The Topological State Derivative: An Optimal Control Perspective on Topology Optimisation.

Area of Science:

• Computational Mathematics
• Optimal Control Theory
• Shape Optimization

Background:

• Standard optimal control theory often deals with fixed domains.
• Topological derivatives are crucial for shape optimization problems.
• Existing methods for topological derivatives can be limited in scope.

Purpose of the Study:

• Introduce the topological state derivative for general topological dilatations.
• Explore its connection to standard optimal control theory.
• Provide a flexible framework for shape optimization.

Main Methods:

• Differentiating shape-dependent state variables with respect to topology.
• Linearizing systems similar to those in optimal control.
• Utilizing Stampacchia-type regularity estimates and asymptotic expansions.
• Handling general dilatations of shapes (curves, surfaces, hypersurfaces).

Main Results:

• Established a link between topological state derivatives and optimal control.
• Demonstrated that the topological state derivative can be computed using a linearized system.
• Showed the flexibility of the approach for various shape perturbations.
• Provided a method to compute first-order topological derivatives of shape functionals.

Conclusions:

• The topological state derivative offers a unified and flexible approach to shape optimization.
• It connects domain variations with optimal control principles.
• The method accommodates complex shape changes beyond simple point perturbations.