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Resolving controls for exact and approximate controllability of viscous Burgers’ equation: The Green’s function approach

Asatur Zh. Khurshudyan

Asatur Zh. Khurshudyan

Department on Dynamics of Deformable Systems and Coupled Fields, Institute of Mechanics, NAS of Armenia, Armenia

Institute of Natural Sciences, Shanghai Jiao, Tong University, Shanghai, P. R. China

E-mail Address: khurshudyan@mechins.sci.am

E-mail Address: khurshudyan@sjtu.edu.cn

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https://doi.org/10.1142/S0129183118500456Cited by:1 (Source: Crossref)

Abstract

In this paper, we consider a nonlinear control problem for one-dimensional viscous Burgers’ equation associated with a controlled linear heat equation by means of the Hopf–Cole transformation. The control is carried out by the time-dependent intensity of a distributed heat source influencing the heat equation. The set of admissible controls consists of compactly supported $L^{\infty}$ functions. Using the Green’s function approach, we analyze the possibilities of exact and approximate establishment of a given terminal state for the associated nonlinear Burgers’ equation within a desired amount of time. It is shown that the exact controllability of the associated Burgers’ equation and the heat equation are equivalent. Furthermore, sufficient conditions for the approximate controllability are derived. The set of resolving controls is constructed in both cases. The determination of the resolving controls providing exact controllability is reduced to an infinite-dimensional system of linear algebraic equations. By means of the heuristic method of resolving control determination, parametric hierarchies of solutions providing approximate controllability are constructed. The results of a numerical simulation supporting the theoretical derivations are discussed.

Keywords:

PACS: 02.30.Jr, 02.30.Yy

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