Bang-Bang Property of Time Optimal Control for a Kind of Microwave Heating Problem

In this paper, the bang–bang property and the existence of the solution for the time optimal control of a specific kind of microwave heating problem are studied. The mathematical model for the time optimal control problem (P) of microwave heating is formulated, in which the governing equations of the controlled system are the weak coupling system of Maxwell equations and the heat equation. Based on a new observability estimate of the heat controlled system, the controllability of the system is derived by proving the null controllability of an equivalent system. The existence of the time optimal control of microwave heating is acquired under some stated assumptions, enabling the bang–bang property of the time optimal control problem (P) to be achieved.

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

[2]  D. Russell,et al.  The Dirichlet-Neumann boundary control problem associated with Maxwell's equations in a cylindrical region , 1986 .

[3]  Lijuan Wang,et al.  The Bang-Bang principle of time optimal controls for the heat equation with internal controls , 2007, Syst. Control. Lett..

[4]  Sohrab Effati,et al.  Time optimal control problem of the heat equation with thermal source , 2014, IMA J. Math. Control. Inf..

[5]  Lijuan Wang,et al.  Bang-bang property for time optimal control of semilinear heat equation , 2014 .

[6]  Viorel Barbu,et al.  Analysis and control of nonlinear infinite dimensional systems , 1993 .

[7]  Wei Wei,et al.  An optimal control problem for microwave heating , 2012 .

[8]  S. S. Krigman,et al.  Boundary controllability of Maxwell's equations with nonzero conductivity inside a cube, I Spectral controllability , 2007 .

[9]  Karl Kunisch,et al.  TIME OPTIMAL CONTROL OF THE HEAT EQUATION WITH POINTWISE CONTROL CONSTRAINTS , 2013 .

[10]  Carleman estimates and controllability results for the one-dimensional heat equation with BV coefficients , 2007 .

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

[12]  N. Weck,et al.  Exact Boundary Controllability of a Maxwell Problem , 2000, SIAM J. Control. Optim..

[13]  K. Kime.  Boundary controllability of Maxwell's equations in a spherical region , 1987 .

[14]  Ekkehard W. Sachs,et al.  Optimal Control of the Sterilization of Prepackaged Food , 1999, SIAM J. Optim..

[15]  Jiongmin Yong,et al.  Optimal Control Theory for Infinite Dimensional Systems , 1994 .

[16]  J. Lagnese,et al.  Exact boundary controllability of Maxwell's equations in a general region , 1989 .

[17]  A. Metaxas.  Foundations of Electroheat : A Unified Approach , 1996 .

[18]  Karl Kunisch,et al.  Bang-bang property of time optimal controls of semilinear parabolic equation , 2015 .

[19]  Serge Nicaise,et al.  Exact Boundary Controllability of Maxwell's Equations in Heterogeneous Media and an Application to an Inverse Source Problem , 2000, SIAM J. Control. Optim..

[20]  F. Tröltzsch.  Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .

[21]  H. Yin.  Regularity of weak solution to Maxwell's equations and applications to microwave heating , 2004 .

[22]  Karl Kunisch,et al.  The bang-bang property of time optimal controls for the Burgers equation , 2014 .

[23]  M. Eller,et al.  Exact Boundary Controllability of Electromagnetic Fields in a General Region , 2002 .

[24]  Gengsheng Wang,et al.  Linfinity-Null Controllability for the Heat Equation and Its Consequences for the Time Optimal Control Problem , 2008, SIAM J. Control. Optim..