# A gradient flow equation for optimal control problems with end-point cost

arXiv: https://arxiv.org/abs/2107.00556 · HTML: https://mlsanity.com/paper/2107.00556/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jul 2021.

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## Abstract (arXiv)

In this paper we consider a control system of the form dot x = F(x)u, linear in the control variable u. Given a fixed starting point, we study a finite-horizon optimal control problem, where we want to minimize a weighted sum of an end-point cost and the squared 2-norm of the control. This functional induces a gradient flow on the Hilbert space of admissible controls, and we prove a convergence result by means of the Lojasiewicz-Simon inequality. Finally, we show that, if we let the weight of the end-point cost tend to infinity, the resulting family of functionals is Gamma-convergent, and it turns out that the limiting problem consists in joining the starting point and a minimizer of the end-point cost with a horizontal length-minimizer path.