  It is nontrivial to achieve exponential stability even for time-invariant
nonlinear systems with matched uncertainties and persistent excitation (PE)
condition. In this paper, without the need for PE condition, we address the
problem of global exponential stabilization of strict-feedback systems with
mismatched uncertainties and unknown yet time-varying control gains. The
resultant control, embedded with time-varying feedback gains, is capable of
ensuring global exponential stability of parametric-strict-feedback systems in
the absence of persistence of excitation. By using the enhanced Nussbaum
function, the previous results are extended to more general nonlinear systems
where the sign and magnitude of the time-varying control gain are unknown. In
particular, the argument of the Nussbaum function is guaranteed to be always
positive with the aid of nonlinear damping design, which is critical to perform
a straightforward technical analysis of the boundedness of the Nussbaum
function. Finally, the global exponential stability of parameter-varying
strict-feedback systems, the boundedness of the control input and the update
rate, and the asymptotic constancy of the parameter estimate are established.
Numerical simulations are carried out to verify the effectiveness and benefits
of the proposed methods.
