Numerical approximation of control problems of non-monotone and non-coercive semilinear elliptic equations

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 ထ (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.The first two authors were partially supported by the Spanish Ministerio de Economía, Industria y Competitividad under project MTM2017-83185-

An approximation theory for the identification of nonlinear distributed parameter systems

An abstract approximation framework for the identification of nonlinear distributed parameter systems is developed. Inverse problems for nonlinear systems governed by strongly maximal monotone operators (satisfying a mild continuous dependence condition with respect to the unknown parameters to be identified) are treated. Convergence of Galerkin approximations and the corresponding solutions of finite dimensional approximating identification problems to a solution of the original finite dimensional identification problem is demonstrated using the theory of nonlinear evolution systems and a nonlinear analog of the Trotter-Kato approximation result for semigroups of bounded linear operators. The nonlinear theory developed here is shown to subsume an existing linear theory as a special case. It is also shown to be applicable to a broad class of nonlinear elliptic operators and the corresponding nonlinear parabolic partial differential equations to which they lead. An application of the theory to a quasilinear model for heat conduction or mass transfer is discussed

A Neumann interface optimal control problem with elliptic PDE constraints and its discretization and numerical analysis

We study an optimal control problem governed by elliptic PDEs with interface, which the control acts on the interface. Due to the jump of the coefficient across the interface and the control acting on the interface, the regularity of solution of the control problem is limited on the whole domain, but smoother on subdomains. The control function with pointwise inequality constraints is served as the flux jump condition which we called Neumann interface control. We use a simple uniform mesh that is independent of the interface. The standard linear finite element method can not achieve optimal convergence when the uniform mesh is used. Therefore the state and adjoint state equations are discretized by piecewise linear immersed finite element method (IFEM). While the accuracy of the piecewise constant approximation of the optimal control on the interface is improved by a postprocessing step which possesses superconvergence properties; as well as the variational discretization concept for the optimal control is used to improve the error estimates. Optimal error estimates for the control, suboptimal error estimates for state and adjoint state are derived. Numerical examples with and without constraints are provided to illustrate the effectiveness of the proposed scheme and correctness of the theoretical analysis.Comment: 31pages, 12 figures, 4 table

Challenges in Optimal Control of Nonlinear PDE-Systems

The workshop focussed on various aspects of optimal control problems for systems of nonlinear partial differential equations. In particular, discussions around keynote presentations in the areas of optimal control of nonlinear/non-smooth systems, optimal control of systems involving nonlocal operators, shape and topology optimization, feedback control and stabilization, sparse control, and associated numerical analysis as well as design and analysis of solution algorithms were promoted. Moreover, also aspects of control of fluid structure interaction problems as well as problems arising in the optimal control of quantum systems were considered

International Conference on Nonlinear Differential Equations and Applications

Dear Participants, Colleagues and Friends It is a great honour and a privilege to give you all a warmest welcome to the first Portugal-Italy Conference on Nonlinear Differential Equations and Applications (PICNDEA). This conference takes place at the Colégio Espírito Santo, University of Évora, located in the beautiful city of Évora, Portugal. The host institution, as well the associated scientific research centres, are committed to the event, hoping that it will be a benchmark for scientific collaboration between the two countries in the area of mathematics. The main scientific topics of the conference are Ordinary and Partial Differential Equations, with particular regard to non-linear problems originating in applications, and its treatment with the methods of Numerical Analysis. The fundamental main purpose is to bring together Italian and Portuguese researchers in the above fields, to create new, and amplify previous collaboration, and to follow and discuss new topics in the area