Domingo A. Tarzia

Domingo A. Tarzia

We consider a bounded domain $Ω$ in $\mathbb{R}^{n}$ whose regular boundary $\partialΩ$ consists of the union of two disjoint portions $Γ_{1}$ and $Γ_{2}$ with $meas(Γ_{1})>0$. The convergence of a family of continuous distributed mixed elliptic optimal control problems (DMEOCPs) $P_α$, governed by elliptic variational equalities (EVE), when the parameter $α$ goes to infinity was studied in Gariboldi-Tarzia, Appl. Math. Optim. (2003). It has been proved that the optimal control (OC), and their corresponding system and adjoint system states (SASSs) are strongly convergent, in adequate functional spaces, to the OC, and the SASSs of another CDMEOPC $P$ governed also by an EVE with a different boundary condition on $Γ_{1}$. We consider the discrete approximations $P_{hα}$ and $P_{h}$ of the OCPs $P_α$ and $P$ respectively, for each $h>0$ and $α>0$, through the finite element method with parameter $h$. We also discrete the EVEs which define the SASSs, and the corresponding cost functional of the DMEOCPs $P_α$ and $P$. The goal is to study the double convergence of this family of discrete DMEOCPs $P_{hα}$ when $α\to +\infty$ and $h\to 0$ simultaneously. We prove the convergence of the discrete OCs, the discrete SASSs of the family $P_{hα}$ to the corresponding to the discrete DMEOCP $P_{h}$ when $α\to +\infty$, for each $h>0$,. We study the convergence of the discrete OCPs $P_{hα}$ and $P_{h}$ when $h\to 0$ obtaining a commutative diagram which relates the continuous and discrete DMEOCPs $P_{hα}$, $P_{h}$, $P_α$ and $P$ by taking the limits $h\to 0$ and $α\to +\infty$ respectively. We also study the double convergence of $P_{hα}$ to $P$ when $(h,α)\to (0,+\infty)$.

Julieta Bollati, Mariela C. Olguin, Domingo A. Tarzia

We consider two steady-state heat conduction systems called, $S$ and $S_α$, in a multidimensional bounded domain $D$ for the Poisson equation with source energy $g$. In one system, we impose mixed boundary conditions (temperature $b$ on the boundary $Γ_1$, heat flux $q$ on $Γ_2$ and an adiabatic condition on $Γ_3$). In the other system, the condition on $Γ_1$ is replaced by a convective heat flux condition with coefficient $α$. For each of these systems, we consider three associated optimization problems $(P_{i})$ and $(P_{iα})$, $i=1,2,3$, where the variable is the source energy $g$, the heat flux $q$ and the environmental temperature $b$, respectively. In the particular case where $D$ is a rectangle, the explicit continuous optimization variables and the corresponding state of the systems are known. In the present work, by using a finite difference scheme, we obtain the discrete systems $({S^h})$ and ${(S^h_α)}$ and discrete optimization problems ${(P^h_i)}$ and ${(P^h_{i α})}$, $i=1,2,3$, where $h$ is the space step in the discretization. Explicit discrete solutions are found, and convergence and estimation errors results are proved when $h$ goes to zero and when $α$ goes to infinity. Moreover, some numerical simulations are provided in order to test theoretical results. Finally, we note that the use of a three-point finite-difference approximation for the Neumann or Robin boundary condition at the boundary improves the global order of convergence from $O(h)$ to $O(h^2)$.

Mariela C. Olguin, Domingo A. Tarzia

The numerical analysis of a family of distributed mixed optimal control problems governed by elliptic variational inequalities (with parameter $α>0$) is obtained through the finite element method when its parameter $h\rightarrow 0$. We also obtain the limit of the discrete optimal control and the associated state system solutions when $α\rightarrow \infty$ (for each $h>0$) and a commutative diagram for two continuous and two discrete optimal control and its associated state system solutions is obtained when $h\rightarrow 0$ and $α\rightarrow \infty$. Moreover, the double convergence is also obtained when $(h, α)\rightarrow(0, \infty)$.

Mariela Olguín, Domingo A. Tarzia

A continuous optimal control problem governed by an elliptic variational inequality was considered in Boukrouche-Tarzia, Comput. Optim. Appl., 53 (2012), 375-392 where the control variable is the internal energy $g$. It was proved the existence and uniqueness of the optimal control and its associated state system. The objective of this work is to make the numerical analysis of the above optimal control problem, through the finite element method with Lagrange's triangles of type 1. We discretize the elliptic variational inequality which define the state system and the corresponding cost functional, and we prove that there exists a discrete optimal control and its associated discrete state system for each positive $h$ (the parameter of the finite element method approximation). Finally, we show that the discrete optimal control and its associated state system converge to the continuous optimal control and its associated state system when the parameter $h$ goes to zero.

Domingo A. Tarzia

We consider a bounded domain D whose regular boundary consists of the union of two portions F1 and F2. The convergence of a family of continuous Neumann boundary mixed elliptic optimal control problems (Pa), governed by elliptic variational equalities, when the parameter a of the family goes to infinity was studied in Gariboldi - Tarzia, Adv. Diff. Eq. Control Processes, 1 (2008), 113-132, being the control variable the heat flux on the boundary F2. It has been proved that the optimal control problem (Pa) are strongly convergent to another optimal control (P) governed also by an elliptic variational equality with a different boundary condition on the portion of the boundary F1. We consider the discrete approximations (Pha) and (Ph) of the optimal control problems (Pa) and (P) respectively, for each h>0, a>0, through the finite element method with Lagrange's triangles of type 1 with parameter h (the longest side of the triangles). We also discrete the elliptic variational equalities which define the system and their adjoint system states, and the corresponding cost functional of the Neumann boundary optimal control problems (Pa) and (P). The goal of this paper is to study the convergence of this family of discrete Neumann boundary mixed elliptic optimal control problems (Pha) when the parameter a goes to infinity. We prove the convergence of the discrete optimal controls, the discrete system and adjoint system states of the family (Pha) to the corresponding to the discrete Neumann boundary mixed elliptic optimal control problem (Ph) when a goes to infinity, for each h>0, in adequate functional spaces. We also study the convergence when h goes to zero and we obtain a commutative diagram which relates the continuous and discrete Neumann boundary mixed elliptic optimal control problems (Pha), (Pa), (Ph) and (P) by taking the limits h goes to zero and a goes to infinity respectively.