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Boundary controllability of a nonlinear elastic body

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Abstract

This paper addresses the exact controllability of vibrations in a three-dimensional Cosserat elastic solid body using mathematical techniques such as operator theory and semigroup methods. The verification of exact (shape) controllability is accomplished through the application of the Hilbert Uniqueness Method, which involves investigating the boundary observability for the dual system. In partial differential equations control theory, the concept of exact observability for the dual system is fundamental to achieving exact controllability, although it differs from the common understanding of controllability. While control theory for systems governed by ordinary differential equations (ODEs) has a relatively formal and standardized approach, systems with distributed parameters, such as the Cosserat medium under consideration, involve a multitude of technical inequalities that must be established. Notably, Cosserat media possess six degrees of freedom for microstructures, in contrast to classical media with only three degrees of freedom. Consequently, exact control is required for all six variables, encompassing three translational and three rotational degrees of freedom, while classical media only necessitate the exact control of three translational variables. The paper concludes with a series of numerical studies utilizing the fast Fourier transform (FFT) and various simulations, which serve to validate the effectiveness of the proposed control scheme.

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Authors and Affiliations

  1. Faculty of Mechanical Engineering, K. N. Toosi University of Technology, Tehran, Iran

    Ali Najafi Ardekany & Zohreh Malek Hosseini

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  1. Ali Najafi Ardekany
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  2. Zohreh Malek Hosseini
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Correspondence to Ali Najafi Ardekany.

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Ardekany, A.N., Hosseini, Z.M. Boundary controllability of a nonlinear elastic body. Acta Mech 235, 3149–3166 (2024). https://doi.org/10.1007/s00707-023-03840-8

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