Resumen de Analysis of Optimal Control of the Mechanical Structures Vibrating in the Viscoelastic Environment

Yiqun Li, Hong Wang, Xiangcheng Zheng

• We investigate an optimal control problem with integral constraints for the dynamic vibration of the mechanical structures (e.g., strings, rods or membranes) in the viscoelastic environment, which is governed by a time-fractional wave equation with space-time dependent fractional order and coefficients. To account for the space-time dependent fractional order and coefficients, we develop an equivalent but more feasible variant of the model combined with compact operator theory and Fredholm alternative to prove the well-posedness of the state equation. In addition, we prove some enhanced mapping properties of the variable-order fractional differential operators, based on which we prove the high-order regularity results of the solution to the state equation. Furthermore, we analyze the adjoint equation derived from the variational inequality, which is a Riemann-Liouville time-fractional equation with the hidden-memory variable order and consequently requires more subtle treatments. We ultimately analyze the well-posedness of the optimal control problem and prove the high-order regularity estimates of its solutions.