Ergodic measure shaping in controlled iterated function systems via optimal transport

We study an invariant-law shaping problem for controlled iterated function systems (IFS) on a Polish state space (X,d). A finite-dimensional parameter u∈U⊂Rm modifies the IFS primitives and induces an invariant law μ(u) as the unique fixed point of an associated Markov operator. The design objective is to select u so that μ(u) matches a prescribed target distribution μtar∈P2(X) in squared 2-Wasserstein distance, possibly with a regularization term R(u). Under standard contraction-type conditions ensuring existence and uniqueness of μ(u), we formulate a computable surrogate based on (i) empirical approximation of μ(u) from finite trajectories and (ii) entropically regularized optimal transport (Sinkhorn loss) between empirical measures. This yields a practical projected-gradient scheme in the parameter space U. Numerical illustrations on controlled affine IFS models show close approximation of a unimodal Gaussian target, while the displayed bimodal example does not recover the separation of the target modes.