Controllability For The Kuramoto-Sivashinsky Equations

The Kuramoto-Sivashinsky (KS) equation was derived independently by Ku-ramoto et al. as a model for phase turbulence in reaction-diffusion systems and by Sivashinsky as a model for plane flame propagation, describing the combined influence of diffusion and thermal conduction of the gas on stability of a plane flame front. This nonlinear partial differential equation describes incipient instabilities in a variety of physical and chemical systems. This equa-tion also arises in the modeling of the ow of a thin film of viscous liquid falling down on an inclined plane subject to an applied electric field. The KS equation has been studied as a prototypical example for an infinite-dimensional dynam-ical system. Compared with the deterministic KS equation, the stochastic KS equation also attracts many authors’attention. The stochastic KS equation appears in the study of dynamic roughening in sputter-eroded surfaces and, in principle, any physical system modeled by the deterministic KS equation in which the relevance of time dependent noise can be argued for.Control theory is certainly, at present, one of the most interdisciplinary areas of research. Control theory arises in most modern applications. The same could be said about the very first technological discoveries of the industrial revolution. On the other hand, control theory has been a discipline where many mathematical ideas and methods have melt to produce a new body of important Mathematics. In the last two decades, the control theory of partial differential equations develops fast, motivated by practical problems, now it attracts many authors’ attention. As far as we know, there are only a few results about the controllability for the KS equations. This paper is devoted to the controllability for the KS equations.In the first part, we investigate the exact controllability to the trajectories for the KS equation. We first establish a new global Carleman estimate for a fourth order parabolic operator which is a one-parameter Carleman estimate. The main difference between this Carleman estimate and the existing Carleman estimate is that the weight function in our Carleman estimate depends on only one parameter. To obtain this estimate, we construct a new weight function. Then, we investigate the null controllability of the linearized KS equation. Applying the Carleman estimate and the energy estimate, we establish the observability estimate for the duality system of the linearized KS equation. Combining this observability estimate and the classical duality argument, we can obtain the null controllability of the linearized KS equation. According to the linearized result and Kakutani’s fixed point Theorem, we prove the local exact controllability to the trajectories of the KS equation.In the second part, we discuss the null controllability for forward and backward linear stochastic KS equations. First we proved the well-posedness of forward and backward linear stochastic KS equations by Galerkin method. Then we establish the Carleman estimates for stochastic fourth order forward and backward parabolic equations. The main difference between the stochas-tic case and the determined case is the influence of the noise. Therefore, the methods that used to establish global Carleman estimates for determined par-tial differential equations can not be applied to stochastic partial differential equations. Here we adopt a new method which has been applied to obtain global Carleman estimates for forward and backward linear stochastic heat equations. Based on these Carleman estimates and the energy estimates for backward and forward linear stochastic KS equations, we can establish the ob- servability estimates for backward and forward linear stochastic KS equations. By means of the observability estimates and Hahn-Banach Theorem, we can obtain the null controllability for forward and backward linear stochastic KS equations.