Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's
Konstantinos Chrysafinos Efthimios N. Karatzas
Discrete & Continuous Dynamical Systems - B 2012, 17(5): 1473-1506 doi: 10.3934/dcdsb.2012.17.1473
A discontinuous Galerkin finite element method for an optimal control problem having states constrained to semilinear parabolic PDE's is examined. The schemes under consideration are discontinuous in time but conforming in space. It is shown that under suitable assumptions, the error estimates of the corresponding optimality system are of the same order to the standard linear (uncontrolled) parabolic problem. These estimates have symmetric structure and are also applicable for higher order elements.
keywords: distributed control discontinuous Galerkin Symmetric error estimates semi-linear parabolic PDE's.
Error estimates for time-discretizations for the velocity tracking problem for Navier-Stokes flows by penalty methods
Konstantinos Chrysafinos
Discrete & Continuous Dynamical Systems - B 2006, 6(5): 1077-1096 doi: 10.3934/dcdsb.2006.6.1077
Semi-discrete in time approximations of the velocity tracking problem are studied based on a pseudo-compressibility approach. Two different methods are used for the analysis of the corresponding optimality system. The first one, the classical penalty formulation, leads to estimates of order $k + \varepsilon$, under suitable regularity assumptions. The estimate is based on previously derived results for the solution of the unsteady Navier-Stokes problem by penalty methods (see e.g. Jie Shen [26]) and the Brezzi-Rappaz-Raviart theory (see e.g. [12]). The second one, based on the artificially compressible optimality system, leads to an improved estimate of the form $k + \varepsilon k$ for the linearized system.
keywords: Navier Stokes equations semidiscrete error estimates. artificial compressibility velocity tracking Optimal control
Analysis and optimal control of some quasilinear parabolic equations
Eduardo Casas Konstantinos Chrysafinos
Mathematical Control & Related Fields 2018, 8(3&4): 607-623 doi: 10.3934/mcrf.2018025

In this paper, we consider optimal control problems associated with a class of quasilinear parabolic equations, where the coefficients of the elliptic part of the operator depend on the state function. We prove existence, uniqueness and regularity for the solution of the state equation. Then, we analyze the control problem. The goal is to get first and second order optimality conditions. To this aim we prove the necessary differentiability properties of the relation control-to-state and of the cost functional.

keywords: Optimal control quasilinear parabolic equations first and second order optimality conditions

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