# American Institute of Mathematical Sciences

March  2014, 4(1): 45-99. doi: 10.3934/mcrf.2014.4.45

## Control of a Korteweg-de Vries equation: A tutorial

 1 Departamento de Matemática, Universidad Técnica Federico Santa María, Avda. España 1680, Casilla 110-V, Valparaíso, Chile

Received  January 2012 Revised  January 2013 Published  December 2013

These notes are intended to be a tutorial material revisiting in an almost self-contained way, some control results for the Korteweg-de Vries (KdV) equation posed on a bounded interval. We address the topics of boundary controllability and internal stabilization for this nonlinear control system. Concerning controllability, homogeneous Dirichlet boundary conditions are considered and a control is put on the Neumann boundary condition at the right end-point of the interval. We show the existence of some critical domains for which the linear KdV equation is not controllable. In despite of that, we prove that in these cases the nonlinearity gives the exact controllability. Regarding stabilization, we study the problem where all the boundary conditions are homogeneous. We add an internal damping mechanism in order to force the solutions of the KdV equation to decay exponentially to the origin in $L^2$-norm.
Citation: Eduardo Cerpa. Control of a Korteweg-de Vries equation: A tutorial. Mathematical Control & Related Fields, 2014, 4 (1) : 45-99. doi: 10.3934/mcrf.2014.4.45
##### References:

show all references

##### References:
 [1] Mohamed Ouzahra. Approximate controllability of the semilinear reaction-diffusion equation governed by a multiplicative control. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021081 [2] Shi'an Wang, N. U. Ahmed. Optimal control and stabilization of building maintenance units based on minimum principle. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1713-1727. doi: 10.3934/jimo.2020041 [3] Simone Cacace, Maurizio Falcone. A dynamic domain decomposition for the eikonal-diffusion equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 109-123. doi: 10.3934/dcdss.2016.9.109 [4] Jamal Mrazgua, El Houssaine Tissir, Mohamed Ouahi. Frequency domain $H_{\infty}$ control design for active suspension systems. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021036 [5] K. Ravikumar, Manil T. Mohan, A. Anguraj. Approximate controllability of a non-autonomous evolution equation in Banach spaces. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 461-485. doi: 10.3934/naco.2020038 [6] Ademir Fernando Pazoto, Lionel Rosier. Uniform stabilization in weighted Sobolev spaces for the KdV equation posed on the half-line. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1511-1535. doi: 10.3934/dcdsb.2010.14.1511 [7] Christophe Zhang. Internal rapid stabilization of a 1-D linear transport equation with a scalar feedback. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021006 [8] Xiaorong Luo, Anmin Mao, Yanbin Sang. Nonlinear Choquard equations with Hardy-Littlewood-Sobolev critical exponents. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021022 [9] Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $G$-expectation framework. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021072 [10] Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437 [11] Wentao Huang, Jianlin Xiang. Soliton solutions for a quasilinear Schrödinger equation with critical exponent. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1309-1333. doi: 10.3934/cpaa.2016.15.1309 [12] Tomoyuki Tanaka, Kyouhei Wakasa. On the critical decay for the wave equation with a cubic convolution in 3D. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021048 [13] Claudianor O. Alves, Giovany M. Figueiredo, Riccardo Molle. Multiple positive bound state solutions for a critical Choquard equation. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021061 [14] Jihoon Lee, Nguyen Thanh Nguyen. Gromov-Hausdorff stability of reaction diffusion equations with Robin boundary conditions under perturbations of the domain and equation. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1263-1296. doi: 10.3934/cpaa.2021020 [15] Jiahui Chen, Rundong Zhao, Yiying Tong, Guo-Wei Wei. Evolutionary de Rham-Hodge method. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3785-3821. doi: 10.3934/dcdsb.2020257 [16] Bruno Premoselli. Einstein-Lichnerowicz type singular perturbations of critical nonlinear elliptic equations in dimension 3. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021069 [17] Yohei Yamazaki. Center stable manifolds around line solitary waves of the Zakharov–Kuznetsov equation with critical speed. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3579-3614. doi: 10.3934/dcds.2021008 [18] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448 [19] Amit Goswami, Sushila Rathore, Jagdev Singh, Devendra Kumar. Analytical study of fractional nonlinear Schrödinger equation with harmonic oscillator. Discrete & Continuous Dynamical Systems - S, 2021  doi: 10.3934/dcdss.2021021 [20] Pavel I. Naumkin, Isahi Sánchez-Suárez. Asymptotics for the higher-order derivative nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021028

2019 Impact Factor: 0.857