American Institute of Mathematical Sciences

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 & 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. doi: 10.1016/j.jde.2015.07.006. Google Scholar [2] L. Bociu and J. P. Zolesio, Hyperbolic equations with mixed boundary conditions: Shape differentiability analysis,, Applied Mathematics & Optimization, (2016), 1. doi: 10.1007/s00245-016-9354-4. Google Scholar [3] H. O. Fattorini, The Cauchy Problem,, Encyclopedia of Mathematics and its Applications, (1983). Google Scholar [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. doi: 10.3792/pja/1195521686. Google Scholar [5] P. Grisvard, Caracterization de quelques espaces d'interpolation,, Arch. Rational Mech. Anal., 25 (1967), 40. doi: 10.1007/BF00281421. Google Scholar [6] T. Kato, Fractional powers of dissipative operators,, J. Math. Soc. Japan, 13 (1961), 246. doi: 10.2969/jmsj/01330246. Google Scholar [7] I. Lasiecka, Unified theory for abstract parabolic boundary problems-a semigroup approach,, Appl. Math. & Optimiz., 6 (1980), 287. doi: 10.1007/BF01442900. Google Scholar [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. Google Scholar [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. doi: 10.1007/BF01442108. Google Scholar [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. doi: 10.1007/BF01448390. Google Scholar [11] I. Lasiecka and R. 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Triggiani, Regularity theory of hyperbolic equations with non-homogeneous Neumann boundary conditions, Part II: General boundary data,, J. Diff. Eqns., 94 (1991), 112. doi: 10.1016/0022-0396(91)90106-J. Google Scholar [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. doi: 10.1016/0022-0396(91)90022-2. Google Scholar [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, (2000). doi: 10.1017/CBO9780511574801.002. Google Scholar [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, 216 (2010), 243. doi: 10.1007/978-3-0348-0069-3_14. Google Scholar [19] J. L. Lions, Controllabilite Exacte es Stabilization de Systemes Distribues,, vol 1, (1988). Google Scholar [20] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I,, Springer-Verlag, (1972). Google Scholar [21] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. II,, Springer-Verlag, (1972). Google Scholar [22] S. Myatake, Mixed problems for hyperbolic equations of second order,, J. Math. Kyoto Univ, 130 (1973), 435. Google Scholar [23] R. Sakamoto, Hyperbolic Boundary Value Problems,, Cambridge University Press, (1982). Google Scholar [24] M. Sova, Cosine operator functions,, Rozprawy Mat, 49 (1966), 1. Google Scholar [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. doi: 10.1002/mma.1670050110. Google Scholar [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. Google Scholar [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. Google Scholar [28] R. Triggiani, Interior and boundary regularity of the wave equation of interior point control,, J. Diff. Eqts, 103 (1993), 394. doi: 10.1006/jdeq.1993.1057. Google Scholar [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. doi: 10.1016/j.jde.2008.02.033. Google Scholar

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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. doi: 10.1016/j.jde.2015.07.006. Google Scholar [2] L. Bociu and J. P. Zolesio, Hyperbolic equations with mixed boundary conditions: Shape differentiability analysis,, Applied Mathematics & Optimization, (2016), 1. doi: 10.1007/s00245-016-9354-4. Google Scholar [3] H. O. Fattorini, The Cauchy Problem,, Encyclopedia of Mathematics and its Applications, (1983). Google Scholar [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. doi: 10.3792/pja/1195521686. Google Scholar [5] P. Grisvard, Caracterization de quelques espaces d'interpolation,, Arch. Rational Mech. Anal., 25 (1967), 40. doi: 10.1007/BF00281421. Google Scholar [6] T. Kato, Fractional powers of dissipative operators,, J. Math. Soc. Japan, 13 (1961), 246. doi: 10.2969/jmsj/01330246. Google Scholar [7] I. Lasiecka, Unified theory for abstract parabolic boundary problems-a semigroup approach,, Appl. Math. & Optimiz., 6 (1980), 287. doi: 10.1007/BF01442900. Google Scholar [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. Google Scholar [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. doi: 10.1007/BF01442108. Google Scholar [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. doi: 10.1007/BF01448390. Google Scholar [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. doi: 10.1090/S0002-9939-1988-0964851-1. Google Scholar [12] I. Lasiecka and R. Triggiani, Exact boundary controllability for the wave equation with Neumann boundary control,, Appl. Math. Optimiz., 19 (1989), 243. doi: 10.1007/BF01448201. Google Scholar [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., 83 (1990), 285. Google Scholar [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. doi: 10.1016/0022-247X(89)90205-9. Google Scholar [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. doi: 10.1016/0022-0396(91)90106-J. Google Scholar [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. doi: 10.1016/0022-0396(91)90022-2. Google Scholar [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, (2000). doi: 10.1017/CBO9780511574801.002. Google Scholar [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, 216 (2010), 243. doi: 10.1007/978-3-0348-0069-3_14. Google Scholar [19] J. L. Lions, Controllabilite Exacte es Stabilization de Systemes Distribues,, vol 1, (1988). Google Scholar [20] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I,, Springer-Verlag, (1972). Google Scholar [21] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. II,, Springer-Verlag, (1972). Google Scholar [22] S. Myatake, Mixed problems for hyperbolic equations of second order,, J. Math. Kyoto Univ, 130 (1973), 435. Google Scholar [23] R. Sakamoto, Hyperbolic Boundary Value Problems,, Cambridge University Press, (1982). Google Scholar [24] M. Sova, Cosine operator functions,, Rozprawy Mat, 49 (1966), 1. Google Scholar [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. doi: 10.1002/mma.1670050110. Google Scholar [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. Google Scholar [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. Google Scholar [28] R. Triggiani, Interior and boundary regularity of the wave equation of interior point control,, J. Diff. Eqts, 103 (1993), 394. doi: 10.1006/jdeq.1993.1057. Google Scholar [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. doi: 10.1016/j.jde.2008.02.033. Google Scholar
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