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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.

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
Full Text: DOI
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