Control of parabolic PDEs with time-varying spatial domain: Czochralski crystal growth process

This paper considers the optimal control problem for a class of convection-diffusion-reaction systems modelled by partial differential equations (PDEs) defined on time-varying spatial domains. The class of PDEs is characterised by the presence of a time-dependent convective-transport term which is associated with the time evolution of the spatial domain boundary. The functional analytic description of the PDE yields the representation of the initial and boundary value problem as a nonautonomous parabolic evolution equation on an appropriately defined infinite-dimensional function space. The properties of the time-varying evolution operator to guarantee existence and well posedness of the initial and boundary value problem are demonstrated which serves as the basis for the optimal control problem synthesis. An industrial application of the crystal temperature regulation problem for the Czochralski crystal growth process is considered and numerical simulation results are provided.

[1]  P. K. C. Wang,et al.  Stabilization and control of distributed systems with time-dependent spatial domains , 1990 .

[2]  Alessandra Lunardi An Introduction to Parabolic Moving Boundary Problems , 2004 .

[3]  Roberto Triggiani,et al.  Boundary feedback stabilizability of parabolic equations , 1980 .

[4]  Denis Dochain,et al.  Dynamical analysis of distributed parameter tubular reactors , 2000, Autom..

[5]  田辺 広城,et al.  Functional analytic methods for partial differential equations , 1997 .

[6]  Philippe Martin,et al.  MOTION PLANNING FOR A NONLINEAR STEFAN PROBLEM , 2003 .

[7]  I. Lasiecka Unified theory for abstract parabolic boundary problems—a semigroup approach , 1980 .

[8]  H. Abou-Kandil,et al.  Matrix Riccati Equations in Control and Systems Theory , 2003, IEEE Transactions on Automatic Control.

[9]  Alain Bensoussan,et al.  Representation and Control of Infinite Dimensional Systems, 2nd Edition , 2007, Systems and control.

[10]  A. M. Meirmanov,et al.  The Stefan Problem , 1992 .

[11]  Antonios Armaou,et al.  Robust control of parabolic PDE systems with time-dependent spatial domains , 2001, Autom..

[12]  Tosio Kato Perturbation theory for linear operators , 1966 .

[13]  Stefano Bonaccorsi,et al.  A Variational Approach to Evolution Problems with Variable Domains , 2001 .

[14]  Marcos Antón Amayuelas The Stefan problem , 2015 .

[15]  Jeffrey J. Derby,et al.  Thermal-capillary analysis of Czochralski and liquid encapsulated Czochralski crystal growth: I. Simulation , 1986 .

[16]  S. Mitter,et al.  Representation and Control of Infinite Dimensional Systems , 1992 .

[17]  John Sylvester,et al.  The heat equation in time dependent domains with insulated boundaries , 2004 .

[18]  F. Flandoli,et al.  Initial boundary value problems and optimal control for nonautonomous parabolic systems , 1991 .

[19]  J. Derby,et al.  Thermal-capillary analysis of Czochralski and liquid encapsulated Czochralski crystal growth. II - Processing strategies , 1986 .

[20]  M. Willis,et al.  ADVANCED PROCESS CONTROL , 2005 .

[21]  J. Derby,et al.  On the dynamics of Czochralski crystal growth , 1987 .

[22]  Jie Zhang,et al.  Stabilization for Markovian jump systems with partial information on transition probability based on free-connection weighting matrices , 2011, Autom..

[23]  Robert McOwen,et al.  Partial differential equations : methods and applications , 1996 .

[24]  Antonio Fasano,et al.  A Free Boundary-Value Problem Related to the Combustion of a Solid , 1985 .

[25]  G. Burton Sobolev Spaces , 2013 .

[26]  Thomas I. Seidman,et al.  Boundary feedback stabilization of a parabolic equation , 1984 .

[27]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[28]  Costas J. Spanos,et al.  Advanced process control , 1989 .

[29]  Giuseppe Savaré,et al.  Parabolic problems with mixed variable lateral conditions: An abstract approach , 1997 .

[30]  P. Christofides,et al.  Crystal temperature control in the Czochralski crystal growth process , 2001 .

[31]  Paolo Acquistapace,et al.  A unified approach to abstract linear nonautonomous parabolic equations , 1987 .

[32]  Peter E. Kloeden,et al.  Pullback attractors for a semilinear heat equation in a non-cylindrical domain , 2008 .

[33]  Denis Dochain,et al.  Sturm-Liouville systems are Riesz-spectral systems , 2003 .

[34]  Alessandra Lunardi,et al.  Smooth solutions to a class of free boundary parabolic problems , 2003 .

[35]  Hans Zwart,et al.  An Introduction to Infinite-Dimensional Linear Systems Theory , 1995, Texts in Applied Mathematics.

[36]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[37]  Peter E. Kloeden,et al.  Pullback attractors for a semilinear heat equation on time-varying domains ✩ , 2009 .

[38]  P. Kokotovic,et al.  The peaking phenomenon and the global stabilization of nonlinear systems , 1991 .

[39]  Jeffrey J. Derby,et al.  Finite-element methods for analysis of the dynamics and control of Czochralski crystal growth , 1987 .