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## On a degenerate Riccati equation ∗ by

Citations: | 1 - 0 self |

### Citations

3123 |
Perturbation theory for linear operators.
- Kato
- 1966
(Show Context)
Citation Context ...ccati equation 1401 Example 4.2 If (A, D(A)) is the infinitesimal generator of an analytic semigroup on Z and if the resolvent of A is a compact operator in Z, then the spectrum of A is discrete (see =-=Kato, 1995-=-). Let us assume that A has no eigenvalue on the imaginary axis. From Kato (1995), pp. 178–182, it follows that the space Z can be decomposed in the form Z = Zs ⊕ Zu, where Zu ∩ D(A) = Zu and Zs ∩ D(A... |

158 | Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Encyclopedia of Mathematics and its Applications - Lasiecka, Triggianni - 2000 |

153 |
Controllability of evolution equations
- Fursikov, Imanuvilov
- 1996
(Show Context)
Citation Context ...nvalues having a real part greater or equal than −α is finite, for any α > 0. Moreover, system (6.6) is exponentially stabilizable with any prescribed decay rate, because it is null controllable (see =-=Fursikov and Imanuvilov, 1996-=-). Without loss of generality we can choose α > 0 so that the pair (A + αI, B) satisfies the assumptions of Theorem 2. Next, by making the change of variable ˆy = e αt y, û = e αt u, for some α > 0, w... |

83 |
Mathematical control theory: an introduction,
- Zabczyk
- 1992
(Show Context)
Citation Context ...ists a control u ∈ L 2 (0, ∞; U) such that J(zz0,u, u) < ∞. If we assume that FCC is valid, then it is easy to see that problem (Pz0) admits a unique solution. In this case, it can also be shown (see =-=Zabczyk, 2008-=-; Bensoussan et al., 1993; Lasiecka and Triggiani, 2000) that inf(Pz0) = 1 2 ( ˜ Pz0, z0)Z for all z0 ∈ Z, where ˜ P is the so-called minimal solution to the algebraic Riccati equation P ∈ L(Z), P = P... |

21 |
On the stabilizability problem in Banach space,”
- Triggiani
- 1975
(Show Context)
Citation Context ... and that ‖e tπsA πsz‖Z ≤ Ce −δt ‖πsz‖Z, 0 πuA − πuBB ∗ π ∗ uPu with δ > 0, C > 0. Combining these two stability results, it can be easily shown that A − BB ∗ P is also exponentially stable (see e.g. =-=Triggiani, 1975-=-). This completes the proof. Assume now that either Z is finite dimensional and that A ∈ L(Z), or that the unbounded operator (A, D(A)) is the infinitesimal generator of an analytic semigroup on Z wit... |

19 | Algebraic Riccati equations with non-smoothing observation arising in hyperbolic and EulerBernoulli boundary control problems, - Flandoli, Lasiecka, et al. - 1988 |

13 | Mesh Independence of Kleinman–Newton Iterations for Riccati Equations in Hilbert Space
- Burns, Sachs, et al.
(Show Context)
Citation Context ... solve equation (1.4) (see Benner and Baur, 2008). The Kleinman-Newton algorithm (see Kleinman, 1968) is still efficient in solving equation (1.4) for large scale equation (see Benner and Baur, 2008; =-=Burns et al., 2008-=-). The inconvenience of that method is that, in order to guarantee the convergence of the Newton method, the initial guess P0 must be chosen so that Ah − BhB ∗ h P0 is exponentially stable. Finding su... |

13 | Boundary feedback stabilization of the two dimensional Navier-Stokes equations with finite dimensional controllers., Discrete Contin - Raymond, Thevenet |

9 | Null controllability with vanishing energy - Priola, Zabczyk |

6 |
Optimal pole shifting for continuous multivariable linear systems
- Amin
- 1985
(Show Context)
Citation Context ...0) and not some other ones. Thus, for clarity we shall write all the proofs. Let us finally mention that the results established in Sections 4 and 5 are well known for finite dimensional systems (see =-=Amin, 1985-=-; Ibbini and Amin, 1993; Zhou et al., 2008). In Section 6, we end the paper by giving a concrete example where results of Section 4 apply. 2. A comparison principle In this section, we recall a compar... |

4 |
Nonlinear feedback stabilization of a two dimensional Burgers equation
- Buchot, Thevenet, et al.
(Show Context)
Citation Context ...Observe that B = B ∗ . The linearized system associated to (6.5) is y ′ = Ay + Bu in (0, ∞), y(0) = y0, (6.6) Let us recall that (A, D(A)) is the infinitesimal generator of an analytic semigroup (see =-=Thevenet et al., 2009-=-) and that the resolvent of (A, D(A)) is compact. Thus, the spectrum of A is discrete and the eigenvalues have finite multiplicity. The number of eigenvalues having a real part greater or equal than −... |

3 | P.: Efficient solution of algebraic Bernoulli equations using Hmatrix arithmetic
- Baur, Benner
- 2007
(Show Context)
Citation Context ...e dimension N can be very large and in that case classical algorithms are not efficient in solving equation (1.4). This is why new algorithms are still developed nowadays to solve equation (1.4) (see =-=Benner and Baur, 2008-=-). The Kleinman-Newton algorithm (see Kleinman, 1968) is still efficient in solving equation (1.4) for large scale equation (see Benner and Baur, 2008; Burns et al., 2008). The inconvenience of that m... |

2 |
An invariant subspace method for large-scale algebraic Riccati equation
- Amodei, Buchot
- 2010
(Show Context)
Citation Context ...group. (1.5) (In Benner and Baur, 2008, the authors consider the case where dim(Z) < ∞.) Indeed, there are specific algorithms to solve equation (1.5) even when the dimension of Z is relatively high (=-=Amodei and Buchot, 2008-=-; Benner and Baur, 2008). (Equation (1.5) corresponds to the algebraic Riccati equation (1.2) when C = 0.) It is well known that there exists a unique solution to equation (1.2) such that A − BB ∗ P g... |

2 |
On an iterative technique for Riccati equations
- Kleinman
- 1968
(Show Context)
Citation Context ...algorithms are not efficient in solving equation (1.4). This is why new algorithms are still developed nowadays to solve equation (1.4) (see Benner and Baur, 2008). The Kleinman-Newton algorithm (see =-=Kleinman, 1968-=-) is still efficient in solving equation (1.4) for large scale equation (see Benner and Baur, 2008; Burns et al., 2008). The inconvenience of that method is that, in order to guarantee the convergence... |

1 | Observateurs et stabilité - Bensoussan - 1987 |

1 |
A state feedback controller with minimum control effort
- Ibbini, Amin
- 1993
(Show Context)
Citation Context ...ome other ones. Thus, for clarity we shall write all the proofs. Let us finally mention that the results established in Sections 4 and 5 are well known for finite dimensional systems (see Amin, 1985; =-=Ibbini and Amin, 1993-=-; Zhou et al., 2008). In Section 6, we end the paper by giving a concrete example where results of Section 4 apply. 2. A comparison principle In this section, we recall a comparison principle for solu... |