On the computation of Nash and Pareto equilibria for some bi-objective control problems for the wave equation
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On the computation of Nash and Pareto equilibria for some bi-objective control problems for the wave equation ´ Pitagoras Pinheiro de Carvalho1 Juan Bautista L´ımaco Ferrel3
2· ´ · Enrique Fernandez-Cara
Received: 24 February 2019 / Accepted: 6 August 2020 / © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper deals with the numerical implementation of a systematic method for solving bi-objective optimal control problems for wave equations. More precisely, we look for Nash and Pareto equilibria which respectively correspond to appropriate noncooperative and cooperative strategies in multi-objective optimal control. The numerical methods described here consist of a combination of the following: finite element techniques for space approximation; finite difference schemes for time discretization; gradient algorithms for the solution of the discrete control problems. The efficiency of the computational methods is illustrated by the results of some numerical experiments. Keywords Wave equation · Finite elements and finite differences · Bi-objective optimal control · Nash and Pareto equilibria Mathematics Subject Classification (2010) 34K35 · 35Q93 · 49J20 · 90C29
Communicated by: Enrique Zuazua Pit´agoras Pinheiro de Carvalho
[email protected] Enrique Fern´andez-Cara [email protected] Juan Bautista L´ımaco Ferrel [email protected] 1
Coord. Matem´atica - Universidade Estadual do Piau´ı, Teresina, PI, Brazil
2
Dpto. E.D.A.N., Universidad de Sevilla, Aptdo. 1160, 41080, Sevilla, Spain
3
Inst. Matem´atica e Estat´ıstica - Universidade Federal Fluminense, RJ, Brazil
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Adv Comput Math
(2020) 46:73
1 Introduction In control theory, an interesting situation arises when several objectives are considered. It can also be expectable to have more than one control acting on the equation. In these cases, we are led to multi-objective control problems. In contrast with the mono-objective case, various strategies for the choice of good controls can be established, depending on the characteristics of the problem. These strategies lead to what we call equilibria that can be cooperative (when the controls collaborate to achieve the goals) and noncooperative (in the opposite case). The related concepts and arguments have their origins mainly in game theory and economics. Thus, for an extremal problem with p cost functionals J1 , . . . , Jp to minimize the noncooperative optimization strategy proposed by Nash in [11] reduces the problem to the search of a set of p players or controls vi such that, at (v1 , . . . , vp ), Ji is optimized with respect to the i-th variable. Obviously, if the Ji are regular enough and no constraint is imposed, the vi can be characterized in terms of the derivatives of the functionals as follows: ∂Ji (v1 , . . . , vp ) = 0, i = 1, . . . , p. ∂vi On the other hand, the cooperative optimization strategy is proposed by Pareto in [12]. It is said that (v1 , . . . , vp ) is optimal in the sense of Pareto if there is no other choice of the vi that i
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