Abstract
In this paper, we study the existence of solutions for 2l order (n × n) cooperative systems governed by Dirichlet and Neumann problems involving hyperbolic operators with an infinite number of variables and with variable coefficients. The necessary and sufficient conditions for optimality of the distributed control with constraints are obtained and the set of inequalities that defining the optimal control of these systems are also obtained.
Highlights
IntroductionThe optimality conditions for systems consisting of only one equation and for n × n systems governed by different types of partial differential equations defined on spaces of functions of infinitely many variables have been discussed for example in [1,2,3,4,5,6,7,8,9,10,11]
We study the existence of solutions for 2l order (n n) cooperative systems governed by Dirichlet and Neumann problems involving hyperbolic operators with an infinite number of variables and with variable coefficients
Making use of the theory of Lions [22] and Berezanskiĭ [23], we consider the optimal control problem of distributed type for 2l order (n × n) cooperative systems governed by Dirichlet and Neumann problems involving hyperbolic operators with an infinite number of variables and with variable coefficients
Summary
The optimality conditions for systems consisting of only one equation and for n × n systems governed by different types of partial differential equations defined on spaces of functions of infinitely many variables have been discussed for example in [1,2,3,4,5,6,7,8,9,10,11]. Optimal control problems for systems involving operators with an infinite number of variables for non-standard functional and time delay have been introduced in [12,13]. Some applications of optimal control problem for systems involving Schrodinger operators are introduced for example in [17,18,19,20,21]. Making use of the theory of Lions [22] and Berezanskiĭ [23], we consider the optimal control problem of distributed type for 2l order (n × n) cooperative systems governed by Dirichlet and Neumann problems involving hyperbolic operators with an infinite number of variables and with variable coefficients.
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