An Efficient Algorithm for an ℓ₁/ℓ₂ Mixed Optimal Control Problem with a Box Constraint and Parallelization

An ℓ₁/ℓ₂ mixed optimal control problem for discrete linear systems is considered in this letter. Compared to conventional LQ-type optimal control problems, which consists of quadratic costs on states and inputs, an ℓ₁ cost on the input are included to induce the sparsity of the control input sequence. An efficient algorithm based on the alternating direction method of multipliers (ADMM) is proposed. In the proposed algorithm, proximal points are computed by the closed form solution. Furthermore, by exploiting the structure of the optimal control cost function, the proposed algorithm is parallelized. Finally, numerical examples illustrate the effectiveness of proposed algorithms.