Optimal Control of Elastic Boundary Value Problems
Summary
Optimal control of elastic boundary value problems investigates how to influence the behaviour of elastic media by adjusting boundary inputs or internal parameters so as to achieve desired mechanical performance. At its core lies a system of partial differential equations describing elastic equilibrium, coupled with a cost functional that quantifies deviation from target states or minimises energy expenditure. Fundamental approaches combine variational methods with control theory, enabling the formulation of existence and uniqueness results, derivation of necessary optimality conditions and development of efficient numerical schemes. Applications span the design of compliant mechanisms, minimisation of stress concentrations in aerospace structures, adaptive support in civil engineering and targeted therapies in soft tissue modelling. Recent advances have deepened understanding of the interplay between material heterogeneity, geometric design and control objectives, revealing new pathways for tailoring elasticity at multiple scales.
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Optimal Control of Elastic Boundary Value Problems publication trend
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Technical terms
Boundary value problem: A mathematical model in which an elastic body is described by equilibrium equations subject to prescribed conditions on its boundary.
Cost functional: A scalar quantity to be minimised or maximised in optimal control, typically measuring energy, displacement error or stress concentration.
Variational inequality: A generalisation of boundary value problems accommodating inequality constraints such as non-penetration or unilateral contact.
Shape derivative: The sensitivity of a functional with respect to infinitesimal changes in domain geometry, used to guide shape optimisation.
Homogenisation: A mathematical technique that replaces a heterogeneous medium by an effective homogeneous one, capturing microstructure effects in a limit model.
References
- Existence of an optimal size of a delaminated rigid inclusion embedded in the Kirchhoff-Love plate. Boundary Value Problems (2015).
- Optimal control for evolutionary imperfect transmission problems. Boundary Value Problems (2015).
- First-Order Shape Derivative of the Energy for Elastic Plates with Rigid Inclusions and Interfacial Cracks. Applied Mathematics & Optimization (2020).
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