1,139 research outputs found
State-constrained optimal control of semilinear elliptic equations with nonlocal radiation interface conditions
We consider a control- and state-constrained optimal control problem
governed by a semilinear elliptic equation with nonlocal interface
conditions. These conditions occur during the modeling of diffuse-gray
conductive-radiative heat transfer. The nonlocal radiation interface
condition and the pointwise state-constraints represent the particular
features of this problem. To deal with the state-constraints, continuity of
the state is shown which allows to derive first-order necessary conditions.
Afterwards, we establish second-order sufficient conditions that account for
strongly active sets and ensure local optimality in an -neighborhood
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
Second order optimality conditions and their role in PDE control
If f : Rn R is twice continuously differentiable, fâ(u) = 0 and fââ(u) is positive definite, then u is a local minimizer of f. This paper surveys the extension of this well known second order suffcient optimality condition to the case f : U R, where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite-dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How second order sufficient optimality conditions must be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled order sufficient optimality condition to the case f : U R, where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite-dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How second order sufficient optimality conditions must be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled?
It turns out that infinite dimensions cause new difficulties that do not occur in finite dimensions. We will be faced with the surprising fact that the space, where fââ(u) exists can be useless to ensure positive definiteness of the quadratic form v fââ(u)v2. In this context, the famous two-norm discrepancy, its consequences, and techniques for overcoming this difficulty are explained. To keep the presentation simple, the theory is developed for problems in function spaces with simple box constraints of the form a = u = Ă. The theory of second order conditions in the control of partial differential equations is presented exemplarily for the nonlinear heat equation. Different types of critical cones are introduced, where the positivity of fââ(u) must be required. Their form depends on whether a so-called Tikhonov regularization term is part of the functional f or not. In this context, the paper contains also new results that lead to quadratic growth conditions in the strong sense.
As a first application of second-order sufficient conditions, the stability of optimal solutions with respect to perturbations of the data of the control problem is discussed. Second, their use in analyzing the discretization of control problems by finite elements is studied. A survey on further related topics, open questions, and relevant literature concludes the paper.The first author was partially supported by the Spanish Ministerio de EconomĂa y Competitividad under project MTM2011-22711, the second author by DFG in the framework of the Collaborative Research Center SFB 910, project B6
Optimal Control of the Thermistor Problem in Three Spatial Dimensions
This paper is concerned with the state-constrained optimal control of the
three-dimensional thermistor problem, a fully quasilinear coupled system of a
parabolic and elliptic PDE with mixed boundary conditions. This system models
the heating of a conducting material by means of direct current. Local
existence, uniqueness and continuity for the state system are derived by
employing maximal parabolic regularity in the fundamental theorem of Pr\"uss.
Global solutions are addressed, which includes analysis of the linearized state
system via maximal parabolic regularity, and existence of optimal controls is
shown if the temperature gradient is under control. The adjoint system
involving measures is investigated using a duality argument. These results
allow to derive first-order necessary conditions for the optimal control
problem in form of a qualified optimality system. The theoretical findings are
illustrated by numerical results
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
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