# American Institute of Mathematical Sciences

March  2016, 5(1): 185-199. doi: 10.3934/eect.2016.5.185

## A matrix-valued generator $\mathcal{A}$ with strong boundary coupling: A critical subspace of $D((-\mathcal{A})^{\frac{1}{2}})$ and $D((-\mathcal{A}^*)^{\frac{1}{2}})$ and implications

 1 Department of Mathematical Sciences, University of Memphis, Memphis, TN 38152, United States

Received  September 2015 Revised  February 2016 Published  March 2016

We study the free dynamic operator $\mathcal{A}$ which arises in the study of a heat-viscoelastic structure model with highly coupled boundary conditions at the interface between the heat domain and the contiguous structure domain. We use Baiocchi's characterization on the interpolation of subspaces defined by a constrained map [1], [16,p 96] to identify a relevant subspace $V_0$ of both $D((-\mathcal{A})^{\frac{1}{2}})$ and $D((-\mathcal{A}^∗)^{\frac{1}{2}})$, which is sufficient to determine the optimal regularity of the interface (boundary) $\to$ interior map $\mathcal{A}^{-1} \mathcal{B}_N$ from the interface to the energy space. Here, $\mathcal{B}_N$ is the (boundary) control operator acting at the interface in the Neumann boundary conditions.
Citation: Roberto Triggiani. A matrix-valued generator $\mathcal{A}$ with strong boundary coupling: A critical subspace of $D((-\mathcal{A})^{\frac{1}{2}})$ and $D((-\mathcal{A}^*)^{\frac{1}{2}})$ and implications. Evolution Equations and Control Theory, 2016, 5 (1) : 185-199. doi: 10.3934/eect.2016.5.185
##### References:
 [1] C. Baiocchi, Un teorema di interpolazione: Applicazioni ai problemi ai limiti per le equazioni a derivate parziali, Ann. Mat. Pura Appl., 73 (1966), 233-251. doi: 10.1007/BF02415089. [2] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, $2^{nd}$ edition, Birkhauser, 2007, 575 pages. doi: 10.1007/978-0-8176-4581-6. [3] S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems: The case $\alpha = 1/2$, Springer-Verlag Lecture Notes in Mathematics, 1354 (1988), 234-256. Proceedings of Seminar on Approximation and Optimization, University of Havana, Cuba (January 1987). doi: 10.1007/BFb0089601. [4] S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems: The case $1/2 \leq \alpha \leq 1$), Pacific J. Math., 136 (1989), 15-55. doi: 10.2140/pjm.1989.136.15. [5] S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Diff. Eqns., 88 (1990), 279-293. doi: 10.1016/0022-0396(90)90100-4. [6] L. De Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rendiconti del Seminario Matematico della Universita di Padova, 34 (1964), 205-223. [7] 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. doi: 10.3792/pja/1195521686. [8] P. Grisvard, Characterization de qualques espaces d' interpolation, Arch. Pat. Mech. Anal., 25 (1967), 40-63. doi: 10.1007/BF00281421. [9] T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan., 13 (1961), 246-274. doi: 10.2969/jmsj/01330246. [10] I. Lasiecka, Unified theory for abstract parabolic boundary problems-a semigroup approach, Appl. Math. & Optimiz., 6 (1980), 287-333. doi: 10.1007/BF01442900. [11] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I, Abstract Parabolic Systems Encyclopedia of Mathematics and Its Applications Series, Cambridge University Press, January 2000. [12] I. Lasiecka and R. Triggiani, Domains of fractional powers of matrix-valued Operators: A general approach, Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, Operator Theory Advances and Applications, W.Arendt, R.Chill and Y.Tomilov, Editors, 250 (2015), 297-309. doi: 10.1007/978-3-319-18494-4_20. [13] I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, Communications on Pure & Applied Analysis, to appear. [14] C. Lebiedzik and R. Triggaini, The optimal interior regularity for the critical case of a clamped thermoelastic system with point control revisited, Modern Aspects of the Theory of PDEs. Vol. 216 of Operator Theory: Advances and Applications, 243-259, Birkhäuser/Springer, Basel, 2011. M. Ruzhansky and J. Wirth, eds. doi: 10.1007/978-3-0348-0069-3_14. [15] J. L. Lions, Especes d'interpolation et domaines de puissances fractionnaires d'openateurs, J. Math Soc., 14 (1962), 233-241. doi: 10.2969/jmsj/01420233. [16] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I, Springer-Verlag (1972), 357 pp. [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [18] R. Triggiani, A heat-viscoelastic structure interaction model with Neumann or Dirichlet boundary control at the interface: optimal regularity, control theoretic implications, Applied Mathematics and Optimization, special issue in memory of A.V.Balakrishnan, to appear.

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##### References:
 [1] C. Baiocchi, Un teorema di interpolazione: Applicazioni ai problemi ai limiti per le equazioni a derivate parziali, Ann. Mat. Pura Appl., 73 (1966), 233-251. doi: 10.1007/BF02415089. [2] A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, $2^{nd}$ edition, Birkhauser, 2007, 575 pages. doi: 10.1007/978-0-8176-4581-6. [3] S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems: The case $\alpha = 1/2$, Springer-Verlag Lecture Notes in Mathematics, 1354 (1988), 234-256. Proceedings of Seminar on Approximation and Optimization, University of Havana, Cuba (January 1987). doi: 10.1007/BFb0089601. [4] S. Chen and R. Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems: The case $1/2 \leq \alpha \leq 1$), Pacific J. Math., 136 (1989), 15-55. doi: 10.2140/pjm.1989.136.15. [5] S. Chen and R. Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Diff. Eqns., 88 (1990), 279-293. doi: 10.1016/0022-0396(90)90100-4. [6] L. De Simon, Un'applicazione della teoria degli integrali singolari allo studio delle equazioni differenziali lineari astratte del primo ordine, Rendiconti del Seminario Matematico della Universita di Padova, 34 (1964), 205-223. [7] 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. doi: 10.3792/pja/1195521686. [8] P. Grisvard, Characterization de qualques espaces d' interpolation, Arch. Pat. Mech. Anal., 25 (1967), 40-63. doi: 10.1007/BF00281421. [9] T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan., 13 (1961), 246-274. doi: 10.2969/jmsj/01330246. [10] I. Lasiecka, Unified theory for abstract parabolic boundary problems-a semigroup approach, Appl. Math. & Optimiz., 6 (1980), 287-333. doi: 10.1007/BF01442900. [11] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I, Abstract Parabolic Systems Encyclopedia of Mathematics and Its Applications Series, Cambridge University Press, January 2000. [12] I. Lasiecka and R. Triggiani, Domains of fractional powers of matrix-valued Operators: A general approach, Operator Semigroups Meet Complex Analysis, Harmonic Analysis and Mathematical Physics, Operator Theory Advances and Applications, W.Arendt, R.Chill and Y.Tomilov, Editors, 250 (2015), 297-309. doi: 10.1007/978-3-319-18494-4_20. [13] I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, Communications on Pure & Applied Analysis, to appear. [14] C. Lebiedzik and R. Triggaini, The optimal interior regularity for the critical case of a clamped thermoelastic system with point control revisited, Modern Aspects of the Theory of PDEs. Vol. 216 of Operator Theory: Advances and Applications, 243-259, Birkhäuser/Springer, Basel, 2011. M. Ruzhansky and J. Wirth, eds. doi: 10.1007/978-3-0348-0069-3_14. [15] J. L. Lions, Especes d'interpolation et domaines de puissances fractionnaires d'openateurs, J. Math Soc., 14 (1962), 233-241. doi: 10.2969/jmsj/01420233. [16] J. L. Lions and E. Magenes, Nonhomogeneous Boundary Value Propblems and Applications, Vol. I, Springer-Verlag (1972), 357 pp. [17] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer Verlag, 1983. doi: 10.1007/978-1-4612-5561-1. [18] R. Triggiani, A heat-viscoelastic structure interaction model with Neumann or Dirichlet boundary control at the interface: optimal regularity, control theoretic implications, Applied Mathematics and Optimization, special issue in memory of A.V.Balakrishnan, to appear.
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