Distributional Uncertainty and Adaptive Decision-Making in System

Complex engineered systems require coordinated design choices across heterogeneous components under multiple conflicting objectives and uncertain specifications. Monotone co-design provides a compositional framework for such problems by modeling each subsystem as a design problem: a feasible relation between provided functionalities and required resources in partially ordered sets. Existing uncertain co-design models rely on interval bounds, which support worst-case reasoning but cannot represent probabilistic risk or multi-stage adaptive decisions. We develop a distributional extension of co-design that models uncertain design outcomes as distributions over design problems and supports adaptive decision processes through Markov-kernel re-parameterizations. Using quasi-measurable and quasi-universal spaces, we show that the standard co-design interconnection operations remain compositional under this richer notion of uncertainty. We further introduce queries and observations that extract probabilistic design trade-offs, including feasibility probabilities, confidence bounds, and distributions of minimal required resources. A task-driven unmanned aerial vehicle case study illustrates how the framework captures risk-sensitive and information-dependent design choices that interval-based models cannot express.