Boundary control problem with an infinite number of variables

Abstract

Abstract. Using a previous result by Gali and El-Saify (1983) and the theory of Kotarski (1989), and Lions (1971), we formulate the boundary control problem for a system governed by Neumann problem involving selfadjoint elliptic operator of 2 th order with an infinite number of variables. The inequalities which characterize the optimal control in terms of the adjoint system are obtained, it is studied in order to construct algorithms attainable to numerical computations for the approximation of the control.2000 Mathematics Subject Classification. 49J20, 93C20.

Boundary Control for 2 × 2 Elliptic Systems with Conjugation Conditions

“…Systems governed by elliptic, parabolic, and hyperbolic operators have been considered, some of which are of distributed type as in [4][5][6][7][8][9][10][11][12], while some others are of boundary type as in [13][14][15][16][17].…”

Optimality Conditions for Infinite Order Hyperbolic Differential System

“…Assume that the state is given by (3) and the cost function is given by (6). Then the optimal control u 0 = (u 0 i ) n i=1 ∈ U is determined by the simultaneous solution of (3) with…”

“…The solution of the stated optimal control problem is equivalent to finding a pair (y 0 , u 0 ) ∈ E, where Y × U = E = E 1 × E 2 × · · · × E n that satisfies Eqs. (3) and minimizes the performance functional (9) subject to the control constraints (10) and the state constraints (11). We formulate the necessary and sufficient conditions of optimality in the following theorem.…”

“…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][4][5][6][7][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.…”

Optimality of the boundary control for n×n parabolic lag system

“…The necessary and sufficient conditions of optimality for systems (n × n systems) governed by different types of partial differential operators defined on spaces of functions of infinitely many variables are discussed in [2][3][4][5][6][7][8][9][10][11][12]. The interest in the study of this class of operators is stimulated by problems in quantum field theory [1].…”