E-commerce product catalogs contain billions of items. Most products have lengthy titles, as sellers pack them with product attributes to improve retrieval, and highlight key product aspects. This results in a gap between such unnatural products titles, and how customers refer to them. It also limits how e-commerce stores can use these seller-provided titles for recommendation, QA, or review summarization. Inspired by recent work on instruction-tuned LLMs, we present InstructPTS, a controllable approach for the task of Product Title Summarization (PTS). Trained using a novel instruction fine-tuning strategy, our approach is able to summarize product titles according to various criteria (e.g. number of words in a summary, inclusion of specific phrases, etc.). Extensive evaluation on a real-world e-commerce catalog shows that compared to simple fine-tuning of LLMs, our proposed approach can generate more accurate product name summaries, with an improvement of over 14 and 8 BLEU and ROUGE points, respectively.