Non-Lipschitz continuous adaptive regulation of nonlinear systems with uncontrollable unstable linearization

Adaptive regulation problem is solved for a class of nonlinear systems whose Jacobian linearization may have uncontrollable modes with eigenvalues lying on the right half-plane. By proposing a continuous version of the LaSalle-Yoshizawa theorem and using the technique of adding a power integrator, we developed a constructive design tool that solves the adaptive regulation problem under continuous framework. We designed a C⁰ adaptive control input, an update law for unknown parameters, and a C¹ control Lyapunov function which is positive definite and proper. A physical example is provided to illustrate the proposed adaptive control schemes.