Poro-Visco-Elasticity in Biomechanics: Optimal Control

Bociu, Lorena; Strikwerda, Sarah

doi:10.1007/978-3-031-04496-0_5

Cited by 2 publications

(1 citation statement)

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“…Therefore, in order to dominate the quadratic term in the proofs, the boundedness with respect to the product of the 𝐿 1 -norms is assumed, see also the related comments in [40,44]. A related assumption (Lipschitz continuity of the derivative) is made in ( 5), (10) in [28] (note that the abstract space 𝑌 therein becomes 𝐿 1 for the examples). However, Assumption 4.1 may be considered to be too restrictive for practical applications of Algorithm 1 because it requires an improvement of the input regularity of the control-to-state operator.…”

Section: Input Regularizationmentioning

confidence: 99%

Input Regularization for Integer Optimal Control in BV with Applications to Control of Poroelastic and Poroviscoelastic Systems

Bociu,

Manns,

Severitt

et al. 2024

Journal of Nonsmooth Analysis and Optimization

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We revisit a class of integer optimal control problems for which a trust-region method has been proposed and analyzed in arXiv:2106.13453v3 [math.OC]. While the algorithm proposed in arXiv:2106.13453v3 [math.OC] successfully solves the class of optimization problems under consideration, its convergence analysis requires restrictive regularity assumptions. There are many examples of integer optimal control problems involving partial differential equations where these regularity assumptions are not satisfied. In this article we provide a way to bypass the restrictive regularity assumptions by introducing an additional partial regularization of the control inputs by means of mollification and proving a $\Gamma$-convergence-type result when the support parameter of the mollification is driven to zero. We highlight the applicability of this theory in the case of fluid flows through deformable porous media equations that arise in biomechanics. We show that the regularity assumptions are violated in the case of poro-visco-elastic systems, and thus one needs to use the regularization of the control input introduced in this article. Associated numerical results show that while the homotopy can help to find better objective values and points of lower instationarity, the practical performance of the algorithm without the input regularization may be on par with the homotopy.

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