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 ∞.
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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)
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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
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DOI: https://doi.org/10.1007/s10589-008-9181-x