Inverse optimality of adaptive control for Korteweg-de Vries-Burgers equation

The Korteweg-de Vries-Burgers (KdVB) equation is one of the simplest nonlinear mathematical models, which is used to model motion of waves in a variety of fluid flow processes. And inverse optimality allows the design of optimal control laws, which may minimize/maximize a physical quantity of interest and which may possess certain robustness margins, without the need to solve a Hamilton-Jacobi-Isaacs partial differential equation (PDE) that may not be possible to solve. Therefore, it is important to study inverse optimal control for KdVB equation. In this paper, it is proved that the boundary control of Balogh and Krstic (IEEE Trans Autom Control 45(9):1739-1745, 2000) is inverse optimal for a meaningful functional. Next, an adaptive boundary control design for KdVB equation with an unknown dissipation coefficient is presented. Furthermore, it is shown that this adaptive control design is also inverse optimal for a meaningful functional. Two examples are given to illustrate the validity of the proposed design.