
The
KalmanYakubovichPopov Theorem in thermoelastic problems 
The KalmanYakubovichPopov Theorem for infinitedimensional
systems: Some new results
a) Bounded control operator, C_{0}semigroups, PritchardSalamon
systems
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regulator problem and operator Riccati equation under
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 Van Keulen, B.: Equivalent conditions for the
solvability of the nonstandard LQproblem for PritchardSalamon
systems. J. Control Optim., 33, 1995, 1326  1356.
 Curtain, R. F.: The KalmanYakubovichPopov Lemma for
PritchardSalamon 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, EulerBernoulli 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 nonstandard LQR problem for boundary
control systems. Rend. Sem. Mat. Univ. Pol. Torino, 56 (4),
1998, 105  114.
 Pandolfi, L.: The KalmanYakubovichPopov 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
KalmanYakubovichPopov 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 infinitedimensional continuous time KalmanYakubovichPopov
inequality. Operator Theory: Advances and Applications, 1, 2005, Birkhäuser Verlag Basel,
1  28.
h) Overview articles
 Pandolfi, L.: The KalmanPopovYakubovich 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 KalmanPopovYakubovich problem. Proc.
Int. Conf. on Mathematics and its Applications, Yagyarta, 1999.

