Abstract
The value function associated with an optimal control problem subject to the Navier–Stokes equations in dimension two is analyzed. Its smoothness is established around a steady state, moreover, its derivatives are shown to satisfy a Riccati equation at the order two and generalized Lyapunov equations at the higher orders. An approximation of the optimal feedback law is then derived from the Taylor expansion of the value function. A convergence rate for the resulting controls and closed-loop systems is demonstrated.
Highlights
In this work we continue our investigations of the value function associated with infinite-horizon optimal control problems of partial differential equations, that we initiated in [15,17]
To the contrary, we show that the value function is smooth and that the HJB equation is satisfied in the strict sense, in a neighborhood of the steady state
We show that the derivatives of the value function, at the steady state, are solutions to an algebraic Riccati equation and to linear equations, called generalized Lyapunov equations, for the higher orders
Summary
In this work we continue our investigations of the value function associated with infinite-horizon optimal control problems of partial differential equations, that we initiated in [15,17]. Densely defined linear operator (A, D(A)) in Y , its adjoint (again considered as an operator in Y ) will be denoted with ( A∗, D( A∗)). Given two multilinears mappings T1 ∈ M(Y k, Z ) and T2 ∈ M(Y , Z ), we denote by T1 ⊗ T2 the bounded multilinear form defined by. We note that as a consequence of Proposition 1, the operator A can be extended to a bounded linear operator from V to V in the following manner: Ay, w V ,V = −ν ∇y, ∇w L2( ) − A0(z, y), w V ,V. Let us note that e : W∞ × L2(0, ∞; U ) → L2(0, ∞; V ) × Y is well-defined by Corollary 3
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