Steering semilinear beam dynamics with memory, delay, and non-instantaneous impulses: Approximate controllability and numerical evidence

Boshra K. Alarfaj; Kawther K. Alarfaj; Fatema Al Najim; Maryam Khalid Alarfaj; Boshra K. Alarfaj; Kawther K. Alarfaj; Fatema Al Najim; Maryam Khalid Alarfaj

doi:10.3934/math.20261401

AIMS Mathematics

2026, Volume 11, Issue 11: 36163-36187. doi: 10.3934/math.20261401

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Research article

Steering semilinear beam dynamics with memory, delay, and non-instantaneous impulses: Approximate controllability and numerical evidence

1.
Department of Statistics and Operations Research, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
2.
Department of Mathematics and Statistics, College of Science, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
3.
School of Computer Science, Faculty of Engineering and Information Technology, University of Technology Sydney, NSW, Australia

Received: 18 August 2026 Revised: 10 October 2026 Accepted: 15 October 2026 Published: 04 November 2026
MSC : 74K10, 93B05, 35L75

A conditional approximate-controllability framework is developed for a semilinear Euler–Bernoulli beam with delay, hereditary memory, and non-instantaneous impulses in a product Hilbert space. The linear steering mechanism uses a short terminal control window and a Tikhonov-regularized controllability Gramian. With this window shorter than the delay and located after the final impulse, the delayed nonlinear contributions are determined by the preceding history. We identify hypotheses under which these residuals vanish and exclude an earlier inappropriate vanishing-control-energy requirement. To provide corresponding numerical evidence, MATLAB R2025b calculations use a spatially distributed Gramian controller for every finite-difference beam mode, coupled with an implicit-midpoint structural scheme, nonlinear delayed forcing, exponentially fading memory, and a finite-duration impulse. The computed displacement, velocity, and full product-energy terminal errors decrease under regularization (from $ 2.2050 $ to $ 3.3683\times10^{-5} $ in the full product norm), while control cost rises. Direct full-grid propagation agrees with the Gramian residual prediction at the reported precision. These calculations support the specified finite-grid, control-affine benchmark; they do not themselves prove the infinite-dimensional theorem.
- semilinear beam equation,
- non-instantaneous impulses,
- approximate controllability,
- time delay,
- semigroup theory,
- memory effects
Citation: Boshra K. Alarfaj, Kawther K. Alarfaj, Fatema Al Najim, Maryam Khalid Alarfaj. Steering semilinear beam dynamics with memory, delay, and non-instantaneous impulses: Approximate controllability and numerical evidence[J]. AIMS Mathematics, 2026, 11(11): 36163-36187. doi: 10.3934/math.20261401

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Abstract

A conditional approximate-controllability framework is developed for a semilinear Euler–Bernoulli beam with delay, hereditary memory, and non-instantaneous impulses in a product Hilbert space. The linear steering mechanism uses a short terminal control window and a Tikhonov-regularized controllability Gramian. With this window shorter than the delay and located after the final impulse, the delayed nonlinear contributions are determined by the preceding history. We identify hypotheses under which these residuals vanish and exclude an earlier inappropriate vanishing-control-energy requirement. To provide corresponding numerical evidence, MATLAB R2025b calculations use a spatially distributed Gramian controller for every finite-difference beam mode, coupled with an implicit-midpoint structural scheme, nonlinear delayed forcing, exponentially fading memory, and a finite-duration impulse. The computed displacement, velocity, and full product-energy terminal errors decrease under regularization (from $ 2.2050 $ to $ 3.3683\times10^{-5} $ in the full product norm), while control cost rises. Direct full-grid propagation agrees with the Gramian residual prediction at the reported precision. These calculations support the specified finite-grid, control-affine benchmark; they do not themselves prove the infinite-dimensional theorem.

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