In this work, decay estimates are derived for the solutions of 1-D linear parabolic PDEs with disturbances at both boundaries and distributed disturbances. The decay estimates are given in the L2 and H1 norms of the solution and discontinuous disturbances are allowed. Although an eigenfunction expansion for the solution is exploited for the proof of the decay estimates, the estimates do not require knowledge of the eigenvalues and the eigenfunctions of the corresponding Sturm–Liouville operator. Examples show that the obtained results can be applied for the stability analysis of parabolic PDEs with nonlocal terms.
Mots clés : Parabolic partial differential equation, input-to-state stability, non-local PDEs, decay estimates, boundary disturbances
@article{COCV_2018__24_4_1511_0,
author = {Karafyllis, Iasson and Krstic, Miroslav},
title = {Decay estimates for {1-D} parabolic {PDES} with boundary disturbances},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {1511--1540},
publisher = {EDP-Sciences},
volume = {24},
number = {4},
year = {2018},
doi = {10.1051/cocv/2018043},
mrnumber = {3922445},
zbl = {1412.35159},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2018043/}
}
TY - JOUR AU - Karafyllis, Iasson AU - Krstic, Miroslav TI - Decay estimates for 1-D parabolic PDES with boundary disturbances JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2018 SP - 1511 EP - 1540 VL - 24 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2018043/ DO - 10.1051/cocv/2018043 LA - en ID - COCV_2018__24_4_1511_0 ER -
%0 Journal Article %A Karafyllis, Iasson %A Krstic, Miroslav %T Decay estimates for 1-D parabolic PDES with boundary disturbances %J ESAIM: Control, Optimisation and Calculus of Variations %D 2018 %P 1511-1540 %V 24 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2018043/ %R 10.1051/cocv/2018043 %G en %F COCV_2018__24_4_1511_0
Karafyllis, Iasson; Krstic, Miroslav. Decay estimates for 1-D parabolic PDES with boundary disturbances. ESAIM: Control, Optimisation and Calculus of Variations, Tome 24 (2018) no. 4, pp. 1511-1540. doi : 10.1051/cocv/2018043. http://www.numdam.org/articles/10.1051/cocv/2018043/
[1] , and , D1-input-to-state stability of a time-varying nonhomogeneous diffusive equation subject to boundary disturbances. Proceedings of the American Control Conference, Montreal, QC (2012) 2978–2983.
[2] , , and , A strict control Lyapunov function for a diffusion equation with time-varying distributed coefficients. IEEE Trans. Autom. Control 58 (2013) 290–303. | DOI | MR | Zbl
[3] and , Elementary Differential Equations and Boundary Value Problems, 6th edn. Wiley (1997). | Zbl
[4] and , On the uniform input-to-state stability of reaction-diffusion systems, in Proceedings of the 49th Conference on Decision and Control, Atlanta, GA, USA (2010) 6547–6552.
[5] and , Local ISS of reaction-diffusion systems. Proceedings of the 18th IFAC World Congress, Milano, Italy (2011) 11018–11023.
[6] and , Input-to-state stability of infinite-dimensional control systems. Math. Control Signals Syst. 25 (2013) 1–35. | DOI | MR | Zbl
[7] and , Input-to-state stability of nonlinear impulsive systems. SIAM J. Control Optim. 51 (2013) 1962–1987. | DOI | MR | Zbl
[8] , Extension of a property of the heat equation to linear thermoelasticity and other theories. Q. Appl. Math. 40 (1982) 319–330. | DOI | MR | Zbl
[9] , A decreasing property of solutions of parabolic equations with applications to thermoelasticity. Q. Appl. Math. 40 (1983) 468–475. | DOI | MR | Zbl
[10] and , Large time behavior for convection-diffusion equations in ℝN with asymptotically constant diffusion. Commun. Partial Differ. Equ. 24 (1999) 1283–1340. | DOI | MR | Zbl
[11] , Finite difference methods for a nonlocal boundary value problem for the heat equation. BIT 31 (1991) 245–261. | DOI | MR | Zbl
[12] and , Galerkin methods for a semilinear parabolic problem with nonlocal boundary conditions. Adv. Comput. Math. 6 (1996) 243–262. | DOI | MR | Zbl
[13] , Monotone decay of solutions of parabolic equations with nonlocal boundary conditions. Q. Appl. Math. 44 (1986) 401–407. | DOI | MR | Zbl
[14] and , Invariant manifolds and the long-time asymptotics of the Navier–Stokes and vorticity equations on ℝ2. Arch. Ration. Mech. Anal. 163 (2002) 209–258. | DOI | MR | Zbl
[15] , and , A description of all possible decay rates for solutions of some semilinear parabolic equations. J. Math. Pures Appl. 103 (2015) 868–899. | DOI | MR | Zbl
[16] ,, and , Infinite-dimensional input-to-state stability and Orlicz spaces. SIAM J. Control Optimiz. 56 (2016) 868–889. | DOI | MR | Zbl
[17] ,, and , On input-to-state-stability and integral input-to-state-stability for parabolic boundary control systems. Proceedings of MTNS 2016 (2016).
[18] , and , Infinite-dimensional feedback systems: the circle criterion and input-to-state stability. Commun. Inf. Syst. 8 (2008) 403–434. | DOI | MR | Zbl
[19] and , Stability and stabilization of nonlinear systems. Series: Communications and Control Engineering. Springer-Verlag, London (2011). | MR | Zbl
[20] and , On the relation of delay equations to first-order hyperbolic partial differential equations. ESAIM: COCV 20 (2014) 894–923. | Numdam | MR | Zbl
[21] and , Input-to state stability with respect to boundary disturbances for the 1-D heat equation, in Proceedings of the 55th IEEE Conference on Decision and Control (2016) 2247–2252.
[22] and , ISS with respect to boundary disturbances for 1-D parabolic PDEs. IEEE Trans. Autom. Control 61 (2016) 3712–3724.. | DOI | MR | Zbl
[23] and , ISS in different norms for 1-D parabolic PDEs with boundary disturbances. SIAM J. Control Optim. 55 (2017) 1716–1751. | DOI | MR | Zbl
[24] and , Boundary Control of PDEs: A Course on Backstepping Designs. SIAM (2008). | DOI | MR | Zbl
[25] , Numerical solution of the heat equation with nonlocal boundary conditions. J. Comput. Appl. Math. 110 (1999) 115–27. | DOI | MR | Zbl
[26] , Infinite-dimensional Lur’e systems: the circle criterion, input-to-state stability and the converging-input-converging-state property, in Proceedings of the 21st International Symposium on Mathematical Theory of Networks and Systems, Groningen, The Netherlands (2014) 1624–1627.
[27] and , Strict Lyapunov functionals for nonlinear parabolic partial differential equations, in Proceedings of the 18th IFAC World Congress, Milan, Italy (2011) 12550–12555.
[28] and , Strict Lyapunov functions for semilinear parabolic partial differential equations. Math. Control Relat. Fields 1 (2011) 231–50. | DOI | MR | Zbl
[29] , Local input-to-state stability: characterizations and counterexamples. Syst. Control Lett. 87 (2016) 23–28. | DOI | MR | Zbl
[30] and , Integral input-to-state stability of bilinear infinite-dimensional systems, in Proceedings of the 53rd IEEE Conference on Decision and Control, Los Angeles, California, USA (2014) 3155–3160. | DOI
[31] and , Construction of Lyapunov functions for interconnected parabolic systems: an iISS approach. SIAM J. Control Optim. 53 (2015) 3364–3382. | DOI | MR | Zbl
[32] and , Restatements of input-to-state stability in infinite dimensions: what goes wrong, in Proceedings of the 22nd International Symposium on Mathematical Theory of Systems and Networks (2016) 667–674.
[33] and , Global converse Lyapunov theorems for infinite-dimensional systems, in Proceedings of the 10th IFAC Symposium on Nonlinear Control Systems (2016) 909–914.
[34] and , Linear Operator Theory in Engineering and Science. Springer (1982). | DOI | MR | Zbl
[35] , On general properties of eigenvalues and eigenfunctions of a Sturm–Liouville operator: comments on “ISS with respect to boundary disturbances for 1-D parabolic PDEs”. IEEE Trans. Autom. Control 62 (2017) 5970–5973. | DOI | MR | Zbl
[36] and , Large time behavior in ℝN for linear parabolic equations with periodic coefficients. Asymptot. Anal. 22 (2000) 51–85. | MR | Zbl
[37] , Asymptotic behavior of solutions of reaction-diffusion equations with nonlocal boundary conditions. J. Comput. Appl. Math. 88 (1998) 225–238. | DOI | MR | Zbl
[38] , Numerical solutions of reaction-diffusion equations with nonlocal boundary conditions. J. Comput. Appl. Math. 136 (2001) 227–243. | DOI | MR | Zbl
[39] and , Decay bounds for solutions of second order parabolic problems and their derivatives. Math. Models Methods Appl. Sci. 5 (1995) 95–110. | DOI | MR | Zbl
[40] and , Decay bounds for solutions of second order parabolic problems and their derivatives II. Math. Inequal. Appl. 7 (2004) 543–549. | MR | Zbl
[41] , and Decay bounds for solutions of second order parabolic problems and their derivatives IV. Appl. Anal.: An Int. J. 85 (2006) 293–302. | DOI | MR | Zbl
[42] , Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York (1983). | DOI | MR | Zbl
[43] and , ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws. Math. Control, Signals Syst. 24 (2012) 111–134. | DOI | MR | Zbl
[44] , Shock Waves and Reaction–Diffusion Equations, 2nd edn. Springer-Verlag, New York (1994). | DOI | MR | Zbl
[45] and , Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations. IEEE Trans. Autom. Control 49 (2004) 2185–2202. | DOI | MR | Zbl
[46] and , Adaptive Control of Parabolic PDEs. Princeton University Press (2010). | DOI | MR | Zbl
[47] , Smooth stabilization implies coprime factorization. IEEE Trans. Autom. Control 34 (1989) 435–443. | DOI | MR | Zbl
[48] , Qualitative properties for solutions of reaction-diffusion parabolic systems and their gradients. Nonlinear Anal. 68 (2008) 1775–1783. | DOI | MR | Zbl
[49] and , Input-to state stability with respect to boundary disturbances for a class of semi-linear parabolic equations. Preprint (2017). | arXiv | MR
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