronic engineering. Such a knowledge makes many of these problems well-posed when the material coefficientsact as the control factors. Physically, this is because the optimal material layouts in space-time generally shape themselves as microstructures.This work is aimed to prepare a conceptual basis for a subsequent investigation of material optimization in dynamics.