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 characterisation 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.

[1]  Dietmar Hömberg,et al.  Optimal Control for the Thermistor Problem , 2009, SIAM J. Control. Optim..

[2]  Bin Liu,et al.  Optimal control problem for stochastic evolution equations in Hilbert spaces , 2010, Int. J. Control.

[3]  Meir Shillor,et al.  Existence of a Solution to the Stefan Problem with Joule′s Heating , 1993 .

[4]  J. Lions.  Optimal Control of Systems Governed by Partial Differential Equations , 1971 .

[6]  Giovanni Cimatti.  Optimal control for the thermistor problem with a current limiting device , 2007, IMA J. Math. Control. Inf..

[7]  D. R. Westbrook,et al.  Numerical solutions of the thermistor equations , 1997 .

[8]  Numerical approximation of the thermistor problem , 2007, 0711.0597.

[9]  Michel Chipot,et al.  The thermistor problem: existence, smoothness, uniqueness, blowup , 1994 .

[10]  Hyung-Chun Lee,et al.  Analysis of Optimal Control Problems for the Two-Dimensional Thermistor System , 2005, SIAM J. Control. Optim..

[12]  Kwok K. Ng,et al.  Complete guide to semiconductor devices , 1995 .

[13]  Moulay Rchid Sidi Ammi,et al.  Optimal Control for a nonlocal parabolic problem resulting from thermistor system , 2007 .

[14]  S. Singh.  Nonlinear Functional Analysis and Its Applications , 1986 .

[15]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[16]  Vladimir A. Protopopescu,et al.  Optimal Control of a Convective Boundary Condition in a Thermistor Problem , 2008, SIAM J. Control. Optim..

[17]  Jaime E. Muñoz Rivera,et al.  Optimal control theory for ambient pollution , 2010, Int. J. Control.

[18]  J. Lions.  Quelques méthodes de résolution de problèmes aux limites non linéaires , 1969 .

[19]  A. Lacey.  Thermal runaway in a non-local problem modelling Ohmic heating. Part II: General proof of blow-up and asymptotics of runaway , 1995, European Journal of Applied Mathematics.

[20]  Delfim F. M. Torres,et al.  Necessary Optimality Conditions for a Dead Oil Isotherm Optimal Control Problem , 2006, math/0612376.

[21]  E. Boschi.  Recensioni: J. L. Lions - Quelques méthodes de résolution des problémes aux limites non linéaires. Dunod, Gauthier-Vi;;ars, Paris, 1969; , 1971 .

[23]  M. Pachter,et al.  Optimal control of partial differential equations , 1980 .

[24]  E. Zeidler.  Nonlinear functional analysis and its applications , 1988 .

[25]  José Francisco Rodrigues,et al.  A NONLINEAR PARABOLIC SYSTEM ARISING IN THERMOMECHANICS AND IN THERMOMAGNETISM , 1992 .

[26]  Ian A. Frigaard,et al.  Temperature surges in current-limiting circuit devices , 1992 .