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Guaranteeing stability of a designed control system is a challenging problemin data-driven control approaches such as Gaussian process (GP)-based control.The reason is that the inequality conditions, which are used in ensuring thestability, should be evaluated for all states in the state space, meaning thatan infinite number of inequalities must be evaluated. Previous researchintroduced the idea of using a finite number of sampled states with the boundsof the stability inequalities near the samples. However, high-order bounds withrespect to the distance between the samples are essential to decrease thenumber of sampling. From the standpoint of control theory, the requirement isnot 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 existingapproaches use first-order bounds. The proposed method obtaining the bounds arewidely applicable to various functions such as polynomials, Gaussian processes,Gaussian mixture models, and sum/product functions of them. Unifying thederived bounds and nonlinear optimal control theory yields a stabilizing(sub-)optimal controller for GP dynamics. A numerical simulation demonstratesthe stability performance of the proposed approach.

Second-order Bounds of Gaussian Kernel-based Functions and its Application to Nonlinear Optimal Control with Stability