January  2022, 42(1): 463-479. doi: 10.3934/dcds.2021124

Exact null-controllability of interconnected abstract evolution equations with unbounded input operators

Holon Institute of Technology, Israel

Received  December 2019 Revised  June 2021 Published  January 2022 Early access  September 2021

The exact null-controllability problem in the class of smooth controls with applications to interconnected systems was considered in [23] for the case of bounded input operators appearing in systems with distributed controls. The current paper constitutes an extension of the [23] for the case of unbounded input operators (with more emphasis on the controllability of interconnected systems). The proofs of the results of [23] for the case of bounded input operators are adopted for the case of unbounded input operators.

Citation: Benzion Shklyar. Exact null-controllability of interconnected abstract evolution equations with unbounded input operators. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 463-479. doi: 10.3934/dcds.2021124
References:
[1]

F. Ammar-KohdjaM. BenabdallahL. González-Burgos and L. de Teresa, Recent results on the controllability of coupled parabolic problems: A survey, Mathematical Control and Related Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[2] S. Avdonin and S. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge, University Press, 1995.   Google Scholar
[3]

A. Balakrishnan, Applied Functional Analysis, Springer-Verlag, New-York Heidelberg Berlin, 1976.  Google Scholar

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P. Balasz, D. Stoeva and J.-P. Antonie, Classification of general sequences by Frame-Related operators, Sampl. Theory Signal Image Process, 10 (2011), 151–170, arXiv: 10090.1496vl [math.FA].  Google Scholar

[5]

R. Bellmann and K. Cooke, Differential-Difference Equations, New York Academic Press London, 1963.  Google Scholar

[6]

A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, 2$^nd$ edition, Systems and Control, Foundations, Applications, Birkhuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[7]

R. Boas, A general moment problem, Amer. J. Math., 63 (1941), 361-370.  doi: 10.2307/2371530.  Google Scholar

[8]

A. Butkovskii, The method of moments in Optimal Control Theory for distributed parameter systems, Automation and Remote Control, 24 (1963), 1217-1225.   Google Scholar

[9]

A. Butkovskii, Characterizations of Distributed Systems, Moscow, Nauka Publisher, 1979 (in Russian). Google Scholar

[10]

H. Fattorini, Exact controllability of linear systems in infinite dimensional spaces, Partial Differential Equations and Related Topics, 446 (1975), 166-183.   Google Scholar

[11]

H. Fattorini and D. Russel, Exact controllability theorems for linear parabolic equations in one space dimensions, Archive for Rational Mechanics and Analysis, 43 (1971), 272-292.  doi: 10.1007/BF00250466.  Google Scholar

[12]

R. Gurtain and D. Salamon, Finite dimensional compensators for infinite dimensional systems with unbounded input operators, SIAM J. Control and Optimization, 24 (1986), 797-816.  doi: 10.1137/0324050.  Google Scholar

[13]

J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York Heidelberg Berlin, 1977.  Google Scholar

[14]

E. Hille and R. Philips, Functional Analysis and Semi-Groups, AMS, 1957.  Google Scholar

[15]

M. Krein, Linear Differential Equations in Banach Spaces, Nauka Publisher, Moscow, 1967.  Google Scholar

[16]

I. Lasiecka, Mathematical Control Theory of Coupled PDEs, CBMS-NSF Regional Conference Series in Applied Mathematics, 75, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898717099.  Google Scholar

[17]

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, 74, Cambridge University Press, Cambridge, 2000.  Google Scholar

[18]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. II, Abstract parabolic systems. Encyclopedia of Mathematics and its Applications, 74, Cambridge University Press, Cambridge, 2000. Google Scholar

[19]

R. Reid, A class of Riesz-Fischer sequences, Proc. Amer. Math. Soc., 123 (1995), 827-829.  doi: 10.1090/S0002-9939-1995-1223519-0.  Google Scholar

[20]

D. Russel, Nonharmonic Fourier series in control theory for distributed parameter systems, J. Math. Anal. Appl., 18 (1967), 542-560.  doi: 10.1016/0022-247X(67)90045-5.  Google Scholar

[21]

D. Salamon, Infinite dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.  Google Scholar

[22]

B. Shklyar, Exact null controllability of abstract differential equations by finite-dimensional control and strongly minimal families of exponentials, Differential Equations and Applications, 3 (2011), 171-188.  doi: 10.7153/dea-03-10.  Google Scholar

[23]

B. Shklyar, Exact null controllability of evolution equations by smooth scalar distributed control and applications to controllability of interconnected systems, Applied Mathematics and Computations, 238 (2014), 444-459.  doi: 10.1016/j.amc.2014.03.093.  Google Scholar

[24]

M. Shubov, Exact controllability of damped Timoshenko beam, J. Math. Contr. Inform., 17 (2000), 375-395.  doi: 10.1093/imamci/17.4.375.  Google Scholar

[25]

F. Tricomi, Integral Equations, Interscience Publisher, Inc., New York, London, 1957.  Google Scholar

[26]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroup, Birkheauser Advanced Texts: Basler Lehrbucher, Birkheauser Verlag, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[27]

G. Weiss, Admissibility of unbounded control operators, SIAM J. Contr. Optimiz., 27 (1989), 527-545.  doi: 10.1137/0327028.  Google Scholar

[28] R. Young, An Introduction to Non-harmonic Fourier Series, Academic Press, New York, 1980.   Google Scholar
[29]

R. Young, On the class of Riesz-Fisher sequences, Proc. Amer. Math. Soc., 126 (1998), 1139-1142.  doi: 10.1090/S0002-9939-98-04416-5.  Google Scholar

show all references

References:
[1]

F. Ammar-KohdjaM. BenabdallahL. González-Burgos and L. de Teresa, Recent results on the controllability of coupled parabolic problems: A survey, Mathematical Control and Related Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.  Google Scholar

[2] S. Avdonin and S. Ivanov, Families of Exponentials. The Method of Moments in Controllability Problems for Distributed Parameter Systems, Cambridge, University Press, 1995.   Google Scholar
[3]

A. Balakrishnan, Applied Functional Analysis, Springer-Verlag, New-York Heidelberg Berlin, 1976.  Google Scholar

[4]

P. Balasz, D. Stoeva and J.-P. Antonie, Classification of general sequences by Frame-Related operators, Sampl. Theory Signal Image Process, 10 (2011), 151–170, arXiv: 10090.1496vl [math.FA].  Google Scholar

[5]

R. Bellmann and K. Cooke, Differential-Difference Equations, New York Academic Press London, 1963.  Google Scholar

[6]

A. Bensoussan, G. Da Prato, M. Delfour and S. Mitter, Representation and Control of Infinite Dimensional Systems, 2$^nd$ edition, Systems and Control, Foundations, Applications, Birkhuser Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4581-6.  Google Scholar

[7]

R. Boas, A general moment problem, Amer. J. Math., 63 (1941), 361-370.  doi: 10.2307/2371530.  Google Scholar

[8]

A. Butkovskii, The method of moments in Optimal Control Theory for distributed parameter systems, Automation and Remote Control, 24 (1963), 1217-1225.   Google Scholar

[9]

A. Butkovskii, Characterizations of Distributed Systems, Moscow, Nauka Publisher, 1979 (in Russian). Google Scholar

[10]

H. Fattorini, Exact controllability of linear systems in infinite dimensional spaces, Partial Differential Equations and Related Topics, 446 (1975), 166-183.   Google Scholar

[11]

H. Fattorini and D. Russel, Exact controllability theorems for linear parabolic equations in one space dimensions, Archive for Rational Mechanics and Analysis, 43 (1971), 272-292.  doi: 10.1007/BF00250466.  Google Scholar

[12]

R. Gurtain and D. Salamon, Finite dimensional compensators for infinite dimensional systems with unbounded input operators, SIAM J. Control and Optimization, 24 (1986), 797-816.  doi: 10.1137/0324050.  Google Scholar

[13]

J. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York Heidelberg Berlin, 1977.  Google Scholar

[14]

E. Hille and R. Philips, Functional Analysis and Semi-Groups, AMS, 1957.  Google Scholar

[15]

M. Krein, Linear Differential Equations in Banach Spaces, Nauka Publisher, Moscow, 1967.  Google Scholar

[16]

I. Lasiecka, Mathematical Control Theory of Coupled PDEs, CBMS-NSF Regional Conference Series in Applied Mathematics, 75, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2002. doi: 10.1137/1.9780898717099.  Google Scholar

[17]

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, 74, Cambridge University Press, Cambridge, 2000.  Google Scholar

[18]

I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. II, Abstract parabolic systems. Encyclopedia of Mathematics and its Applications, 74, Cambridge University Press, Cambridge, 2000. Google Scholar

[19]

R. Reid, A class of Riesz-Fischer sequences, Proc. Amer. Math. Soc., 123 (1995), 827-829.  doi: 10.1090/S0002-9939-1995-1223519-0.  Google Scholar

[20]

D. Russel, Nonharmonic Fourier series in control theory for distributed parameter systems, J. Math. Anal. Appl., 18 (1967), 542-560.  doi: 10.1016/0022-247X(67)90045-5.  Google Scholar

[21]

D. Salamon, Infinite dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.  doi: 10.2307/2000351.  Google Scholar

[22]

B. Shklyar, Exact null controllability of abstract differential equations by finite-dimensional control and strongly minimal families of exponentials, Differential Equations and Applications, 3 (2011), 171-188.  doi: 10.7153/dea-03-10.  Google Scholar

[23]

B. Shklyar, Exact null controllability of evolution equations by smooth scalar distributed control and applications to controllability of interconnected systems, Applied Mathematics and Computations, 238 (2014), 444-459.  doi: 10.1016/j.amc.2014.03.093.  Google Scholar

[24]

M. Shubov, Exact controllability of damped Timoshenko beam, J. Math. Contr. Inform., 17 (2000), 375-395.  doi: 10.1093/imamci/17.4.375.  Google Scholar

[25]

F. Tricomi, Integral Equations, Interscience Publisher, Inc., New York, London, 1957.  Google Scholar

[26]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroup, Birkheauser Advanced Texts: Basler Lehrbucher, Birkheauser Verlag, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[27]

G. Weiss, Admissibility of unbounded control operators, SIAM J. Contr. Optimiz., 27 (1989), 527-545.  doi: 10.1137/0327028.  Google Scholar

[28] R. Young, An Introduction to Non-harmonic Fourier Series, Academic Press, New York, 1980.   Google Scholar
[29]

R. Young, On the class of Riesz-Fisher sequences, Proc. Amer. Math. Soc., 126 (1998), 1139-1142.  doi: 10.1090/S0002-9939-98-04416-5.  Google Scholar

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