#### Prof. Dr. Arnd Rösch

Nonlinear Optimization

University of Duisburg-Essen

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- arnd[dot]roesch[at]uni-due[dot]de
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**A priori error estimates for the finite element approximation of a nonsmooth optimal control problem governed by a coupled semilinear pde-ode system**

Holtmannspotter, M. and Rösch, A.*SIAM Journal on Control and Optimization*59 (2021)In this paper we investigate a priori error estimates for finite element discretization of a simplified semilinear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of an elliptic PDE which has to be fulfilled at almost all times coupled with a nonsmooth, semilinear ODE that has to hold true in almost all points in space. The system is discretized by a constant discontinuous Galerkin method in time and usual conforming linear finite elements in space. For the uncontrolled equation, we prove linear convergence in time and an order of O (h 3 2 e) for the discretization error in space. Our main result regarding the optimal control problem is the uniform convergence of dG(0)cG(1)-discrete controls to l in H1 { 0} (0, T;L2(Ω). Error estimates for the controls are established via a quadratic growth condition. Numerical experiments are added to illustrate the proven rates of convergence. © 2021 Society for Industrial and Applied Mathematics Publications. All rights reserved.view abstract 10.1137/20M1331263 **A priori error estimates for the space-time finite element discretization of an optimal control problem governed by a coupled linear pde-ode system**

Holtmannspötter, M. and Rösch, A.*Mathematical Control and Related Fields*11 (2021)In this paper we investigate a priori error estimates for the space-time Galerkin finite element discretization of an optimal control problem governed by a simplified linear gradient enhanced damage model. The model equations are of a special structure as the state equation consists of an elliptic PDE which has to be fulfilled at almost all times coupled with an ODE that has to hold true in almost all points in space. The state equation is discretized by a piecewise constant discontinuous Galerkin method in time and usual con-forming linear finite elements in space. For the discretization of the control we employ the same discretization technique which turns out to be equivalent to a variational discretization approach. We provide error estimates of optimal order both for the discretization of the state equation as well as for the optimal control. Numerical experiments are added to illustrate the proven rates of convergence. © 2021, American Institute of Mathematical Sciences. All rights reserved.view abstract 10.3934/MCRF.2021014 **No-gap second-order optimality conditions for optimal control of a non-smooth quasilinear elliptic equation**

Clason, C. and Nhu, V.H. and Rösch, A.*ESAIM - Control, Optimisation and Calculus of Variations*27 (2021)This paper deals with second-order optimality conditions for a quasilinear elliptic control problem with a nonlinear coefficient in the principal part that is finitely PC2 (continuous and C2 apart from finitely many points). We prove that the control-To-state operator is continuously differentiable even though the nonlinear coefficient is non-smooth. This enables us to establish "no-gap"second-order necessary and sufficient optimality conditions in terms of an abstract curvature functional, i.e., for which the sufficient condition only differs from the necessary one in the fact that the inequality is strict. A condition that is equivalent to the second-order sufficient optimality condition and could be useful for error estimates in, e.g., finite element discretizations is also provided. © EDP Sciences, SMAI 2021.view abstract 10.1051/cocv/2020092 **Numerical approximation of control problems of non-monotone and non-coercive semilinear elliptic equations**

Casas, E. and Mateos, M. and Rösch, A.*Numerische Mathematik*149 (2021)We analyze the numerical approximation of a control problem governed by a non-monotone and non-coercive semilinear elliptic equation. The lack of monotonicity and coercivity is due to the presence of a convection term. First, we study the finite element approximation of the partial differential equation. While we can prove existence of a solution for the discrete equation when the discretization parameter is small enough, the uniqueness is an open problem for us if the nonlinearity is not globally Lipschitz. Nevertheless, we prove the existence and uniqueness of a sequence of solutions bounded in L∞(Ω) and converging to the solution of the continuous problem. Error estimates for these solutions are obtained. Next, we discretize the control problem. Existence of discrete optimal controls is proved, as well as their convergence to solutions of the continuous problem. The analysis of error estimates is quite involved due to the possible non-uniqueness of the discrete state for a given control. To overcome this difficulty we define an appropriate discrete control-to-state mapping in a neighbourhood of a strict solution of the continuous control problem. This allows us to introduce a reduced functional and obtain first order optimality conditions as well as error estimates. Some numerical experiments are included to illustrate the theoretical results. © 2021, The Author(s).view abstract 10.1007/s00211-021-01222-7 **Optimal control of a non-smooth quasilinear elliptic equation**

Clason, C. and Nhu, V.H. and Rösch, A.*Mathematical Control and Related Fields*11 (2021)This work is concerned with an optimal control problem governed by a non-smooth quasilinear elliptic equation with a nonlinear coefficient in the principal part that is locally Lipschitz continuous and directionally but not Gâteaux differentiable. This leads to a control-to-state operator that is directionally but not Gâteaux differentiable as well. Based on a suitable regu-larization scheme, we derive C-and strong stationarity conditions. Under the additional assumption that the nonlinearity is a P C1 function with countably many points of nondifferentiability, we show that both conditions are equiva-lent. Furthermore, under this assumption we derive a relaxed optimality system that is amenable to numerical solution using a semi-smooth Newton method. This is illustrated by numerical examples. © 2021, American Institute of Mathematical Sciences. All rights reserved.view abstract 10.3934/MCRF.2020052 **Analysis of control problems of nonmontone semilinear elliptic equations**

Casas, E. and Mateos, M. and Rösch, A.*ESAIM - Control, Optimisation and Calculus of Variations*26 (2020)In this paper we study optimal control problems governed by a semilinear elliptic equation. The equation is nonmonotone due to the presence of a convection term, despite the monotonocity of the nonlinear term. The resulting operator is neither monotone nor coervive. However, by using conveniently a comparison principle we prove existence and uniqueness of solution for the state equation. In addition, we prove some regularity of the solution and differentiability of the relation control-to-state. This allows us to derive first and second order conditions for local optimality. © EDP Sciences, SMAI 2020.view abstract 10.1051/cocv/2020032 **Error estimates for semilinear parabolic control problems in the absence of Tikhonov term**

Casas, E. and Mateos, M. and Rösch, A.*SIAM Journal on Control and Optimization*57 (2019)In this paper, we analyze optimal control problems of semilinear parabolic equations, where the controls are distributed and depend only on time. Box constraints for the controls are imposed and the cost functional does not involve the control itself, only the associated state. We prove second order optimality conditions for local strong minimizers, which are used to derive error estimates in the numerical approximation. First we estimate the difference between the discrete and continuous optimal states. In the last part, under an additional assumption on the optimal adjoint state, we prove error estimates for the controls and improve the estimates for the states. © 2019 Society for Industrial and Applied Mathematics.view abstract 10.1137/18M117220X **Superconvergent Graded Meshes for an Elliptic Dirichlet Control Problem**

Apel, T. and Mateos, M. and Pfefferer, J. and Rösch, A.*Lecture Notes in Computational Science and Engineering*128 (2019)Superconvergent discretization error estimates can be obtained when the solution is smooth enough and the finite element meshes enjoy some structural properties. The simplest one is that any two adjacent triangles form a parallelogram. Existing results on finite element estimates on superconvergent meshes are reviewed, which can be used for numerical analysis of Dirichlet control problems. Moreover, an error estimate is given for a variational normal derivative which is of higher order on superconvergence meshes. Graded meshes can be used as a remedy of the reduced convergence order in the case of quasi-uniform meshes when elliptic boundary value problems with singularities in the vicinity of corners are treated. Discretization error estimates on graded meshes are reviewed. Depending on the construction, graded meshes may or may not have superconvergence properties. The discretization error of an elliptic Dirichlet control problem is discussed in the case of superconvergent graded meshes. Results of a paper in preparation are announced, where error estimates for Dirichlet optimal control problems on superconvergent graded meshes will be shown. © Springer Nature Switzerland AG 2019.view abstract 10.1007/978-3-030-14244-5_1 **Error estimates for dirichlet control problems in polygonal domains: Quasi-uniform meshes**

Apel, T. and Mateos, M. and Pfefferer, J. and Rösch, A.*Mathematical Control and Related Fields*8 (2018)The paper deals with finite element approximations of elliptic Dirichlet boundary control problems posed on two-dimensional polygonal domains. Error estimates are derived for the approximation of the control and the state variables. Special features of unconstrained and control constrained problems as well as general quasi-uniform meshes and superconvergent meshes are carefully elaborated. Compared to existing results, the convergence rates for the control variable are not only improved but also fully explain the observed orders of convergence in the literature. Moreover, for the first time, results in nonconvex domains are provided. © 2018, American Institute of Mathematical Sciences. All rights reserved.view abstract 10.3934/mcrf.2018010 **Improved approximation rates for a parabolic control problem with an objective promoting directional sparsity**

Casas, E. and Mateos, M. and Rösch, A.*Computational Optimization and Applications*(2018)We discretize a directionally sparse parabolic control problem governed by a linear equation by means of control approximations that are piecewise constant in time and continuous piecewise linear in space. By discretizing the objective functional with the help of appropriate numerical quadrature formulas, we are able to show that the discrete optimal solution exhibits a directional sparse pattern alike the one enjoyed by the continuous solution. Error estimates are obtained and a comparison with the cases of having piecewise approximations of the control or a semilinear state equation are discussed. Numerical experiments that illustrate the theoretical results are included. © 2018 Springer Science+Business Media, LLC, part of Springer Natureview abstract 10.1007/s10589-018-9979-0 **Finite element approximation of sparse parabolic control problems**

Casas, E. and Mateos, M. and Rösch, A.*Mathematical Control and Related Fields*7 (2017)We study the finite element approximation of an optimal control problem governed by a semilinear partial differential equation and whose objective function includes a term promoting space sparsity of the solutions. We prove existence of solution in the absence of control bound constraints and provide the adequate second order sufficient conditions to obtain error estimates. Full discretization of the problem is carried out, and the sparsity properties of the discrete solutions, as well as error estimates, are obtained. © 2017, American Institute of Mathematical Sciences. All rights reserved.view abstract 10.3934/mcrf.2017014 **Mass lumping for the optimal control of elliptic partial differential equations**

Rösch, A. and Wachsmuth, G.*SIAM Journal on Numerical Analysis*55 (2017)The finite element discretization of a control constrained elliptic optimal control problem is studied. Control and state are discretized by higher order finite elements. The inequality constraints are only posed in the Lagrange points. The computational effort is significantly reduced by a new mass lumping strategy. The main contribution is the derivation of new a priori error estimates up to order h4 on locally refined meshes. Moreover, we propose a new algorithmic strategy to obtain such highly accurate results. The theoretical findings are illustrated by numerical examples. © 2017 Society for Industrial and Applied Mathematics.view abstract 10.1137/16M1074473 **Pontryagin’s Principle for Optimal Control Problem Governed by 3D Navier–Stokes Equations**

Kien, B.T. and Rösch, A. and Wachsmuth, D.*Journal of Optimization Theory and Applications*173 (2017)This paper deals with the Pontryagin maximum principle for optimal control problems governed by 3D Navier–Stokes equations with pointwise control constraint. The obtained result is proved by using some results on regularity of solutions of the Navier–Stokes equations and techniques of optimal control theory. © 2017, Springer Science+Business Media New York.view abstract 10.1007/s10957-017-1081-8 **Reliable a posteriori error estimation for state-constrained optimal control**

Rösch, A. and Siebert, K.G. and Steinig, S.*Computational Optimization and Applications*(2017)We derive a reliable a posteriori error estimator for a state-constrained elliptic optimal control problem taking into account both regularisation and discretisation. The estimator is applicable to finite element discretisations of the problem with both discretised and non-discretised control. The performance of our estimator is illustrated by several numerical examples for which we also introduce an adaptation strategy for the regularisation parameter. © 2017 Springer Science+Business Media New Yorkview abstract 10.1007/s10589-017-9908-7 **Second-order optimality conditions for boundary control problems with mixed pointwise constraints**

Son, N.H. and Kien, B.T. and Rösch, A.*SIAM Journal on Optimization*26 (2016)This paper deals with second-order optimality conditions for a boundary control problem which is governed by semilinear elliptic equations with mixed pointwise state-control constraints. We show that in some cases, there is no gap between second-order necessary optimality conditions and second-order sufficient optimality conditions. In addition, second-order sufficient optimality conditions for the problem where the objective function does not depend on control variables are also discussed. © 2016 Society for Industrial and Applied Mathematics.view abstract 10.1137/15M1033629 **Finite element discretization of state-constrained elliptic optimal control problems with semilinear state equation**

Neitzel, I. and Pfefferer, J. and Rösch, A.*SIAM Journal on Control and Optimization*53 (2015)We study a class of semilinear elliptic optimal control problems with pointwise state constraints. The purpose of this paper is twofold. First, we present convergence results for the finite element discretization of this problem class similarly to known results with finite-dimensional control space, thus extending results that are-for control functions-only available for linear-quadratic convex problems. We rely on a quadratic growth condition for the continuous problem that follows from second order sufficient conditions. Second, we show that the second order sufficient conditions for the continuous problem transfer to its discretized version. This is of interest, for example, when considering questions of local uniqueness of solutions or the convergence of solution algorithms such as the SQP method. © 2015 Society for Industrial and Applied Mathematics.view abstract 10.1137/140960645 **Introduction to the special issue for EUCCO 2013**

Benner, P. and Herzog, R. and Hinze, M. and Rösch, A. and Schiela, A. and Schulz, V.*Computational Optimization and Applications*62 (2015)view abstract 10.1007/s10589-015-9776-y **Lower semicontinuity of the solution map to a parametric elliptic optimal control problem with mixed pointwise constraints**

Kien, B.T. and Nhu, V.H. and Rösch, A.*Optimization*64 (2015)This paper studies the solution stability of a parametric optimal control problem governed by single linear elliptic equations with mixed control-state constraints and convex cost functions. By reducing the problem to a parametric programming problem and a parametric variational inequality, we obtain sufficient conditions under which the solution map to an elliptic optimal control problem is lower semicontinuous. © 2014 Taylor & Francis.view abstract 10.1080/02331934.2013.853060 **On the regularity of the solutions of dirichlet optimal control problems in polygonal domains**

Apel, T. and Mateos, M. and Pfefferer, J. and Rösch, A.*SIAM Journal on Control and Optimization*53 (2015)A linear quadratic Dirichlet control problem governed by an elliptic equation posed on a possibly nonconvex polygonal domain is analyzed. Detailed regularity results are provided in classical Sobolev (Slobodetski) spaces. In particular, it is proved that in the presence of pointwise control constraints, the optimal control is continuous despite the nonconvexity of the domain. © 2015 Society for Industrial and Applied Mathematics.view abstract 10.1137/140994186 **Regularization in Sobolev Spaces with Fractional Order**

Aßmann, U. and Rösch, A.*Numerical Functional Analysis and Optimization*36 (2015)We study the minimization of a quadratic functional where the Tichonov regularization term is an Hs-norm with a fractional s > 0. Moreover, pointwise bounds for the unknown solution are given. A multilevel approach as an equivalent norm concept is introduced. We show higher regularity of the solution of the variational inequality. This regularity is used to show the existence of regular Lagrange multipliers in function space. The theory is illustrated by two applications: a Dirichlet boundary control problem and a parameter identification problem. Copyright © Taylor & Francis Group, LLC 2015.view abstract 10.1080/01630563.2014.970644 **Second-Order Necessary Optimality Conditions for a Class of Optimal Control Problems Governed by Partial Differential Equations with Pure State Constraints**

Kien, B.T. and Nhu, V.H. and Rösch, A.*Journal of Optimization Theory and Applications*165 (2015)Based on some tools of variation analysis, we deal with first- and second-order necessary optimality conditions for a class of optimal control problems governed by semilinear elliptic equations and stationary Navier-Stokes equations with pure state constraints. To do this, we first derive optimality conditions for an abstract optimal control problem and then apply the obtained results to derive second-order necessary optimality conditions for semilinear elliptic optimal control problems as well as optimal control problems governed by stationary Navier-Stokes equations. © 2014 Springer Science+Business Media New York.view abstract 10.1007/s10957-014-0628-1 **A posteriori error analysis of optimal control problems with control constraints**

Kohls, K. and Rösch, A. and Siebert, K.G.*SIAM Journal on Control and Optimization*52 (2014)We derive a unifying framework for the a posteriori error analysis of control constrained linear-quadratic optimal control problems. We consider finite element discretizations with discretized and nondiscretized control. A fundamental error equivalence drastically simplifies the a posteriori error analysis for optimal control problems. It basically remains to apply error estimators for the linear state and adjoint problem. We give several examples, including stabilized discretizations, and investigate the quality of the estimators and the performance of the adaptive iteration by selected numerical experiments. © 2014 Society for Industrial and Applied Mathematics.view abstract 10.1137/130909251 **Identification of an unknown parameter function in the main part of an elliptic partial differential equation**

Aßmann, U. and Rösch, A.*Zeitschrift fur Analysis und ihre Anwendung*32 (2013)The identification of an unknown parameter function in the main part of an elliptic partial differential equation is studied. We use a Tichonov regularization with an Hs-norm and s > 0. Moreover, pointwise bounds for the unknown parameter are assumed. Existence of solutions is shown and necessary optimality conditions are established. The main contribution is the discussion of second-order sufficient optimality conditions. Here, we get a size condition of the parameter s. © European Mathematical Society.view abstract 10.4171/ZAA/1479 **A priori error estimates for a state-constrained elliptic optimal control problem**

Rösch, A. and Steinig, S.*ESAIM: Mathematical Modelling and Numerical Analysis*46 (2012)We examine an elliptic optimal control problem with control and state constraints in 3. An improved error estimate of (h s)with 3/4s1 is proven for a discretisation involving piecewise constant functions for the control and piecewise linear for the state. The derived order of convergence is illustrated by a numerical example. © EDP Sciences, SMAI, 2012.view abstract 10.1051/m2an/2011076 **A-posteriori error estimates for optimal control problems with state and control constraints**

Rösch, A. and Wachsmuth, D.*Numerische Mathematik*120 (2012)We discuss the full discretization of an elliptic optimal control problem with pointwise control and state constraints. We provide the first reliable a-posteriori error estimator that contains only computable quantities for this class of problems. Moreover, we show, that the error estimator converges to zero if one has convergence of the discrete solutions to the solution of the original problem. The theory is illustrated by numerical tests. © 2011 Springer-Verlag.view abstract 10.1007/s00211-011-0422-z **Finite element error estimates for Neumann boundary control problems on graded meshes**

Apel, T. and Pfefferer, J. and Rösch, A.*Computational Optimization and Applications*52 (2012)A specific elliptic linear-quadratic optimal control problem with Neumann boundary control is investigated. The control has to fulfil inequality constraints. The domain is assumed to be polygonal with reentrant corners. The asymptotic behaviour of two approaches to compute the optimal control is discussed. In the first the piecewise constant approximations of the optimal control are improved by a postprocessing step. In the second the control is not discretized; instead the first order optimality condition is used to determine an approximation of the optimal control. Although the quality of both approximations is in general affected by corner singularities a convergence order of 3/2 can be proven provided that the mesh is sufficiently graded. © 2011 Springer Science+Business Media, LLC.view abstract 10.1007/s10589-011-9427-x **Regularization for semilinear elliptic optimal control problems with pointwise state and control constraints**

Krumbiegel, K. and Neitzel, I. and Rösch, A.*Computational Optimization and Applications*52 (2012)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 regularization parameter are derived. © 2010 Springer Science+Business Media, LLC.view abstract 10.1007/s10589-010-9357-z **On saturation effects in the Neumann boundary control of elliptic optimal control problems**

Mateos, M. and Rösch, A.*Computational Optimization and Applications*49 (2011)A Neumann boundary control problem for a linear-quadratic elliptic optimal control problem in a polygonal domain is investigated. The main goal is to show an optimal approximation order for discretized problems after a postprocessing process. It turns out that two saturation processes occur: The regularity of the boundary data of the adjoint is limited if the largest angle of the polygon is at least 2π/3. Moreover, piecewise linear finite elements cannot guarantee the optimal order, if the largest angle of the polygon is greater than π/2. We will derive error estimates of order h α with α [1,2] depending on the largest angle and properties of the finite elements. Finally, numerical test illustrates the theoretical results. © 2009 Springer Science+Business Media, LLC.view abstract 10.1007/s10589-009-9299-5 **Semi-smooth Newton method for an optimal control problem with control and mixed control-state constraints**

Rösch, A. and Wachsmuth, D.*Optimization Methods and Software*26 (2011)A class of optimal control problems for a linear parabolic partial differential equation with control and mixed control-state constraints is considered. For this problem, a projection formula is derived that is equivalent to the necessary optimality conditions. As a main result, the superlinear convergence of a semi-smooth Newton method is shown. Moreover, we show the numerical treatment and several numerical experiments. © 2011 Taylor & Francis.view abstract 10.1080/10556780903548257 **A priori error analysis for linear quadratic elliptic neumann boundary control problems with control and State Constraints**

Krumbiegel, K. and Meyer, C. and Rösch, A.*SIAM Journal on Control and Optimization*48 (2010)In this paper we consider a state-constrained optimal control problem with boundary control, where the state constraints are imposed only in an interior subdomain. Our goal is to derive a priori error estimates for a finite element discretization with and without additional regularization. We will show that the separation of the set where the control acts and the set where the state constraints are given improves the approximation rates significantly. The theoretical results are illustrated by numerical computations. © 2010 Society for Industrial and Applied Mathematics.view abstract 10.1137/090746148 **Error estimates for joint Tikhonov and Lavrentiev regularization of constrained control problems**

Lorenz, D.A. and Rösch, A.*Applicable Analysis*89 (2010)We consider joint Tikhonov and Lavrentiev regularization of control problems with pointwise control- and state-constraints. We derive error estimates for the error which is introduced by the Tikhonov regularization. With the help of these results we show that, if the solution of the unconstrained problem has no active constraints, the same holds for the Tikhonov regularized solution if the regularization parameter is small enough and a certain source condition is fulfilled. © 2010 Taylor & Francis.view abstract 10.1080/00036811.2010.496360 **Lipschitz stability for elliptic optimal control problems with mixed control-state constraints**

Alt, W. and Griesse, R. and Metla, N. and Rösch, A.*Optimization*59 (2010)A family of linear-quadratic optimal control problems with pointwise mixed state-control constraints governed by linear elliptic partial differential equations is considered. All data depend on a vector parameter of perturbations. Lipschitz stability with respect to perturbations of the optimal control, the state and adjoint variables, and the Lagrange multipliers is established. © 2010 Taylor & Francis.view abstract 10.1080/02331930902863749

#### boundary controls

#### finite element method

#### inverse problems

#### nonlinear optimization

#### optimal control systems