Riccati equations for strongly stabilizable bounded linear systems
The authors study the Riccati equation \[ A^* Xx+ XAx- (B^* X+ NC)^*\;R^{-1}(B^* X+ NC)x+ C^* QCx= 0.\tag{1} \] It is assumed that it generates a \(C_0\) semigroup \(T(t)\). \(B\) is a linear operator between Hilbert spaces \(U\) and \(Z\), where \(U\) is separable, \(C\) between \(Z\) and \(Y\), where \(Y\) is also a separable Hilbert space. In the last ten years several papers appeared considering spectral factorization of this type of Riccati equation. All of them allow the operators \(B\) and \(C\) to be unbounded. Here, the authors consider only the special case when they are bounded, but they relax other conditions assumed by other authors and also obtain much stronger results. They address existence and uniqueness of strongly stabilizable self-adjoint solutions. As the author state, the more recent papers on strong stabilizability and detectability obtained only sufficient conditions for the existence of solutions. The authors conditions are now necessary and sufficient. The present study is influenced by the papers of Weiss. The class of problems which is studied here arises in mathematical modelling of large structures with collocated actuators and sensors, and in general studies of structural dynamics. The authors consider a vibrating square plate with two parallel edges simply supported, a third one is clamped and the remaining free edge is subject to a control force and a control torque. They quote a paper by Y. You, where it was shown that \(A\) is a closed, densely defined skew-adjoint operator with compact support and compact resolvent. The reviewer comments that this is a true statement for most structural dynamics operators without internal dissipation. The system is reduced by well known procedure to a first-order vector formulation \[ dz/dt= Az(t)+ Bu(t),\quad y(t)= B^*z(t). \] It is also shown that \(A\)-\(BB^*\) generates a strongly stable contraction semigroup \(T_B(t)\). The authors proceed to discuss some properties of linear bounded operators related to the input-output schemes in control theory, such as well-posedness and shift invariance, and particulary the property of generating a strongly stable \(C_0\) semigroup. The concept of \(\tau\) concatenation \(\diamondsuit_\tau\) is defined \[ u\diamondsuit_\tau v= u(t)\quad\text{for }0\leq t<\tau,\quad =v(t-\tau)\quad\text{for }\tau\leq t. \] Now, the authors define a family \(F\) of extended input-output maps such that \(F_{t+\tau}(u\diamondsuit_\tau v)= F_t u\diamondsuit_\tau(\Psi_t \Phi_\tau u+ F_t v)\), with \(F_0= 0\). Here \(\Psi\) is the extended output map, while \(\Phi\) is a bounded operator obeying: \(\Phi_{t+\tau} u\diamondsuit_\tau v= T(t) \Phi_t u+\Phi_tv\), and \(T(t)\) is the previously stated semigroup of bounded linear operators. Against this background, the authors introduce the Popov function \(\Pi(j\omega)\), which plays an important role in the following arguments. For the Riccati equation (1) the Popov function is defined by \[ \Pi(j\omega)= R+ NG(j\omega)+ G(j\omega)^* N^*+ G(j\omega)^* QG(j\omega), \] where \(G(j\omega)= C(j\omega I- A)^{-1}B\). The main theorem of this part of the paper asserts that there exists a unique strongly stabilizing solution of the Riccati equation discussed here if and only if the associated Popov function admits a Wiener-Hopf factorization. This theorem is used to show the existence of \(J\)-spectral factorization and also the existence of stabilizing solutions for the standard form of \(LQ\) Riccati equation. These are important results concerning stabilization of linear control systems in Hilbert spaces.
strongly stabilizable self-adjoint solutions
scientific article; zbMATH DE number 1325616