# Finite-Horizon Optimal Control and Stabilization of Time-Scalable Systems

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### Alex Fax and Richard Murray

2000 Conference on Decision and Control

In this paper, we consider the optimal control of time-scalable systems. The time-scaling property is shown to convert the PDE associated with the Hamilton-Jacobi-Bellman (HJB) equation to a purely spatial PDE. Solution of this PDE yields the value function at a fixed time, and that solutio n can be scaled to find the value function at any point in time. Furthermore, in certain

cases the

unscaled control law stabilizes the system, and the unscaled value function acts

as a

Lyapunov function for that system. For the example of the nonholonomic integrator, this PDE is solved, and the resulting optimal trajectories coincide with the known solution to that problem.

- Conference

Submission: http://www.cds.caltech.edu/~murray/preprints/fm00a-cdc.pdf

- Project(s): Template:HTDB funding::AFOSR