In this paper, sparse elliptic PDE-constrained optimization problems with \(L^{1-2}\) -control cost ( \(L^{1-2}\) -EOCP) are considered. To induce control sparsity, traditional finite element models usually use \(L^{1}\) -control cost to induce sparsity, and in practical problems, many non-convex regularization terms are more capable of inducing sparsity than convex regularization terms, for example, in finite-dimensional problems, the sparsity of \(l_{1-2}\) -norm induced solutions is stronger than \(l_{1}\) -norm. Inspired by the finite-dimensional problems, we extend the \(L^{1-2}\) -regularization technique to infinite-dimensional elliptic PDE-constrained optimization problems. Unlike finite-dimensional problems where the \(l_{1-2}\) -norm is greater than or equal to 0, the conclusion does not hold for the \(L^{1-2}\) -control cost in the infinite-dimensional sense. To overcome these difficulties, an inexact difference of convex functions algorithm with sieving strategy (s-iDCA) is proposed for solving \(L^{1-2}\) -EOCP where the corresponding subproblems are solved by an inexact heterogeneous alternating direction method of multipliers (ihADMM) algorithm. Furthermore, by using the particular structure of the \(L^{1-2}\) -EOCP and constructing a new energy function and using its KŁ property, the global convergence results of the DCA algorithm are given. Numerical experiments show that our proposed s-iDCA algorithm is effective and that the model with the \(L^{1-2}\) -regularization term is stronger than the \(L^{1}\) -regularization term in terms of the sparsity of the induced solutions.
