Prospective memory (PM) tasks require remembering to perform a deferred action and can be associated with predictable contexts. We present a theory and computational model, prospective memory decision control (PMDC), of the cognitive processes by which context supports PM. Under control conditions, participants completed lexical decisions. Under PM conditions, participants had the additional PM task of responding to letter strings containing certain syllables. Stimuli were presented in one of two colors, with color potentially changing after each set of four trials. A pretrial colored fixation was presented before each set. Under control and PM standard conditions, fixation color was meaningless. Under PM context conditions, fixation color indicated whether a PM target could occur within the next set. We replicated prior findings of higher PM accuracy for context compared to standard conditions, and the expected variation in PM costs (slowed lexical decisions) as a function of context relevance. PMDC, which formalizes PM as a process of evidence accumulation among ongoing and PM task responses, accounted for the impact of context on PM costs and accuracy via proactive and reactive cognitive control. Increased ongoing task thresholds and decreased PM thresholds in relevant contexts indicated proactive control. With context provision, PM accumulation rates on PM trials increased, as did inhibition of accumulation to competing responses, indicating reactive control. Although an observed capacity-sharing effect explained some portion of PM costs, we found no evidence that participants redirected more capacity from the ongoing to the PM task when contextually cued to relevant contexts.