The problem of static state feedback H-infinity control with various regional stability constraints for the closed-loop system is considered. A general framework for solving this problem is established using the Lagrange multiplier method in conjunction with generalised Lyapunov theory and Riccati-equation-based H-infinity optimization characterization. Necessary conditions are given for the existence of the desired optimal feedback gains. Furthermore, as the simplest regional stability case, the stability-degree problem is studied with two different approaches, one an indirect overbounding method that extends the standard H-infinity problem formulation with an cu shift and another that uses the direct optimization of the H-infinity performance measure with stability degree as a constraint under the general framework. The conservativeness of the former approach and the generality of the latter approach are clearly demonstrated. Finally, the theory is illustrated with examples and the computational issues are discussed. It is shown that the successive approximation technique of Liu and of Liu and Yedavalli can be used to partially solve the problem.