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### Open Access Journals

DCDS-B

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.

DCDS-B

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.

MCRF

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.

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