Null controllability for degenerate parabolic equations with a nonlocal space term

We consider two degenerate heat equations with a nonlocal space term, studying, in particular, their null controllability property. To this aim, we first consider the associated nonhomogeneous degenerate heat equations: we study their well posedness, the Carleman estimates for the associated adjoint problems and, finally, the null controllability. Then, as a consequence, using the Kakutani’s fixed point Theorem, we deduce the null controllability property for the initial nonlocal problems.

[1]  Angelo Favini,et al.  Degenerate differential equations in Banach spaces , 1998 .

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

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

[4]  Sebastian Aniţa,et al.  Analysis and Control of Age-Dependent Population Dynamics , 2010 .

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

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

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

[8]  E. Fernández-Cara,et al.  Global Carleman estimates for solutions of parabolic systems defined by transposition and some applications to controllability , 2006 .

[9]  Jacques Simeon,et al.  Compact Sets in the Space L~(O, , 2005 .

[10]  Genni Fragnelli,et al.  Null controllability for a singular heat equation with a memory term , 2020, Electronic Journal of Qualitative Theory of Differential Equations.

[11]  Mamadou Gueye.  Insensitizing controls for the Navier-Stokes equations , 2013 .

[12]  E. Fernández-Caraa,et al.  Local exact controllability of the Navier – Stokes system ✩ , 2004 .

[13]  Piermarco Cannarsa,et al.  Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form , 2008 .

[14]  Takéo Takahashi,et al.  Local controllability to stationary trajectories of a Burgers equation with nonlocal viscosity , 2018 .

[15]  G. Fragnelli,et al.  Carleman Estimates and Controllability for a Degenerate Structured Population Model , 2020, Applied Mathematics & Optimization.

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

[17]  Umberto Biccari,et al.  Null Controllability of Linear and Semilinear Nonlocal Heat Equations with an Additive Integral Kernel , 2019, SIAM J. Control. Optim..

[18]  Two severely ill-posed linear parabolic problems , 2011 .

[19]  David E. Edmunds,et al.  Spectral Theory and Differential Operators , 1987, Oxford Scholarship Online.

[20]  Controllability of degenerate parabolic equation with memory , 2020 .

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

[22]  Enrique Zuazua,et al.  Null Controllability of Linear Heat and Wave Equations with Nonlocal Spatial Terms , 2016, SIAM J. Control. Optim..

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

[24]  M. Grinfeld,et al.  Local vs. non-local interactions in population dynamics , 1989 .

[25]  Tsuyoshi Murata,et al.  {m , 1934, ACML.

[26]  À. Calsina.  NON-LOCAL REACTION-DIFFUSIO N EQUATIONS MODELLIN G PREDATOR-PREY COEVOLUTIO N , 1994 .

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

[28]  Qualitative properties for a pseudo-parabolic equation with nonlocal reaction term , 2019, Boundary Value Problems.

[29]  E. Zuazua,et al.  Internal Controllability for Parabolic Systems Involving Analytic Non-local Terms , 2018, Chinese Annals of Mathematics, Series B.

[30]  Mircea Sofonea,et al.  Elements of Nonlinear Analysis , 2013 .

[31]  Dimitri Mugnai,et al.  Control of Degenerate and Singular Parabolic Equations , 2021 .