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 ${\mathcal{E}}_{min}\left(T\right)$, as terminal time $T\downarrow 0$. Key use is made of the underlying analyticity of the semigroup generated by the elastic operator $\mathcal{A}$, as well as of the explicit characterization of its domain of definition. We ultimately find that the blowup rate for ${\mathcal{E}}_{min}\left(T\right)$, as $T$ goes to zero, depends on the extent of structural damping.

@article{ASNSP_2003_5_2_3_601_0, author = {Avalos, George and Lasiecka, Irena}, 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}, publisher = {Scuola normale superiore}, volume = {Ser. 5, 2}, number = {3}, year = {2003}, pages = {601-616}, zbl = {1170.35470}, mrnumber = {2020861}, language = {en}, url = {http://www.numdam.org/item/ASNSP_2003_5_2_3_601_0} }

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://www.numdam.org/item/ASNSP_2003_5_2_3_601_0/

[1] 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 2059787 | Zbl 1054.74032

- ,[2] “Representation and Control of Infinite Dimensional Systems", Volume II, Birkhäuser, Boston, 1995. | MR 1246331 | Zbl 1117.93002

- - - ,[3] Proof of extension of two conjectures on structural damping for elastic systems: The case $\frac{1}{2}\le \alpha \le 1$, Pacific J. Math. 136 n. 1 (1989), 15-55. | MR 971932 | Zbl 0633.47025

- ,[4] Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Differential Equations 88 (1990), 279-293. | MR 1081250 | Zbl 0717.34066

- ,[5] “An Introduction to Infinite Dimensional Analysis", Scuola Normale Superiore, 2001. | MR 2010852 | Zbl 1065.46001

,[6] 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 1944159 | Zbl 1042.47032

,[7] “Stochastic Equations in Infinite Dimensions", Cambridge University Press, Cambridge, 1992. | MR 1207136 | Zbl 0761.60052

- ,[8] 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 1944161 | Zbl 1039.35134

,[9] Regularity of the minimum time function and minimum energy problems, in SIAM J. Control Optim. 37 n. 4 (1999), 1195-1221. | MR 1691938 | Zbl 0958.49014

- ,[10] Regularity of solutions of second order Hamilton-Jacobi equations and applications to a control problem, Comm. Partial Differential Equations 20 (1995), 775-926. | MR 1326907 | Zbl 0842.49021

,[11] “Table of Integrals, Series, and Products”, Academic Press, San Diego, 1980. | MR 1398882 | Zbl 0918.65001

- ,[12] Exact controllability of the wave equation with Neumann boundary control, Appl. Math. Optim. 28 (1993), 243-290. | MR 974187 | Zbl 0666.49012

- ,[13] Exact null controllability of structurally damped and thermo-elastic parabolic models, Rend. Mat. Acc. Lincei, (9) 9 (1998), 43-69. | MR 1669244 | Zbl 0935.93010

- ,[14] “Control Theory for Partial Differential Equations: Continuous and Approximation Theories", Cambridge University Press, New York, 2000. | Zbl 0961.93003

- ,[15] Schauder estimates for a class of degenerate elliptic and parabolic operators with unbounded coefficients in ${\mathbb{R}}^{n}$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (IV) 24 (1997), 133-164. | Numdam | MR 1475774 | Zbl 0887.35062

,[16] Observation and prediction for the heat equation, J. Math. Anal. Appl. 28 (1969), 303-312. | MR 247301 | Zbl 0183.38201

- ,[17] “Semigroups of Operators and Applications to Partial Differential Equations”, Springer-Verlag, New York, 1983. | MR 710486 | Zbl 0516.47023

,[18] Null controllability with vanishing energy, SIAM J. Control Optim. 42 n. 3 (2003), pp. 1013-1032. | MR 2002145 | Zbl 1050.93010

- ,[19] Controllability and stabilizability theroy for linear partial differential equations: recent progress and open questions, SIAM Rev. 20 (1978), 639-739. | MR 508380 | Zbl 0397.93001

,[20] How fast are violent controls?, Math. Control Signals Systems 1 (1988), 89-95. | MR 923278 | Zbl 0663.49018

,[21] Two results on exact boundary control of parabolic equations, Appl. Math. Optim. 11 (year?), 145-152. | MR 743923 | Zbl 0562.49003

,[22] How fast are violent controls, II?, Math. Control Signals Systems 9 (1997), 327-340. | MR 1450356 | Zbl 0906.93007

- ,[23] Optimal estimates of minimal norm controls in exact nullcontrollability ot two non-classical abstract parabolic equations, Adv. Differential Equations 8 (2003), 189-229. | MR 1948044 | Zbl pre02002528

,