Fixed-Time State Transfer via Pontryagin Extremals

Abstract

This paper concerns the problem of fixed-time transition between two states of nonlinear systems (i.e., the point-to-point steering problem). We propose a formulation applicable to a broad class of nonlinear systems, and show that it is theoretically complete in the sense that it admits a solution if and only if the target state is reachable from the initial state. When the target state is reachable, we prove that a solution can always be constructed by concatenation of two Pontryagin extremals, one generated by the original dynamics $f$ from the initial state and the other generated by the inverted dynamics $-f$ from the terminal state. This allows the problem to be formulated as a two-point boundary value problem (TPBVP) of extremals, where the solution existence to the formulated TPBVP is equivalent to that of the original problem. The theoretical developments are applied to curves with prescribed curvature bounds in $\mathbb{R}^3$, thereby extending the recent works on Dubins car to dimension three. We prove that to construct a curvature-bounded path in $\mathbb{R}^3$ with prescribed length and boundary conditions, it suffices to consider the trajectories that are concatenations of CSC, CCC, their subsegments, and H, where C denotes a circular arc with maximum curvature, S a straight line segment, and H a certain class of helicoidal arcs with constant curvature. Numerical demonstrations are conducted on a nonlinear dynamics example, and on curvature-bounded paths in $\mathbb{R}^2$ and $\mathbb{R}^3$.