2024/02/09 by Kumar, Pardeep, Mohan, Manil T.
#Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2402.06335
In this article, we discuss the local exact controllability to trajectories of the following convective Brinkman-Forchheimer (CBF) equations (or damped Navier-Stokes equations) defined in a bounded domain Ω ⊂ℝd (d=2,3) with smooth boundary: \frac∂\boldsymbolu∂ t-μΔ\boldsymbolu+(\boldsymbolu⋅∇)\boldsymbolu+α\boldsymbolu+β|\boldsymbolu|2\boldsymbolu+∇ p=\boldsymbolf+\boldsymbolϑ, ∇⋅\boldsymbolu=0, where the control \boldsymbolϑ is distributed in a subdomain ω⊂ Ω, and the parameters α,β,μ>0 are constants. We first present global Carleman estimates and observability inequality for the adjoint problem of a linearized version of CBF equations by using a global Carleman estimate for the Stokes system. This allows us to obtain its null controllability at any time T>0. We then use the inverse mapping theorem to deduce local results concerning the exact controllability to the trajectories of CBF equations.