Optimality conditions for boundary control of non‐well‐posed elliptic equations

In this paper, we discuss a boundary optimal control problem governed by a class of non‐well‐posed semi‐linear elliptic equations. By considering a well‐posed penalization problem and taking limit in the optimality system for penalization problem, we obtain the necessary optimality conditions for optimal pairs of initial control problem. Copyright © 2009 John Wiley & Sons, Ltd.     

Necessary and sufficient conditions of optimality for an optimal control problem with non‐local boundary conditions and quadratic functional

In this paper, the optimal control problem with a quadratic functional for Helmholtz equation with non‐local boundary conditions is considered. Necessary and sufficient conditions of optimality are obtained on the basis of which the existence and uniqueness of a solution to the optimal problem are proved. Copyright © 2013 John Wiley & Sons, Ltd.     

Second-order necessary optimality conditions for a discrete optimal control problem with nonlinear state equations

In this article, we study second-order necessary optimality conditions for a discrete optimal control problem with a nonconvex cost function, nonlinear state equations and mixed constraints. In order to achieve these conditions, we first establish an abstract result on the second-order necessary optimality conditions for a mathematical programming problem and then we derive the second-order necessary optimality conditions for a discrete optimal control problem. The main result of this article is illustrated by two examples.     

The problem of starting control with two intermediate moments of observation in the boundary value problem for the hyperbolic equation

In this article, a starting control problem with two intermediate moments of observation in the mixed linear problem for the second-order hyperbolic equation with quadratic functional is studied. The necessary and sufficient optimality conditions are obtained in the form of variational inequality.     

A priori and a posteriori error estimates of finite‐element approximations for elliptic optimal control problem with measure data

We analyze both a priori and a posteriori error analysis of finite‐element method for elliptic optimal control problems with measure data in a bounded convex domain in (d = 2or3). The solution of the state equation of such type of problems exhibits low regularity due to the presence of measure data, which introduces some difficulties for both theory and numerics of the finite‐element method. We first prove the existence, uniqueness, and regularity of the solution to the optimal control problem. To discretize the control problem, we use continuous piecewise linear elements for the approximations of the state and co‐state variables, whereas piecewise constant functions are used for the control variable. We derive a priori error estimates of order for the state, co‐state, and control variables in the L²‐norm. Further, global a posteriori upper bounds for the state, co‐state, and control variables in the L²‐norm are established. Moreover, local lower bounds for the errors in the state and co‐state variables and a global lower bound for the error in the control variable are obtained in the case of two space dimensions (d = 2). Numerical experiments are provided, which support our theoretical results.