Deterministic impulse control problems: Two discrete approximations of the quasi-variational inequality

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• 作者：Naï ; ma El Farouq ^{Naima.ElFarouq@univ-bpclermont.fr" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Infinite horizon impulse control ; Hamilton&ndash ; Jacobi quasi-variational inequality ; Viscosity solution ; Discrete approximations ; Convergence ; Convergence rate
• 刊名：Journal of Computational and Applied Mathematics
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：309
• 期：Complete
• 页码：200-218
• 全文大小：480 K

文摘

In this paper, we study a deterministic infinite horizon, mixed continuous and impulse control problem in 26af94a3b933860109a2" title="Click to view the MathML source">Rⁿ${\mathbb{R}}^n$, with general impulses, and cost of impulses. We assume that the cost of impulses is a positive function. We prove that the value function of the control problem is the unique viscosity solution of the related first order Hamilton–Jacobi quasi-variational inequality.¹ We then propose time discretization schemes of this QVI, where we consider two approximations of the “Hamiltonian hH$hH$”, including a natural one. We prove that the approximate value function u^h$u^h$ exists, that it is the unique solution of the approximate QVI and that it forms a uniformly bounded and uniformly equicontinuous family. We also prove that the approximate value function converges locally uniformly, towards the value function of the control problem, when the discretization step a27dc" title="Click to view the MathML source">h$h$ goes to zero; the rate of convergence is proved to be in 26ab085d249d30d9574d" title="Click to view the MathML source">h^σ$h^{\sigma }$, where 0<σ<1/2$0<\sigma <1/2$.