In this paper we consider a linear KdV equation with a transport term posed on a finite interval with the boundary conditions considered by Colin and Ghidaglia. The main results concern the behavior of the cost of null controllability with respect to the dispersion coefficient when the control acts on the left endpoint. In particular, for any final time we prove that this cost grows exponentially as the dispersion coefficient vanishes and the transport coefficient is negative.
E.Cerpa and E.Crépeau, Boundary controllability for the nonlinear Korteweg–de Vries equation on any critical domain, Ann. Inst. H. Poincaré Anal. Non Linéaire26(2) (2009), 457–475.
2.
E.Cerpa, I.Rivas and B.-Y.Zhang, Boundary controllability of the Korteweg–de Vries equation on a bounded domain, SIAM J. Control Optim.51(4) (2013), 2976–3010.
3.
T.Colin and J.-M.Ghidaglia, Un problème mixte pour l’équation de Korteweg–de Vries sur un intervalle borné, C. R. Acad. Sci. Paris Sér. I Math.324(5) (1997), 599–603.
4.
T.Colin and J.-M.Ghidaglia, An initial-boundary value problem for the Korteweg–de Vries equation posed on a finite interval, Adv. Differential Equations6(12) (2001), 1463–1492.
5.
J.-M.Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, Vol. 136, American Mathematical Society, Providence, RI, 2007.
6.
J.-M.Coron and S.Guerrero, Singular optimal control: A linear 1-D parabolic–hyperbolic example, Asymptot. Anal.44(3–4) (2005), 237–257.
7.
L.Evans, Partial Differential Equations, Graduate Studies in Mathematics, Vol. 19, American Mathematical Society, Providence, RI, 1998.
8.
A.V.Fursikov and O.Yu.Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, Vol. 34, Global Analysis Research Center, Research Institute of Mathematics, Seoul National Univ., Seoul, 1996.
9.
O.Glass and S.Guerrero, Some exact controllability results for the linear KdV equation and uniform controllability in the zero-dispersion limit, Asymptot. Anal.60(1-2) (2008), 61–100.
10.
O.Glass and S.Guerrero, Uniform controllability of a transport equation in zero diffusion–dispersion limit, Math. Models Methods Appl. Sci.19(9) (2009), 1567–1601.
11.
S.Guerrero and G.Lebeau, Singular optimal control for a transport–diffusion equation, Comm. Partial Differential Equations32(10–12) (2007), 1813–1836.
12.
J.-P.Guilleron, Null controllability of a linear KdV equation on an interval with special boundary conditions, Math. Control Signals Syst.26(3) (2014), 375–401.
13.
E.Kramer, I.Rivas and B.-Y.Zhang, Well-posedness of a class of non-homogeneous boundary value problems of the Korteweg–de Vries equation on a finite domain, ESAIM Control Optim. Calc. Var.19(2) (2013), 358–384.
14.
J.-L.Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev.30(1) (1988), 1–68.
15.
L.Rosier, Exact boundary controllability for the Korteweg–de Vries equation on a bounded domain, ESAIM Control Optim. Calc. Var.2 (1997), 33–55.
16.
L.Rosier, Control of the surface of a fluid by a wavemaker, ESAIM Control Optim. Calc. Var.10(3) (2004), 346–380.
