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\(H^\infty\)-control theory of fluid dynamics. (English) Zbl 0919.93026

In recent years the \(H^\infty\) functional setting has been gaining importance in control theory, as the closely related research of operator theory in Hardy spaces has steadily progressed. The authors develop in this paper an \(H^\infty\) theory for some control problems in fluid dynamics, in particular for linearized Navier-Stokes equations.
The authors begin by examining the input-output system in a real Hilbert space \(H\): \[ d/dt(y(t))+\nu Ay(t)+ F(y(t))= B_2(u(t))+ B_1(w(t)),\;z(t)= C_1y(t)+ D_{12}u(t),\;t\in R^+,\tag{1} \] \(y(0)= y_0\), \(\nu>0\), and all operators in these equations are linear. \(u(t)\) is the control input, \(w(t)\) the external input. Since the range of the measurement (or observation) operator \(C_1\) is a Hilbert space with a negative index, both pointwise (for example Dirac delta) and distributive measurements are admissible. The authors introduce a Hilbert space \(V\) continuously imbedded in \(H\), with \(V\subset H\subset V^*\). The linear operator \(A\) is symmetric and positive definite. The operator \(F: V\rightarrow V^*\) is defined in terms of triples \(b(y,z,w)\), obeying certain norm inequalities. One of the main results states that after decomposing the operator \(F\) into \({\mathcal F}+ F_0\), where \(F_0\in L(V,V^*)\), and assuming that a specific multiple of a triple function \(b(.,.,.)\) bounds above \(| y|^2\) (omitting details for brevity), we can conclude that \(\nu A+ F_0\) generates a \(C^0\) semigroup on \(H\). This implies existence of operators \(D\) and \(K\) such that \(-(\nu A+ F_0+ B_2D)\) and \(-(\nu A+ F_0+ KC_1)\) generate stable semigroups on \(H\).
The crucial statement in most \(H^\infty\) control theory papers concerns the following property of Hardy classes: The \(L^2\) norm of the control gain operator in the time domain is equal to the Hardy \(H^\infty\) norm of the transfer operator in the frequency domain. This permits the authors to announce their main result: Suppose that system (1) has a \(\gamma\)-suboptimal solution (it is smaller than \(\gamma\)). Then there exist a neighborhood \(\Sigma_\mu: \{\| y\|<\mu, \mu> 0\}\) and a unique map \(G\in C^1(\Sigma_\mu, V)\) such that \[ 2((\nu Ax+ Fx), G(x))- \gamma^{-2}\| B_1* G|^2W+ | B_2* G|^2_U- | C_1x|^2_Z= 0 \] for all \(x\) in \(\Sigma_\mu\). The control \(u= -B_2* G(y)\) asymptotically stabilizes the original system on \(\Sigma_\mu\).

MSC:

93B36 \(H^\infty\)-control
93C20 Control/observation systems governed by partial differential equations
35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
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