Petruchio's intentions in taming Katherina are, on the surface, to marry her and win her dowry. However, his motivations are more complex than this. Petruchio is a proud and arrogant man who believes that women are inferior to men. He sees Katherina as a challenge, and he takes pleasure in the fact that she is difficult to tame. He enjoys the power that he has over her, and he feels a sense of accomplishment when he finally succeeds in making her obedient.
On another level, Petruchio's intentions in taming Katherina can be seen as a way of controlling his own anger and aggression. Petruchio is a hot-tempered man who is prone to violence. He uses his relationship with Katherina as a way to channel his negative emotions. By controlling Katherina, he is able to control himself.
Ultimately, Petruchio's intentions in taming Katherina are complex and contradictory. He both loves and hates her. He wants to control her, but he also wants to protect her. He is drawn to her strength, but he is also afraid of it. In the end, Petruchio's intentions are fulfilled, but at the cost of Katherina's happiness.