J. Math. Fluid Mech. 20 (2018), 199–211
2017 Springer International Publishing (Outside USA)
Journal of Mathematical
Existence of Optimal Controls for Compressible Viscous Flow
Stefan Doboszczak, Manil T. Mohan and Sivaguru S. Sritharan
Abstract. We formulate a control problem for a distributed parameter system where the state is governed by the compressible
Navier–Stokes equations. Introducing a suitable cost functional, the existence of an optimal control is established within
the framework of strong solutions in three dimensions.
Mathematics Subject Classiﬁcation. 35Q30, 49J20, 93C20.
Keywords. Compressible Navier–Stokes equations, optimal control, state constraint, weak-strong uniqueness.
This paper is devoted to an optimal control problem for the compressible Navier–Stokes equations in the
following basic form,
L(ρ, u, U)dt =: J
(ρu)+div(ρu ⊗ u)+∇p =divS + ρf + ρU.
The objective is to determine a distributed control (body force) U that realizes the minimum of J . Such
problems appear in compressible ﬂuid dynamics where the cost functional J may include contributions
from kinetic energy, vorticity, drag, etc. Control of such ﬂows are of relevance, for example, in aerodynamic
Important early works on control problems in ﬂuids are due to Fursikov  and Lions . The current
literature on existence of optimal controls for the incompressible Navier–Stokes equations is extensive, see
for instance [9,16,17,25–27,29]. For viscous compressible ﬂow the theory is not as established. Controls for
one-dimensional unsteady compressible ﬂow linearized about a constant state are considered in  and ,
and the two-dimensional linearized problem is studied in . A two-dimensional unsteady compressible
control problem in the half-space is considered in a formal way in  and , where some numerical
schemes are introduced. To the best of our knowledge, existence of optimal controls for the nonlinear
unsteady compressible problem has not been fully resolved.
Our approach in this paper follows elements of [19,20,29], where penalty terms are introduced in the
cost functional to handle problems with state constraint, as well as [15,16,28], where the notion of generic
solvability is introduced for control problems to get around a lack of uniqueness for the Navier–Stokes
equations. In addition, we make use of the weak-strong uniqueness result of , ensuring that strong
solutions are unique within the class of weak solutions provided some extra regularity conditions hold.
This paper is organized as follows. In Sects. 1.1 and 1.2 the governing equations are introduced. In
Sect. 2 we describe the notion of weak solutions along with known existence and uniqueness results.