On the Uniform Controllability of the Inviscid and Viscous Burgers-$\alpha$ Systems

Abstract

This work is devoted to prove the global uniform exact controllability of the inviscid and viscous Burgers-$\alpha$ systems. The state $y$ is the solution to a regularized Burgers equation, where the transport velocity $z$ consists of a filtered version of $y$ - specifically $z=(Id-{\alpha }^2{\partial }_{xx}^2{)}^{-1}y$ with $\alpha >0$ being a small parameter - in place of $y$. First, a global uniform exact controllability result for the nonviscous Burgers-$\alpha$ system with three scalar controls is obtained, using the return method. Then, global exact controllability to constant states of the viscous system is deduced from a local exact controllability result and a global approximate controllability result for smooth initial and target states.

25 pages

References in corpus (4)

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• A Regularization of Burgers Equation using a Filtered Convective Velocity
• Uniform local null control of the Leray-$\alpha$ model
• On the control of the Burgers-alpha model