Abstract
Optimal Control of the Two-Dimensional Evolutionary Navier--Stokes Equations with Measure Valued Controls
Highlights
In this paper we investigate the following optimal control problem (P)
| y\bfu (x, t) yd(x, t)| 2 dx dt, where Uad = \{ u \in L\infty (0, T ; M(\omega )) : \| u(t)\| \bfM (\omega ) \leq \gama for a.a. t \in (0, T )\} with 0 < \gama < \infty, and y and u are related by the Navier--Stokes system
On the one hand there it is the genuine interest in low-order regularity of the controls, on the other hand it relates to their sparsity promoting structure
Summary
\left\{ \partialy \partialt - \nu \Delta y + (y \cdot \nabla )y + \nabla \frakp = f0 + \chi \omega u in Q = \Omega \times I, div y = 0 in Q, y = 0 on \Sigma = \Gama \times I, y(0) = y0 in \Omega. I = (0, T ) with 0 < T < \infty , \Omega denotes a bounded domain in \BbbR 2 with a C3 boundary \Gama , and \omega is a relatively closed subset of \Omega. Regarding the state equation, \nu > 0 is the kinematic viscosity coefficient, \chi \omega u denotes the extension of u by zero outside \omega , and f0 is a given element of Lq(I, W - 1,p(\Omega )) with W - 1,p(\Omega ) = W - 1,p(\Omega ) \times W - 1,p(\Omega ), where (1.2). \ast Received by the editors July 8, 2020; accepted for publication (in revised form) March 12, 2021; published electronically June 17, 2021
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