Vanishing impulse cost in the quasi-variational inequality for ergodic impulse control

Article type: Research Article

Authors: Perthame, B.

Affiliations: Centre de Mathématiques Appliquées, Ecole Normale Supérieure, 75230 Paris Cedex 05, France

Abstract: We consider the impulse control of a reflected diffusion. It has been proved that the long run average cost for this problem solves the ergodic Quasi-Variational inequality (Q.V.I.) \begin{equation}\left\{\begin{array}{l}\int_{\Omega}\triangledown u_{k}\triangledown (v-u_{k})dx\geq {\int_{\Omega}}(f-\lambda _{k})\cdot(v-u_{k})dx,\\\forall_{v}\in H^{1}(\Omega),\,v\leq k+Mu_{k}\,\mathrm{a.e.;}\quad u_{k}\inH^{1}(\Omega),\,u_{k}\leq k+Mu_{k}\,\mathrm{a.e.,}\quad \int_{\Omega}u_{k}dx=0,\end{array}\right.\end{equation} Mu(x)=infess{c0(ξ)+u(x+ξ);ξ≥0,x+ξ∈Ω}. We prove uniform bounds in H1∩L∞ on uk and we show that, extracting a subsequence if necessary, (uk,λk) converge as k→0 to a solution of (1)0. We also study the uniqueness of (u0,λ0), and we prove that it is false in general although the complete sequence λk converges to the maximal λ0 such that (1)0 admits a solution.

DOI: 10.3233/ASY-1988-1103

Journal: Asymptotic Analysis, vol. 1, no. 1, pp. 13-21, 1988