A tree structure algorithm for optimal control problems with state constraints

TL;DR

This paper introduces a tree structure algorithm for solving optimal control problems with state constraints, emphasizing convergence, efficiency, and reduced computational complexity without relying on a fixed space grid.

Contribution

The paper develops a novel tree-based dynamic programming method that handles state constraints effectively and reduces computational costs through pruning and no need for a predefined space grid.

Findings

The algorithm converges for a discrete time approximation of the value function.

It reduces CPU time and memory compared to classical grid-based methods.

The method effectively incorporates state constraints by pruning branches in the tree.

Abstract

We present a tree structure algorithm for optimal control problems with state constraints. We prove a convergence result for a discrete time approximation of the value function based on a novel formulation of the constrained problem. Then the Dynamic Programming approach is developed by a discretization in time leading to a tree structure in space derived by the controlled dynamics, in this construction the state constraints are taken into account to cut several branches of the tree. Moreover, an additional pruning allows for the reduction of the tree complexity as for the case without state constraints. Since the method does not use an a priori space grid, no interpolation is needed for the reconstruction of the value function and the accuracy essentially relies on the time step $h$. These features permit a reduction in CPU time and in memory allocations. The synthesis of optimal feedback controls is based on the values on the tree and an interpolation on the values obtained on the tree will be necessary if a different discretization in the control space is adopted, e.g. to improve the accuracy of the method in the reconstruction of the optimal trajectories. Several examples show how this algorithm can be applied to problems in low dimension and compare it to a classical DP method on a grid.