Uniqueness of the Riccati operator of the non-standard ARE of a third order dynamics with boundary control

Lasiecka, Irena; Triggiani, Roberto

doi:10.2478/candc-2022-0013

Abstract

The Moore-Gibson-Thompson [MGT] dynamics is considered. This third order in time evolution arises within the context of acoustic wave propagation with applications in high frequency ultrasound technology. The optimal boundary feedback control is constructed in order to have on-line regulation. The above requires wellposedness of the associated Algebraic Riccati Equation. The paper by Lasiecka and Triggiani (2022) recently contributed a comprehensive study of the Optimal Control Problem for the MGT-third order dynamics with boundary control, over an infinite time-horizon. A critical missing point in such a study is the issue of uniqueness (within a specific class) of the corresponding highly non-standard Algebraic Riccati Equation. The present note resolves this problem in the positive, thus completing the study of Lasiecka and Triggiani (2022) with the final goal of having on line feedback control, which is also optimal.

References

Barbu, V., Lasiecka, I. and Triggiani, R. (2006) Tangential Boundary Stabilization of Navier-Stokes Equations. Memoirs AMS 181, 852, 128.10.1090/memo/0852
Search in Google Scholar Back to article
Bongarti, M., Lasiecka, I. and Rodrigues, J. H. (2022) Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity. Discrete and Continuous Dynamical Systems, Series S, doi: 10.3934/dcdss.2022020
Open DOI Search in Google Scholar Back to article
Bongarti, M., Lasiecka, I. and Triggiani, R. (2022) The SMGTJ equation from the boundary: regularity and stabilization. Applicable Analysis, to appear. doi: 10.1080/00036811.2021.1999420
Open DOI Search in Google Scholar Back to article
Bucci, F. and Lasiecka, I. (2019) Feedback control of the acoustic pressure in ultrasonic propagation. Optimization, 68, 10, 1811–1854.10.1080/02331934.2018.1504051
Search in Google Scholar Back to article
Clason, C. and Kaltenbacher, B. (2015) Avoiding degeneracy in the Westervelt equation by state constrained optimal control. Evol. Equ. Control Theory 2, 2, 281–300.
Search in Google Scholar Back to article
Clason, C., Kaltenbacher, B. and Veljović, S. (2009) Boundary Optimal Control of the Westervelt and the Kuznetsov equations. J. Math. Anal. Appl. 356, 738–751.
Search in Google Scholar Back to article
Christov, C. I. and Jordan, P. M. (2005) Heat Conduction Paradox Involving Second-Sound Propagation in Moving Media. Physical Review Letters 94, 15.10.1103/PhysRevLett.94.15430115904151
Search in Google Scholar Back to article
Datko, R. (1970) Extending a theorem of Lyapunov to Hilbert spaces. J. Math. Anal. & Appl. 32, 610–616.
Search in Google Scholar Back to article
Jordan, P. (2004) An analytical study of Kuznetsov’s equation: diffusive solitons, shock formation, and solution bifurcation. Physics Letters A, 326, 77–84.10.1016/j.physleta.2004.03.067
Search in Google Scholar Back to article
Jordan, P. M. (2014) Second-sound phenomena in inviscid, thermally relaxing gases. Discrete Contin. Dyn. Syst. Ser. B 19 7, 2189–2205.
Search in Google Scholar Back to article
Jordan, P. M. (2016) The effects of coupling on finite-amplitude acoustic traveling waves in thermoviscous gases: Blackstock’s models. Evol. Equ. Control Theory 5 3, 383–397.10.3934/eect.2016010
Search in Google Scholar Back to article
Kaltenbacher, B., Lasiecka, I. and Marchand, R. (2011) Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equations arising in high intensity ultrasound. Control and Cybernetics 40, 4, 971–988.
Search in Google Scholar Back to article
Kaltenbacher, B., Lasiecka, I. and Pospieszalska, M. (2012) Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound. Math. Methods in the Applied Sciences, 22.11, 1250035.10.1142/S0218202512500352
Search in Google Scholar Back to article
Kaltenbacher, B. (2015) Mathematics of nonlinear acoustics. Evol. Equ. Control Theory 4, 447–491.
Search in Google Scholar Back to article
Lasiecka, I., Lukes, D. and Pandolfi, L. (1995) Input dynamics and nonstandard Riccati equations with applications to boundary control of damped wave and plate equations. J. Optimiz. Theory Appl. 84, 3, 549–574.
Search in Google Scholar Back to article
Lasiecka, I., Pandolfi, L. and Triggiani, R. (1997) A singular control approach to highly damped second-order abstract equations and applications. Appl. Math. Optim. 36, 67–107.
Search in Google Scholar Back to article
Lasiecka, I. and Triggiani, R. (2000) Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Vol I: Abstract Parabolic Systems; Vol II: Abstract Hyperbolic Systems over a Finite Time Horizon. Encyclopedia of Mathematics and Its Applications Series. Cambridge University Press, January 2000.
Search in Google Scholar Back to article
Lasiecka, I. and Triggiani, R. (2022) Optimal feedback arising in a third order dynamics with boundary controls and infinite horizon. J. Opt. Th. & Appl., DOI 10.1007/s10957–022–02017–y.
Open DOI Search in Google Scholar Back to article
Louis, D. and Wexler, D. (1991) The Hilbert space regulator and the operator Riccati equation under stabilizability. Annales de la Societé Scientifique de Bruxelles 105, 157–165.
Search in Google Scholar Back to article
Moore, F. K. and Gibson, W. E. (1960) Propagation of weak disturbances in a gas subject to relaxation effects. J. Aero/Space Sci., 27, 117–127.
Search in Google Scholar Back to article
Marchand, R., McDevitt, T. and Triggiani, R. (2012) An abstract semigroup approach to the third-order MGT equation arising in high-intensity ultrasound: Structural decomposition, spectral analysis, exponential stability. Math. Methods in the Applied Sciences, 35: 1896–1929.10.1002/mma.1576
Search in Google Scholar Back to article
Nikolic, V. and Kaltenbacher, B. (2017) Sensitivity analysis for shape optimization of a focusing acoustic lens in lithotripsy. Appl. Math. Optim. 76, 2, 261–301.
Search in Google Scholar Back to article
Stokes, G. G. (1851) An examination of the possible effect of the radiation of heat on the propagation of sound. Philosophical Magazine Series, 1(4), 305–317.10.1080/14786445108646736
Search in Google Scholar Back to article
Straughan, B. (2010) Acoustic waves in Cattaneo-Christov gas. Physics Letters A 374, 2667-2669.10.1016/j.physleta.2010.04.054
Search in Google Scholar Back to article
Thompson, P. A. (1972) Compressible-Fluid Dynamics. McGraw-Hill, New York.10.1115/1.3422684
Search in Google Scholar Back to article
Triggiani, R. (1994a) Optimal boundary control and new Riccati equations for highly damped second-order equations. Differential Integral Equations 7, 1109–1144.
Search in Google Scholar Back to article
Triggiani, R. (1994b) An optimal quadratic boundary control problem for wave and plate-like equations with high internal damping: An abstract approach. Marcel Dekker Lecture Notes Pure Appl. Math. 165, 215–271. International Conference on Optimal Control for Partial Differential Equations, University of Trento, January 1993.10.1201/9781482277654-89
Search in Google Scholar Back to article

Uniqueness of the Riccati operator of the non-standard ARE of a third order dynamics with boundary control

Abstract

Paradigm

My account