Is the husband sinful if he does not approve his wife's travel to perform the Hajj pilgrimage with her brother, in the event that he asks her to dedicate herself to the care of their children who require continuous care?

A husband may not prevent his wife from performing the obligatory Hajj if she finds a mahram. If he prevents her, he is sinful because he is preventing her from an obligatory duty. If he does not permit her to perform the obligatory Hajj, she may perform it without his permission. As for a supererogatory Hajj, he may prevent her from it. The presence of children is not an excuse that permits the husband to prevent his wife from Hajj if he can care for the children or hire someone to care for them.