Regulators and academics are increasingly interested in the causal effect that algorithmic actions of a digital platform have on user consumption. In pursuit of estimating this effect from observational data, we identify a set of assumptions that permit causal identifiability without assuming randomized platform actions. This approach is applicable to systems that rely on (deterministic) machine-learning-powered predictions, such as recommendation algorithms. The key novelty of our approach is to explicitly model the dynamics of consumption over time, exploiting the repeated interaction digital platforms have with their participants. By viewing the platform as a controller acting on a dynamical system, we are able to show that exogenous variation in consumption and appropriately responsive algorithmic control actions are sufficient for identifying the causal effect of interest. We complement our claims with an analysis of finite sample estimators and empirical investigations. More broadly, our results deriving identifiability conditions tailored to digital platform settings illustrate a fruitful interplay of control theory and causal inference.