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SUMMARY:Maurizio Falcone (Sapienza Università di Roma\, Italy)
DTSTART:20210928T140000Z
DTEND:20210928T150000Z
DTSTAMP:20260423T021030Z
UID:SNAP/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/SNAP/9/">Dyn
 amic Programming on a tree for the approximation of finite horizon optimal
  control problems</a>\nby Maurizio Falcone (Sapienza Università di Roma\,
  Italy) as part of Seminars on Numerics and Applications\n\n\nAbstract\nTh
 e classical Dynamic Programming (DP) approach to optimal control problems 
 is based on the characterization of the value function as the unique visco
 sity solution of a Hamilton-Jacobi-Bellman (HJB) equation [2]. The DP sche
 me for the numerical approximation of viscosity solutions of those equatio
 ns is typically based on a time discretization coupled with a projection o
 n a fixed space triangulation of the numerical domain [3]. The time discre
 tization is obtained by a one-step scheme for the dynamics and the project
 ion is based on a polynomial interpolation. This approach allows to get a 
 synthesis of optimal controls in feedback form and is very powerful for no
 nlinear optimal control problems in low dimension although general converg
 ence results are valid in any dimension. The computational cost is severe 
 in high dimension and several methods have been proposed to mitigate the "
 curse of dimensionality" of DP schemes\, e.g. static and dynamic domain de
 composition\, fast-marching and fast-sweeping methods\, discrete represent
 ation formulas (when available)\, see [3] and the references therein.<br /
 >\nWe present a new approach for finite horizon optimal control problems [
 1\, 4] where we compute the value function on a tree structure generated b
 y the time discrete dynamics avoiding the construction of a space grid/tri
 angulation to solve the HJB equation. This drops the computational cost of
  space interpolation although the tree mantains a perfect matching with th
 e discrete dynamics. We prove first order convergence to the value functio
 n for a first order discretization of the dynamics. We will also discuss e
 xtensions to high-order schemes and to problems with state constraints als
 o showing some numerical tests.<br />\n<i>Works in collaboration with A. A
 lla (PUC\, Rio de Janeiro) and L. Saluzzi (Sapienza\, Roma).</i><br />\n<b
 >References</b><br />\n[1] A. Alla\, M. Falcone and L. Saluzzi. An efficie
 nt DP algorithm on a tree-structure for finite horizon optimal control pro
 blems\, SIAM Journal on Scientific Computing\, (41) 4\, 2019\, A2384-A2406
 <br />\n[2] M. Bardi\, I. Capuzzo-Dolcetta\, Optimal Control and Viscosity
  Solutions of Hamilton-Jacobi-Bellman Equations\, Birkhäuser\, Basel\, 19
 97.<br />\n[3] M. Falcone\, R. Ferretti\, Semi-Lagrangian Approximation Sc
 hemes for Linear and Hamilton-Jacobi Equations\, Society for Industrial an
 d Applied Mathematics\, Philadelphia\, 2013.<br />\n[4] L. Saluzzi\, A. Al
 la and M. Falcone. Error estimates for a tree structure algorithm for dyna
 mic programming equations\, submitted\, 2018 https://arxiv.org/abs/1812.11
 194\n
LOCATION:https://researchseminars.org/talk/SNAP/9/
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