Stability and distributed control of degenerate diffusion equations

Abstract This paper is concerned with stability and control of the parabolic p-Laplace equation. The autonomous equation is shown to be asymptotically stable, while the stronger property of exponential stability is guaranteed by the presence of lower-order terms satisfying a suitable growth condition. On the basis of these results, the problem of reference tracking using a distributed control input is investigated and, in particular, two approaches are discussed: finite-time stabilization and quadratic optimal control. Numerical simulations are provided to support and illustrate the theoretical results.

[1]  M. Krstić,et al.  Backstepping observers for a class of parabolic PDEs , 2005, Syst. Control. Lett..

[2]  A. S. Kalashnikov.  Some problems of the qualitative theory of non-linear degenerate second-order parabolic equations , 1987 .

[3]  Existence and regularity of a global attractor for doubly nonlinear parabolic equations. , 2002 .

[4]  T. Kilpeläinen,et al.  On the Dirichlet boundary value problem for a degenerate parabolic equation , 1996 .

[5]  Andrea Cristofaro,et al.  Robust tracking control for a class of perturbed and uncertain reaction-diffusion equations , 2014 .

[6]  Yury Orlov,et al.  Tracking Control of the Uncertain Heat and Wave Equation via Power-Fractional and Sliding-Mode Techniques , 2011, SIAM J. Control. Optim..

[7]  On the Time Derivative in an Obstacle Problem , 2010, 1005.2096.

[8]  Christophe Prieur,et al.  Global Stabilization of a Korteweg-De Vries Equation With Saturating Distributed Control , 2016, SIAM J. Control. Optim..

[9]  Tommaso Leonori,et al.  Asymptotic behavior for solutions of parabolic equations with natural growth terms and irregular data , 2006, Asymptot. Anal..

[10]  Henrik Anfinsen,et al.  Estimating the left boundary condition of coupled 1-D linear hyperbolic PDEs from right boundary sensing , 2016, 2016 European Control Conference (ECC).

[11]  Emilia Fridman,et al.  An LMI approach to Hinfinity boundary control of semilinear parabolic and hyperbolic systems , 2009, Autom..

[12]  An Lê,et al.  Eigenvalue problems for the p-Laplacian , 2006 .

[13]  Stephan Luckhaus,et al.  Quasilinear elliptic-parabolic differential equations , 1983 .

[14]  Christophe Prieur,et al.  Boundary observers for linear and quasi-linear hyperbolic systems with application to flow control , 2013, Autom..

[15]  Mahamadi Warma,et al.  The Fractional Relative Capacity and the Fractional Laplacian with Neumann and Robin Boundary Conditions on Open Sets , 2015 .

[16]  J. Vázquez,et al.  Large-time geometric properties of solutions of the evolution p-Laplacian equation , 2006 .

[17]  H. Sira-Ramírez.  Distributed sliding mode control in systems described by linear and quasilinear partial differential equations , 1989 .

[18]  Sebastian Schwarzacher,et al.  On Regularity of the Time Derivative for Degenerate Parabolic Systems , 2015, SIAM J. Math. Anal..

[19]  Existence and convergence theorems for doubly nonlinear partial differential equations of elliptic-parabolic type , 1990 .

[20]  R. G. Casten,et al.  Stability Properties of Solutions to Systems of Reaction-Diffusion Equations , 1977 .

[21]  Marco Degiovanni,et al.  Nontrivial Solutions for p-Laplace Equations with Right-Hand Side Having p-Linear Growth at Infinity , 2005 .

[22]  P. Pucci,et al.  Nonexistence for p-Laplace equations with singular weights , 2010 .

[23]  Miroslav Krstic,et al.  Adaptive control of PDES , 2007 .

[24]  Andrea Cristofaro,et al.  Lyapunov Stability Results for the Parabolic p-Laplace Equation , 2018, 2018 European Control Conference (ECC).

[25]  J. Heinonen.  Lectures on Lipschitz analysis , 2005 .

[26]  R. Manzo,et al.  On optimality conditions for optimal control problem in coefficients for Δp-Laplacian , 2014 .

[27]  Petri Juutinen,et al.  Principal eigenvalue of a very badly degenerate operator and applications , 2007 .

[28]  G. Barles,et al.  EXISTENCE AND COMPARISON RESULTS FOR FULLY NONLINEAR DEGENERATE ELLIPTIC EQUATIONS WITHOUT ZEROTH-ORDER TERM* , 2001 .

[29]  Nathaniel Chafee,et al.  Asymptotic behavior for solutions of a one-dimensional parabolic equation with homogeneous Neumann boundary conditions , 1975 .

[30]  Jesús Ildefonso Díaz Díaz,et al.  On a nonlinear parabolic problem arising in some models related to turbulent flows , 1994 .