  Guaranteeing stability of a designed control system is a challenging problem
in data-driven control approaches such as Gaussian process (GP)-based control.
The reason is that the inequality conditions, which are used in ensuring the
stability, should be evaluated for all states in the state space, meaning that
an infinite number of inequalities must be evaluated. Previous research
introduced the idea of using a finite number of sampled states with the bounds
of the stability inequalities near the samples. However, high-order bounds with
respect to the distance between the samples are essential to decrease the
number of sampling. From the standpoint of control theory, the requirement is
not only evaluating stability but also simultaneously designing a controller.
This paper overcomes theses two issues to stabilize GP-based dynamical systems.
Second-order bounds of the stability inequalities are derived whereas existing
approaches use first-order bounds. The proposed method obtaining the bounds are
widely applicable to various functions such as polynomials, Gaussian processes,
Gaussian mixture models, and sum/product functions of them. Unifying the
derived bounds and nonlinear optimal control theory yields a stabilizing
(sub-)optimal controller for GP dynamics. A numerical simulation demonstrates
the stability performance of the proposed approach.
