December  2016, 5(4): 489-514. doi: 10.3934/eect.2016016

Sharp regularity theory of second order hyperbolic equations with Neumann boundary control non-smooth in space

1. 

Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152

Received  August 2016 Revised  August 2016 Published  October 2016

The purpose of this paper is to complement available literature on sharp regularity theory of second order mixed hyperbolic problem of Neumann type [13,15,26] with a series of new results in the case--so far rather unexplored--where the Neumann boundary term (input, control) possesses a regularity below $L^2(\Gamma)$ in space on the boundary $\Gamma$. We concentrate on the cases $H^{-\frac{1}{2}}(\Gamma))$, $H^{-\beta}(\Gamma))$, $H^{-1}(\Gamma))$, $\beta$ being a distinguished parameter of the problem. Our present results are consistent with the sharp result of [13,15,26] (obtained through a pseudo-differential/micro-local analysis approach), whose philosophy is expressed by a gain of $\beta$ in space regularity in going from the boundary control to the position in the interior. A number of physically relevant illustrations are given.
Citation: Roberto Triggiani. Sharp regularity theory of second order hyperbolic equations with Neumann boundary control non-smooth in space. Evolution Equations and Control Theory, 2016, 5 (4) : 489-514. doi: 10.3934/eect.2016016
References:
[1]

L. Bociu and J. P. Zolesio, A pseudo-extractor approach to hidden boundary regularity for the wave equation with Neumann boundary condition, Journal of Differential Equations, 259 (2015), 5688-5708. doi: 10.1016/j.jde.2015.07.006.

[2]

L. Bociu and J. P. Zolesio, Hyperbolic equations with mixed boundary conditions: Shape differentiability analysis, Applied Mathematics & Optimization, (2016), 1-24. doi: 10.1007/s00245-016-9354-4.

[3]

H. O. Fattorini, The Cauchy Problem, Encyclopedia of Mathematics and its Applications, Addison-Wesley, 1983.

[4]

D. Fujiwawa, Concrete characterization of the domain of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad, 43 (1967), 82-86. doi: 10.3792/pja/1195521686.

[5]

P. Grisvard, Caracterization de quelques espaces d'interpolation, Arch. Rational Mech. Anal., 25 (1967), 40-63. doi: 10.1007/BF00281421.

[6]

T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan, 13 (1961), 246-274. doi: 10.2969/jmsj/01330246.

[7]

I. Lasiecka, Unified theory for abstract parabolic boundary problems-a semigroup approach, Appl. Math. & Optimiz., 6 (1980), 287-333. doi: 10.1007/BF01442900.

[8]

I. Lasiecka, J. L. Lions and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.

[9]

I. Lasiecka and R. Triggiani, A cosine operator approach to modelling $L_2(0,T; L_2(\Omega))$ boundary input hyperbolic equations, Appl. Math. Optimiz., 7 (1981), 35-93. doi: 10.1007/BF01442108.

[10]

I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under $L_2(0,T; L_2(\Gamma))$-Dirichlet boundary terms, Appl. Math. Optimiz., 10 (1983), 275-286. doi: 10.1007/BF01448390.

[11]

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, Proceedings American Mathematical Society, 104 (1988), 745-755. doi: 10.1090/S0002-9939-1988-0964851-1.

[12]

I. Lasiecka and R. Triggiani, Exact boundary controllability for the wave equation with Neumann boundary control, Appl. Math. Optimiz., 19 (1989), 243-290. (Also, preliminary version in Springer-Verlag Lecture Notes, 100 (1987), 316-371.) doi: 10.1007/BF01448201.

[13]

I. Lasiecka and R. Triggiani, Sharp regularity for mixed second order hyperbolic equations of Neumann type Part I: The $L_2$-boundary case, Annali Matem. Pura Appl., (IV) CLVII (1990), 285-367. (Announcements in Accad. Lincei, 83 (1989), 109-113, Classe di Scienze Matematiche, Rome, Italy, and Springer-Verlag Lecture Notes, 114.)

[14]

I. Lasiecka and R. Triggiani, Trace regularity of the solutions of the wave equation with homogeneous Neumann boundary conditions and compactly supported data, J. Math. Anal. Appl., 141 (1989), 49-71. doi: 10.1016/0022-247X(89)90205-9.

[15]

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.

[16]

I. Lasiecka and R. Triggiani, Exact Controllability and uniform stabilization of Kirchoff plates with boundary controls only on $\Delta w|_{\Sigma},$ J. Diff. Eqts., 93 (1991), 62-101. doi: 10.1016/0022-0396(91)90022-2.

[17]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories II, Abstract Hyperbolic Systems over a Finite Time Horizon, Encyclopedia of Mathematics and Its Applications Series, Cambridge University Press, January 2000. doi: 10.1017/CBO9780511574801.002.

[18]

C. Lebiedzik and R. Triggiani, The optimal interior regularity for the critical case of a clamped thermoelastic system with point control revisited, Modern Aspects of the Theory of Partial Differential Equations, Operator Theory Advances and Applications, M. Ruzhansky, Jens Wirth Editors, Birkhawser, 216 (2010), 243-259. doi: 10.1007/978-3-0348-0069-3_14.

[19]

J. L. Lions, Controllabilite Exacte es Stabilization de Systemes Distribues, vol 1, Masson, 1988.

[20]

J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I, Springer-Verlag, 1972.

[21]

J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. II, Springer-Verlag, 1972.

[22]

S. Myatake, Mixed problems for hyperbolic equations of second order, J. Math. Kyoto Univ, 130 (1973), 435-487.

[23]

R. Sakamoto, Hyperbolic Boundary Value Problems, Cambridge University Press, London/New York, 1982.

[24]

M. Sova, Cosine operator functions, Rozprawy Mat, 49 (1966), 1-47.

[25]

W. Symes, A trace theorem for solutions of the wave equation, and the remote determination of acoustic sources, Mathematical Methods in the Applied Sciences, 5 (1983), 131-152. doi: 10.1002/mma.1670050110.

[26]

D. Tataru, On the regularity of boundary traces for the wave equation, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 26 (1998), 185-206.

[27]

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.

[28]

R. Triggiani, Interior and boundary regularity of the wave equation of interior point control, J. Diff. Eqts, 103 (1993), 394-420. doi: 10.1006/jdeq.1993.1057.

[29]

R. Triggiani, The critical case of clamped thermoelastic systems with interior point control: optimal interior and boundary regularity results, J. Diff. Eqts., 245 (2008), 3764-3805. doi: 10.1016/j.jde.2008.02.033.

show all references

References:
[1]

L. Bociu and J. P. Zolesio, A pseudo-extractor approach to hidden boundary regularity for the wave equation with Neumann boundary condition, Journal of Differential Equations, 259 (2015), 5688-5708. doi: 10.1016/j.jde.2015.07.006.

[2]

L. Bociu and J. P. Zolesio, Hyperbolic equations with mixed boundary conditions: Shape differentiability analysis, Applied Mathematics & Optimization, (2016), 1-24. doi: 10.1007/s00245-016-9354-4.

[3]

H. O. Fattorini, The Cauchy Problem, Encyclopedia of Mathematics and its Applications, Addison-Wesley, 1983.

[4]

D. Fujiwawa, Concrete characterization of the domain of fractional powers of some elliptic differential operators of the second order, Proc. Japan Acad, 43 (1967), 82-86. doi: 10.3792/pja/1195521686.

[5]

P. Grisvard, Caracterization de quelques espaces d'interpolation, Arch. Rational Mech. Anal., 25 (1967), 40-63. doi: 10.1007/BF00281421.

[6]

T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan, 13 (1961), 246-274. doi: 10.2969/jmsj/01330246.

[7]

I. Lasiecka, Unified theory for abstract parabolic boundary problems-a semigroup approach, Appl. Math. & Optimiz., 6 (1980), 287-333. doi: 10.1007/BF01442900.

[8]

I. Lasiecka, J. L. Lions and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl., 65 (1986), 149-192.

[9]

I. Lasiecka and R. Triggiani, A cosine operator approach to modelling $L_2(0,T; L_2(\Omega))$ boundary input hyperbolic equations, Appl. Math. Optimiz., 7 (1981), 35-93. doi: 10.1007/BF01442108.

[10]

I. Lasiecka and R. Triggiani, Regularity of hyperbolic equations under $L_2(0,T; L_2(\Gamma))$-Dirichlet boundary terms, Appl. Math. Optimiz., 10 (1983), 275-286. doi: 10.1007/BF01448390.

[11]

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, Proceedings American Mathematical Society, 104 (1988), 745-755. doi: 10.1090/S0002-9939-1988-0964851-1.

[12]

I. Lasiecka and R. Triggiani, Exact boundary controllability for the wave equation with Neumann boundary control, Appl. Math. Optimiz., 19 (1989), 243-290. (Also, preliminary version in Springer-Verlag Lecture Notes, 100 (1987), 316-371.) doi: 10.1007/BF01448201.

[13]

I. Lasiecka and R. Triggiani, Sharp regularity for mixed second order hyperbolic equations of Neumann type Part I: The $L_2$-boundary case, Annali Matem. Pura Appl., (IV) CLVII (1990), 285-367. (Announcements in Accad. Lincei, 83 (1989), 109-113, Classe di Scienze Matematiche, Rome, Italy, and Springer-Verlag Lecture Notes, 114.)

[14]

I. Lasiecka and R. Triggiani, Trace regularity of the solutions of the wave equation with homogeneous Neumann boundary conditions and compactly supported data, J. Math. Anal. Appl., 141 (1989), 49-71. doi: 10.1016/0022-247X(89)90205-9.

[15]

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.

[16]

I. Lasiecka and R. Triggiani, Exact Controllability and uniform stabilization of Kirchoff plates with boundary controls only on $\Delta w|_{\Sigma},$ J. Diff. Eqts., 93 (1991), 62-101. doi: 10.1016/0022-0396(91)90022-2.

[17]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories II, Abstract Hyperbolic Systems over a Finite Time Horizon, Encyclopedia of Mathematics and Its Applications Series, Cambridge University Press, January 2000. doi: 10.1017/CBO9780511574801.002.

[18]

C. Lebiedzik and R. Triggiani, The optimal interior regularity for the critical case of a clamped thermoelastic system with point control revisited, Modern Aspects of the Theory of Partial Differential Equations, Operator Theory Advances and Applications, M. Ruzhansky, Jens Wirth Editors, Birkhawser, 216 (2010), 243-259. doi: 10.1007/978-3-0348-0069-3_14.

[19]

J. L. Lions, Controllabilite Exacte es Stabilization de Systemes Distribues, vol 1, Masson, 1988.

[20]

J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I, Springer-Verlag, 1972.

[21]

J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. II, Springer-Verlag, 1972.

[22]

S. Myatake, Mixed problems for hyperbolic equations of second order, J. Math. Kyoto Univ, 130 (1973), 435-487.

[23]

R. Sakamoto, Hyperbolic Boundary Value Problems, Cambridge University Press, London/New York, 1982.

[24]

M. Sova, Cosine operator functions, Rozprawy Mat, 49 (1966), 1-47.

[25]

W. Symes, A trace theorem for solutions of the wave equation, and the remote determination of acoustic sources, Mathematical Methods in the Applied Sciences, 5 (1983), 131-152. doi: 10.1002/mma.1670050110.

[26]

D. Tataru, On the regularity of boundary traces for the wave equation, Annali della Scuola Normale Superiore di Pisa-Classe di Scienze, 26 (1998), 185-206.

[27]

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.

[28]

R. Triggiani, Interior and boundary regularity of the wave equation of interior point control, J. Diff. Eqts, 103 (1993), 394-420. doi: 10.1006/jdeq.1993.1057.

[29]

R. Triggiani, The critical case of clamped thermoelastic systems with interior point control: optimal interior and boundary regularity results, J. Diff. Eqts., 245 (2008), 3764-3805. doi: 10.1016/j.jde.2008.02.033.

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