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 71630303X&_mathId=si1.gif&_user=111111111&_pii=S037704271630303X&_rdoc=1&_issn=03770427&md5=9b7bc9d91e7526af94a3b933860109a2" 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 71630303X&_mathId=si2.gif&_user=111111111&_pii=S037704271630303X&_rdoc=1&_issn=03770427&md5=9f4ac6782a8eacfcce49db885c001f2b" title="Click to view the MathML source">hH$hH$”, including a natural one. We prove that the approximate value function 71630303X&_mathId=si3.gif&_user=111111111&_pii=S037704271630303X&_rdoc=1&_issn=03770427&md5=2efc991ca35643808565cb7f7ffab19b" title="Click to view the MathML source">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 71630303X&_mathId=si4.gif&_user=111111111&_pii=S037704271630303X&_rdoc=1&_issn=03770427&md5=237429f0de9ebf37c4f8121f658a27dc" title="Click to view the MathML source">h$h$ goes to zero; the rate of convergence is proved to be in 71630303X&_mathId=si5.gif&_user=111111111&_pii=S037704271630303X&_rdoc=1&_issn=03770427&md5=139815241af826ab085d249d30d9574d" title="Click to view the MathML source">h^σ$h^{\sigma }$, where 71630303X&_mathId=si6.gif&_user=111111111&_pii=S037704271630303X&_rdoc=1&_issn=03770427&md5=237adc6a2ad17255edbcf9c022219cb5" title="Click to view the MathML source">0<σ<1/2$0<\sigma <1/2$.