Abstract
Summary This paper is concerned with the time optimal control problem governed by the internal controlled Kuramoto–Sivashinsky–Korteweg-de Vries equation, which describes many physical processes in motion of turbulence and other unstable process systems. We prove the existence of optimal controls with the help of the Carleman inequality, which has been widely used to obtain the local controllability or null controllability of parabolic differential systems. More precisely, with the help of the Carleman inequality, we obtain a relationship between the null controllability and time optimal control problem. Moreover, we give the bang-bang principle for an optimal control of our original problem by using the one of approximate problems. This method is new for time optimal control problems. The bang-bang principle established here seems also to be new for fourth-order parabolic differential equations. Copyright © 2015 John Wiley & Sons, Ltd.
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