Which term describes an ongoing illness that can be controlled but has no known cure?

The term that accurately describes an ongoing illness that can be managed but is not curable is "chronic illness." Chronic illnesses are typically long-lasting conditions that can persist for years or even a lifetime. Examples include diabetes, hypertension, and arthritis. These illnesses often require continued medical attention, lifestyle adjustments, and management strategies to control symptoms and maintain the patient's quality of life.

Acute illnesses, in contrast, are conditions that typically develop suddenly and are often severe but are usually short-lived, such as a cold or a sprained ankle. Terminal illnesses refer to conditions that are expected to lead to death relatively soon, such as advanced cancer, and do not fit the description of being manageable without a cure. Sudden illnesses might imply a prompt onset, but they lack the chronic aspect of prolonged management without a cure. Thus, chronic illness is the term that best fits the definition provided in the question.