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Dynamic programming of the Navier-Stokes equations. (English) Zbl 0737.49021

The author studies a class of optimal control problems for viscous incompressible flows. His formulation allows him to treat several problems e.g., optimal acceleration of an obstacle in viscous incompressible fluid, control of flow in a bounded container (i.e., driving the flow to a desired one using, for example, blowing and suction on the boundary) and minimization of dissipation in channel flows.
The author states an existence theorem for an optimal control and then proceeds to study properties of the value function; he shows that the value function is in fact a viscosity solution of the Hamilton-Jacobi- Bellman equation associated with the control problem considered. The author finally derives the Pontryagin maximum principle for the Navier- Stokes equations and establishes the verification theorem for the Hamilton-Jacobi-Bellman equation.
Reviewer: A.J.Meir (Auburn)

MSC:

49L20 Dynamic programming in optimal control and differential games
49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
35Q30 Navier-Stokes equations
49K20 Optimality conditions for problems involving partial differential equations
35B37 PDE in connection with control problems (MSC2000)
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