Boundary Control for 2 × 2 Elliptic Systems with Conjugation Conditions

Abstract

In this paper, we consider 2 × 2 non-cooperative elliptic system involving Laplace operator defined on bounded, continuous and strictly Lipschitz domain of R n . First we prove the existence and uniqueness for the state of the system under conjugation conditions; then we discuss the existence of the optimal control of boundary type with Neumann conditions, and we find the set of equations and inequalities that characterize it.

Optimal control for cooperative systems involving fractional Laplace operators

“…Optimal control for partial differential equations (PDEs) has been widely studied in many fields such as biology, ecology, economics, engineering, and finance [5-10, 18, 22, 24, 25, 30, 34, 37]. These results have been expanded in [12,14,15,29,[31][32][33] to cooperative and noncooperative systems. The fractional optimal control problems are the generalization of standard optimal control problems.…”

Distributed Control for Non-Cooperative Systems Governed by Time-Fractional Hyperbolic Operators

“…These foundational works have provided crucial insights into the fundamental principles governing such systems. Expanding upon this groundwork, subsequent investigations [16,17] have further elucidated the nuanced dynamics of cooperative and non-cooperative systems. By delving deeper into the interplay of various factors and exploring novel methodologies, these studies have enriched our understanding and paved the way for advancements in optimal control theory for PDE-driven systems.…”

Distributed Control for Cooperative Elliptic Systems Under Conjugation Conditions

“…The control problems described by either infinite order operators or operators with an infinite number of variables have been discussed by Gali et al [3][4][5][6][7][8]. These results have been extended in [1,2,10,13,18] to cooperative and noncooperative systems. In [19][20][21], Sergienko and Deineka introduced control problems of distributed systems with conjugation conditions and quadratic cost functions.…”

“…Let us suppose that(2.8) holds and the cost function is given by(2.11), then the distributed control u is characterized by: together with (2.9) and (2.10) ,where p(u) = { (u), (u)} is the adjoint state.ProofThe optimal control u is characterized by [see11,13]…”