On convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints

• Autorzy: Neitzel I. , Troltzsch F.
• Języki publikacji: EN

Abstrakty

EN

Moreau-Yosida and Lavrentiev type regularization methods are considered for nonlinear optimal control problems governed by semilinear parabolic equations with bilateral pointwise control and state constraints. The convergence of optimal controls of the regularized problems is studied for regularization parameters tending to infinity or zero, respectively. In particular, the strong convergence of global and local solutions is addressed. Moreover, strong regularity of the Lavrentiev-regularized optimality system is shown under certain assumptions, which, in particular, allows to show that locally optimal solutions of the Lavrentiev regularized problems are locally unique. This analysis is based on a second-order sufficient optimality condition and a separation assumption on almost active sets.

Słowa kluczowe

EN

optimal control; semilinear parabolic equation; pointwise state constraints; Moreau-Yosida regularization; Lavrentiev regularization; convergence; strong regularity; local uniqueness

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2008
• Tom: Vol. 37, no 4
• Strony: 1013--1043
• Opis fizyczny: Bibliogr. 26 poz.

Twórcy

autor

Neitzel I.

autor

Troltzsch F.

• Technische Universitat Berlin, Fakultat II - Mathematik und Naturwissenschaften, Str. des 17. Juni 136, D-10623 Berlin, Germany,

Bibliografia

• ALT, W. (1990) The Lagrange-Newton method for infinite dimensional optimization problems. Numerical Functional Analysis and Optimization 11, 201-224.
• ALT, W., GRIESSE, R., METLA, N. and RÖSCH, A. (2006) Lipschitz stability for elliptic optimal control problems with mixed control-state constraints. Submitted.
• BERGOUNIOUX, M., ITO, K. and KUNISCH, K. (1999) Primal-dual strategy for constrained optimal control problems. SIAM J. Control Optim. 37, 1176-1194.
• CASAS, E. (1997) Pontryagin’s principle for state-constrained boundary control problems of semilinear parabolic equations. SIAM J. Control Optim. 35, 1297-1327.
• CASAS, E. and TRÖLTZSCH, F. (2002) Error estimates for the finite-element approximation of a semilinear elliptic control problem. Control and Cybernetics 31, 695-712.
• DI BENEDETTO, E. (1986) On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients. Ann. Scuola Sup. Pisa, Ser. I, 13, 487-535.
• DONTCHEV, A. L. (1996) Local analysis of a newton-type method based on partial linearization. In: J. Renegar, M. Shub and S. Smale, eds., Proceedings of the AMS-SIAM summer seminar in applied mathematics on mathematics of numerical analysis: Real number algorithms. AMS Lectures in Appl. Math. 32, 295-306.
• GRIESSE, R. (2006) Lipschitz stability of solutions to some state-constrained elliptic optimal control problems. ZAMM 25 (4), 435-455.
• GRIESSE, R., METLA, N. and RÖSCH, A. (2008) Convergence analysis of the SQP method for nonlinear mixed-constrained elliptic optimal control problems. ZAMM 88 (10), 776-792.
• HINTERMÜLLER, M., ITO, K. and KUNISCH, K. (2003) The primal-dual active set strategy as a semismooth Newton method. SIAM J. Optim. 13, 865-888.
• HlNTERMÜLLER, M., TRÖLTZSCH, F. and YOUSEPT, I. (2008) Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems. Numerische Mathematik 108 (4), 571-603.
• HINZE, M. and MEYER, C. (2007) Stability of infinite dimensional control problems with point-wise state constraints. Technical Report WIAS.
• ITO, K. and KUNISCH, K. (2003) Semi-smooth Newton methods for state-constrained optimal control problems. Systems and Control Letters 50, 221-228.
• JOSEPHY, N.H. (1979) Newton’s method for generalized equations. Technical Summary Report 1965. Mathematics Research Center, University of Wisconsin, Madison, WI.
• MALANOWSKI, K. (2001) Stability and sensitivity analysis for optimal control problems with control-state constraints. Dissertationes mathematicae (Rozprawy Matematyczne) 394.
• MEYER, C. and TRÖLTZSCH, F. (2005) On an elliptic optimal control problem with pointwise mixed control-state constraints. In: A. Seeger, ed., Recent Advances in Optimization. Proceedings of the 12th French-German-Spanish Conference on Optimization held in Avignon, September 20-24, 2004 • Lectures Notes in Economics and Mathematical Systems. Springer-Verlag.
• MEYER, C., RÖSCH, A. and TRÖLTZSCH, F. (2006) Optimal control of PDEs with regularized pointwise state constraints. COAP 33, 209-228.
• MEYER, C. and YOUSEPT, I. (2008) Regularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions. COAP. online first, doi 10.1007/sl0589-007-9151-8.
• PRÜFERT, U. and TRÖLTZSCH, F. (2007) An interior point method for a parabolic optimal control problem with regularized pointwise state constraints. ZAMM 87(8-9), 564-589.
• RAYMOND, J.-P. and TRÖLTZSCH, F. (2000) Second order sufficient optimality conditions for nonlinear parabolic control problems with state constraints. Discrete and Continuous Dynamical Systems 6, 431-450.
• ROBINSON, S.M. (1980) Strongly regular generalized equations. Mathematics of Operation Research 5, 43-62.
• RÖSCH, A. and TRÖLTZSCH, F. (2007) On regularity of solutions and Lagrange multipliers of optimal control problems for semilinear equations with mixed pointwise control-state constraints. SIAM J. Control Optim. 46, 1098-1115.
• TRÖLTZSCH, F. (2000) Lipschitz stability of solutions of linear-quadratic parabolic control problems with respect to perturbations. Dyn. Contin. Discrete Impulsive Syst. 7, 289-306.
• TRÖLTZSCH, F. (2005) Optimale Steuerung partieller Differentialgleichungen. Theorie, Verfahren und Anwendungen. Vieweg, Wiesbaden.
• TRÖLTZSCH, F. and YOUSEPT, I. (2008) A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints. COAP. online first, doi 10.1007/sW589-007-9114-0.
• ZOWE, J. and KURCYUSZ, S. (1979) Regularity and stability for the mathematical programming problem in Banach spaces. Appl. Math. Optimization 5, 49-62.