This paper focuses on the stability analysis and control of adaptive switching systems by establishing Lyapunov-based logic switching rules. Through considering performance-index-based adaptive switching control under a general framework, sufficient conditions are proposed and proved to ensure global generalized exponential stability and global finite-time stability. By taking these conditions as guidelines for designing control laws and logic switching rules, we explore their applications in controlling a class of lower-triangular nonlinear systems with an unknown control direction.