Numerical Solutions of Hamilton-Jacobi Inequalities by Constrained Gaussian Process Regression
Kenji Fujimoto, Hirofumi Beppu, Yuji Takaki · SICE Journal of Control Measurement and System Integration · 2018
This paper proposes numerical solutions of Hamilton-Jacobi inequalities based on constrained Gaussian process regression. While Gaussian process regression is a tool to estimate an unknown function from its input and output data conventionally, the proposed method applies it to solving a known partial differential inequality. This is done by generating sample data pairs of states and corresponding values of the unknown function satisfying the inequality. A formal algorithm to execute such a procedure to obtain the probability of a solution to the Hamilton-Jacobi inequality is proposed. In addition, a nonstationary covariance function is introduced to increase the accuracy of the solutions and to reduce the computational cost. Furthermore, its hyper parameters are optimized using an empirical gradient method.