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All issues Volume 28 (2022) ESAIM: COCV, 28 (2022) 69 References
Open Access
Issue
ESAIM: COCV
Volume 28, 2022
Article Number 69
Number of page(s) 25
DOI https://doi.org/10.1051/cocv/2022067
Published online 09 November 2022
  1. M. Akian, S. Gaubert and A. Lakhoua, The max-plus finite element method for solving deterministic optimal control problems: basic properties and convergence analysis. SIAM J. Control Optim. 47 (2008) 817–848. [CrossRef] [MathSciNet] [Google Scholar]
  2. A. Alla, M. Falcone and D. Kalise, An efficient policy iteration algorithm for dynamic programming equations. SIAM J. Sci. Comput. 37 (2015) 181–200. [Google Scholar]
  3. A. Alla, M. Falcone and L. Saluzzi, An efficient DP algorithm on a tree-structure for finite horizon optimal control problems. SIAM J. Sci. Comput. 41 (2019) A2384–A2406. [CrossRef] [Google Scholar]
  4. A. Alla, M. Falcone and L. Saluzzi, High-order approximation of the finite horizon control problem via a tree structure algorithm. IFAC-PapersOnLine 52 (2019) 19–24. [CrossRef] [MathSciNet] [Google Scholar]
  5. A. Alla, M. Falcone and L. Saluzzi, A tree structure algorithm for optimal control problems with state constraints. Rend. Matemat. Sue Applic. 41 (2020) 193–221. [Google Scholar]
  6. A. Alla, M. Falcone and S. Volkwein, Error analysis for POD approximations of infinite horizon problems via the dynamic programming approach. SIAM J. Control Optim. 55 (2017) 3091–3115. [CrossRef] [MathSciNet] [Google Scholar]
  7. A. Alla and L. Saluzzi, A HJB-POD approach for the control of nonlinear PDEs on a tree structure. Appl. Numer. Math. 155 (2020) 192–207. [CrossRef] [MathSciNet] [Google Scholar]
  8. A. Alla and L. Saluzzi, Feedback reconstruction techniques for optimal control problems on a tree structure, Preprint arXiv:2210.02375 (2022). [Google Scholar]
  9. M. Assellaou, O. Bokanowski and A. Desilles, H. Zidani, Value function and optimal trajectories for a maximum running cost control problem with state constraints. Application to an abort landing problem. ESAIM: Math. Model. Numer. Anal 52 (2018) 305–335. [CrossRef] [EDP Sciences] [MathSciNet] [Google Scholar]
  10. A. Bachouch, C. Huré, N. Langrené and H. Pham, Deep neural networks algorithms for stochastic control problems on finit hotizone: numerical applications. Methodol. Comput. Appl. Probab. 24 (2022) 143–178. [CrossRef] [MathSciNet] [Google Scholar]
  11. M. Bardi and I. Capuzzo-Dolcetta, Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations. Birkhäuser, Basel (1997). [Google Scholar]
  12. R. Bellman, Dynamic Programming. Princeton University Press, Princeton, NJ (1957). [Google Scholar]
  13. P. Benner, Z. Bujanovič, P. Kürschner and J. Saak, A numerical comparison of different solvers for large-scale, continuous-time algebraic Riccati equations and LQR problems. SIAM J. Sci. Comput. 42 (2020) A957–A996. [CrossRef] [Google Scholar]
  14. D. Bini, B. Iannazzo and B. Meini, Numerical Solution of Algebraic Riccati Equations, Fundam. Algorithms, SIAM, Philadelphia (2012). [Google Scholar]
  15. S. Cacace, E. Cristiani, M. Falcone and A. Picarelli, A patchy dynamic programming scheme for a class of Hamilton-Jacobi- Bellman equations. SIAM J. Sci. Comput. 34 (2012) A2625–A2649. [CrossRef] [Google Scholar]
  16. P. Cannarsa and C. Sinestrari, Semiconcave functions, Hamilton-Jacobi equations and optimal control problems. Birkhäuser Boston (2004). [CrossRef] [Google Scholar]
  17. I. Capuzzo Dolcetta, On a discrete approximation of the Hamilton-Jacobi equation of dynamic programming. Appl. Math. Optim. 10 (1983) 367–377. [CrossRef] [MathSciNet] [Google Scholar]
  18. I. Capuzzo Dolcetta and H. Ishii, Approximate solutions of the Bellman equation of deterministic control theory. Appl. Math. Optim. 11 (1984) 161–181. [CrossRef] [MathSciNet] [Google Scholar]
  19. J. Darbon and S. Osher, Splitting enables overcoming the curse of dimensionality. Splitting methods in communication, imaging, science, and engineering, Sci. Comput., Springer, Cham (2016) 427–432. [Google Scholar]
  20. J. Darbon and S. Osher, Algorithms for overcoming the curse of dimensionality for certain Hamilton-Jacobi equations arising in control theory and elsewhere. Res. Math. Sci. 3 (2016) 19–26. [CrossRef] [Google Scholar]
  21. S. Dolgov, D. Kalise and K. Kunisch, Tensor decompositions for high-dimensional Hamilton-Jacobi-Bellman equations. SIAM J. Sci. Comput. 43 (2021) A1625–A1650. [CrossRef] [Google Scholar]
  22. S. Dolgov, D. Kalise and L. Saluzzi, Data-driven Tensor Train Gradient Cross Approximation for Hamilton-Jacobi-Bellman Equations. Preprint arXiv:2205.05109 (2022). [Google Scholar]
  23. M. Falcone, A numerical approach to the infinite horizon problem of deterministic control theory, Appl. Math. Optim. 15 (1987) 1–13. [CrossRef] [MathSciNet] [Google Scholar]
  24. M. Falcone and R. Ferretti, Discrete time high-order schemes for viscosity solutions of Hamilton-Jacobi-Bellman equations. Numer. Math. 67 (1994) 315–344. [CrossRef] [MathSciNet] [Google Scholar]
  25. M. Falcone and R. Ferretti, Semi-Lagrangian Approximation Schemes for Linear and Hamilton-Jacobi equations. SIAM (2013). [Google Scholar]
  26. M. Falcone and T. Giorgi, An approximation scheme for evolutive Hamilton-Jacobi equations, in W.M. McEneaney, G. Yin and Q. Zhang (eds.), “Stochastic Analysis, Control, Optimization and Applications: A Volume in Honor of W.H. Fleming”. Birkhaäuser (1999) 289–303. [Google Scholar]
  27. M. Falcone, P. Lanucara and A. Seghini, A splitting algorithm for Hamilton-Jacobi-Bellman equations. Appl. Numer. Math. 15 (1994) 207–218. [CrossRef] [MathSciNet] [Google Scholar]
  28. A. Festa, Domain decomposition based parallel Howard’s algorithm. Math. Comput. Simul. 147 (2018) 121–139. [CrossRef] [Google Scholar]
  29. W.H. Fleming and H.M. Soner, Controlled Markov processes and viscosity solutions. Springer–Verlag, New York (1993). [Google Scholar]
  30. J. Garcke and A. Kröner, Suboptimal feedback control of PDEs by solving HJB equations on adaptive sparse grids. J. Sci. Comput. 70 (2017) 1–28. [CrossRef] [MathSciNet] [Google Scholar]
  31. L. Grüne and J. Pannek, Nonlinear Model Predictive Control. Springer (2011). [CrossRef] [Google Scholar]
  32. L. Grüne, Dynamic programming, optimal control and model predictive control. Handbook of Model Predictive Control. Control Eng., Birkhäuser/Springer, Cham (2019), pp. 29–52. [Google Scholar]
  33. J. Han, A. Jentzen and E. Weinan, Solving high-dimensional partial differential equations using deep learning. Proc. Natl. Acad. Sci. USA 115 (2018) 8505–8510. [CrossRef] [MathSciNet] [PubMed] [Google Scholar]
  34. M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, vol. 23 of Optimization with PDE Constraints. Mathematical Modelling: Theory and Applications. Springer–Verlag (2009). [Google Scholar]
  35. C. Huré, H. Pham, A. Bachouch and N. Langrené, Deep neural networks algorithms for stochastic control problems on finite horizon: Convergence analysis. SINUM 59 (2021) 525–557. [CrossRef] [Google Scholar]
  36. D. Kalise and K. Kunisch, Polynomial approximation of high-dimensional Hamilton-Jacobi-Bellman equations and applications to feedback control of semilinear parabolic PDEs. SIAM J. Sci. Comput. 40 (2018) A629–A652. [CrossRef] [Google Scholar]
  37. K. Kunisch, S. Volkwein and L. Xie, HJB-POD based feedback design for the optimal control of evolution problems. SIAM J. Appl. Dyn. Syst. 4 (2004) 701–722. [CrossRef] [MathSciNet] [Google Scholar]
  38. R.J. Leveque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems. SIAM Book (2007). [CrossRef] [Google Scholar]
  39. W.M. McEneaney, Convergence rate for a curse-of-dimensionality-free method for Hamilton-Jacobi-Bellman PDEs represented as maxima of quadratic forms. SIAM J. Control Optim. 48 (2009) 2651–2685. [CrossRef] [MathSciNet] [Google Scholar]
  40. W.M. McEneaney, A curse-of-dimensionality-free numerical method for solution of certain HJB PDEs. SIAM J. Control Optim. 46 (2007) 1239–1276. [CrossRef] [MathSciNet] [Google Scholar]
  41. C. Navasca and A.J. Krener, Patchy solutions of Hamilton-Jacobi-Bellman partial differential equations, in A. Chiuso et al. (eds.), vol. 364 of Modeling, Estimation and Control, Lecture Notes in Control and Information Sciences (2007), pp. 251–270. [Google Scholar]
  42. S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces. Springer (2003). [CrossRef] [Google Scholar]
  43. L. Saluzzi, A Tree-Structure Algorithm for Optimal Control Problems via Dynamic Programming, PhD thesis, Gran Sasso Science Institute (2020) https://iris.gssi.it/handle/20.500.12571/10021. [Google Scholar]
  44. J.A. Sethian, Level set methods and fast marching methods. Cambridge University Press (1999). [Google Scholar]
  45. V. Simoncini, Analysis of the rational Krylov subspace projection method for large-scale algebraic Riccati equations. SIAM J. Matrix Anal. Appl. 37 (2016) 1655–1674. [CrossRef] [MathSciNet] [Google Scholar]
  46. V. Simoncini, D.B. Szyld and M. Monsalve, On two numerical methods for the solution of large-scale algebraic Riccati equations. IMA J. Numer. Anal. 34 (2014) 904–920. [CrossRef] [MathSciNet] [Google Scholar]
  47. S. Volkwein, Model Reduction using Proper Orthogonal Decomposition, Lecture Notes, University of Konstanz (2013). [Google Scholar]
  48. I. Yegorov, P.M. Dower and L. Grüne, A characteristics based curse-of-dimensionality-free approach for approximating control Lyapunov functions and feedback stabilization. Proceedings of the 23rd International Symposium on Mathematical Theory of Networks and Systems Hong Kong University of Science and Technology, Hong Kong, July 16–20 (2018) 342–349. [Google Scholar]

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