Optimal control of a class of linear nonautonomous parabolic PDE via two-parameter semigroup representation

This paper considers the two-parameter semigroup representation of a class of parabolic partial differential equation (PDE) with time and spatially dependent coefficients. The properties of the PDE which are necessary for the initial and boundary value problem to be posed as a linear nonautonomous evolution equation on an appropriately defined infinite-dimensional function space are presented. Using these properties, the associated nonautonomous operator generates a two-parameter semigroup which yields the generalized solution of the initial and boundary value problem. The explicit expression of the two-parameter semigroup is provided and enables the application of optimal control theory for infinite dimensional systems.

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

[2]  Robert A. Brown,et al.  Theory of transport processes in single crystal growth from the melt , 1988 .

[3]  H. Noda,et al.  Optimal operation of a catalytic tubular reactor with fouling catalyst by coke deposition , 1975 .

[4]  Ruth F. Curtain,et al.  On Stabilizability of Linear Spectral Systems via State Boundary Feedback , 1985 .

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

[6]  Abdul-Majid Wazwaz,et al.  Partial differential equations : methods and applications , 2002 .

[7]  Cheng-Zhong Xu,et al.  On spectrum and Riesz basis assignment of infinite dimensional linear systems by bounded linear feedbacks , 1995, Proceedings of 1995 34th IEEE Conference on Decision and Control.

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

[9]  Jinwen Chen,et al.  Review on criteria to ensure ideal behaviors in trickle-bed reactors , 2009 .

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

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

[12]  Sun Shun-Hua,et al.  On Spectrum Distribution of Completely Controllable Linear Systems , 1981 .

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

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

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

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

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

[18]  G. Burton.  Sobolev Spaces , 2013 .

[19]  W. Marsden.  I and J , 2012 .

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

[21]  Daniel B. Henry.  Geometric Theory of Semilinear Parabolic Equations , 1989 .

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