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<p>In this paper, we study a&#xa0;time-optimal boundary control problem for the evolution of thin film thickness governed by a&#xa0;time-fractional biharmonic (fourth-order) parabolic equation with a&#xa0;Caputo time derivative. The model accounts for the interplay between surface-tension effects and memory-driven anomalous diffusion, both of which play a&#xa0;key role in the dynamics of thin film growth. By employing a&#xa0;spectral decomposition method, we represent the solution in terms of eigenfunctions and Mittag-Leffler functions, and reformulate the control problem as a&#xa0;Volterra integral equation of the second kind. We construct admissible control functions that drive the system to a&#xa0;prescribed average thickness in minimal time, and derive explicit estimates for this time in terms of the model parameters. The results highlight the novelty of the fractional-order extension and extend the classical time-optimal control framework for the biharmonic equation to the subdiffusive regime, emphasizing how fractional memory affects controllability of thin film systems.</p>

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Time-Optimal Control of Thin-Film Growth under a Time-Fractional fourth-order parabolic Equation with Caputo Derivative

  • 期刊论文

  • 发表日期：

    2026年4月13日

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  • Farrukh Dekhkonov

摘要

In this paper, we study a time-optimal boundary control problem for the evolution of thin film thickness governed by a time-fractional biharmonic (fourth-order) parabolic equation with a Caputo time derivative. The model accounts for the interplay between surface-tension effects and memory-driven anomalous diffusion, both of which play a key role in the dynamics of thin film growth. By employing a spectral decomposition method, we represent the solution in terms of eigenfunctions and Mittag-Leffler functions, and reformulate the control problem as a Volterra integral equation of the second kind. We construct admissible control functions that drive the system to a prescribed average thickness in minimal time, and derive explicit estimates for this time in terms of the model parameters. The results highlight the novelty of the fractional-order extension and extend the classical time-optimal control framework for the biharmonic equation to the subdiffusive regime, emphasizing how fractional memory affects controllability of thin film systems.