In this paper,the non-stationary incompressible fluid flows governed by the Navier-Stokes equations are studied in a bounded domain.This study focuses on the time-fractional Navier-Stokes equations in the optimal cont...In this paper,the non-stationary incompressible fluid flows governed by the Navier-Stokes equations are studied in a bounded domain.This study focuses on the time-fractional Navier-Stokes equations in the optimal control subject,where the control is distributed within the domain and the time-fractional derivative is proposed as Riemann-Liouville sort.In addition,the control object is to minimize the quadratic cost functional.By using the Lax-Milgram lemma with the assistance of the fixed-point theorem,we demonstrate the existence and uniqueness of the weak solution to this system.Moreover,for a quadratic cost functional subject to the time-fractional Navier-Stokes equations,we prove the existence and uniqueness of optimal control.Also,via the variational inequality upon introducing the adjoint linearized system,some inequalities and identities are given to guarantee the first-order necessary optimality conditions.A direct consequence of the results obtained here is that when a→1,the obtained results are valid for the classical optimal control of systems governed by the Navier-Stokes equations.展开更多
基金The authors extend their appreciation to the Deanship of Scientific Research at King Khalid University for funding their work through General Research Project under grant number(GRP-114-41).
文摘In this paper,the non-stationary incompressible fluid flows governed by the Navier-Stokes equations are studied in a bounded domain.This study focuses on the time-fractional Navier-Stokes equations in the optimal control subject,where the control is distributed within the domain and the time-fractional derivative is proposed as Riemann-Liouville sort.In addition,the control object is to minimize the quadratic cost functional.By using the Lax-Milgram lemma with the assistance of the fixed-point theorem,we demonstrate the existence and uniqueness of the weak solution to this system.Moreover,for a quadratic cost functional subject to the time-fractional Navier-Stokes equations,we prove the existence and uniqueness of optimal control.Also,via the variational inequality upon introducing the adjoint linearized system,some inequalities and identities are given to guarantee the first-order necessary optimality conditions.A direct consequence of the results obtained here is that when a→1,the obtained results are valid for the classical optimal control of systems governed by the Navier-Stokes equations.
