Ordinary Differential Equations (ODEs) are a general form of differential equations. This mathematical format is often used to represent the dynamic behavior of physical systems such as control systems and chemical processes. Linear ODEs can usually be solved analytically while nonlinear ODEs may need numerical methods to obtain approximate solutions. There are also various developments for validated simulation of nonlinear ODEs such as explicit and implicit guaranteed Runge-Kutta integration schemes. The implicit ones are mainly based on zonotopic computations using affine arithmetics. It allows to compute the reachability of a nonlinear ODE with a zonotopic set as its initial value. In this paper, we propose a new validated approach to solve nonlinear ODEs with a polytopic set as the initial value using an indirectly implemented polytopic set computation technique.
Alexandre dit Sandretto, J., & Wan, J. (2018). Reachability analysis of nonlinear odes using polytopic based validated runge-kutta. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 11123 LNCS, pp. 1–14). Springer Verlag. https://doi.org/10.1007/978-3-030-00250-3_1