Abstract
In this article, an optimal control problem subject to a semilinear elliptic equation and mixed control-state constraints is investigated. The problem data depends on certain parameters. Under an assumption of separation of the active sets and a second-order sufficient optimality condition, Bouligand-differentiability (B-differentiability) of the solutions with respect to the parameter is established. Furthermore, an adjoint update strategy is proposed which yields a better approximation of the optimal controls and multipliers than the classical Taylor expansion, with remainder terms vanishing in L ∞.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Adams, R.A.: Sobolev Spaces. Academic Press, San Diego (1978)
Alt, W., Griesse, R., Metla, N., Rösch, A.: Lipschitz stability for elliptic optimal control problems with mixed control-state constraints (2006, submitted)
Bonnans, F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, Berlin (2000)
Dontchev, A.: Implicit function theorems for generalized equations. Math. Program. 70, 91–106 (1995)
Griesse, R., Grund, T., Wachsmuth, D.: Update strategies for perturbed nonsmooth equations. Optim. Methods Softw., to appear
Hintermüller, M., Tröltzsch, F., Yousept, I.: Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems. Numer. Math. 108(4), 571–603 (2008)
Hintermüller, M., Yousept, I.: A sensitivity-based extrapolation technique for the numerical solution of state-constrained optimal control problems (2007, submitted)
Malanowski, K.: Bouligand differentiability of solutions to parametric optimal control problems. Numer. Funct. Anal. Optim. 22(7/8), 973–990 (2001)
Malanowski, K.: Stability and sensitivity analysis for optimal control problems with control-state constraints. Dissertationes Math. (Rozprawy Mat.) 394, 51 (2001)
Malanowski, K.: Remarks on differentiability of metric projections onto cones of nonnegative functions. J. Convex Anal. 10(1), 285–294 (2003)
Malanowski, K.: Stability and sensitivity analysis for linear-quadratic optimal control subject to state constraints. Optimization 56(4), 463–478 (2007)
Malanowski, K.: Sufficient optimality conditions in stability analysis for state-constrained optimal control. Appl. Math. Optim. 55(2), 255–271 (2007)
Malanowski, K., Tröltzsch, F.: Lipschitz stability of solutions to parametric optimal control for parabolic equations. Z. Anal. Anwend. 18, 469–489 (1999)
Malanowski, K., Tröltzsch, F.: Lipschitz stability of solutions to parametric optimal control for parabolic equations. J. Anal. Appl. 18(2), 469–489 (1999)
Meyer, C., Prüfert, U., Tröltzsch, F.: On two numerical methods for state-constrained elliptic control problems. Optim. Methods Softw. 22(6), 871–899 (2007)
Meyer, C., Tröltzsch, F.: On an elliptic optimal control problem with pointwise mixed control-state constraints. In: Recent Advances in Optimization. Lecture Notes in Economics and Mathematical Systems, vol. 563, pp. 187–204. Springer, Berlin (2006)
Rösch, A., Tröltzsch, F.: On the regularity of solutions and Lagrange multipliers of optimal control problems for semilinear equations with mixed pointwise control-state constraints. SIAM J. Control Optim. 46(3), 1098–1115 (2007)
Rösch, A., Wachsmuth, D.: Semi-smooth Newton’s method for an optimal control problem with control and mixed control-state constraints. Matheon Preprint 415 (2007, submitted)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Griesse, R., Wachsmuth, D. Sensitivity analysis and the adjoint update strategy for an optimal control problem with mixed control-state constraints. Comput Optim Appl 44, 57–81 (2009). https://doi.org/10.1007/s10589-008-9181-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-008-9181-x