Controllability for a scalar conservation law with nonlocal velocity

Abstract In this paper, we study the state controllability and nodal profile controllability for a scalar conservation law, with a nonlocal velocity, that models a highly re-entrant manufacturing system as encountered in semi-conductor production. We first prove a local state controllability result, i.e., there exists a control that drives the solution from any given initial data to any desired final data in a certain time period, provided that the initial and final data are both close to a given equilibrium ρ ¯ ⩾ 0 . We also obtain a global state controllability result for the same system, where there is no limitation on the distance between the initial and final data. Finally, we prove a nodal profile controllability result, i.e., there exists a control under which the solution starts from any initial data reaches exactly any given out-flux over a fixed time period.

[1]  M. Kawski,et al.  Analysis of a conservation law modeling a highly re-entrant manufacturing system , 2009, 0907.1274.

[2]  T. Mexia,et al.  Author ' s personal copy , 2009 .

[3]  Ta-Tsien Li,et al.  Exact Boundary Controllability for Quasi-Linear Hyperbolic Systems , 2002, SIAM J. Control. Optim..

[4]  A. Bressan,et al.  Vanishing Viscosity Solutions of Nonlinear Hyperbolic Systems , 2001, math/0111321.

[5]  Zhiqiang Wang,et al.  Global exact boundary controllability for first order quasilinear hyperbolic systems of diagonal form , 2007 .

[6]  Frédérique Clément,et al.  Multiscale Modeling of Follicular Ovulation as a Reachability Problem , 2006, Multiscale Model. Simul..

[7]  Olivier Glass,et al.  On the controllability of the 1-D isentropic Euler equation , 2007 .

[8]  Kevin Zumbrun,et al.  On a nonlocal dispersive equation modeling particle suspensions , 1999 .

[9]  A. Bressan.  Hyperbolic systems of conservation laws : the one-dimensional Cauchy problem , 2000 .

[10]  F. Ancona,et al.  On the Attainable Set for Scalar Nonlinear Conservation Laws with Boundary Control , 1998 .

[11]  Irena Lasiecka,et al.  Control Methods in PDE-Dynamical Systems , 2007 .

[12]  Tai-Ping Liu,et al.  Well‐posedness theory for hyperbolic conservation laws , 1999 .

[13]  Frédérique Clément,et al.  Multi-scale modeling of the follicle selection process in the ovary. , 2005, Mathematical biosciences.

[14]  Michael Herty,et al.  Flow control in gas networks: Exact controllability to a given demand , 2011 .

[15]  T. Horsin,et al.  On the controllability of the burger equation , 1998 .

[16]  M. Herty,et al.  Control of the Continuity Equation with a Non Local Flow , 2009, 0902.2623.

[17]  Christian A. Ringhofer,et al.  Control of Continuum Models of Production Systems , 2010, IEEE Transactions on Automatic Control.

[18]  Axel Klar,et al.  Existence of Solutions for Supply Chain Models Based on Partial Differential Equations , 2007, SIAM J. Math. Anal..

[19]  Jean-Michel Coron,et al.  Exact Boundary Controllability for 1-D Quasilinear Hyperbolic Systems with a Vanishing Characteristic Speed , 2009, SIAM J. Control. Optim..

[20]  P. LeFloch.  Hyperbolic Systems of Conservation Laws , 2002 .

[21]  J. Coron.  Control and Nonlinearity , 2007 .

[22]  D. Russell.  Controllability and Stabilizability Theory for Linear Partial Differential Equations: Recent Progress and Open Questions , 1978 .

[23]  Ta-Tsien Li.  Global classical solutions for quasilinear hyperbolic systems , 1994 .

[24]  Jean-Michel Coron,et al.  Local controllability of a 1-D tank containing a fluid modeled by the shallow water equations , 2002 .

[25]  Günter Leugering,et al.  Global boundary controllability of the de St. Venant equations between steady states , 2003 .

[26]  Christian A. Ringhofer,et al.  A Model for the Dynamics of large Queuing Networks and Supply Chains , 2006, SIAM J. Appl. Math..

[27]  José Carlos Goulart de Siqueira,et al.  Differential Equations , 1919, Nature.

[28]  Christian A. Ringhofer,et al.  A Continuum Model for a Re-entrant Factory , 2006, Oper. Res..

[29]  Tatsien Li,et al.  Boundary value problems for quasilinear hyperbolic systems , 1985 .

[30]  Hongjie Dong.  Well-posedness for a transport equation with nonlocal velocity , 2008 .

[31]  Zhiqiang Wang,et al.  Exact Controllability for Nonautonomous First Order Quasilinear Hyperbolic Systems* , 2006 .

[32]  Andrew J. Majda,et al.  A simple one-dimensional model for the three-dimensional vorticity equation , 1985 .

[33]  Yin Zong-fang.  Controllability and Observability of Weakly Discontinuous Solutions for First Order Quasilinear Hyperbolic Systems , 2010 .

[34]  Zhiqiang Wang,et al.  Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system , 2010, 1003.4411.