Optimal Control of Nonclassical Diffusion Equations with Memory
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Optimal Control of Nonclassical Diffusion Equations with Memory Nguyen Duong Toan1
Received: 12 February 2019 / Accepted: 16 January 2020 © Springer Nature B.V. 2020
Abstract In this paper we consider an optimal control problem of nonclassical diffusion equations with memory. We investigate the existence and uniqueness of optimal solutions. The necessary and sufficient optimality conditions are also studied. The main novelty of our result is to establish the optimality conditions for the parabolic optimal control problem with memory. Keywords Nonclassical diffusion equations · Optimal control · Necessary optimality conditions · Sufficient optimality conditions Mathematics Subject Classification 35B41 · 35Q30 · 37L30 · 35D05
1 Introduction Optimal control plays an essential role in the modern control theory and has a wider application in modern engineering. In recent years, the optimal control problem for the parabolic equations have been extensively studied, see e.g. [9, 13–15, 26] and references therein. In particular, the second-order optimal conditions for parabolic control problems have also been investigated in a sequence of articles [4, 13, 15, 16, 25, 32, 38, 39, 41]. The focus of this paper is on the optimal control problems with memory, which have been considered over the last few years. In 2013, Cannarsa et al. [11] investigated the existence and the regularity of solutions for Bolza optimal control problems in infinite dimension governed by a class of semi-linear evolution equations. In 2014, Confortola and Mastrogiacomo only pointed out the standard synthesis of the optimal control for stochastic heat equation with memory (see [17]). In [21], Dahl et al. derived necessary and sufficient maximum principles for the stochastic optimal control problem with noisy memory. Hwang [23] showed the existence of optimal control and derived a necessary condition for the optimal
B N.D. Toan
[email protected]; [email protected]
1
Department of Mathematics, Haiphong University, 171 Phan Dang Luu, Kien An, Haiphong, Vietnam
N.D. Toan
control problems by the von Kármán system with a long memory. Besides, in a framework for memory, we refer to [12, 13, 30]. To the best of our knowledge, there are only a few works on the optimal control problems of the parabolic equations with memory, focusing on the necessary/sufficient optimality conditions. The purposes of this article are to prove the existence of optimal solutions and to establish necessary/sufficient optimality conditions to optimal control problems of the nonclassical diffusion equation with memory. The main difficulty of this work is the appearance ´∞ of the convolution term (or memory term) − 0 κ(s)y(t − s)ds as in (1.1). Moreover, we consider the problem under a general assumption on the memory kernel κ (as in [5, 19]). Notice that the convolution term takes into account the influence of the past history of u on its future evolution, providing a more accurate description of the diffusive process in certain materials, such as high viscosity liqui
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