Abstract and Applied Analysis

Singular estimates and uniform stability of coupled systems of hyperbolic/parabolic PDEs

F. Bucci, I. Lasiecka, and R. Triggiani

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Abstr. Appl. Anal., Volume 7, Number 4 (2002), 169-237.

First available in Project Euclid: 14 April 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35M10: Equations of mixed type 35B37
Secondary: 49J20: Optimal control problems involving partial differential equations 93D15: Stabilization of systems by feedback 93C20: Systems governed by partial differential equations


Bucci, F.; Lasiecka, I.; Triggiani, R. Singular estimates and uniform stability of coupled systems of hyperbolic/parabolic PDEs. Abstr. Appl. Anal. 7 (2002), no. 4, 169--237. doi:10.1155/S1085337502000763. https://projecteuclid.org/euclid.aaa/1050348463

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  • G. Avalos, Optimal control for a coupled hyperbolic-parabolic system in arising structural acoustics, Ph.D. thesis, University of Virginia, 1995.
  • ––––, Sharp regularity estimates for solutions of the wave equation and their traces with prescribed Neumann data, Appl. Math. Optim. 35 (1997), no. 2, 203–219.
  • G. Avalos and I. Lasiecka, Differential Riccati equation for the active control of a problem in structural acoustics, J. Optim. Theory Appl. 91 (1996), no. 3, 695–728.
  • A. Bensoussan, G. Da Prato, M. C. Delfour, and S. K. Mitter, Representation and Control of Infinite-Dimensional Systems, Vol. II, Birkhäuser, Massachusetts, 1993.
  • S. P. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. Russell on structural damping for elastic systems: The case $\alpha = \frac{1}{2}$, Proceedings of the Seminar on Approximation and Optimization (Cuba, 1987), Lecture Notes in Mathematics, vol. 1354, Springer-Verlag, 1987, pp. 234–256.
  • ––––, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math. 136 (1989), no. 1, 15–55.
  • ––––, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Differential Equations 88 (1990), no. 2, 279–293.
  • E. K. Dimitriadis, C. R. Fuller, and C. A. Rogers, Piezoelectric actuators for distributed vibration excitation of thin plates, J. Vibration and Acoustic 113 (1991), 100–107.
  • L. Gearhart, Spectral theory for contraction semigroups on Hilbert space, Trans. Amer. Math. Soc. 236 (1978), 385–394.
  • P. Grisvard, Caractérisation de quelques espaces d'interpolation, Arch. Rational Mech. Anal. 25 (1967), 40–63 (French).
  • ––––, Elliptic Problems in Nonsmooth Domains, Monographs and Studies in Mathematics, vol. 24, Pitman, Massachusetts, 1985.
  • D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics, vol. 840, Springer-Verlag, Berlin, 1981.
  • I. Herbst, The spectrum of Hilbert space semigroups, J. Operator Theory 10 (1983), no. 1, 87–94.
  • J. S. Howland, On a theorem of Gearhart, Integral Equations Operator Theory 7 (1984), no. 1, 138–142.
  • T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.
  • I. Lasiecka, Mathematical Control Theory of Coupled PDE Systems, NSF-CMBS Lecture Notes, SIAM, Pennsylvania, 2001.
  • I. Lasiecka, J.-L. Lions, and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl. (9) 65 (1986), no. 2, 149–192.
  • I. Lasiecka and R. Triggiani, Optimal control and differential Riccati equations under singular estimates for $e^{At}B$ in the absence of analyticity, to appear in Advances in Dynamics and Control.
  • ––––, Regularity of hyperbolic equations under ${L_2}(0,{T};{L_2}(\Gamma ))$-Dirichlet boundary terms, Appl. Math. Optim. 10 (1983), no. 3, 275–286.
  • ––––, Trace regularity of the solutions of the wave equation with homogeneous Neumann boundary conditions and data supported away from the boundary, J. Math. Anal. Appl. 141 (1989), no. 1, 49–71.
  • ––––, Sharp regularity theory for second order hyperbolic equations of Neumann type. I. ${L_2}$ nonhomogeneous data, Ann. Mat. Pura Appl. (4) 157 (1990), 285–367.
  • ––––, Regularity theory of hyperbolic equations with nonhomogeneous Neumann boundary conditions. II. General boundary data, J. Differential Equations 94 (1991), no. 1, 112–164.
  • ––––, Recent Advances in Regularity of Second Order Hyperbolic Mixed Problems, and Applications. Dynamics Reported, Expositions in Dynamical Systems. New Series, vol. 3, Springer-Verlag, Berlin, 1994.
  • ––––, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Vol. I: Abstract Parabolic Systems, Cambridge University Press, Cambridge, 2000.
  • ––––, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. Vol. II: Abstract Hyperbolic-like Systems over a Finite Time Horizon, Cambridge University Press, Cambridge, 2000.
  • ––––, Optimal control and algebraic Riccati equations under singular estimates for $e^{At}B$ in the absence of analyticity, Differential Equations and Optimal Control, Marcel Dekker Lecture Notes in Pure and Applied Mathematics, vol. 225, Marcel Dekker, 2001, pp. 193–220.
  • J.-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications. Vol. I, Springer-Verlag, New York, 1972.
  • ––––, Non-Homogeneous Boundary Value Problems and Applications. Vol. II, Springer-Verlag, New York, 1972.
  • R. H. Martin, Jr., Nonlinear Operators and Differential Equations in Banach Spaces, Pure and Applied Mathematics, Wiley & Sons, New York, 1976.
  • L. Monauni, Exponential decay of solutions to Cauchy's abstract problem as determined by the extended spectrum of the dynamical operator, Tech. Report LIDS-P-947, Department of Electrical Engineering, Massachusetts Institute of Technology, USA, 1979, Tech. Report No. 90, Control Theory Center, University of Warwick, England, 1980.
  • P. M. Morse and K. Ingard, Theoretical Acoustics, McGraw-Hill, New York, 1968.
  • A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983.
  • J. Prüss, On the spectrum of ${C_0}$-semigroups, Trans. Amer. Math. Soc. 284 (1984), no. 2, 847–857.
  • D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 26 (1998), no. 1, 185–206.
  • R. Triggiani, Min-Max game theory and optimal control with indefinite cost under a singular estimate for $e^{At}B$ in the absence of analyticity, invited paper in special volume of Birkhäuser Verlag in memory of B. Terreni; A. Lorenzi, and B. Ruf, eds., to appear.
  • ––––, Regularity with interior point control. II. Kirchhoff equations, J. Differential Equations 103 (1993), no. 2, 394–420.
  • J. Zabczyk, Mathematical Control Theory: An Introduction, Birkhäuser, Massachusetts, 1992.