Numerical approximation for a time optimal control problems governed by semi-linear heat equations

Abstract

In this paper, we study the optimal time for a time optimal control problem (P), governed by an internally controlled semi-linear heat equation. By projecting the original problem via the finite element method, we obtain another time optimal control problem (P h ) governed by a semi-linear system of ordinary differential equations. Here, h is the mesh sizes of the finite element spaces. The purpose of this study is to approach the optimal time for the problem (P) through the optimal time for the problem (P h ). We obtain error estimates between the optimal times in terms of h. MSC: 35K05; 49J20

Error Estimates for Space-Time Discretization of Parabolic Time-Optimal Control Problems with Bang-Bang Controls

“…However, in the particular case considered there the linearized Slater condition holds uniformly for the discrete problem, and this would also suffice for our argument; we also refer to [2,Section 5.6] for a generalization of [22] to fully discrete problems. For the case of a distributed control with the variational control discretization we can improve the result of [18] (see also [38] for a semilinear state equation) and obtain an optimal rate O(k + h 2 ). While the corresponding result from [37] with an explicit control discretization requires certain conditions (H1) and (H2), which so far could only be verified in very special situations, we assume a condition on the set of switchings which can be justified from practical observations; see Corollary 4.19.…”

“…Even though time-optimal control problems have been extensively studied, there are a few publications concerning the discretization of this problem class in the context of parabolic equations. In [30,22,23,37,38,18,31,21] the state equation is discretized in space only; see also the introduction of [5] for a detailed comparison. To the best of our knowledge, the only paper considering a full space-time discretization is [5] by the authors.…”

GMRES based numerical simulation and parallel implementation of multicomponent multiphase flow in porous media

“…Modelling multicomponent multiphase fluid (oil and gas) flow in porous media (in oil reservoirs) is relevant and, at the same time, a complex problem of hydrodynamic simulation. To solve such problems, various methods and schemes are used (Aceto et al, 2006;Ahmed, 2006;Borisov et al, 2013;Chen, 2006;Chen et al, 2006;Edwards et al, 2018;Javidi & Ahmad, 2013;Imankulov et al 2018;Zheng & Yin, 2014), some of which are iterative methods for solving linear systems (Lacroix et al, 2003;Mittal & Al-Kurdi, 2002;Vabishchevich & Vasilyeva, 2011;Wang et al, 2013).…”

Identification of the initial temperature from the given temperature data at the left end of a rod

“…Zheng, Guo and Ali [5] have investigated the stability of the optimization problem for a multidimensional heat equation. Zheng and Yin [6] have studied the optimal time for the time optimal control problem governed by an internally controlled semi-linear heat equation. In [7][8][9], the inverse problems with different controls and cost functions have been investigated.…”