The correct answer is option 4: extend the territorial boundaries of the English East India Company.

The Doctrine of Lapse was a policy implemented by the British East India Company in the 19th century. According to this policy, if a ruler of a Princely State died without a male heir, the state would be annexed by the British and become a part of their territory. This policy was introduced by Lord Dalhousie, the Governor-General of India from 1848 to 1856.

The purpose of the Doctrine of Lapse was to expand the territorial boundaries of the British East India Company. By annexing the Princely States that had no male heir, the British aimed to consolidate their control and increase their influence in India. This policy was also driven by the desire to weaken the power of the Indian rulers and establish direct British rule over the Indian population.

It is important to note that while the Doctrine of Lapse did have an impact on the control of rebellions in the Princely States, this was not its primary objective. Similarly, the policy was not aimed at extending the military strength of the British or regulating landlords.