# Optimal actuator design for vibration control based on LQR performance and shape calculus

## TL;DR

This paper develops a method to optimize actuator shapes for vibration control by integrating LQR performance criteria with shape calculus, enabling efficient design of actuators for beam models.

## Contribution

It introduces a novel framework combining shape calculus and topological derivatives to optimize actuator shape based on LQR performance in infinite-dimensional systems.

## Key findings

- Optimal actuator shapes improve vibration suppression.
- Level-set method effectively computes shape derivatives.
- Framework applicable to various structural models.

## Abstract

Optimal actuator design for a vibration control problem is calculated. The actuator shape is optimized according to the closed-loop performance of the resulting linear-quadratic regulator and a penalty on the actuator size. The optimal actuator shape is found by means of shape calculus and a topological derivative of the linear-quadratic regulator (LQR) performance index. An abstract framework is proposed based on the theory for infinite-dimensional optimization of both the actuator shape and the associated control problem. A numerical realization of the optimality condition is presented for the actuator shape using a level-set method for topological derivatives. A Numerical example illustrating the design of actuator for Euler-Bernoulli beam model is provided.