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 and 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.

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

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

[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].

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[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.

[27]

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

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

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.

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

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

[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].

[5]

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

[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.

[7]

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

[8]

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

[9]

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

[10]

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

[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.

[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.

[13]

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

[14]

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

[15]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[25]

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

[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.

[27]

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

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

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