arXiv:2610.19976v1 Announce Type: cross Abstract: We present a novel class of minimum energy estimation problems for positive linear systems with linear costs subject to coupled process and measurement disturbances, where the latter is bounded by the state and the former is bounded by the noise. Through dynamic programming we show that the problem admits explicit solutions. Remarkably, the Luenberger observer structure of the optimal estimator emerges from the optimization itself, with a switching gain that inherits the sparsity of the disturbance constraints, and it is obtained from a differential equation in the finite horizon case. In the infinite horizon case, an algebraic equation characterizes the limit of the finite-horizon solution and yields a stationary estimator with a static gain. We introduce two distinct notions of detectability to characterize when the static gain stabilizes the error dynamics while preserving their positivity, and analyze under which conditions the stationary solution can be formulated as a linear program. We further establish that a disturbance perturbation of finite L1-norm yields an estimation error of finite L1-norm. The framework is illustrated on a large-scale irrigation network.