The finite element approximation of optimal control problems for semilinear elliptic partial differential equation is considered, where the control belongs to a finite-dimensional set and state constraints are given in finitely many points of the domain. Under the standard linear independency condition on the active gradients and a strong second-order sufficient optimality condition, optimal error estimates are derived for locally optimal controls.

The goal of this paper is to derive some error estimates for the numerical discretization of some optimal control problems governed by semilinear elliptic equations with bound constraints on the control and a finitely number of equality and inequality state constraints. We prove some error estimates for the optimal controls in the $L^{\infty }$ norm and we also obtain error estimates for the Lagrange multipliers associated to the state constraints as well as for the optimal states and optimal adjoint states....

The goal of this paper is to derive some error estimates for the numerical discretization of some optimal control problems governed by semilinear elliptic equations with bound constraints on the control and a finitely number of equality and inequality state constraints. We prove some error estimates for the optimal controls in the L∞ norm and we also obtain error estimates for the Lagrange multipliers associated to the state constraints as well as for the optimal states and optimal adjoint states. ...

Rate-independent problems are considered, where the stored energy density is a function of the gradient. The stored energy density may not be quasiconvex and is assumed to grow linearly. Moreover, arbitrary behaviour at infinity is allowed. In particular, the stored energy density is not required to coincide at infinity with a positively 1-homogeneous function. The existence of a rate-independent process is shown in the so-called energetic formulation.

We set a coupled boundary value problem between two domains of different dimension. The first one is the unit cube of Rn, n C [2,3], with a crack and the second one is the crack. this problem comes from Ciarlet et al. (1989), that obtained an analogous coupled problem. We show that the solution has singularities due to the crack. As in Grisvard (1989), we adapt the Hilbert uniqueness method of J.-L. Lions (1968,1988) in order to obtain the exact controllability of the associated wave equation with...

This paper is concerned with the global exact controllability of the semilinear heat equation (with nonlinear terms involving the state and the gradient) completed with boundary conditions of the form $\frac{\partial y}{\partial n}+f\left(y\right)=0$. We consider distributed controls, with support in a small set. The null controllability of similar linear systems has been analyzed in a previous first part of this work. In this second part we show that, when the nonlinear terms are locally Lipschitz-continuous and slightly superlinear, one...

In this paper, we consider probability measures μ and ν on a d-dimensional sphere in ${𝐑}^{d+1},d\ge 1,$ and cost functions of the form $c(𝐱,𝐲)=l\left(\frac{{|𝐱-𝐲|}^2}{2}\right)$ that generalize those arising in geometric optics where $l\left(t\right)=-logt.$ We prove that if μ and ν vanish on $(d-1)$-rectifiable sets, if |l'(t)|>0,${lim}_{t\to 0^{+}}l\left(t\right)=+\infty ,$ and $g\left(t\right):=t(2-t){\left(l^{\text{'}}\left(t\right)\right)}^2$ is monotone then there exists a unique optimal map To that transports μ onto $\nu ,$ where optimality is measured against c. Furthermore, ${inf}_{𝐱}|T_o𝐱-𝐱|>0.$ Our approach is based on direct variational arguments. In the special case when $l\left(t\right)=-logt,$ existence of optimal maps...

Here we present an approximation method for a rather broad class of first order variational problems in spaces of piece-wise constant functions over triangulations of the base domain. The convergence of the method is based on an inequality involving $W^{-1,p}$ norms obtained by Nečas and on the general framework of Γ-convergence theory.