Error estimates for the approximation of the velocity tracking problem with Bang-Bang controls

Casas, Eduardo; Chrysafinos, Konstantinos

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 4, pp. 1267-1291.

The velocity tracking problem for the evolutionary Navier–Stokes equations in 2d is studied. The controls are of distributed type but the cost functional does not involve the usual quadratic term for the control. As a consequence the resulting controls can be of bang-bang type. First and second order necessary and sufficient conditions are proved. A fully-discrete scheme based on discontinuous (in time) Galerkin approach combined with conforming finite element subspaces in space, is proposed and analyzed. Provided that the time and space discretization parameters, $\tau$ and $h$ respectively, satisfy $\tau \le C{h}^2$, then $L^2$ error estimates are proved for the difference between the states corresponding to locally optimal controls and their discrete approximations.

Classification : 49J20, 65M60, 49K20, 35K55, 65N30
Mots clés : Evolution Navier–Stokes equations, optimal control, bang-bang controls, a priori error estimates

F. Abergel and R. Temam, On some control problems in fluid mechanics. Theor. Comput. Fluid Dyn. 1 (1990) 303–325. | DOI | Zbl

E. Casas, An optimal control problem governed by the evolution Navier–Stokes equations, in Optimal Control of Viscous Flows, edited by S. Sritharan. Frontiers in Applied Mathematics. SIAM, Philadelphia, (1998). | MR

E. Casas, Second order analysis for bang-bang control problems of PDEs. SIAM J. Control Optim. 50 (2012) 2356–2372. | DOI | MR | Zbl

E. Casas and K. Chrysafinos, A discontinuous Galerkin time-stepping scheme for the velocity tracking problem. SIAM J. Numer. Anal. 50 (2012) 2281–2306. | DOI | MR | Zbl

E. Casas and K. Chrysafinos, Error estimates for the discretization of the velocity tracking problem. Numer. Math. 130 (2015) 615–643. | DOI | MR | Zbl

E. Casas, and K. Chrysafinos, Analysis of the velocity tracking control problem for the 3d evolutionary Navier–Stokes equations. SIAM J. Control Optim. 54 (2016) 99–128. | DOI | MR | Zbl

E. Casas and F. Troltzsch, Second order analysis for optimal control problems: Improving results expected from from abstract theory. SIAM J. Optim. 22 (2012) 261–279. | DOI | MR | Zbl

E. Casas and F. Troltzsch, Second order optimality conditions and their role in pde control. Jahresbericht der Deutschen Mathematiker-Vereinigung 117 (2015) 3–44. | DOI | MR | Zbl

E. Casas, M. Mateos and J.-P. Raymond, Error estimates for the numerical approximation of a distributed control problem for the steady-state navier-stokes equations. SIAM J. Control Optim. 46 (2007) 952–982. | DOI | MR | Zbl

E. Casas, C. Ryll and F. Tröltzsch, Second Order and Stability Analysis for Optimal Sparse Control of the FitzHugh-Nagumo Equation. SIAM J. Control Optim. 53 (2015) 2168–2202. | DOI | MR | Zbl

K. Chrysafinos, Convergence of discontinuous Galerkin approximations of an optimal control problem associated to semilinear parabolic pde’s. ESAIM: M2AN 44 (2010) 189–206. | DOI | Numdam | MR | Zbl

K. Deckelnick and M. Hinze, Semidiscretization and error estimates for distributed control of the instationary Navier–Stokes equations. Numer. Math. 97 (2004) 297–320. | DOI | MR | Zbl

J. Douglas Jr., T. Dupont and L. Wahlbin, The stability in $L^q$ of the $L^2$ projection into finite element function spaces. Numer. Math. 23 (1975) 193–197. | DOI | MR | Zbl

J.C. Dunn, On second order sufficient optimality conditions for structured nonlinear programs in infinite-dimensional function spaces. In Mathematical Programming with Data Perturbations, edited by A. Fiacco. Marcel Dekker (1998) 83–107. | MR | Zbl

P. Girault and P. Raviart, Finite Element Methods for Navier–Stokes Equations. Theory and Algorithms. Springer-Verlag, Berlin, Heidelberg, New-York, Tokyo (1986). | MR | Zbl

M.D. Gunzburger, Perspectives in flow control and optimization. Advances in Design and Control SIAM, Philadelphia (2003). | MR | Zbl

M.D. Gunzburger and S. Manservisi, The velocity tracking problem for navier-stokes flows with bounded distributed control. SIAM J. Control Optim. 37 (1999) 1913–1945. | DOI | MR | Zbl

M.D. Gunzburger and S. Manservisi, Analysis and approximation of the velocity tracking problem for Navier–Stokes flows with distributed control. SIAM J. Numer. Anal. 37 (2000) 1481–1512. | DOI | MR | Zbl

M. Hinze, A Variational discretization concept in control constrained optimization: The linear quadratic case. Comput. Optim. Appl. 30 (2005) 45–61. | DOI | MR | Zbl

M. Hinze and K. Kunisch, Second order methods for optimal control of time-dependent fluid flow. SIAM J. Control Optim. 40 (2001) 925–946. | DOI | MR | Zbl

O. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow. English translation, second edition. Gordon and Breach, New York (1969). | MR | Zbl

O. Ladyzhenskaya, V.A. Solonnikov and N.N. Ural’tseva, Linear and Quasilinear Equations of Parabolic Type. American Mathematical Society (1988). | MR | Zbl

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires. Dunod, Paris (1969). | MR | Zbl

D. Meidner and B. Vexler, A priori error estimates for the space-time finite element discretization of parabolic optimal control problems. Part I: Problems without control constraints. SIAM J. Control Optim. 47 (2008) 1150–1177. | DOI | MR | Zbl

D. Meidner and B. Vexler, A priori error estimates for the space-time finite element discretization of parabolic optimal control problems. Part II: Problems with control constraints. SIAM J. Control Optim. 47 (2008) 1301–1329. | DOI | MR | Zbl

I. Neitzel and B. Vexler, A priori error estimates for space-time finite element discretization of semilinear parabolic optimal control problems. Numer. Math. 120 (2012) 345–386. | DOI | MR | Zbl

P.A Raviart and J.M. Thomas, Introduction à L’analyse Numérique des Equations aux Dérivées Partielles. Masson, Paris (1983). | MR | Zbl

A. Rösch, Error estimates for parabolic optimal control problems with control constraints. Z. Anal. Anwendungen 23 (2004) 353–376 | DOI | MR | Zbl

V. Solonnikov, Estimates for solutions of nonstationary Navier-Stokes equaions. J. Soviet. Math. 8 (1977) 213–317. | DOI | Zbl

S.S. Sritharan, Optimal control of viscous flow. SIAM, Philadelphia (1998). | MR | Zbl

R. Temam, Navier–Stokes Equations. North-Holland, Amsterdam (1979). | MR | Zbl

F. Tröltzsch and D. Wachsmuth, Second-order suficcient optimality conditions for the optimal control of Navier–Stokes equations. ESAIM: COCV 12 (2006) 93–119. | Numdam | MR | Zbl