Linear Quadratic Regulation of Kuramoto-Sivashinsky PDE with Point Actuation

We consider the nonlinear Kuramoto-Sivashinsky equation and its linear part on a finite interval subject to periodic boundary conditions. The linear part can have a finite number of unstable eigenvalues so we assume that there are point actuators that allow a linear feedback to move all the unstable eigenvalues into the open left half plane. Such a linear feedback law is found by the well-known technique of Linear Quadratic Regulation (LQR). This leads to a new Riccati partial differetial equation for quadratic Fredholm kernel of the optimal cost. From this quadratic kernel we obtain the linear kernel of the optimal feedback. We prove that this feedback moves all the unstable eigenvalues into the open left half plane. But it has little effect on the open loop eigenvalues that were already stable. This linear feedback locally stabilizes the nonlinear Kuramoto-Sivashinsky equation but nonLinear nonQuadratic Regulation (nLnQR), which we discuss in this paper, can be used to find a cubic feedback that stabilizes it faster and/or with less control energy.