# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2021124
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## Exact null-controllability of interconnected abstract evolution equations with unbounded input operators

 Holon Institute of Technology, Israel

Received  December 2019 Revised  June 2021 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, doi: 10.3934/dcds.2021124
##### References:
 [1] F. Ammar-Kohdja, M. Benabdallah, L. GonzÃ¡lez-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

show all references

##### References:
 [1] F. Ammar-Kohdja, M. Benabdallah, L. GonzÃ¡lez-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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