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Optimal chattering controls for viscous flow. (English) Zbl 0867.49004
The authors consider a semilinear control system of the form $y'(t)+Ay(t)+B(y(t))= {\mathcal N}(y(t),u(t)),\quad y(0)=\zeta\tag{1}$ in a Banach space $$E$$. The application in mind is the Navier-Stokes equations for the velocity $$y(t)(x)=y(t,x)$$ of a viscous fluid in a domain $$\Omega\subseteq\mathbb{R}^m$$, $$m=2,3$$; here, $$E=X^2(\Omega)=$$ closure of the divergence-free vectors in $$L^2(\Omega;\mathbb{R}^m)$$, $$A=-P\Delta$$ ($$\Delta$$ the $$m$$-dimensional Laplacian with Dirichlet boundary condition, $$P$$ the projection from $$L^2(\Omega;\mathbb{R}^m)$$ into $$X^2(\Omega)$$), $$B(y)=P(y\cdot\nabla)y$$ and $${\mathcal N}(y,u)$$ is an operator introducing the control $$u(t)$$ into the equation. The assumptions on (1) are geared to accomodate this example. It is well known by control theorists that systems like (1) where control appears nonlinearly (or where $$\mathcal N$$ is linear in the control but the control set $$U$$ is not convex) may lack existence of optimal controls in certain control problems; a practical example is that of stabilization of fluid flow by suction-blowing with a finite number of actuators that can only switch between two extreme positions. In the finite-dimensional case, an existence fix inspired in Young’s measures in calculus of variations was implemented by J. Warga [J. Math. Anal. Appl. 4, 129-145 (1962; Zbl 0102.31802)], and consists in replacing ordinary controls $$u(t)$$ by relaxed controls, $$\mu(t,d\mu)$$, where for each $$t$$, $$\mu(t,du)$$ is a probability measure in the control set $$U$$. This concept has been generalized by various authors to infinite-dimensional systems; equation (1) is replaced by $y'(t)+Ay(t)+B(y(t))= \int_U{\mathcal N}(y(t),u)\mu(t,du),\quad y(0)=\zeta\tag{2}$ under various assumptions on the control set (see for instance N. U. Ahmed [SIAM J. Control Optimization 21, 953-967 (1983; Zbl 0524.49008)]). One extension that does not require much of $$U$$ but uses infinitely additive measures is in H. O. Fattorini [SIAM J. Control Optimization 32, No. 2, 311-331 (1994; Zbl 0796.93112)] and other papers of the same author; for the Navier-Stokes equations the extension is in H. O. Fattorini and S. S. Sritharan [Nonlinear Anal., Theory Methods Appl. 25, No. 8, 763-797 (1995)]. Essential requirements of any such extension are relaxation theorems insuring that trajectories $$y(t,\mu)$$ corresponding to relaxed controls can be uniformly approximated by trajectories $$y(t,u)$$ corresponding to ordinary controls, which is the subject of this paper. The proofs follows previous work on infinite-dimensional systems, but handling of the Navier-Stokes nonlinearity requires new technical means.

##### MSC:
 49J20 Existence theories for optimal control problems involving partial differential equations 76D05 Navier-Stokes equations for incompressible viscous fluids 93C20 Control/observation systems governed by partial differential equations 76M30 Variational methods applied to problems in fluid mechanics 93C25 Control/observation systems in abstract spaces
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