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**Approximate solutions of the Bellman equation of deterministic control theory.**
*(English)*
Zbl 0553.49024

This paper considers an infinite horizon discounted optimal control problem and its time discretized approximation. The rate of convergence of approximate solutions to the exact solution is of order \(\gamma\) /2, assuming the exact solution is Hölder continuous with exponent \(0<\gamma \leq 1\). The notion of viscosity solution for the optimal control problem is used in making these estimates. The convergence rate of the approximate solutions is shown to be of order 1 provided these approximation solutions satisfy a semi-concavity assumption. The optimal controls of the approximate problem converge to an optimal relaxed control for the original problem.

Reviewer: S.Lenhart

### MSC:

49M25 | Discrete approximations in optimal control |

35D99 | Generalized solutions to partial differential equations |

49L20 | Dynamic programming in optimal control and differential games |

49J20 | Existence theories for optimal control problems involving partial differential equations |

### Keywords:

infinite horizon discounted optimal control problem; time discretized approximation; viscosity solution; convergence rate
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\textit{I. Capuzzo Dolcetta} and \textit{H. Ishii}, Appl. Math. Optim. 11, 161--181 (1984; Zbl 0553.49024)

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### References:

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