April  2019, 39(4): 1685-1730. doi: 10.3934/dcds.2019074

Riccati equations for linear Hamiltonian systems without controllability condition

Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, CZ-61137 Brno, Czech Republic

Dedicated to the memory of Professor Russell A. Johnson.

Received  August 2017 Revised  July 2018 Published  January 2019

Fund Project: This research was supported by the Czech Science Foundation under grant GA16-00611S

In this paper we develop new theory of Riccati matrix differential equations for linear Hamiltonian systems, which do not require any controllability assumption. When the system is nonoscillatory, it is known from our previous work that conjoined bases of the system with eventually the same image form a special structure called a genus. We show that for every such a genus there is an associated Riccati equation. We study the properties of symmetric solutions of these Riccati equations and their connection with conjoined bases of the system. For a given genus, we pay a special attention to distinguished solutions at infinity of the associated Riccati equation and their relationship with the principal solutions at infinity of the system in the considered genus. We show the uniqueness of the distinguished solution at infinity of the Riccati equation corresponding to the minimal genus. This study essentially extends and completes the work of W. T. Reid (1964, 1972), W. A. Coppel (1971), P. Hartman (1964), W. Kratz (1995), and other authors who considered the Riccati equation and its distinguished solution at infinity for invertible conjoined bases, i.e., for the maximal genus in our setting.

Citation: Peter Šepitka. Riccati equations for linear Hamiltonian systems without controllability condition. Discrete & Continuous Dynamical Systems - A, 2019, 39 (4) : 1685-1730. doi: 10.3934/dcds.2019074
References:
[1]

J. Allwright and R. Vinter, Second order conditions for periodic optimal control problems, Control Cybernet., 34 (2005), 617-643.

[2] F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York - London, 1964.
[3] R. Bellman, Introduction to the Mathematical Theory of Control Processes. Vol. I: Linear Equations and Quadratic Criteria, Mathematics in Science and Engineering, Vol. 40, Academic Press, New York, NY, 1967.
[4]

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Second Edition, Springer-Verlag, New York, NY, 2003.

[5] D. S. Bernstein, Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory, Princeton University Press, Princeton, 2005.
[6]

S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Reprint of the 1991 corrected reprint of the 1979 original, Classics in Applied Mathematics, Vol. 56, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898719048.ch0.

[7]

W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin - New York, 1971.

[8]

Z. Došlá and O. Došlý, Quadratic functionals with general boundary conditions, Appl. Math. Optim., 36 (1997), 243-262. doi: 10.1007/s002459900062.

[9]

P. Hartman, Ordinary Differential Equations, John Wiley, New York, 1964.

[10]

M. R. Hestenes, Quadratic control problems, J. Optim. Theory Appl., 17 (1975), 1-42. doi: 10.1007/BF00933915.

[11]

R. Hilscher and V. Zeidan, Applications of time scale symplectic systems without normality, J. Math. Anal. Appl., 340 (2008), 451-465. doi: 10.1016/j.jmaa.2007.07.077.

[12]

R. Hilscher and V. Zeidan, Time scale embedding theorem and coercivity of quadratic functionals, Analysis (Munich), 28 (2008), 1-28. doi: 10.1524/anly.2008.0900.

[13]

R. Hilscher and V. Zeidan, Riccati equations for abnormal time scale quadratic functionals, J. Differential Equations, 244 (2008), 1410-1447. doi: 10.1016/j.jde.2007.10.012.

[14]

R. JohnsonC. Núñez and R. Obaya, Dynamical methods for linear Hamiltonian systems with applications to control processes, J. Dynam. Differential Equations, 25 (2013), 679-713. doi: 10.1007/s10884-013-9300-y.

[15]

R. Johnson, R. Obaya, S. Novo, C. Núñez and R. Fabbri, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control, Developments in Mathematics, Vol. 36, Springer, Cham, 2016. doi: 10.1007/978-3-319-29025-6.

[16]

R. E. Kalman and R. S. Bucy, New results in linear filtering and prediction theory, Trans. ASME Ser. D J. Basic Engrg., 83 (1961), 95-108. doi: 10.1115/1.3658902.

[17]

W. Kratz, Quadratic Functionals in Variational Analysis and Control Theory, Akademie Verlag, Berlin, 1995.

[18]

W. Kratz, Definiteness of quadratic functionals, Analysis, 23 (2003), 163-183. doi: 10.1524/anly.2003.23.2.163.

[19]

B. P. Molinari, Nonnegativity of a quadratic functional, SIAM J. Control, 13 (1975), 792-806. doi: 10.1137/0313046.

[20]

W. T. Reid, Riccati matrix differential equations and non-oscillation criteria for associated linear differential systems, Pacific J. Math., 13 (1963), 665-685. doi: 10.2140/pjm.1963.13.665.

[21]

W. T. Reid, Principal solutions of nonoscillatory linear differential systems, J. Math. Anal. Appl., 9 (1964), 397-423. doi: 10.1016/0022-247X(64)90026-5.

[22]

W. T. Reid, Ordinary Differential Equations, John Wiley & Sons, Inc., New York –London –Sydney, 1971.

[23] W. T. Reid, Riccati Differential Equations, Academic Press, New York –London, 1972.
[24]

W. T. Reid, Sturmian Theory for Ordinary Differential Equations, Springer-Verlag, New York –Berlin –Heidelberg, 1980.

[25]

V. Růžičková, Discrete Symplectic Systems and Definiteness of Quadratic Functionals, Ph.D thesis, Masaryk University, Brno, 2006. Available from https://is.muni.cz/th/p9iz7/?lang=en.

[26]

P. Šepitka, Theory of Principal Solutions at Infinity for Linear Hamiltonian Systems, Ph.D thesis, Masaryk University, Brno, 2014. Available from https://is.muni.cz/th/vqad7/?lang=en.

[27]

P. Šepitka, Genera of conjoined bases for (non)oscillatory linear Hamiltonian systems: extended theory, submitted, 2017.

[28]

P. Šepitka and R. Šimon Hilscher, Minimal principal solution at infinity for nonoscillatory linear Hamiltonian systems, J. Dynam. Differential Equations, 26 (2014), 57-91. doi: 10.1007/s10884-013-9342-1.

[29]

P. Šepitka and R. Šimon Hilscher, Principal solutions at infinity of given ranks for nonoscillatory linear Hamiltonian systems, J. Dynam. Differential Equations, 27 (2015), 137-175. doi: 10.1007/s10884-014-9389-7.

[30]

P. Šepitka and R. Šimon Hilscher, Principal and antiprincipal solutions at infinity of linear Hamiltonian systems, J. Differential Equations, 259 (2015), 4651-4682. doi: 10.1016/j.jde.2015.06.027.

[31]

P. Šepitka and R. Šimon Hilscher, Genera of conjoined bases of linear Hamiltonian systems and limit characterization of principal solutions at infinity, J. Differential Equations, 260 (2016), 6581-6603. doi: 10.1016/j.jde.2016.01.004.

[32]

P. Šepitka and R. Šimon Hilscher, Reid's construction of minimal principal solution at infinity for linear Hamiltonian systems, in Differential and Difference Equations with Applications (eds. S. Pinelas, Z. Došlá, O. Došlý and P. E. Kloeden), Proceedings of the International Conference on Differential & Difference Equations and Applications (Amadora, 2015), Springer Proceedings in Mathematics & Statistics, Vol. 164, Springer, (2016), 359–369. doi: 10.1007/978-3-319-32857-7_34.

[33]

R. Šimon Hilscher, On general Sturmian theory for abnormal linear Hamiltonian systems, in Dynamical Systems, Differential Equations and Applications (eds. W. Feng, Z. Feng, M. Grasselli, A. Ibragimov, X. Lu, S. Siegmund and J. Voigt), Proceedings of the 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications (Dresden, 2010), Discrete Contin. Dynam. Systems, Suppl. 2011, American Institute of Mathematical Sciences (AIMS), (2011), 684–691.

[34]

J. L. Speyer and D. H. Jacobson, Primer on Optimal Control Theory, Advances in Design and Control, Vol. 20, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898718560.

[35]

G. Stefani and P. Zezza, Constrained regular LQ-control problems, SIAM J. Control Optim., 35 (1997), 876-900. doi: 10.1137/S0363012995286848.

[36]

M. Wahrheit, Eigenvalue problems and oscillation of linear Hamiltonian systems, Int. J. Difference Equ., 2 (2007), 221-244.

[37]

V. Zeidan, Sufficiency criteria via focal points and via coupled points, SIAM J. Control Optim., 30 (1992), 82-98. doi: 10.1137/0330006.

[38]

V. Zeidan, The Riccati equation for optimal control problems with mixed state-control constraints: necessity and sufficiency, SIAM J. Control Optim., 32 (1994), 1297-1321. doi: 10.1137/S0363012992233640.

[39]

V. Zeidan, New second-order optimality conditions for variational problems with C2-Hamiltonians, SIAM J. Control Optim., 40 (2001), 577-609. doi: 10.1137/S0363012999358725.

show all references

References:
[1]

J. Allwright and R. Vinter, Second order conditions for periodic optimal control problems, Control Cybernet., 34 (2005), 617-643.

[2] F. V. Atkinson, Discrete and Continuous Boundary Problems, Academic Press, New York - London, 1964.
[3] R. Bellman, Introduction to the Mathematical Theory of Control Processes. Vol. I: Linear Equations and Quadratic Criteria, Mathematics in Science and Engineering, Vol. 40, Academic Press, New York, NY, 1967.
[4]

A. Ben-Israel and T. N. E. Greville, Generalized Inverses: Theory and Applications, Second Edition, Springer-Verlag, New York, NY, 2003.

[5] D. S. Bernstein, Matrix Mathematics. Theory, Facts, and Formulas with Application to Linear Systems Theory, Princeton University Press, Princeton, 2005.
[6]

S. L. Campbell and C. D. Meyer, Generalized Inverses of Linear Transformations, Reprint of the 1991 corrected reprint of the 1979 original, Classics in Applied Mathematics, Vol. 56, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2009. doi: 10.1137/1.9780898719048.ch0.

[7]

W. A. Coppel, Disconjugacy, Lecture Notes in Mathematics, Vol. 220, Springer-Verlag, Berlin - New York, 1971.

[8]

Z. Došlá and O. Došlý, Quadratic functionals with general boundary conditions, Appl. Math. Optim., 36 (1997), 243-262. doi: 10.1007/s002459900062.

[9]

P. Hartman, Ordinary Differential Equations, John Wiley, New York, 1964.

[10]

M. R. Hestenes, Quadratic control problems, J. Optim. Theory Appl., 17 (1975), 1-42. doi: 10.1007/BF00933915.

[11]

R. Hilscher and V. Zeidan, Applications of time scale symplectic systems without normality, J. Math. Anal. Appl., 340 (2008), 451-465. doi: 10.1016/j.jmaa.2007.07.077.

[12]

R. Hilscher and V. Zeidan, Time scale embedding theorem and coercivity of quadratic functionals, Analysis (Munich), 28 (2008), 1-28. doi: 10.1524/anly.2008.0900.

[13]

R. Hilscher and V. Zeidan, Riccati equations for abnormal time scale quadratic functionals, J. Differential Equations, 244 (2008), 1410-1447. doi: 10.1016/j.jde.2007.10.012.

[14]

R. JohnsonC. Núñez and R. Obaya, Dynamical methods for linear Hamiltonian systems with applications to control processes, J. Dynam. Differential Equations, 25 (2013), 679-713. doi: 10.1007/s10884-013-9300-y.

[15]

R. Johnson, R. Obaya, S. Novo, C. Núñez and R. Fabbri, Nonautonomous Linear Hamiltonian Systems: Oscillation, Spectral Theory and Control, Developments in Mathematics, Vol. 36, Springer, Cham, 2016. doi: 10.1007/978-3-319-29025-6.

[16]

R. E. Kalman and R. S. Bucy, New results in linear filtering and prediction theory, Trans. ASME Ser. D J. Basic Engrg., 83 (1961), 95-108. doi: 10.1115/1.3658902.

[17]

W. Kratz, Quadratic Functionals in Variational Analysis and Control Theory, Akademie Verlag, Berlin, 1995.

[18]

W. Kratz, Definiteness of quadratic functionals, Analysis, 23 (2003), 163-183. doi: 10.1524/anly.2003.23.2.163.

[19]

B. P. Molinari, Nonnegativity of a quadratic functional, SIAM J. Control, 13 (1975), 792-806. doi: 10.1137/0313046.

[20]

W. T. Reid, Riccati matrix differential equations and non-oscillation criteria for associated linear differential systems, Pacific J. Math., 13 (1963), 665-685. doi: 10.2140/pjm.1963.13.665.

[21]

W. T. Reid, Principal solutions of nonoscillatory linear differential systems, J. Math. Anal. Appl., 9 (1964), 397-423. doi: 10.1016/0022-247X(64)90026-5.

[22]

W. T. Reid, Ordinary Differential Equations, John Wiley & Sons, Inc., New York –London –Sydney, 1971.

[23] W. T. Reid, Riccati Differential Equations, Academic Press, New York –London, 1972.
[24]

W. T. Reid, Sturmian Theory for Ordinary Differential Equations, Springer-Verlag, New York –Berlin –Heidelberg, 1980.

[25]

V. Růžičková, Discrete Symplectic Systems and Definiteness of Quadratic Functionals, Ph.D thesis, Masaryk University, Brno, 2006. Available from https://is.muni.cz/th/p9iz7/?lang=en.

[26]

P. Šepitka, Theory of Principal Solutions at Infinity for Linear Hamiltonian Systems, Ph.D thesis, Masaryk University, Brno, 2014. Available from https://is.muni.cz/th/vqad7/?lang=en.

[27]

P. Šepitka, Genera of conjoined bases for (non)oscillatory linear Hamiltonian systems: extended theory, submitted, 2017.

[28]

P. Šepitka and R. Šimon Hilscher, Minimal principal solution at infinity for nonoscillatory linear Hamiltonian systems, J. Dynam. Differential Equations, 26 (2014), 57-91. doi: 10.1007/s10884-013-9342-1.

[29]

P. Šepitka and R. Šimon Hilscher, Principal solutions at infinity of given ranks for nonoscillatory linear Hamiltonian systems, J. Dynam. Differential Equations, 27 (2015), 137-175. doi: 10.1007/s10884-014-9389-7.

[30]

P. Šepitka and R. Šimon Hilscher, Principal and antiprincipal solutions at infinity of linear Hamiltonian systems, J. Differential Equations, 259 (2015), 4651-4682. doi: 10.1016/j.jde.2015.06.027.

[31]

P. Šepitka and R. Šimon Hilscher, Genera of conjoined bases of linear Hamiltonian systems and limit characterization of principal solutions at infinity, J. Differential Equations, 260 (2016), 6581-6603. doi: 10.1016/j.jde.2016.01.004.

[32]

P. Šepitka and R. Šimon Hilscher, Reid's construction of minimal principal solution at infinity for linear Hamiltonian systems, in Differential and Difference Equations with Applications (eds. S. Pinelas, Z. Došlá, O. Došlý and P. E. Kloeden), Proceedings of the International Conference on Differential & Difference Equations and Applications (Amadora, 2015), Springer Proceedings in Mathematics & Statistics, Vol. 164, Springer, (2016), 359–369. doi: 10.1007/978-3-319-32857-7_34.

[33]

R. Šimon Hilscher, On general Sturmian theory for abnormal linear Hamiltonian systems, in Dynamical Systems, Differential Equations and Applications (eds. W. Feng, Z. Feng, M. Grasselli, A. Ibragimov, X. Lu, S. Siegmund and J. Voigt), Proceedings of the 8th AIMS Conference on Dynamical Systems, Differential Equations and Applications (Dresden, 2010), Discrete Contin. Dynam. Systems, Suppl. 2011, American Institute of Mathematical Sciences (AIMS), (2011), 684–691.

[34]

J. L. Speyer and D. H. Jacobson, Primer on Optimal Control Theory, Advances in Design and Control, Vol. 20, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2010. doi: 10.1137/1.9780898718560.

[35]

G. Stefani and P. Zezza, Constrained regular LQ-control problems, SIAM J. Control Optim., 35 (1997), 876-900. doi: 10.1137/S0363012995286848.

[36]

M. Wahrheit, Eigenvalue problems and oscillation of linear Hamiltonian systems, Int. J. Difference Equ., 2 (2007), 221-244.

[37]

V. Zeidan, Sufficiency criteria via focal points and via coupled points, SIAM J. Control Optim., 30 (1992), 82-98. doi: 10.1137/0330006.

[38]

V. Zeidan, The Riccati equation for optimal control problems with mixed state-control constraints: necessity and sufficiency, SIAM J. Control Optim., 32 (1994), 1297-1321. doi: 10.1137/S0363012992233640.

[39]

V. Zeidan, New second-order optimality conditions for variational problems with C2-Hamiltonians, SIAM J. Control Optim., 40 (2001), 577-609. doi: 10.1137/S0363012999358725.

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