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Uniting two Control Lyapunov Functions for affine systems

We consider the problem of piecing together two control Lyapunov functions (CLFs). The first CLF characterizes a local controllability property toward the origin, whereas the second CLF satisfies a global controllability property with respect to a compact set. We give a sufficient condition to express explicitly a solution to this uniting problem. This sufficient condition is shown to be always satisfied for a simple chain of integrator. In a second part, we show how this uniting CLF problem can be useful to solve the problem of piecing together two stabilizing control laws.