Viscosity solutions of dynamic-programming equations for the optimal control of the two-dimensional Navier-Stokes equations (Q699799)

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.