1,472 research outputs found
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-
Superconvergence for Neumann boundary control problems governed by semilinear elliptic equations
This paper is concerned with the discretization error analysis of semilinear
Neumann boundary control problems in polygonal domains with pointwise
inequality constraints on the control. The approximations of the control are
piecewise constant functions. The state and adjoint state are discretized by
piecewise linear finite elements. In a postprocessing step approximations of
locally optimal controls of the continuous optimal control problem are
constructed by the projection of the respective discrete adjoint state.
Although the quality of the approximations is in general affected by corner
singularities a convergence order of is proven for domains
with interior angles smaller than using quasi-uniform meshes. For
larger interior angles mesh grading techniques are used to get the same order
of convergence
Sufficient conditions for unique global solutions in optimal control of semilinear equations with nonlinearity
We consider a semilinear elliptic optimal control problem possibly
subject to control and/or state constraints. Generalizing previous work we
provide a condition which guarantees that a solution of the necessary first
order conditions is a global minimum. A similiar result also holds at the
discrete level where the corresponding condition can be evaluated explicitly.
Our investigations are motivated by G\"unter Leugering, who raised the question
whether our previous results can be extended to the nonlinearity
. We develop a corresponding analysis and present several
numerical test examples demonstrating its usefulness in practice
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