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Conference Papers Year : 2023

An Adaptive Multi-Level Max-Plus Method for Deterministic Optimal Control Problems

Marianne Akian
  • Function : Author
Stéphane Gaubert
Shanqing Liu
  • Function : Author

Abstract

We introduce a new numerical method to approximate the solution of a finite horizon deterministic optimal control problem. We exploit two Hamilton-Jacobi-Bellman PDE, arising by considering the dynamics in forward and backward time. This allows us to compute a neighborhood of the set of optimal trajectories, in order to reduce the search space. The solutions of both PDE are successively approximated by max-plus linear combinations of appropriate basis functions, using a hierarchy of finer and finer grids. We show that the sequence of approximate value functions obtained in this way does converge to the viscosity solution of the HJB equation in a neighborhood of optimal trajectories. Then, under certain regularity assumptions, we show that the number of arithmetic operations needed to compute an approximate optimal solution of a $d$-dimensional problem, up to a precision $\varepsilon$, is bounded by $O(C^d (1/\varepsilon) )$, for some constant $C>1$, whereas ordinary grid-based methods have a complexity in$O(1/\varepsilon^{ad}$) for some constant $a>0$.

Dates and versions

hal-04406978 , version 1 (19-01-2024)

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Marianne Akian, Stéphane Gaubert, Shanqing Liu. An Adaptive Multi-Level Max-Plus Method for Deterministic Optimal Control Problems. IFAC 2023 - The 22nd World Congress of the International Federation of Automatic Control, Jul 2023, Yokohama, Japan. ⟨hal-04406978⟩
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