Global ISS for the Viscous Burgers' Equation With Dirichlet Boundary Disturbances

In this note, we study the global input-to-state stability (ISS) of the viscous Burgers' equation with respect to Dirichlet boundary disturbances. More specifically, under the assumption that the reaction coefficient is strictly less than <tex-math notation="LaTeX">π ²</tex-math> , we establish first the global ISS in the spatial <tex-math notation="LaTeX">L^∞</tex-math> -norm for the viscous Burgers' equation with Dirichlet boundary disturbances. Then, under the assumption that the reaction coefficient is nonpositive (and respectively strictly negative), we establish the global ISS in the spatial <tex-math notation="LaTeX">Lq</tex-math> -norm for the viscous Burgers' equation whenever <tex-math notation="LaTeX">q∈ [2,+∞)</tex-math> (and respectively <tex-math notation="LaTeX">q∈ [1,+∞)</tex-math> ). The obtained results provide a positive answer to the open question raised by A. Mironchenko and C. Prieur, "Input-to-state stability of infinite-dimensional systems: recent results and open questions," SIAM Rev., vol. 62, no. 3, pp. 529–614, 2019, regarding the global ISS of the viscous Burgers' equation with Dirichlet boundary disturbances. The main technique employed in this note is the so-called Stampacchia's truncation, as well as approximations of Lyapunov functionals while establishing the ISS in the spatial <tex-math notation="LaTeX">Lq</tex-math> -norm with <tex-math notation="LaTeX">q∈ [1,2)</tex-math> , which can be seen as a generalized Lyapunov method. Numerical simulations are conducted to illustrate the obtained results.