# American Institute of Mathematical Sciences

doi: 10.3934/eect.2022007
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## From low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae

 1 Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, USA 2 Department of Mathematics and Statistics, Haverford College, Haverford, PA 19041, USA

*Corresponding author: Xiang Wan

Received  August 2021 Revised  December 2021 Early access February 2022

Fund Project: The first author is supported by NSF grant DMS-1713506

We consider the linear third order (in time) PDE known as the SMGTJ-equation, defined on a bounded domain, under the action of either Dirichlet or Neumann boundary control $g$. Optimal interior and boundary regularity results were given in [1], after [41], when $g \in L^2(0, T;L^2(\Gamma)) \equiv L^2(\Sigma)$, which, moreover, in the canonical case $\gamma = 0$, were expressed by the well-known explicit representation formulae of the wave equation in terms of cosine/sine operators [19], [17], [24,Vol Ⅱ]. The interior or boundary regularity theory is however the same, whether $\gamma = 0$ or $0 \neq \gamma \in L^{\infty}(\Omega)$, since $\gamma \neq 0$ is responsible only for lower order terms. Here we exploit such cosine operator based-explicit representation formulae to provide optimal interior and boundary regularity results with $g$ "smoother" than $L^2(\Sigma)$, qualitatively by one unit, two units, etc. in the Dirichlet boundary case. To this end, we invoke the corresponding results for wave equations, as in [17]. Similarly for the Neumann boundary case, by invoking the corresponding results for the wave equation as in [22], [23], [37] for control smoother than $L^2(0, T;L^2(\Gamma))$, and [44] for control less regular in space than $L^2(\Gamma)$. In addition, we provide optimal interior and boundary regularity results when the SMGTJ equation is subject to interior point control, by invoking the corresponding wave equations results [42], [24,Section 9.8.2].

Citation: Roberto Triggiani, Xiang Wan. From low to high-and lower-optimal regularity of the SMGTJ equation with Dirichlet and Neumann boundary control, and with point control, via explicit representation formulae. Evolution Equations and Control Theory, doi: 10.3934/eect.2022007
##### References:
 [1] M. Bongarti, I. Lasiecka and R. Triggiani, The SMGTJ equation from the boundary: Regularity and stabilization, Applicable Analysis (2021): 1-39. doi: 10.1080/00036811.2021.1999420. [2] F. Bucci and L. Pandolfi, On the regularity of solutions to the Moore-Gibson-Thompson equation: A perspective via wave equations with memory, J. Evol. Equ., 20 (2020), 837-867.  doi: 10.1007/s00028-019-00549-x. [3] F. Bucci and M. Eller, The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation, C. R. Math. Acad. Sci. Paris, 359 (2021), 881-903.  doi: 10.5802/crmath.231. [4] I. Christov, private communication. [5] G. Da Prato and E. Giusti, Una caratterizzazione dei generatori di funzioni coseno astratte, Boll. Un. Mat. Ital., 22 (1967), 357-362. [6] H. O. Fattorini, The Cauchy Problem, Encyclopedia of Mathematics and its Applications, Addison-Wesley, 1983,636 pp. [7] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, Amsterdam, North-Holland, 1985. [8] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad., 43 (1967), 82-86. [9] P. Jordan, An analytical study of Kuznetsov's equation: Diffusive solitons, shock formation, and solution bifurcation, Phys. Lett. A, 326 (2004), 77-84.  doi: 10.1016/j.physleta.2004.03.067. [10] P. Jordan, private communication. [11] B. Kaltenbacher, I. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control Cybernet., 40 (2011), 971-988. [12] B. Kaltenbacher, I. Lasiecka and M. Pospieszalska., Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 1250035, 34 pp. doi: 10.1142/S0218202512500352. [13] J. Kisyński, On second order Cauchy's problem in a Banach space, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Plys., 18 (1970), 371-374. [14] J. Kisyński, On cosine operator functions and one-parameter groups of operators, Studia Math., 44 (1972), 93-105.  doi: 10.4064/sm-44-1-93-105. [15] J. Kisyński, Semi-groups of operators and some of their applications to partial differential equations, Control Theory and Topics in Functional Analysis (Internat. Sem., Internat. Centre Theoret. Phys., Trieste, 1974), Vienna, International Atomic Energy Agency, 3 (1976), 305–405. [16] I. Lasiecka, Boundary control of parabolic systems, Appl. Math. Optim., 4 (1977/78), 301-327.  doi: 10.1007/BF01442147. [17] I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192. [18] I. Lasiecka and R. Triggiani, A cosine operator approach to modeling $L_2(0, T; L_2(\Omega))$ boundary input hyperbolic equations, Appl. Math. Optim., 7 (1981), 35-93.  doi: 10.1007/BF01442108. [19] I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under $L_2(0, T; L_2(\Omega))$-Dirichlet boundary terms, Appl. Math. Optimiz., 10 (1983), 275-286.  doi: 10.1007/BF01448390. [20] I. Lasiecka and R. Triggiani, Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations, J. Diff. Eqns., 47 (1983), 246-272.  doi: 10.1016/0022-0396(83)90036-0. [21] I. Lasiecka and R. Triggiani, A lifting theorem for the time regularity of solutions to abstract equations with unbounded operators and applications to hyperbolic equations, Proc. Amer. Math. Soc., 104 (1988), 745-755.  doi: 10.1090/S0002-9939-1988-0964851-1. [22] I. Lasiecka and R. Triggiani, Sharp regularity for mixed second order hyperbolic equations of Neumann type Part Ⅰ: The $L_2$-boundary case, Ann. Mat. Pura Appl., 157 (1990), 285-367.  doi: 10.1007/BF01765322. [23] I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations with non-homogeneous Neumann boundary conditions, Part Ⅱ: General boundary data, J. Diff. Eqns., 94 (1991), 112-164.  doi: 10.1016/0022-0396(91)90106-J. [24] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Vol Ⅰ: Abstract Parabolic Systems (644 pp.); Vol Ⅱ: Abstract Hyperbolic Systems over a Finite Time Horizon (422 pp.), Encyclopedia of Mathematics and Its Applications Series, 74. Cambridge University Press, Cambridge, 2000. [25] I. Lasiecka, R. Triggiani and X. Wan, Abstract representation of the SMGTJ equation under rough boundary controls: optimal interior regularity, Submitted. [26] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications I, Springer-Verlag, New York-Heidelberg, 1972. [27] S. Liu and R. Triggiani, Boundary control and boundary inverse theory for non-homogeneous second-order hyperbolic equations: A common Carleman estimates approach, HCDTE Lecture Notes, AIMS Ser. Appl. Math., 6 (2013), 227-343. [28] S. Liu and R. Triggiani, Inverse problem for a linearized Jordan–Moore–Gibson–Thompson equation, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, 10 (2014), 305-351.  doi: 10.1007/978-3-319-11406-4_15. [29] A. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), 607-675.  doi: 10.1002/cpa.3160280504. [30] R. Marchand, T. McDevitt and R. Triggiani, An abstract semigroup approach to the third-order MGT equation arising in high-intensity ultrasound: Structural decomposition, spectral analysis, exponential stability, Math. Methods Appl. Sci., 35 (2012), 1896-1929.  doi: 10.1002/mma.1576. [31] G. Métivier, The block structure condition for symmetric hyperbolic systems, Bull. London Math. Soc., 32 (2000), 689-702.  doi: 10.1112/S0024609300007517. [32] F. K. Moore and W. E. Gibson, Propagation of weak disturbances in a gas subject to relaxation effects, J. Aero/Space Sci., 27 (1960), 117-127.  doi: 10.2514/8.8418. [33] R. Sakamoto, Mixed problems for hyperbolic equations, I and II, J. Math. Kyoto Univ., 10 (1970), 349-373.  doi: 10.1215/kjm/1250523767. [34] R. Sakamoto, Hyperbolic Boundary Value Problems, Translated from the Japanese by Katsumi Miyahara, Cambridge University Press, Cambridge-New York, 1982. [35] M. Sova, Cosine operator functions, Rozprawy Mat., 49 (1966), 47 pp. [36] G. G. Stokes, An examination of the possible effect of the radiation of heat on the propagation of sound, Philosophical magazine, 1 (1851), 305-317.  doi: 10.1017/CBO9780511702266.005. [37] D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 185-206. [38] P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, 1972. [39] C. C. Travis and G. Webb, Second order differential equations in Banach space, Nonlinear Equations in Abstract Spaces, Academic Press, (1978), 331–361. [40] R. Triggiani, A cosine operator approach to modeling boundary input problems for hyperbolic systems, Springer-Verlag Lecture Notes in Control and Information Sciences, 6 (1978), 380-390. [41] R. Triggiani, Sharp interior and boundary regularity of the SMGTJ–equation with Dirichlet or Neumann boundary control, Semigroup of Operators – Theory and Applications, Springer, 325 (2020), 379–426. doi: 10.1007/978-3-030-46079-2_22. [42] R. Triggiani, Interior and boundary regularity of the wave equations with point control, Diff. Int. Eqns., 6 (1993), 111-129. [43] R. Triggiani, Three optimal results, Presentation at Conference Optimal Control for Evolutionary PDEs, Cortona, Italy, (slides of presentation available upon request), (2016), 20–24. [44] R. Triggiani, Sharp regularity theory of second order hyperbolic equations with Neumann boundary control non-smooth in space, Evol. Equ. Control Theory, 5 (2016), 489-514.  doi: 10.3934/eect.2016016.

show all references

##### References:
 [1] M. Bongarti, I. Lasiecka and R. Triggiani, The SMGTJ equation from the boundary: Regularity and stabilization, Applicable Analysis (2021): 1-39. doi: 10.1080/00036811.2021.1999420. [2] F. Bucci and L. Pandolfi, On the regularity of solutions to the Moore-Gibson-Thompson equation: A perspective via wave equations with memory, J. Evol. Equ., 20 (2020), 837-867.  doi: 10.1007/s00028-019-00549-x. [3] F. Bucci and M. Eller, The Cauchy-Dirichlet problem for the Moore-Gibson-Thompson equation, C. R. Math. Acad. Sci. Paris, 359 (2021), 881-903.  doi: 10.5802/crmath.231. [4] I. Christov, private communication. [5] G. Da Prato and E. Giusti, Una caratterizzazione dei generatori di funzioni coseno astratte, Boll. Un. Mat. Ital., 22 (1967), 357-362. [6] H. O. Fattorini, The Cauchy Problem, Encyclopedia of Mathematics and its Applications, Addison-Wesley, 1983,636 pp. [7] H. O. Fattorini, Second Order Linear Differential Equations in Banach Spaces, Amsterdam, North-Holland, 1985. [8] D. Fujiwara, Concrete characterization of the domains of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad., 43 (1967), 82-86. [9] P. Jordan, An analytical study of Kuznetsov's equation: Diffusive solitons, shock formation, and solution bifurcation, Phys. Lett. A, 326 (2004), 77-84.  doi: 10.1016/j.physleta.2004.03.067. [10] P. Jordan, private communication. [11] B. Kaltenbacher, I. Lasiecka and R. Marchand, Wellposedness and exponential decay rates for the Moore-Gibson-Thompson equation arising in high intensity ultrasound, Control Cybernet., 40 (2011), 971-988. [12] B. Kaltenbacher, I. Lasiecka and M. Pospieszalska., Well-posedness and exponential decay of the energy in the nonlinear Jordan-Moore-Gibson-Thompson equation arising in high intensity ultrasound, Math. Models Methods Appl. Sci., 22 (2012), 1250035, 34 pp. doi: 10.1142/S0218202512500352. [13] J. Kisyński, On second order Cauchy's problem in a Banach space, Bull. Acad. Polon. Sci. Sér. Sci. Math. Astr. Plys., 18 (1970), 371-374. [14] J. Kisyński, On cosine operator functions and one-parameter groups of operators, Studia Math., 44 (1972), 93-105.  doi: 10.4064/sm-44-1-93-105. [15] J. Kisyński, Semi-groups of operators and some of their applications to partial differential equations, Control Theory and Topics in Functional Analysis (Internat. Sem., Internat. Centre Theoret. Phys., Trieste, 1974), Vienna, International Atomic Energy Agency, 3 (1976), 305–405. [16] I. Lasiecka, Boundary control of parabolic systems, Appl. Math. Optim., 4 (1977/78), 301-327.  doi: 10.1007/BF01442147. [17] I. Lasiecka, J.-L. Lions and R. Triggiani, Nonhomogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192. [18] I. Lasiecka and R. Triggiani, A cosine operator approach to modeling $L_2(0, T; L_2(\Omega))$ boundary input hyperbolic equations, Appl. Math. Optim., 7 (1981), 35-93.  doi: 10.1007/BF01442108. [19] I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under $L_2(0, T; L_2(\Omega))$-Dirichlet boundary terms, Appl. Math. Optimiz., 10 (1983), 275-286.  doi: 10.1007/BF01448390. [20] I. Lasiecka and R. Triggiani, Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations, J. Diff. Eqns., 47 (1983), 246-272.  doi: 10.1016/0022-0396(83)90036-0. [21] I. Lasiecka and R. Triggiani, A lifting theorem for the time regularity of solutions to abstract equations with unbounded operators and applications to hyperbolic equations, Proc. Amer. Math. Soc., 104 (1988), 745-755.  doi: 10.1090/S0002-9939-1988-0964851-1. [22] I. Lasiecka and R. Triggiani, Sharp regularity for mixed second order hyperbolic equations of Neumann type Part Ⅰ: The $L_2$-boundary case, Ann. Mat. Pura Appl., 157 (1990), 285-367.  doi: 10.1007/BF01765322. [23] I. Lasiecka and R. Triggiani, Regularity theory of hyperbolic equations with non-homogeneous Neumann boundary conditions, Part Ⅱ: General boundary data, J. Diff. Eqns., 94 (1991), 112-164.  doi: 10.1016/0022-0396(91)90106-J. [24] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Vol Ⅰ: Abstract Parabolic Systems (644 pp.); Vol Ⅱ: Abstract Hyperbolic Systems over a Finite Time Horizon (422 pp.), Encyclopedia of Mathematics and Its Applications Series, 74. Cambridge University Press, Cambridge, 2000. [25] I. Lasiecka, R. Triggiani and X. Wan, Abstract representation of the SMGTJ equation under rough boundary controls: optimal interior regularity, Submitted. [26] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Problems and Applications I, Springer-Verlag, New York-Heidelberg, 1972. [27] S. Liu and R. Triggiani, Boundary control and boundary inverse theory for non-homogeneous second-order hyperbolic equations: A common Carleman estimates approach, HCDTE Lecture Notes, AIMS Ser. Appl. Math., 6 (2013), 227-343. [28] S. Liu and R. Triggiani, Inverse problem for a linearized Jordan–Moore–Gibson–Thompson equation, New Prospects in Direct, Inverse and Control Problems for Evolution Equations, 10 (2014), 305-351.  doi: 10.1007/978-3-319-11406-4_15. [29] A. Majda and S. Osher, Initial-boundary value problems for hyperbolic equations with uniformly characteristic boundary, Comm. Pure Appl. Math., 28 (1975), 607-675.  doi: 10.1002/cpa.3160280504. [30] R. Marchand, T. McDevitt and R. Triggiani, An abstract semigroup approach to the third-order MGT equation arising in high-intensity ultrasound: Structural decomposition, spectral analysis, exponential stability, Math. Methods Appl. Sci., 35 (2012), 1896-1929.  doi: 10.1002/mma.1576. [31] G. Métivier, The block structure condition for symmetric hyperbolic systems, Bull. London Math. Soc., 32 (2000), 689-702.  doi: 10.1112/S0024609300007517. [32] F. K. Moore and W. E. Gibson, Propagation of weak disturbances in a gas subject to relaxation effects, J. Aero/Space Sci., 27 (1960), 117-127.  doi: 10.2514/8.8418. [33] R. Sakamoto, Mixed problems for hyperbolic equations, I and II, J. Math. Kyoto Univ., 10 (1970), 349-373.  doi: 10.1215/kjm/1250523767. [34] R. Sakamoto, Hyperbolic Boundary Value Problems, Translated from the Japanese by Katsumi Miyahara, Cambridge University Press, Cambridge-New York, 1982. [35] M. Sova, Cosine operator functions, Rozprawy Mat., 49 (1966), 47 pp. [36] G. G. Stokes, An examination of the possible effect of the radiation of heat on the propagation of sound, Philosophical magazine, 1 (1851), 305-317.  doi: 10.1017/CBO9780511702266.005. [37] D. Tataru, On the regularity of boundary traces for the wave equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 26 (1998), 185-206. [38] P. A. Thompson, Compressible-Fluid Dynamics, McGraw-Hill, New York, 1972. [39] C. C. Travis and G. Webb, Second order differential equations in Banach space, Nonlinear Equations in Abstract Spaces, Academic Press, (1978), 331–361. [40] R. Triggiani, A cosine operator approach to modeling boundary input problems for hyperbolic systems, Springer-Verlag Lecture Notes in Control and Information Sciences, 6 (1978), 380-390. [41] R. Triggiani, Sharp interior and boundary regularity of the SMGTJ–equation with Dirichlet or Neumann boundary control, Semigroup of Operators – Theory and Applications, Springer, 325 (2020), 379–426. doi: 10.1007/978-3-030-46079-2_22. [42] R. Triggiani, Interior and boundary regularity of the wave equations with point control, Diff. Int. Eqns., 6 (1993), 111-129. [43] R. Triggiani, Three optimal results, Presentation at Conference Optimal Control for Evolutionary PDEs, Cortona, Italy, (slides of presentation available upon request), (2016), 20–24. [44] R. Triggiani, Sharp regularity theory of second order hyperbolic equations with Neumann boundary control non-smooth in space, Evol. Equ. Control Theory, 5 (2016), 489-514.  doi: 10.3934/eect.2016016.
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