Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 3, pp. 601-616.

The null controllability problem for a structurally damped abstract wave equation -often referred to in the literature as a structurally damped equation- is considered with a view towards obtaining optimal rates of blowup for the associated minimal energy function min (T), as terminal time T0. Key use is made of the underlying analyticity of the semigroup generated by the elastic operator 𝒜, as well as of the explicit characterization of its domain of definition. We ultimately find that the blowup rate for min (T), as T goes to zero, depends on the extent of structural damping.

Classification: 35, 93
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     title = {Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
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Avalos, George; Lasiecka, Irena. Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 2 (2003) no. 3, pp. 601-616. http://archive.numdam.org/item/ASNSP_2003_5_2_3_601_0/

[1] G. Avalos - I. Lasiecka, A note on the null controllability of thermoelastic plates and singularity of the associated minimal energy function, preprint Scuola Normale Superiore, Pisa, Giugno, 2002. | MR | Zbl

[2] A. Bensoussan - G. Da Prato - M. C. Delfour - S. K. Mitter, “Representation and Control of Infinite Dimensional Systems", Volume II, Birkhäuser, Boston, 1995. | MR | Zbl

[3] S. Chen - R. Triggiani, Proof of extension of two conjectures on structural damping for elastic systems: The case 1 2α1, Pacific J. Math. 136 n. 1 (1989), 15-55. | MR | Zbl

[4] S. Chen - R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Differential Equations 88 (1990), 279-293. | MR | Zbl

[5] G. Da Prato, “An Introduction to Infinite Dimensional Analysis", Scuola Normale Superiore, 2001. | MR | Zbl

[6] G. Da Prato, Bounded perturbations of Ornstein Uhlenbeck semigroups, Evolution Equations Semigroups and Functional Analysis, Vol 50, In: “The series Progress in Nonlinear Differential Equations and Their Applications", Birkhauser, 2002, pp. 97-115. | MR | Zbl

[7] G. Da Prato - J. Zabczyk, “Stochastic Equations in Infinite Dimensions", Cambridge University Press, Cambridge, 1992. | MR | Zbl

[8] M. Fuhrman, On a class of quasi-linear equations in infinite-dimensional spaces, Evolution Equations Semigroups and Functional Analysis, Vol 50, In: “The series Progress in Nonlinear Differential Equations and Their Applications", Birkhauser, 2002, pp. 137-155. | MR | Zbl

[9] F. Gozzi - P. Loretti, Regularity of the minimum time function and minimum energy problems, in SIAM J. Control Optim. 37 n. 4 (1999), 1195-1221. | MR | Zbl

[10] F. Gozzi, Regularity of solutions of second order Hamilton-Jacobi equations and applications to a control problem, Comm. Partial Differential Equations 20 (1995), 775-926. | MR | Zbl

[11] I. S. Gradshteyn - I. M. Ryzhik, “Table of Integrals, Series, and Products”, Academic Press, San Diego, 1980. | MR | Zbl

[12] I. Lasiecka - R. Triggiani, Exact controllability of the wave equation with Neumann boundary control, Appl. Math. Optim. 28 (1993), 243-290. | MR | Zbl

[13] I. Lasiecka - R. Triggiani, Exact null controllability of structurally damped and thermo-elastic parabolic models, Rend. Mat. Acc. Lincei, (9) 9 (1998), 43-69. | MR | Zbl

[14] I. Lasiecka - R. Triggiani, “Control Theory for Partial Differential Equations: Continuous and Approximation Theories", Cambridge University Press, New York, 2000. | Zbl

[15] A. Lunardi, Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in n , Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 24 (1997), 133-164. | Numdam | MR | Zbl

[16] V. J. Mizel - T. I. Seidman, Observation and prediction for the heat equation, J. Math. Anal. Appl. 28 (1969), 303-312. | MR | Zbl

[17] A. Pazy, “Semigroups of Operators and Applications to Partial Differential Equations”, Springer-Verlag, New York, 1983. | MR | Zbl

[18] E. Priola - J. Zabczyk, Null controllability with vanishing energy, SIAM J. Control Optim. 42 n. 3 (2003), pp. 1013-1032. | MR | Zbl

[19] D. L. Russell, Controllability and stabilizability theroy for linear partial differential equations: recent progress and open questions, SIAM Rev. 20 (1978), 639-739. | MR | Zbl

[20] T. I. Seidman, How fast are violent controls?, Math. Control Signals Systems 1 (1988), 89-95. | MR | Zbl

[21] T. I. Seidman, Two results on exact boundary control of parabolic equations, Appl. Math. Optim. 11 (year?), 145-152. | MR | Zbl

[22] T. I. Seidman - J. Yong, How fast are violent controls, II?, Math. Control Signals Systems 9 (1997), 327-340. | MR | Zbl

[23] R. Triggiani, Optimal estimates of minimal norm controls in exact nullcontrollability ot two non-classical abstract parabolic equations, Adv. Differential Equations 8 (2003), 189-229. | MR