Abstract
In this study, a distributed optimal control problem for n × n cooperative hyperbolic systems with infinite order operators and Dirichlet conditions are considered. The existence and uniqueness of the state of these systems are proved. The necessary and sufficient conditions for optimality of distributed control with constraints are found, and the set of equations and inequalities that defining the optimal control of these systems is also obtained. Finally, some examples for the control problem without constraints are given.
Highlights
The earliest theory of optimal control was introduced by Lions [1]
Based on the theories proposed by Lions [1] and Dubinskii [18] [19] [20], the distributed control problem with Dirichlet conditions for 2 × 2 non-cooperative hyperbolic systems involving infinite order operators was discussed in a previous study [17]; in this study, we extend this problem to n × n cooperative hyperbolic
The Sobolev spaces of infinite order operators, which are used in this study, have already been presented in Reference [17]
Summary
The earliest theory of optimal control was introduced by Lions [1]. Majority of the research in this field has focused on discussing the optimal control problem by using several operator types (such as elliptic, parabolic, or hyperbolic operators) [2] [3] [4]. The discussion was extended to systems involving different types of operators (such as infinite order [5]-[11] or infinite number of variables [12] [13] [14]). Based on the theories proposed by Lions [1] and Dubinskii [18] [19] [20], the distributed control problem with Dirichlet conditions for 2 × 2 non-cooperative hyperbolic systems involving infinite order operators was discussed in a previous study [17]; in this study, we extend this problem to n × n cooperative hyperbolic. The Sobolev spaces of infinite order operators, which are used in this study, have already been presented in Reference [17].
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