Abstract
In this work, the elliptic 2times 2 cooperative systems involving fractional Laplace operators are studied. Due to the nonlocality of the fractional Laplace operator, we reformulate the problem into a local problem by an extension problem. Then, the existence and uniqueness of the weak solution for these systems are proved. Hence, the existence and optimality conditions are deduced.
Highlights
Nonlocal operators have been a useful area of investigation in different branches of mathematics such as operator theory and harmonic analysis
5 Summary and conclusion In the present work, we investigate the optimal control problem for 2 × 2 cooperative systems involving the fractional Laplace operator, wherein these systems are subject to the zero Dirichlet condition
Due to the difficulty arising from the nonlocality of the fractional Laplace operator, we follow the Caffarelli and Silvestre technique to extend our problem to local cooperative systems
Summary
Nonlocal operators have been a useful area of investigation in different branches of mathematics such as operator theory and harmonic analysis. Optimal control for partial differential equations (PDEs) has been widely studied in many fields such as biology, ecology, economics, engineering, and finance [5,6,7,8,9,10, 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. In [19], the distributed control for a time-fractional differential system involving a Schrödinger operator is studied, and the optimality conditions are derived. Via the Lax–Milgram lemma, we are able to prove the existence and uniqueness of the weak solution for the local system For both local and nonlocal systems, the optimality conditions are derived via the Lions technique.
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