Optimality Conditions for Semilinear Parabolic Equations with Controls in Leading Term

An optimal control problem for semilinear parabolic partial differential equations is considered. The control variable appears in the leading term of the equation. Necessary conditions for optimal controls are established by the method of homogenizing spike variation. Results for problems with state constraints are also stated.Comment: 22 apges, 1 figure

Optimal control of semilinear elliptic equations in measure spaces

Optimal control problems in measure spaces governed by semilinear elliptic equations are considered. First order optimality conditions are derived and structural properties of their solutions, in particular sparsity, are discussed. Necessary and sufficient second order optimality conditions are obtained as well. On the basis of the sufficient conditions, stability of the solutions is analyzed. Highly nonlinear terms can be incorporated by utilizing an L∞(Ω) regularity result for solutions of the first order necessary optimality conditions.This author’s research was supported by Spanish Ministerio de Economía y Competitividad under project MTM2011-22711

First- and second-order optimality conditions for a class of optimal control problems with quasilinear elliptic equations

A class of optimal control problems for quasilinear elliptic equations is considered, where the coefficients of the elliptic differential operator depend on the state function. First- and second-order optimality conditions are discussed for an associated control-constrained optimal control problem. Main emphasis is laid on second-order sufficient optimality conditions. To this aim, the regularity of the solutions to the state equation and its linearization is studied in detail and the Pontryagin maximum principle is derived. One of the main difficulties is the nonmonotone character of the state equation

Optimal distributed control of a nonlocal convective Cahn-Hilliard equation by the velocity in 3D

In this paper we study a distributed optimal control problem for a nonlocal convective Cahn--Hilliard equation with degenerate mobility and singular potential in three dimensions of space. While the cost functional is of standard tracking type, the control problem under investigation cannot easily be treated via standard techniques for two reasons: the state system is a highly nonlinear system of PDEs containing singular and degenerating terms, and the control variable, which is given by the velocity of the motion occurring in the convective term, is nonlinearly coupled to the state variable. The latter fact makes it necessary to state rather special regularity assumptions for the admissible controls, which, while looking a bit nonstandard, are however quite natural in the corresponding analytical framework. In fact, they are indispensable prerequisites to guarantee the well-posedness of the associated state system. In this contribution, we employ recently proved existence, uniqueness and regularity results for the solution to the associated state system in order to establish the existence of optimal controls and appropriate first-order necessary optimality conditions for the optimal control problem

Characterization of maximum hands-off control

Maximum hands-off control aims to maximize the length of time over which zero actuator values are applied to a system when executing specified control tasks. To tackle such problems, recent literature has investigated optimal control problems which penalize the size of the support of the control function and thereby lead to desired sparsity properties. This article gives the exact set of necessary conditions for a maximum hands-off optimal control problem using an $L_0$-(semi)norm, and also provides sufficient conditions for the optimality of such controls. Numerical example illustrates that adopting an $L_0$ cost leads to a sparse control, whereas an $L_1$-relaxation in singular problems leads to a non-sparse solution.Comment: 6 page

Mixed and discontinuous finite volume element schemes for the optimal control of immiscible flow in porous media

We introduce a family of hybrid discretisations for the numerical approximation of optimal control problems governed by the equations of immiscible displacement in porous media. The proposed schemes are based on mixed and discontinuous finite volume element methods in combination with the optimise-then-discretise approach for the approximation of the optimal control problem, leading to nonsymmetric algebraic systems, and employing minimum regularity requirements. Estimates for the error (between a local reference solution of the infinite dimensional optimal control problem and its hybrid approximation) measured in suitable norms are derived, showing optimal orders of convergence