Non-convex optimization problems for maximum hands-off control

TL;DR

This work addresses the problem of obtaining maximum hands-off (sparse) control for continuous-time linear systems by introducing a broad class of non-convex penalty-based approximations to the $L^0$ objective. It proves that these non-convex formulations are always equivalent to the $L^0$ optimum, even when the usual $L^1$ surrogate fails, by imposing additively separable, symmetry- and magnitude-compatible penalties $\phi$. A difference-of-convex (DC) programming approach is developed to compute solutions: time is discretized, the problem is cast as $J(z)=g(z)-h(z)$ with $g,h$ convex, and the DC algorithm is employed, with convex subproblems solved via ADMM or CVX. Numerical examples (e.g., a double-integrator) validate that non-convex penalties recover the maximum hands-off control and can outperform the $L^1$ surrogate in sparsity, albeit with higher computation times. The results extend sparse control design beyond convex relaxations, providing a practical framework for exact sparsity in continuous-time systems and outlining directions for faster algorithms and penalty selection.

Abstract

The maximum hands-off control is the optimal solution to the L0 optimal control problem. It has the minimum support length among all feasible control inputs. To avoid computational difficulties arising from its combinatorial nature, the convex approximation method that replaces the L0 norm by the L1 norm in the cost function has been employed on standard. However, this approximation method does not necessarily obtain the maximum hands-off control. In response to this limitation, this paper newly introduces a non-convex approximation method and formulates a class of non-convex optimal control problems that are always equivalent to the maximum hands-off control problem. Based on the results, this paper describes the computation method that quotes algorithms designed for the difference of convex functions optimization. Finally, this paper confirms the effectiveness of the non-convex approximation method with a numerical example.

Figures (3)

• Figure 1: Examples of non-convex penalty $\psi$ (dotted line) and the function $\phi$ (solid line). The $L^p$ penalty with $(p,\lambda)=(0.5, 0.8)$ (top left); The MCP with $(\lambda, \alpha) = (0.25, 2)$ (middle left); The SCAD with $(\lambda, \alpha)=(0.25, 3)$ (bottom left); The LSP with $(\lambda, \alpha) = (0.5/\log(1+1/\alpha), 10^{-6})$ (top right); capped $L^1$ penalty with $(\lambda, \alpha) = (0.8, 0.5)$ (middle right); The $L^1/L^2$ penalty with $\lambda=0.6$ (bottom right).
• Figure 2: Optimal solutions to various non-convex optimal control problems equivalent to the maximum hands-off control, and the $L^1$ optimal control for comparison. The $L^1$ optimal control (top left); The $L^p$ optimal control with $(p, \lambda) = (0.5, 0.8)$ (middle left); The MCP optimal control with $(\lambda, \alpha) = (1, 0.5)$ (bottom left); The SCAD optimal control with $(\lambda, \alpha) = (0.25, 3)$ (top right); The LSP optimal control with $(\lambda, \alpha) = (0.1 / \log(1 + 1/\alpha), 10^{-6})$ (middle right); The $L^1/L^2$ optimal control with $\lambda = 0.1$ (bottom right).
• Figure 3: State trajectories, each formed by the optimal solutions in Fig. \ref{['fig:ex1_control']}. The solid lines and the dotted lines show the state variables $x_1$ and $x_2$ on $[0, T]$, respectively.

Theorems & Definitions (8)

• Remark 1
• Remark 2
• Lemma 1: Theorem 8.2, HerLas
• Theorem 1: bang-off-bang property
• Lemma 2: Theorem 3, ITKKTAC18
• Theorem 2: existence and equivalence
• Remark 3
• Proposition 1