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

• Yiqun Li ^[1] ; Hong Wang ^[2] ; Xiangcheng Zheng ^[3]
  1. [1] Wuhan University

     Wuhan University

     China

     
  2. [2] University of South Carolina

     University of South Carolina

     Estados Unidos

     
  3. [3] Shandong University

     Shandong University

     China

     

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• Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 24, Nº 5, 2025
• Idioma: inglés
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• Resumen

  • 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.