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This paper discusses generalized weak rigidity, and aims to apply the theory to formation control problems with a gradient flow law in the 2-and 3-dimensional spaces. The generalized weak rigidity theory is utilized in order that a desired formation is characterized by desired inter-agent distances and angles. As the first result of its applications, the paper provides analysis of local stability for weakly rigid formation systems in the 2-and 3-dimensional spaces. Then, as the second result of its applications, if there are three agents in the 2-dimensional space, almost globally exponential stabilization of weakly rigid formation systems is ensured. Through numerical simulations, the validity of analyses is illustrated.