Large language models (LLMs), or foundation models (FMs), are pretrained transformers that coherently complete sentences auto-regressively. In this paper, we show that LLMs can control simplified space systems after some additional training, called fine-tuning. We look at relatively small language models, ranging between 7 and 13 billion parameters. We focus on four problems: a three-dimensional spring toy problem, low-thrust orbit transfer, low-thrust cislunar control, and powered descent guidance. The fine-tuned LLMs are capable of controlling systems by generating sufficiently accurate outputs that are multi-dimensional vectors with up to 10 significant digits. We show that for several problems the amount of data required to perform fine-tuning is smaller than what is generally required of traditional deep neural networks (DNNs), and that fine-tuned LLMs are good at generalizing outside of the training dataset. Further, the same LLM can be fine-tuned with data from different problems, with only minor performance degradation with respect to LLMs trained for a single application. This work is intended as a first step towards the development of a general space systems controller.