Null controllability of degenerate parabolic equation with memory

In this paper, we analyze the null controllability property for a degenerate parabolic equation involving memory terms with a locally distributed control. We first derive a null controllability result for a nonhomogeneous degenerate heat equation via new Carleman estimates with weighted time functions that do not blow up at t = 0. Then this result is successfully used with a classical fixed point to obtain null controllability for the initial memory system.

[1]  Diego Pallara,et al.  Degenerate Self-adjoint Evolution Equations on the Unit Interval , 1998 .

[2]  Xu Zhang,et al.  Controllability of Evolution Equations with Memory , 2017, SIAM J. Control. Optim..

[3]  M. Grasselli,et al.  Abstract nonlinear Volterra integrodifferential equations with nonsmooth kernels , 1991 .

[4]  Piermarco Cannarsa,et al.  Global Carleman Estimates for Degenerate Parabolic Operators With Applications , 2016 .

[5]  S. Kakutani.  A generalization of Brouwer’s fixed point theorem , 1941 .

[6]  Oleg Yu. Imanuvilov,et al.  Controllability of Evolution equations , 1996 .

[7]  Jiongmin Yong,et al.  Exact Controllability of the Heat Equation with Hyperbolic Memory Kernel , 2005 .

[8]  Luciano Pandolfi,et al.  Heat equation with memory: Lack of controllability to rest , 2009 .

[9]  Mimmo Iannelli,et al.  Controllability of the heat equation with memory , 2000, Differential and Integral Equations.

[10]  L. Maniar,et al.  Carleman Estimates and null controllability of coupled degenerate systems , 2011, 1111.3665.

[11]  G. Fragnelli,et al.  Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations , 2015, 1508.04014.

[12]  G. Fragnelli,et al.  Carleman estimates and observability inequalities for parabolic equations with interior degeneracy , 2012, 1211.6151.

[13]  J. M. Rivera,et al.  Exact boundary controllability in thermoelasticity with memory , 2003 .

[14]  E. Zuazua,et al.  Null Controllability for Wave Equations with Memory , 2015, 1507.02359.

[15]  P. Cannarsa,et al.  Carleman estimates for degenerate parabolic operators with applications to null controllability , 2006 .

[16]  P. Martinez,et al.  Carleman estimates for one-dimensional degenerate heat equations , 2006 .

[17]  Krishnan Balachandran,et al.  On a class of non-linear parabolic control systems with memory effects , 2008, Int. J. Control.

[18]  E. Zuazua,et al.  Null controllability of a system of viscoelasticity with a moving control , 2013, 1303.3452.

[19]  J. Flores,et al.  Null controllability of one dimensional degenerate parabolic equations with first order terms , 2019, 1907.03321.

[20]  Sergio Guerrero,et al.  Remarks on non controllability of the heat equation with memory , 2013 .

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

[22]  Muming Zhang,et al.  On the Controllability of a Class of Degenerate Parabolic Equations with Memory , 2018 .

[23]  Q. Tao,et al.  On the null controllability of heat equation with memory , 2016 .

[24]  Sergio Guerrero,et al.  Global Carleman Inequalities for Parabolic Systems and Applications to Controllability , 2006, SIAM J. Control. Optim..

[25]  Krishnan Balachandran,et al.  Null controllability of nonlinear heat equation with memory effects , 2009 .

[26]  J. Lions,et al.  Non-homogeneous boundary value problems and applications , 1972 .

[27]  Hang Gao,et al.  Interior approximate and null controllability of the heat equation with memory , 2014, Comput. Math. Appl..

[28]  M. Fadili,et al.  Null controllability of $$\varvec{n}$$n-coupled degenerate parabolic systems with $$\varvec{m}$$m-controls , 2017 .

[29]  Patrick Martinez,et al.  Carleman Estimates for a Class of Degenerate Parabolic Operators , 2008, SIAM J. Control. Optim..

[30]  G. Fragnelli,et al.  Carleman estimates for parabolic equations with interior degeneracy and Neumann boundary conditions , 2015, Journal d'Analyse Mathématique.