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| The
Kalman-Yakubovich-Popov Theorem in thermo-elastic problems |
The Kalman-Yakubovich-Popov Theorem for infinite-dimensional
systems: Some new results
a) Bounded control operator, C0-semigroups, Pritchard-Salamon
systems
- Louis, J. Cl. and D. Wexler: The Hilbert space
regulator problem and operator Riccati equation under
stabilizability. Ann. Soc. Sci. Bruxelles, Ser. 1, 105, 1991, 137
-- 165.
- Van Keulen, B.: Equivalent conditions for the
solvability of the nonstandard LQ-problem for Pritchard-Salamon
systems. J. Control Optim., 33, 1995, 1326 -- 1356.
- Curtain, R. F.: The Kalman-Yakubovich-Popov Lemma for
Pritchard-Salamon systems. System & Control Lett., 27, 1996, 67
-- 72.
- Curtain, R. F. and J. C. Oostveen: The Popov criterion
for strongly stable distributed parameter systems. Int. J.
Control, 74 (3), 2001, 265 -- 280.
b) Parabolic systems, holomorphic semigroups, heat equation
- Pandolfi, L.: Dissipativity and Lur'e problem for
parabolic boundary control systems. SIAM J. Control Optimiz.,
36, 1998, 2061 -- 2081.
- Bucci, F.: Frequency domain stability of nonlinear
feedback systems with unbounded input operator. Dynamics of
Continuous, Discrete and Impulsive Systems, 7, 2000, 351 -- 368.
c) First order hyperbolic equations - Triggiani, R.: An optimal control problem with unbounded control operator
and unbounded observation operator where the algebraic Riccati equation is satisfied as a
Lyapunov equation. Appl. Math. Lett., 10 (2), 1997, 95 -- 102.
d) Second order hyperbolic systems, Euler-Bernoulli equation,
Kirchhoff equation, Schrödinger equation, string and membrane
equation
- Lasiecka, I. and R. Triggiani: Algebraic Riccati equations
arising in boundary / point control: A review of theoretical and
numerical results. In: Perspective in control theory,
Jacubczyk, B., Malanowski, K., Raspondek, W. eds., Birkhäuser,
Boston, 1990. Part 1: Continuous case, 175 -- 210; Part 2:
Approximation theory, 211 -- 235.
- Bucci, F.: The non-standard LQR problem for boundary
control systems. Rend. Sem. Mat. Univ. Pol. Torino, 56 (4),
1998, 105 -- 114.
- Pandolfi, L.: The Kalman-Yakubovich-Popov Theorem for
stabilizable hyperbolic boundary control systems. Integral
Equations Operator Theory, 34, 1999, 478 -- 493.
- Barbu, V., Lasiecka, I. and R. Triggiani: Extended algebraic Riccati
equations in the abstract hyperbolic case. Nonlinear. Anal. 40,
2000, 105 -- 129.
e) Nonstandard Riccati equations arising in boundary control
problems governed by damped wave and plate equations; Hammerstein
integral equations with weekly singular kernels
- Flandoli, F.: Riccati equations arising in a boundary
control problem with distributed parameters. SIAM Journal on
Control and Optimization, 22, 1984, 76 -- 86.
- Lasiecka, I., D. Lukas and L. Pandolfi: Input dynamics and
nonstandard Riccati equations with applications to boundary
control of damped wave and plate equations. Journal of
Optimization Theory and Applications, 84 (3), 1995, 549 -- 574.
f) Discrete time distributed systems, approximation theory for
boundary control systems
- Helton, J. W.: A spectral factorization approach to the
distributed stable regulator problem; the algebraic Riccati equation. SIAM Journal on
Control and Optimization, 14, 1976, 639 -- 661.
- Malinen, J.: Discrete time H^\infty algebraic Riccati equations. Techn. Report A 428,
Institute of Mathematics, Helsinki University, Finland, 2000.
- Arov, D. Z., Kaashoek, M. A., and D. R. Pik: The
Kalman-Yakubovich-Popov inequality and infinite dimensional
discrete time dissipative systems. J. Operator Theory, 2005, to
appear.
g) Generalized (possibly unbounded) solutions of the KYP
inequality
- Arov, D. Z. and O. J. Staffans: The infinite-dimensional continuous time Kalman-Yakubovich-Popov
inequality. Operator Theory: Advances and Applications, 1, 2005, Birkhäuser Verlag Basel,
1 -- 28.
h) Overview articles
- Pandolfi, L.: The Kalman-Popov-Yakubovich Theorem: an overview
and new results for hyperbolic control systems. Nonlinear Analysis, Methods & Applications,
30 (2), 1997, 735 -- 745.
- Pandolfi, L.: Recent results on the Kalman-Popov-Yakubovich problem. Proc.
Int. Conf. on Mathematics and its Applications, Yagyarta, 1999.
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