  We propose a technique for the design and analysis of adaptation algorithms
in dynamical systems. The technique applies both to systems with conventional
Lyapunov-stable target dynamics and to ones of which the desired dynamics
around the target set is nonequilibrium and in general unstable in the Lyapunov
sense. Mathematical models of uncertainties are allowed to be nonlinearly
parametrized, smooth, and monotonic functions of linear functionals of the
parameters. We illustrate with applications how the proposed method leads to
control algorithms. In particular we show that the mere existence of nonlinear
operator gains for the desired dynamics guarantees that system solutions are
bounded, reach a neighborhood of the target set, and mismatches between the
modeled uncertainties and uncertainty compensator vanish with time. The
proposed class of algorithms can also serve as parameter identification
procedures. In particular, standard persistent excitation suffices to ensure
exponential convergence of the estimated to the actual values of the
parameters. When a weak, nonlinear version of the persistent excitation
condition is satisfied, convergence is asymptotic. The approach extends to a
broader class of parameterizations where the monotonicity restriction holds
only locally. In this case excitation with oscillations of sufficiently high
frequency ensure convergence.
