Piecewise constant optimal controls and minimal acoustic transmission

Abstract

The existence of completely piecewise constant optimal controls in the wide bandwidth, normal incidence, minimal acoustic transmission problem [MIN(γ2)] for layers of viscoelastic materials is examined numerically. It has been found that such solutions exist if the Hamiltonian is linear in the controls and the dynamic compressibility is defined in terms of fractional derivatives which approximate a generalized Maxwell model of the dilatation modulus. The low‐frequency dynamic compressibility and the low‐ to high‐frequency step of the compressibility in the relaxation region are assumed to be the controls. Calculations are presented that show the resulting optimal, multilayered structures. For shallow slopes of the compressibility the optimal controls are partially continuous (singular). As the slope increases, the controls become piecewise constant. Piecewise constant optimal controls are the exception. Their appearance seems to be restricted to relatively simple definitions of the control functions. Their occurrence may also be limited to the MIN(γ2) problem in the set of three basic problems defined by transmission, reflection, and absorption.