Sufficient Optimality Conditions for the Moreau-yosida-type Regularization Concept Applied to Semilinear Elliptic Optimal Control Problems with Pointwise State Constraints∗
نویسندگان
چکیده
We develop sufficient optimality conditions for a Moreau-Yosida regularized optimal control problem governed by a semilinear elliptic PDE with pointwise constraints on the state and the control. We make use of the equivalence of a setting of Moreau-Yosida regularization to a special setting of the virtual control concept, for which standard second order sufficient conditions have been shown. Moreover, we present a numerical example, solving a Moreau-Yosida regularized model problem with an SQP method. MSC: 49K20, 49M25, 49M29 ∗Accepted for publication in revised form on September 1, 2010. †[email protected] Weierstrass Institute for Applied Mathematics and Stochastics, Nonlinear Optimization and Inverse Problems, Mohrenstrasse 39, D-10117 Berlin; ‡[email protected] Technische Universität Berlin, Fakultät II Mathematik und Naturwissenschaften, Str. des 17. Juni 136, D-10623 Berlin §[email protected] Universität Duisburg-Essen, Department of Mathematics, Forsthausweg 2, D-47057 Duisburg 222 Annals of the Academy of Romanian Scientists Series on Mathematics and its Applications ISSN 2066 6594 Volume 2, Number 2 / 2010 Sufficient optimality conditions for Moreau-Yosida-type regularization 223 keywords: Optimal control, semilinear elliptic equation, state constraints, regularization, Moreau-Yosida approximation, virtual control, sufficient optimality conditions
منابع مشابه
Control and Cybernetics on Convergence of Regularization Methods for Nonlinear Parabolic Optimal Control Problems with Control and State Constraints * By
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 o...
متن کاملRegularization for semilinear elliptic optimal control problems with pointwise state and control constraints
In this paper a class of semilinear elliptic optimal control problem with pointwise state and control constraints is studied. We show that sufficient second order optimality conditions for regularized problems with small regularization parameter can be obtained from a second order sufficient condition assumed for the unregularized problem. Moreover, error estimates with respect to the regulariz...
متن کاملOn convergence of regularization methods for nonlinear parabolic optimal control problems with control and state constraints
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 in nity or zero, respectively. In particular, the strong convergence of...
متن کاملSecond Order Sufficient Optimality Conditions for Parabolic Optimal Control Problems with Pointwise State Constraints
In this paper we study optimal control problems governed by semilinear parabolic equations where the spatial dimension is two or three. Moreover, we consider pointwise constraints on the control and on the state. We formulate first order necessary and second order sufficient optimality conditions. We make use of recent results regarding elliptic regularity and apply the concept of maximal parab...
متن کاملSufficient Second-Order Optimality Conditions for Semilinear Control Problems with Pointwise State Constraints
Abstract. Second-order sufficient optimality conditions are established for the optimal control of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state. In contrast to former publications on this subject, the cone of critical directions is the smallest possible in the sense that the second-order sufficient conditions are the closest to the associat...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2010