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Why did Han dynasty emperors fight the Xiongnu?

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Wide autumn valley with golden grass in the foreground and low, tree-covered mountains of the Greater Khingan Range in the distance under a clear sky. — Inner Mongolia Da Hinggan (Greater Khingan) Range, southeast of Hailar, Inner Mongolia Autonomous Region, China.

Han dynasty emperors fought the Xiongnu primarily to secure the empire's northern frontier and protect Chinese territory from persistent nomadic incursions and violence.12

The motivations for these conflicts included:

Defense and Security: The Xiongnu were a powerful tribal confederation whose mounted archers frequently raided northern China. Han rulers sought to push the Xiongnu back into Central Asia to end these threats and restore the integrity of the Great Wall system.21
Commercial Interests: Beyond defensive measures, the Han government recognized the economic advantages of controlling trade routes. By expanding into Central Asia, they aimed to export surplus Chinese products—such as silk—in exchange for animals and other goods, eventually facilitating the development of the Silk Road.1
Strategic Influence: The Han sought to weaken the Xiongnu by forming alliances with their enemies and establishing Chinese prestige among the oasis communities surrounding the Takla Makan Desert, thereby denying the Xiongnu local support.12

While early Han governments initially attempted to maintain peace through tribute and marriage alliances, Emperor Wudi later adopted a more aggressive military policy to neutralize the Xiongnu threat permanently.12