Time domain inverse problems in nonlinear systems using collocation & radial basis functions

In this study, we consider ill-posed time-domain inverse problems for dynamical systems with various boundary conditions and unknown controllers. Dynamical systems characterized by a system of second-order nonlinear ordinary differential equations (ODEs) are recast into a system of nonlinear first order ODEs in mixed variables. Radial Basis Functions (RBFs) are assumed as trial functions for the mixed variables in the time domain. A simple collocation method is developed in the time-domain, with Legendre-Gauss-Lobatto nodes as RBF source points as well as collocation points. The duffing optimal control problem with various prescribed initial and final conditions, as well as the orbital transfer Lambert's problem are solved by the proposed RBF collocation method as examples. It is shown that this method is very simple, efficient and very accurate in obtaining the solutions, with an arbitrary solution as the initial guess. Since methods such as the Shooting Method and the Pseudo-spectral Method can be unstable and require an accurate initial guess, the proposed method is advantageous and has promising applications in optimal control and celestial mechanics. The extension of the present study to other optimal control problems, and other orbital transfer problems with perturbations, will be pursued in our future studies.