28 research outputs found
Optimal control for the thermistor problem
This paper is concerned with the state-constrained optimal control of the two-dimensional thermistor problem, a quasi-linear coupled system of a parabolic and elliptic PDE with mixed boundary conditions. This system models the heating of a conducting material by means of direct current. Existence, uniqueness and continuity for the state system are derived by employing maximal elliptic and parabolic regularity. By similar arguments the linearized state system is discussed, while the adjoint system involving measures is investigated using a duality argument. These results allow to derive first-order necessary conditions for the optimal control problem
Optimal Control of Nonlocal Thermistor Equations
We are concerned with the optimal control problem of the well known nonlocal
thermistor problem, i.e., in studying the heat transfer in the resistor device
whose electrical conductivity is strongly dependent on the temperature.
Existence of an optimal control is proved. The optimality system consisting of
the state system coupled with adjoint equations is derived, together with a
characterization of the optimal control. Uniqueness of solution to the
optimality system, and therefore the uniqueness of the optimal control, is
established. The last part is devoted to numerical simulations.Comment: Submitted 21-March-2012; revised 11-June-2012; accepted 13-June-2012;
for publication in the International Journal of Contro
On maximal parabolic regularity for non-autonomous parabolic operators
We consider linear inhomogeneous non-autonomous parabolic problems associated
to sesquilinear forms, with discontinuous dependence of time. We show that for
these problems, the property of maximal parabolic regularity can be
extrapolated to time integrability exponents . This allows us to prove
maximal parabolic -regularity for discontinuous non-autonomous
second-order divergence form operators in very general geometric settings and
to prove existence results for related quasilinear equations
Existence result of the global attractor for a triply nonlinear thermistor problem
We study the existence and uniqueness of a bounded weak solution for a triply
nonlinear thermistor problem in Sobolev spaces. Furthermore, we prove the
existence of an absorbing set and, consequently, the universal attractor.Comment: This is a 19 pages preprint of a paper whose final and definite form
is published in 'Moroccan J. of Pure and Appl. Anal. (MJPAA)', ISSN: Online
2351-8227 -- Print 2605-636
Optimal Control of the Thermistor Problem in Three Spatial Dimensions
This paper is concerned with the state-constrained optimal control of the
three-dimensional thermistor problem, a fully quasilinear coupled system of a
parabolic and elliptic PDE with mixed boundary conditions. This system models
the heating of a conducting material by means of direct current. Local
existence, uniqueness and continuity for the state system are derived by
employing maximal parabolic regularity in the fundamental theorem of Pr\"uss.
Global solutions are addressed, which includes analysis of the linearized state
system via maximal parabolic regularity, and existence of optimal controls is
shown if the temperature gradient is under control. The adjoint system
involving measures is investigated using a duality argument. These results
allow to derive first-order necessary conditions for the optimal control
problem in form of a qualified optimality system. The theoretical findings are
illustrated by numerical results
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An effective bulk-surface thermistor model for large-area organic light-emitting diodes
The existence of a weak solution for an effective system of partial differential equations describing the electrothermal behavior of large-area organic light-emitting diodes (OLEDs) is proved. The effective system consists of the heat equation in the three-dimensional bulk glass substrate and two semi-linear equations for the current flow through the electrodes coupled to algebraic equations for the continuity of the electrical fluxes through the organic layers. The electrical problem is formulated on the (curvilinear) surface of the glass substrate where the OLED is mounted. The source terms in the heat equation are due to Joule heating and are hence concentrated on the part of the boundary where the current-flow equation is posed. The existence of weak solutions to the effective system is proved via Schauder's fixed-point theorem. Moreover, since the heat sources are a priori only in , the concept of entropy solutions is used
Optimal elliptic regularity at the crossing of a material interface and a Neumann boundary edge
We investigate optimal elliptic regularity of anisotropic div-grad operators in three dimensions at the crossing of a material interface and an edge of the spatial domain on the Neumann boundary part within the scale of Sobolev spaces