Optimal Control of Shear-Thickening Flows

Abstract

We study optimal control problems of systems describing the flow of incompressible shear-thickening fluids. We prove existence of solutions and derive necessary optimality conditions under precise restrictions on the optimal control.

Distributed control for multistate modified Navier-Stokes equations

“…Due Proposition 2.6, the state equation (3.2) admits a unique solutionȳ ∈ V α . Similarly, due Proposition 3.9 in [2], ifū satisfies (3.6) then the adjoint system (3.3) admits a unique solutionp in Hȳ α . It follows that, if we suppose thatλ = 0 thenp ≡ 0 is the (unique) solution of (3.3) leading to a contradiction with the nontriviality condition (3.1).…”

“…Their qualification is guaranteed under the (C) property, or under a precise condition on the optimal control. This last result was obtained in [2] using a different approach and seems to be new in the sense that no constraint on the size of admissible controls is needed.…”

“…System (1.1) may then exhibit non-uniqueness of weak solutions and the control-to-state mapping u −→ y u can be multivalued. Combining the methods developed in [1,4,5] together with explicite estimates established in [2], we derive optimality conditions in a nonqualified form in both two-dimensional and three-dimensional cases. Their qualification is guaranteed under the (C) property, or under a precise condition on the optimal control.…”

Hölder continuity of velocity gradients for shear-thinning fluids under perfect slip boundary conditions

“…These results are closely related to the regularity of coefficients in the main part of the associated differential operators and enables to derive corresponding optimality conditions, as is done for example in [17]. We refer also to [1] and [2] as an example of the work where author is dealing with the lack of the regularity result.…”

“…such that Φ ∈ C 1,1 ((0, ∞)) ∩ C 1 ([0, ∞)), Φ(0) = 0 and there exist p ∈ (1,2] and 0 < c 1 ≤ c 2 such that for all A, B ∈ R 2×2…”