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Linear operator inequalities for strongly stable weakly regular linear systems. (English) Zbl 1114.93029

Weakly regular linear systems introduced by M. Weiss and G. Weiss [Math. Control Signals Syst. 10, No. 4, 287–330 (1997; Zbl 0884.49021)] form a large subclass of well-posed (in the sense of D. Salamon [Trans. Am. Math. Soc. 300, 383–431 (1987; Zbl 0623.93040)]) linear systems, which includes many systems described by partial differential equations with boundary control and point observations as well as by delay equations with delayed observations and control actions. They are characterized by the property that the transfer function \(G\) has a weak limit at \(+\infty\). Recently O. Staffans and G. Weiss [Trans. Am. Math. Soc. 354, No. 8, 3229–3262 (2002; Zbl 0996.93012)] showed that the system operator (in general, unbounded) of a weakly regular linear system can be naturally splitted into four blocks \(A,B,C,D\) (the generating operators of a system).
In the paper under review, the following optimal control problem is considered. Let \({\mathcal X,U,Y}\) be separable Hilbert spaces; \(u(t), y(t)\) be the input and output functions for a strongly stable, weakly regular linear system \(\Sigma\) with generating operators \(A,B,C,0\). Given the operators \(R=R^*\in {\mathcal L(U)}, Q=Q^*\in {\mathcal L(Y)}, N\in {\mathcal L(Y,U)}\) and the initial state \(x_0\in {\mathcal X}\), find the input function \(u^{\text{ opt}}\in L_2(0,\infty;{\mathcal U})\) that minimizes the cost functional \[ J(x_0,u)=\int\limits_0^{\infty}\left\langle\left[\begin{matrix} Q & N^* \cr N & R \end{matrix}\right]\left[\begin{matrix} y(t)\cr u(t)\end{matrix}\right],\left[\begin{matrix} y(t)\cr u(t)\end{matrix}\right]\right\rangle_{\mathcal Y\times U}\,dt \] associated with \(\Sigma\).
This problem is splitted into two subproblems:
(1) the existence of a spectral factorization \(\Pi(i\omega)=\Xi(i\omega)^*\Xi(i\omega)\) (a.e. \(\omega\in {\mathbb R}\)) for the Popov function \(\Pi: i{\mathbb R}\rightarrow {\mathcal L(U)}\), associated with \(\Sigma\) and given by \[ \Pi(i\omega)=R+NG(i\omega)+G(i\omega)^*N^*+G(i\omega)^*QG(i\omega), \] where the spectral factor \(\Xi\in H_{\infty}({\mathcal L(U)})\) is outer;
(2) the existence of a certain extended output map.
The author considers the singular case \(\Pi(i\omega)\geq 0\); the case \(\Pi(i\omega)\geq\varepsilon I\) for some \(\varepsilon>0\) and a.e. \(\omega\in {\mathbb R}\) was treated by M. Weiss and G. Weiss (op. cit.), and by O. Staffans [SIAM J. Control Optim. 36, No. 4, 1268–1292 (1998; Zbl 0919.93040), ibid. 37, No. 1, 131–164 (1999; Zbl 0955.49018)]. Sufficient conditions are found for the solvability of both of these subproblems.
These results are applied to various examples of heat equation systems satisfying a positive-real condition, and to a certain class of retarded systems.

MSC:

93B28 Operator-theoretic methods
93D10 Popov-type stability of feedback systems
93C20 Control/observation systems governed by partial differential equations
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