Necessary and sufficient conditions for optimal controls in viscous flow problems

Fattorini, H. O.; Sritharan, S. S.

doi:10.1017/s0308210500028444

Cited by 54 publications

(57 citation statements)

References 20 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Optimal control for the Navier-Stokes equations has been the subject of extensive study in recent years and much progress has been made both mathematically and computationally; see, e.g., [AT], [FS1], [FS2], [FS3], [Fu1], [Fu2], [Fu3], [Gun], [GHS1], [GHS2], [GHS3], [HS], [HY1], [HY2], [HYR], [Li], [S1], and [S2]. In this work we confine ourselves to optimal Dirichlet control problems for the steadystate Navier-Stokes equations.…”

Section: Introductionmentioning

confidence: 99%

“…For instance, one often attempts, through the suction and injection of fluid through orifices on the boundary to reduce the drag on a body moving through a fluid. Optimal Dirichlet control problems for time-dependent Navier-Stokes equations were studied in [FGH] for general Dirichlet controls and in [FS1], [FS2], [FS3] and [S1], [S2] for Dirichlet controls in a special case, namely, when the control is of the separation-of-variable type. Optimal Dirichlet control problems for steady-state Navier-Stokes equations were studied in [GHS2], [GHS3], and [HS].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A Penalized Neumann Control Approach for Solving an Optimal Dirichlet Control Problem for the Navier--Stokes Equations

Hou¹,

Ravindran²

1998

SIAM J. Control Optim.

View full text Add to dashboard Cite

We introduce a penalized Neumann boundary control approach for solving an optimal Dirichlet boundary control problem associated with the two-or three-dimensional steady-state Navier-Stokes equations. We prove the convergence of the solutions of the penalized Neumann control problem, the suboptimality of the limit, and the optimality of the limit under further restrictions on the data. We describe the numerical algorithm for solving the penalized Neumann control problem and report some numerical results.

show abstract

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

A Penalized Neumann Control Approach for Solving an Optimal Dirichlet Control Problem for the Navier--Stokes Equations

Hou¹,

Ravindran²

1998

SIAM J. Control Optim.

View full text Add to dashboard Cite

show abstract

“…For the steady state Navier-Stokes system, complete and systematic mathematical and numerical analyses of optimal control problems of different types (e.g., having Dirichlet, Neumann, and distributed controls and also finite-dimensional controls) were given in [15,16,17,18]. Mathematical treatments of optimal control problems for the time-dependent Navier-Stokes system were given in [2], [6,7,8,9,10,11,12,13], [20], and [24,25,26,27]. In [6], free convection problems with boundary heat flux controls were considered; the existence of optimal solutions was proved and necessary conditions that characterize optimal controls and states were derived.…”

Section: Introductionmentioning

confidence: 99%

“…Various optimal control problems involving both distributed and boundary controls were considered in [2], although detailed proofs were provided only for the case of distributed controls. In [7,8,9,10] and [24,25,26,27] extensive studies of optimal control problems were given for Dirichlet controls in a special case, namely, when the control is of the separation-of-variable type.…”

Section: Introductionmentioning

confidence: 99%

Boundary Value Problems and Optimal Boundary Control for the Navier--Stokes System: the Two-Dimensional Case

Fursikov¹,

Gunzburger²,

Hou³

1998

SIAM J. Control Optim.

145

View full text Add to dashboard Cite

Abstract. We study optimal boundary control problems for the two-dimensional Navier-Stokes equations in an unbounded domain. Control is effected through the Dirichlet boundary condition and is sought in a subset of the trace space of velocity fields with minimal regularity satisfying the energy estimates. An objective of interest is the drag functional. We first establish three important results for inhomogeneous boundary value problems for the Navier-Stokes equations; namely, we identify the trace space for the velocity fields possessing finite energy, we prove the existence of a solution for the Navier-Stokes equations with boundary data belonging to the trace space, and we identify the space in which the stress vector (along the boundary) of admissible solutions is well defined. Then, we prove the existence of an optimal solution over the control set. Finally, we justify the use of Lagrange multiplier principles, derive an optimality system of equations in the weak sense from which optimal states and controls may be determined, and prove that the optimality system of equations satisfies in appropriate senses a system of partial differential equations with boundary values.Key words. optimal control, Navier-Stokes equations, boundary value problem, drag minimization AMS subject classifications. 76D05, 49J20, 49K20, 35K50PII. S0363012994273374 1. Introduction. Optimal control problems for fluid flows have been a subject of interest to experimenters and designers since at least the time of Prandtl. In more recent times, they have also become of substantial interest to mathematicians and computational scientists. For the steady state Navier-Stokes system, complete and systematic mathematical and numerical analyses of optimal control problems of different types (e.g., having Dirichlet, Neumann, and distributed controls and also finite-dimensional controls) were given in [15,16,17,18]. Mathematical treatments of optimal control problems for the time-dependent Navier-Stokes system were given in [2], [6,7,8,9,10,11,12,13], [20], and [24, 25, 26, 27]. In [6], free convection problems with boundary heat flux controls were considered; the existence of optimal solutions was proved and necessary conditions that characterize optimal controls and states were derived. In [11,12,13], the existence of optimal distributed controls was shown, an optimality system of equations was derived, and the question of the uniqueness

show abstract

“…The questzon of the unzgueness is not treated; in general (in nonlinear problems) the answer is expected to be negative (for the same Ternark see for example Fattorzni et a1 [11]- [12] and…”

Section: Remarkmentioning

confidence: 99%

On some control problems in a shallow-water model

Belmiloudi¹

Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No.01CH37228)

View full text Add to dashboard Cite

Proposed by SIAM Journal on Control and OptimizationInternational audienceWe develop methods of control for the modelling of long waves occurring in shallow water zones. The observation is the variability of the sea level. It is deduced from satellite measurements. The equations satisfied by the depth and the depth averaged velocity are of nonlinear shallow-water type. The result of existence and uniqueness of the solution for the direct problem is given in the case of Dirichlet non-homogeneous boundary conditions. We establish, by means of minimizing sequences, the existence of an optimal control in the case of small data and a very viscous fluid. In order to characterize it, we build a sequence of control problems corresponding to a linearization of the problem. We obtain the necessary conditions of optimality. The optimality conditions for the nonlinear problem is obtained as the limit of the penalization. Our approach can be used to develop a numerical schem

show abstract

Necessary and sufficient conditions for optimal controls in viscous flow problems

Cited by 54 publications

References 20 publications

A Penalized Neumann Control Approach for Solving an Optimal Dirichlet Control Problem for the Navier--Stokes Equations

A Penalized Neumann Control Approach for Solving an Optimal Dirichlet Control Problem for the Navier--Stokes Equations

Boundary Value Problems and Optimal Boundary Control for the Navier--Stokes System: the Two-Dimensional Case

On some control problems in a shallow-water model

Contact Info

Product

Resources

About