Viscosity solutions of dynamic-programming equations for the optimal control of the two-dimensional Navier-Stokes equations. (English) Zbl 1005.49028

Gozzi, Fausto; Sritharan, S. S.; Świech, Andrzej

Arch. Ration. Mech. Anal. 163, No. 4, 295-327 (2002).

The authors consider the abstract model of the Navier-Stokes equations \[ \begin{aligned} X'(s) &= - {\mathbf A} X(s) - {\mathbf B}(X(s), X(s)) + {\mathbf f}(s, {\mathbf a}(s)) \quad ((t, T] \times {\mathbf H}) \\ X(t) &= {\mathbf x} \in {\mathbf H}\end{aligned} \] where \({\mathbf H}\) is the closure of the set of all solenoidal vectors in \(L^2(\Omega;\mathbb{R}^2)\) \((\Omega\) a bounded 2-dimensional domain) and the control \({\mathbf a}(\cdot)\) ranges over a strategy set \(\mathcal U.\) Minimization over \({\mathbf a}(\cdot)\) of a cost functional \[ J(t, {\mathbf x}; {\mathbf a}) = \int_t^T \ell(s, X(s), {\mathbf a}(s)) ds + g(X(T)) \] leads to the Hamilton-Jacobi equation for the value function \[ {\mathcal V}(t, {\mathbf x}) = \inf_{{\mathbf a(\cdot}) \in {\mathcal U}} J(t, {\mathbf x}; {\mathbf a}). \] The main contribution of this paper is the proof of global unique solvability of the Hamilton-Jacobi equation.

Reviewer: Hector O.Fattorini (Los Angeles)

MSC:

49L25	Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
76D05	Navier-Stokes equations for incompressible viscous fluids
76D55	Flow control and optimization for incompressible viscous fluids
49L20	Dynamic programming in optimal control and differential games
49K20	Optimality conditions for problems involving partial differential equations

Keywords:

dynamic programming; Hamilton-Jacobi equations; viscosity solutions; optimal control of two dimensional flow