Optimal actuator design based on shape calculus

TL;DR

This work addresses optimal actuator design for linear diffusion by casting actuator placement and sizing as shape and topology optimization problems. It derives shape and topological derivatives for two cost functionals tied to closed-loop linear-quadratic performance, and implements gradient-based and level-set algorithms to realize optimal actuators. The approach yields nontrivial multi-component actuators and demonstrates robustness across initial data sets, with numerical tests in 1D and 2D showing clear performance gains over heuristic designs. The framework provides a principled route to actuator design in distributed parameter systems and suggests extensions to robust control and semilinear dynamics.

Abstract

An approach to optimal actuator design based on shape and topology optimisation techniques is presented. For linear diffusion equations, two scenarios are considered. For the first one, best actuators are determined depending on a given initial condition. In the second scenario, optimal actuators are determined based on all initial conditions not exceeding a chosen norm. Shape and topological sensitivities of these cost functionals are determined. A numerical algorithm for optimal actuator design based on the sensitivities and a level-set method is presented. Numerical results support the proposed methodology.

Figures (11)

• Figure 1: Test 1. Left: different single-component actuators with different centers have been spanned over the domain, locating the minimum value of $\mathcal{J}_1$ for the center at $x=0.5$. Center: starting from an initial guess for the actuator far from $0.5$, the gradient-based approach of Algorithm \ref{['alg:shape']} locates the optimal position in the middle. Right: as the actuator moves towards the center in the subsequent iterations of Algorithm \ref{['alg:shape']}, the value $\mathcal{J}_1$ decays until reaching a stationary point.
• Figure 2: Test 2. Left: inspecting different values of $\mathcal{J}_1$ by spanning actuators with different centers, the optimal center location is found to be close to $0.2$ . Center: the gradient-based approach steers the initial actuator to the optimal position. Right: the value $\mathcal{J}_1$ decays until reaching a stationary point, which coincides with the minimum for the first plot on the left.
• Figure 3: Test 3. (a) Initial condition $y_0(x)=\max(\sin(3\pi x),0)^2$. (b) Optimal actuator for $\alpha=10^3$, without initialization via increasing penalization. (c) Optimal actuator for $\alpha=10^{-1}$, subsequently used in the quadratic penalty approach. (d) Optimal actuator for $\alpha=10^3$, via increasing penalization.
• Figure 4: Test 3. Level set method implemented in Algorithm \ref{['alg:topo']}. Left: starting from an initial actuator, the topological derivative of the cost is computed and an updated actuator is obtained. The new shape is evaluated according to its closed-loop performance. If the update is rejected, the parameter $\beta_n$ is reduced. Middle: the level-set approach generates an update of the actuator shape based on the information from $\psi_h^{n}$, $\beta_n$ and $g_{\omega_n}$. Right: This iterative loop generates a decay in the total cost $J_1$, (which accounts for both the closed-loop performance of the actuator and its volume constraint).
• Figure 5: Test 4. (a) Initial condition $y_0(x)=\sin(3\pi x)^2\chi_{\{x<2/3\}}(x)$. (b) Optimal actuator for $\alpha=10^4$, without initialization via increasing penalization. (c) Optimal actuator for $\alpha=10^{-1}$, subsequently used in the quadratic penalty approach. (d) Optimal actuator for $\alpha=10^4$, via increasing penalization.

Theorems & Definitions (45)

• Remark 2.1
• Lemma 2.2
• Proof 1
• Lemma 2.3
• Proof 2
• Lemma 2.4
• Proof 3
• Remark 2.5
• Definition 3.1
• Lemma 3.2