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 & 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. doi: 10.1007/BF02415089. Google Scholar

[2]

A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems,, $2^{nd}$ edition, (2007). doi: 10.1007/978-0-8176-4581-6. Google Scholar

[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. doi: 10.1007/BFb0089601. Google Scholar

[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. doi: 10.2140/pjm.1989.136.15. Google Scholar

[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. doi: 10.1016/0022-0396(90)90100-4. Google Scholar

[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. Google Scholar

[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. doi: 10.3792/pja/1195521686. Google Scholar

[8]

P. Grisvard, Characterization de qualques espaces d' interpolation,, Arch. Pat. Mech. Anal., 25 (1967), 40. doi: 10.1007/BF00281421. Google Scholar

[9]

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

[10]

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

[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, (2000). Google Scholar

[12]

I. Lasiecka and R. Triggiani, Domains of fractional powers of matrix-valued Operators: A general approach,, Operator Semigroups Meet Complex Analysis, 250 (2015), 297. doi: 10.1007/978-3-319-18494-4_20. Google Scholar

[13]

I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay,, Communications on Pure & Applied Analysis, (). Google Scholar

[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, (2011), 243. doi: 10.1007/978-3-0348-0069-3_14. Google Scholar

[15]

J. L. Lions, Especes d'interpolation et domaines de puissances fractionnaires d'openateurs,, J. Math Soc., 14 (1962), 233. doi: 10.2969/jmsj/01420233. Google Scholar

[16]

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

[17]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[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, (). Google Scholar

show all references

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. doi: 10.1007/BF02415089. Google Scholar

[2]

A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems,, $2^{nd}$ edition, (2007). doi: 10.1007/978-0-8176-4581-6. Google Scholar

[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. doi: 10.1007/BFb0089601. Google Scholar

[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. doi: 10.2140/pjm.1989.136.15. Google Scholar

[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. doi: 10.1016/0022-0396(90)90100-4. Google Scholar

[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. Google Scholar

[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. doi: 10.3792/pja/1195521686. Google Scholar

[8]

P. Grisvard, Characterization de qualques espaces d' interpolation,, Arch. Pat. Mech. Anal., 25 (1967), 40. doi: 10.1007/BF00281421. Google Scholar

[9]

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

[10]

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

[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, (2000). Google Scholar

[12]

I. Lasiecka and R. Triggiani, Domains of fractional powers of matrix-valued Operators: A general approach,, Operator Semigroups Meet Complex Analysis, 250 (2015), 297. doi: 10.1007/978-3-319-18494-4_20. Google Scholar

[13]

I. Lasiecka and R. Triggiani, Heat-structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay,, Communications on Pure & Applied Analysis, (). Google Scholar

[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, (2011), 243. doi: 10.1007/978-3-0348-0069-3_14. Google Scholar

[15]

J. L. Lions, Especes d'interpolation et domaines de puissances fractionnaires d'openateurs,, J. Math Soc., 14 (1962), 233. doi: 10.2969/jmsj/01420233. Google Scholar

[16]

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

[17]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Springer Verlag, (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar

[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, (). Google Scholar

[1]

Irena Lasiecka, Roberto Triggiani. Heat--structure interaction with viscoelastic damping: Analyticity with sharp analytic sector, exponential decay, fractional powers. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1515-1543. doi: 10.3934/cpaa.2016001

[2]

Demetris Hadjiloucas. Stochastic matrix-valued cocycles and non-homogeneous Markov chains. Discrete & Continuous Dynamical Systems - A, 2007, 17 (4) : 731-738. doi: 10.3934/dcds.2007.17.731

[3]

Yongge Tian. A survey on rank and inertia optimization problems of the matrix-valued function $A + BXB^{*}$. Numerical Algebra, Control & Optimization, 2015, 5 (3) : 289-326. doi: 10.3934/naco.2015.5.289

[4]

Daniel Alpay, Eduard Tsekanovskiĭ. Subclasses of Herglotz-Nevanlinna matrix-valued functtons and linear systems. Conference Publications, 2001, 2001 (Special) : 1-13. doi: 10.3934/proc.2001.2001.1

[5]

George Avalos, Roberto Triggiani. Rational decay rates for a PDE heat--structure interaction: A frequency domain approach. Evolution Equations & Control Theory, 2013, 2 (2) : 233-253. doi: 10.3934/eect.2013.2.233

[6]

Miaomiao Cai, Li Ma. Moving planes for nonlinear fractional Laplacian equation with negative powers. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4603-4615. doi: 10.3934/dcds.2018201

[7]

Qiang Du, M. D. Gunzburger, L. S. Hou, J. Lee. Analysis of a linear fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - A, 2003, 9 (3) : 633-650. doi: 10.3934/dcds.2003.9.633

[8]

Jiang-Xia Nan, Deng-Feng Li. Linear programming technique for solving interval-valued constraint matrix games. Journal of Industrial & Management Optimization, 2014, 10 (4) : 1059-1070. doi: 10.3934/jimo.2014.10.1059

[9]

Anthony Tongen, María Zubillaga, Jorge E. Rabinovich. A two-sex matrix population model to represent harem structure. Mathematical Biosciences & Engineering, 2016, 13 (5) : 1077-1092. doi: 10.3934/mbe.2016031

[10]

Francis C. Motta, Patrick D. Shipman. Informing the structure of complex Hadamard matrix spaces using a flow. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 2349-2364. doi: 10.3934/dcdss.2019147

[11]

John Cleveland. Basic stage structure measure valued evolutionary game model. Mathematical Biosciences & Engineering, 2015, 12 (2) : 291-310. doi: 10.3934/mbe.2015.12.291

[12]

Roberto Triggiani, Jing Zhang. Heat-viscoelastic plate interaction: Analyticity, spectral analysis, exponential decay. Evolution Equations & Control Theory, 2018, 7 (1) : 153-182. doi: 10.3934/eect.2018008

[13]

Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424

[14]

Eugenio Aulisa, Akif Ibragimov, Emine Yasemen Kaya-Cekin. Fluid structure interaction problem with changing thickness beam and slightly compressible fluid. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1133-1148. doi: 10.3934/dcdss.2014.7.1133

[15]

Grégoire Allaire, Alessandro Ferriero. Homogenization and long time asymptotic of a fluid-structure interaction problem. Discrete & Continuous Dynamical Systems - B, 2008, 9 (2) : 199-220. doi: 10.3934/dcdsb.2008.9.199

[16]

Serge Nicaise, Cristina Pignotti. Asymptotic analysis of a simple model of fluid-structure interaction. Networks & Heterogeneous Media, 2008, 3 (4) : 787-813. doi: 10.3934/nhm.2008.3.787

[17]

Igor Kukavica, Amjad Tuffaha. Solutions to a fluid-structure interaction free boundary problem. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1355-1389. doi: 10.3934/dcds.2012.32.1355

[18]

Antonio Greco, Antonio Iannizzotto. Existence and convexity of solutions of the fractional heat equation. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2201-2226. doi: 10.3934/cpaa.2017109

[19]

Fausto Ferrari, Michele Miranda Jr, Diego Pallara, Andrea Pinamonti, Yannick Sire. Fractional Laplacians, perimeters and heat semigroups in Carnot groups. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 477-491. doi: 10.3934/dcdss.2018026

[20]

Angkana Rüland, Mikko Salo. Quantitative approximation properties for the fractional heat equation. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019027

2018 Impact Factor: 1.048

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]