2018/07/29 by Alessandro Alla, Alla, Alessandro, Maurizio Falcone +3 · 1 citation
Engineering · Environmental Science · #Advanced Control Systems Optimization #FOS: Mathematics #Numerical Analysis (math.NA) #Water resources management and optimization #Water-Energy-Food Nexus Studies
paper · pdf · doi:10.48550/arxiv.1807.11008
openalex publication_date 2018/07/29 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The classical Dynamic Programming (DP) approach to optimal control problems\nis based on the characterization of the value function as the unique viscosity\nsolution of a Hamilton-Jacobi-Bellman (HJB) equation. The DP scheme for the\nnumerical approximation of viscosity solutions of Bellman equations is\ntypically based on a time discretization which is projected on a fixed\nstate-space grid. The time discretization can be done by a one-step scheme for\nthe dynamics and the projection on the grid typically uses a local\ninterpolation. Clearly the use of a grid is a limitation with respect to\npossible applications in high-dimensional problems due to the curse of\ndimensionality. Here, we present a new approach for finite horizon optimal\ncontrol problems where the value function is computed using a DP algorithm on a\ntree structure algorithm (TSA) constructed by the time discrete dynamics. In\nthis way there is no need to build a fixed space triangulation and to project\non it. The tree will guarantee a perfect matching with the discrete dynamics\nand drop off the cost of the space interpolation allowing for the solution of\nvery high-dimensional problems. Numerical tests will show the effectiveness of\nthe proposed method.\n