A lifting theorem for the time regularity of solutions to abstract equations with unbounded operators and applications to hyperbolic equations
Authors:
I. Lasiecka and R. Triggiani
Journal:
Proc. Amer. Math. Soc. 104 (1988), 745755
MSC:
Primary 34G10; Secondary 35L10, 47A50, 47D05
MathSciNet review:
964851
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Abstract: We consider the solution operator corresponding to the abstract equation on a reflexive Banach space , where the linear operator is the infinitesimal generator of a (strongly continuous) group of bounded operators on , and is a generally unbounded linear operator with being another reflexive Banach space (without loss of generality we take to be boundedly invertible). Let be given. We prove the following theorem: if is continuous , then in fact continuous , a lifting regularity theorem in the time variable. Moreover, we show by a parabolic example with nonhomogeneous term in the Dirichlet boundary conditions that the theorem fails to be true, if is merely a s.c. semigroup even if holomorphic. Applications of the theorem include mixed hyperbolic problems, including second order scalar hyperbolic equations defined on an open bounded domain , with nonhomogeneous term of class acting in the Dirichlet or in the Neumann boundary conditions. In the former case, the theorem recovers the authors' original procedure which yielded optimal regularity results for this dynamics [LT.2]; in the latter, the theorem improves upon results of LionsMagenes [LM.1, vol. II]. Extension to is also studied.
 [C.1]
S. Chang, Ph. D. dissertation, Mathematics Department, Univ. of Florida, 1984.
 [CL.1]
S.
Chang and I.
Lasiecka, Riccati equations for nonsymmetric and nondissipative
hyperbolic systems with 𝐿₂boundary controls, J. Math.
Anal. Appl. 116 (1986), no. 2, 378–414. MR 842807
(87k:49008), http://dx.doi.org/10.1016/S0022247X(86)800051
 [DaPLT.1]
G.
Da Prato, I.
Lasiecka, and R.
Triggiani, A direct study of the Riccati equation arising in
hyperbolic boundary control problems, J. Differential Equations
64 (1986), no. 1, 26–47. MR 849662
(87i:93068), http://dx.doi.org/10.1016/00220396(86)900690
 [DS.1]
N. Dunford and J. T. Schwartz, Linear operators. Part I, Interscience, New York, 1958.
 [FLT.1]
F. Flandoli, I. Lasiecka and R. Triggiani, Algebraic Riccati equations with nonsmoothing observation arising in hyperbolic and EulerBernoulli equations, Ann. Mat. Pura Appl. (to appear).
 [H.1]
Lop
Fat Ho, Observabilité frontière de
l’équation des ondes, C. R. Acad. Sci. Paris Sér.
I Math. 302 (1986), no. 12, 443–446 (English,
with French summary). MR 838598
(87d:93017)
 [HP.1]
Einar
Hille and Ralph
S. Phillips, Functional analysis and semigroups, American
Mathematical Society Colloquium Publications, vol. 31, American
Mathematical Society, Providence, R. I., 1957. rev. ed. MR 0089373
(19,664d)
 [L.1]
J.L.
Lions, Contrôle des systèmes distribués
singuliers, Méthodes Mathématiques de
l’Informatique [Mathematical Methods of Information Science],
vol. 13, GauthierVillars, Montrouge, 1983 (French). MR 712486
(85c:93002)
 [L.2]
JacquesLouis
Lions, Contrôlabilité exacte des systèmes
distribués, C. R. Acad. Sci. Paris Sér. I Math.
302 (1986), no. 13, 471–475 (French, with
English summary). MR 838402
(87e:93051), http://dx.doi.org/10.1007/BFb0007542
 [L.3]
J.L.
Lions, Optimal control of systems governed by partial differential
equations., Translated from the French by S. K. Mitter. Die
Grundlehren der mathematischen Wissenschaften, Band 170, SpringerVerlag,
New YorkBerlin, 1971. MR 0271512
(42 #6395)
 [L.4]
JacquesLouis
Lions, Un résultat de régularité pour
l’opérateur
∂²/∂𝑡²+Δ², Current topics in
partial differential equations, Kinokuniya, Tokyo, 1986,
pp. 247–260 (French). MR
1112149
 [LLT.1]
I.
Lasiecka, J.L.
Lions, and R.
Triggiani, Nonhomogeneous boundary value problems for second order
hyperbolic operators, J. Math. Pures Appl. (9) 65
(1986), no. 2, 149–192. MR 867669
(88c:35092)
 [LM.1]
J. L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications, vols. I, II, SpringerVerlag, Berlin and New York, 1972.
 [LT.1]
I.
Lasiecka and R.
Triggiani, A cosine operator approach to modeling
𝐿₂(0,𝑇;\𝐿₂(Γ))—boundary
input hyperbolic equations, Appl. Math. Optim. 7
(1981), no. 1, 35–93. MR 600559
(82b:35097), http://dx.doi.org/10.1007/BF01442108
 [LT.2]
I.
Lasiecka and R.
Triggiani, Regularity of hyperbolic equations under
𝐿₂(0,𝑇;𝐿₂(Γ))Dirichlet
boundary terms, Appl. Math. Optim. 10 (1983),
no. 3, 275–286. MR 722491
(85j:35111), http://dx.doi.org/10.1007/BF01448390
 [LT.3]
I.
Lasiecka and R.
Triggiani, Riccati equations for hyperbolic partial differential
equations with
𝐿₂(0,𝑇;𝐿₂(Γ))Dirichlet
boundary terms, SIAM J. Control Optim. 24 (1986),
no. 5, 884–925. MR 854062
(87k:93057), http://dx.doi.org/10.1137/0324054
 [LT.4]
, Uniform exponential energy decay of wave equation in a bounded region with feedback control in the Dirichlet boundary conditions, J. Differential Equations 65 (1986), 340390.
 [LT.5]
, Hyperbolic equations with nonhomogeneous Neumann boundary terms. Part I: Regularity, preprint 1983.
 [LT.6]
, Sharp regularity results for hyperbolic equations of Neumann type, 1987.
 [LT.7]
, Regularity theory for a class of Euler Bernoulli equations: a cosine operator approach, Boll. Un. Mat. Ital. (to appear).
 [LT.8]
I.
Lasiecka and R.
Triggiani, Exact controllability of the EulerBernoulli equation
with controls in the Dirichlet and Neumann boundary conditions: a
nonconservative case, SIAM J. Control Optim. 27
(1989), no. 2, 330–373. MR 984832
(90c:93010), http://dx.doi.org/10.1137/0327018
 [LT.9]
, Infinite horizon quadratic cost problems for boundary control problems, Proc. 20th CDC Conference, pp. 10051010, Los Angeles, December 1987.
 [R.1]
Jeffrey
Rauch, \cal𝐿₂ is a continuable initial condition for
Kreiss’ mixed problems, Comm. Pure Appl. Math.
25 (1972), 265–285. MR 0298232
(45 #7284)
 [R.2]
David
L. Russell, Controllability and stabilizability theory for linear
partial differential equations: recent progress and open questions,
SIAM Rev. 20 (1978), no. 4, 639–739. MR 508380
(80c:93032), http://dx.doi.org/10.1137/1020095
 [K.1]
HeinzOtto
Kreiss, Initial boundary value problems for hyperbolic
systems, Comm. Pure Appl. Math. 23 (1970),
277–298. MR 0437941
(55 #10862)
 [T.1]
Roberto
Triggiani, A cosine operator approach to modelling boundary input
hyperbolic systems, Optimization techniques (Proc. 8th IFIP Conf.,
Würzburg, 1977) Springer, Berlin, 1978, pp. 380–390.
Lecture Notes in Control and Informat. Sci., Vol. 6. MR 0502082
(58 #19248)
 [T.2]
, Exact boundary controllability on for the wave equation with Dirichlet control acting on a portion of the boundary, and related problems, Appl. Math. Optim. (to appear), SpringerVerlag Lecture Notes in Control Sciences, vol. 102, pp. 292332, Proceedings of 3rd International Conference, Vorau, Austria, July 612, 1986.
 [T.3]
R.
Triggiani, Wave equation on a bounded domain with boundary
dissipation: an operator approach, Operator methods for optimal
control problems (New Orleans, La., 1986), Lecture Notes in Pure and Appl.
Math., vol. 108, Dekker, New York, 1987, pp. 283–310. MR 920578
(89d:35104)
 [W.1]
George
Weiss, Admissibility of unbounded control operators, SIAM J.
Control Optim. 27 (1989), no. 3, 527–545. MR 993285
(90c:93060), http://dx.doi.org/10.1137/0327028
 [C.1]
 S. Chang, Ph. D. dissertation, Mathematics Department, Univ. of Florida, 1984.
 [CL.1]
 S. Chang and I. Lasiecka, Riccati equations for nonsymmetric and nondissipative hyperbolic systems, J. Math. Anal. Appl. 115 (1986), 378414. MR 842807 (87k:49008)
 [DaPLT.1]
 G. Da Prato, I. Lasiecka and R. Triggiani, A direct study of the Riccati equation arising in hyperbolic boundary control problems, J. Differential Equations 64 (1986), 2647. MR 849662 (87i:93068)
 [DS.1]
 N. Dunford and J. T. Schwartz, Linear operators. Part I, Interscience, New York, 1958.
 [FLT.1]
 F. Flandoli, I. Lasiecka and R. Triggiani, Algebraic Riccati equations with nonsmoothing observation arising in hyperbolic and EulerBernoulli equations, Ann. Mat. Pura Appl. (to appear).
 [H.1]
 F. L. Ho, Observabilité frontiere de l'equation des ondes, C. R. Acad. Sci. Paris 302 (1986). MR 838598 (87d:93017)
 [HP.1]
 E. Hille and R. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, R.I., 1957. MR 0089373 (19:664d)
 [L.1]
 J. L. Lions, Controle des systemes distribues singuliers, GauthierVillars, 1983. MR 712486 (85c:93002)
 [L.2]
 , Controlabilité exacte des systemes distribues, C. R. Acad. Sci. Paris 302 (1986), 471475. MR 838402 (87e:93051)
 [L.3]
 , Optimal control of systems governed by partial differential equations, SpringerVerlag, Berlin and New York, 1971. MR 0271512 (42:6395)
 [L.4]
 J.L. Lions, Un resultat de regularité (paper dedicated to S. Mizohata), Current Topics in Partial Differential Equations, Y. Ohya et al., Eds., Kinokuniya, Tokyo, 1986. MR 1112149
 [LLT.1]
 I. Lasiecka, J. L. Lions and R. Triggiani, Non homogeneous boundary value problems for second order hyperbolic operators, J. Math. Pures Appl. 69 (1986), 149192. MR 867669 (88c:35092)
 [LM.1]
 J. L. Lions and E. Magenes, Nonhomogeneous boundary value problems and applications, vols. I, II, SpringerVerlag, Berlin and New York, 1972.
 [LT.1]
 I. Lasiecka and R. Triggiani, A cosine operator approach to modeling boundary input hyperbolic equations, Appl. Math. Optim. 7 (1981), 3593. MR 600559 (82b:35097)
 [LT.2]
 , Regularity of hyperbolic equations under boundary terms, Appl. Math. Optim. 10 (1983), 275286. MR 722491 (85j:35111)
 [LT.3]
 , Riccati equations for hyperbolic partial differential equations with Dirichlet boundary terms, SIAM J. Control Optim. 24 (1986), 884926. MR 854062 (87k:93057)
 [LT.4]
 , Uniform exponential energy decay of wave equation in a bounded region with feedback control in the Dirichlet boundary conditions, J. Differential Equations 65 (1986), 340390.
 [LT.5]
 , Hyperbolic equations with nonhomogeneous Neumann boundary terms. Part I: Regularity, preprint 1983.
 [LT.6]
 , Sharp regularity results for hyperbolic equations of Neumann type, 1987.
 [LT.7]
 , Regularity theory for a class of Euler Bernoulli equations: a cosine operator approach, Boll. Un. Mat. Ital. (to appear).
 [LT.8]
 , Exact controllability of the EulerBernoulli equation with controls in the Dirichlet and Neumann boundary conditions: a nonconservative case, SIAM J. Control & Optim. (to appear). MR 984832 (90c:93010)
 [LT.9]
 , Infinite horizon quadratic cost problems for boundary control problems, Proc. 20th CDC Conference, pp. 10051010, Los Angeles, December 1987.
 [R.1]
 J. Rauch, is a continuable initial condition for Kreiss' mixed problems, Comm. Pure Appl. Math. 25 (1972), 265285. MR 0298232 (45:7284)
 [R.2]
 D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions, SIAM Rev. 20 (1978), 639740. MR 508380 (80c:93032)
 [K.1]
 H. O. Kreiss, Initial boundary value problem for hyperbolic systems, Comm. Pure Appl. Math. 13 (1970), 277298. MR 0437941 (55:10862)
 [T.1]
 R. Triggiani, A cosine operator approach to modeling boundary inputs problems for hyperbolic systems, Lecture Notes in Math., vol. 6, SpringerVerlag, Berlin and New York, 1978, pp. 380390. MR 0502082 (58:19248)
 [T.2]
 , Exact boundary controllability on for the wave equation with Dirichlet control acting on a portion of the boundary, and related problems, Appl. Math. Optim. (to appear), SpringerVerlag Lecture Notes in Control Sciences, vol. 102, pp. 292332, Proceedings of 3rd International Conference, Vorau, Austria, July 612, 1986.
 [T.3]
 , Wave equation on a bounded domain with boundary dissipation: an operator approach, J. Math. Anal. Appl. (to appear); also, Lecture Notes in Pure and Applied Mathematics, vol. 108, pp. 283310; in Operator Methods for Optimal Control Problems (Sung J. Lee, Ed.), Marcel Dekker (1987); also in Recent Advances in Communication and Control Theory, honoring the sixtieth anniversary of A. V. Balakrishnan (R. E. Kalman and G. I. Marchuk, Eds.), Optimization Software (New York, 1987), pp. 262286. MR 920578 (89d:35104)
 [W.1]
 G. Weiss, Admissibility of unbounded control operators, preprint. MR 993285 (90c:93060)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939198809648511
PII:
S 00029939(1988)09648511
Article copyright:
© Copyright 1988
American Mathematical Society
