Control theory for the Burgers equation: Agrachev-Sarychev approach

2017/12/28 by Shirikyan, Armen · 4 citations
#35Q35 #93B05 #93C20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1712.09807

Abstract

This paper is devoted to a description of a general approach introduced by Agrachev and Sarychev in 2005 for studying some control problems for Navier-Stokes equations. The example of a 1D Burgers equation is used to illustrate the main ideas. We begin with a short discussion of the Cauchy problem and establish a continuity property for the resolving operator. We next turn to the property of approximate controllability and prove that it can be achieved by a two-dimensional external force. Finally, we investigate a stronger property, when the approximate controllability and the exact controllability of finite-dimensional functionals are proved simultaneously.

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• Small-time global controllability of a class of bilinear fourth-order parabolic equations
• Small-time approximate controllability for the nonlinear complex Ginzburg-Landau equation with bilinear control
• Global controllability of the Cahn-Hilliard equation
• Controllability of Boussinesq flows driven by finite-dimensional and physically localized forces

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