Optimality Conditions for Infinite Order Hyperbolic Differential System
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DOI: 10.1007/s10450‐006‐0002‐1
OPTIMALITY CONDITIONS FOR INFINITE ORDER HYPERBOLIC DIFFERENTIAL SYSTEM H. A. EL SAIFY Abstract. In this paper, first, we establish the solvability of (n × n)-systems of hyperbolic type with inhomogeneous mixed Neumann conditions involving operator of infinite order. Using theorems of J. L. Lions, quadratic boundary control problem for this system is considered. The necessary and sufficient conditions for the optimality of the control is obtained and the set of inequalities that characterize these conditions is formulated. Finally, by applying the Dubovitskii– Milyutin theorem of W. Kotarski, necessary and sufficient conditions of optimality are derived for the same problem, where the performance index is more general than the quadratic one and has an integral form.
Introduction Optimal control problems governed by partial differential equations are a field of an active research. Most of the work is done to derive necessary or sufficient optimality conditions of first and second order. The necessary and sufficient conditions of optimality for systems governed by different types of partial differential operators defined on spaces of functions of a finite number of variables have already been considered by Lions [12]. The optimal control problem of systems governed by different types of operators defined on spaces of functions of an infinite number of variables are initiated and proved, e.g., in [3–8], we have obtained the set of inequalities that characterize the optimal control for systems or (n × n)-systems governed by elliptic, parabolic, and hyperbolic equations of infinite number of variables with Dirichlet and Neumann conditions. The questions treated in this paper are related to the above results but in a different direction, by taking the case of operator of infinite order in a finite dimension. Thus, we derive optimality conditions for quadratic boundary control problem of n × n hyperbolic differential equations of infinite order. 2000 Mathematics Subject Classification. 49J20, 49K20, 93C20, 35R15. Key words and phrases. Operator of infinite order, hyperbolic system, Neumann conditions, boundary control, optimality conditions.
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H. A. EL SAIFY
A set of inequalities that characterizes this optimal control is obtained and this set is studied in order to construct algorithms attainable to numerical computations for the approximation of the control. Finally, by applying the generalizations of the Dubovitskii–Mulyutin theorem [2], we obtain necessary and sufficient conditions for optimality of the same problem, where the performance index is more general than the quadratic one and has an integral form and control-state constraints. This paper is organized as follows. In Sec. 1, we introduce spaces of functions of infinite order. Then we consider the Cartesian product of these spaces. In Sec. 2, we formulate some facts and results which enable us to state our problem. In Sec. 3, we prove the existence and uniqueness of the soluti
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