On the determination of the principal coefficient from boundary measurements in a KdV equation

Abstract

This paper concerns the inverse problem of retrieving the principal coefficient in a Korteweg-de Vries (KdV) equation from boundary measurements of a single solution. The Lipschitz stability of this inverse problem is obtained using a new global Carleman estimate for the linearized KdV equation. The proof is based on the Bukhgeȋm-Klibanov method.

Local exact controllability to the trajectories of the Korteweg–de Vries–Burgers equation on a bounded domain with mixed boundary conditions

“…In order to apply the results obtained in the previous sections we consider the following change of variable. Let us set y − y =: z and use this equality in (2), where y solves (3). It is easy to verify that z satisfies                z t + z xxx − ν(t)z xx + (zy) x + zz x = v1 ω in Q, z(0, t) = z(L, t) = 0 on (0, T ), z x (0, t) = z x (L, t) on (0, T ), z(•, 0) = y 0 − y 0 in (0, L).…”

“…The estimate (9) allows us to prove a null controllability result for the linear system (7) with right-hand side satisfying suitable decreasing properties near t = T . Theorem 1.1 will be proved using the same approach as in [21,2,7]. ii) Then establish the local exact controllability to the trajectories for the KdVB equation.…”

The Bubnov–Galerkin method in control problems for bilinear systems

“…As usual, ∂Ω denotes the boundary of region Ω, n is the vector of its external normal. Examples of operator D u [·] can be found, for instance, in [1,4,5,[8][9][10][11][12][13].…”

“…A relatively comprehensive list of references can be found in the book [5]. Other control problems for bilinear systems described by ordinary differential equations can be found, e.g., in [6,7]; partial differential equations, in [8][9][10][11][12][13].…”

“…In [13], the required control function occurs in the coefficients of the state function of Schrödinger equation, while in [9,12] it occurs in that of the first derivative of the state function of the wave equation. An interesting control problem for the coefficient of a Korteweg-de Vries equation which is nonlinear in the state function but linear in the control function has been considered in [11]. In [1], classical variational calculus methods are used to study different problems of the theory of elasticity, when state equations are linear with respect to the state function and nonlinear with respect to the control function.…”