"My Last Duchess" is all about power: the political and social power wielded by the speaker (the Duke) and his attempt to control the domestic sphere (his marriage) in the same way that he rules his lands. He rules with an iron fist. The Duke views everything that he possesses and everyone with whom he interacts as an opportunity to expand his power base. Wives need to be dominated; servants need to understand his authority; and fancy objects in his art gallery display his influence to the world – if he decides to show them. Kindness, joy, and emotion are all threats to his tyrannical power.
The Duke of Ferrara sets himself up to have his power threatened, because he never communicates directly with people about his expectations for their behavior.