  We consider the problem of nonlinear system identification when prior
knowledge is available on the region of attraction (ROA) of an equilibrium
point. We propose an identification method in the form of an optimization
problem, minimizing the fitting error and guaranteeing the desired stability
property. The problem is approached by joint identification the dynamics and a
Lyapunov function verifying the stability property. In this setting, the
hypothesis set is a reproducing kernel Hilbert space, and with respect to each
point of the given subset of the ROA, the Lie derivative inequality of the
Lyapunov function imposes a constraint. The problem is a non-convex
infinite-dimensional optimization with infinite number of constraints. To
obtain a tractable formulation, only a suitably designed finite subset of the
constraints are considered. The resulting problem admits a solution in form of
a linear combination of the sections of the kernel and its derivatives. An
equivalent optimization problem with a quadratic cost function subject to
linear and bilinear constraints is derived. A suitable change of variable gives
a convex reformulation of the problem. To reduce the number of hyperparameters,
the optimization problem is adapted to the case of diagonal kernels. The method
is demonstrate by means of an example.
