Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

Finite-Horizon Linear-Quadratic Optimal Control with General Boundary Conditions

  • Augusto FerranteEmail author
  • Lorenzo Ntogramatzidis
Living reference work entry

Latest version View entry history

DOI: https://doi.org/10.1007/978-1-4471-5102-9_202-2


The linear-quadratic (LQ) problem is the prototype of a large number of optimal control problems, including the fixed endpoint, the point-to-point, and several H2H control problems, as well as the dual counterparts. In the past 50 years, these problems have been addressed using different techniques, each tailored to their specific structure. It is only in the last 10 years that it was recognized that a unifying framework is available. This framework hinges on formulae that parameterize the solutions of the Hamiltonian differential equation in the continuous-time case and the solutions of the extended symplectic system in the discrete-time case. Whereas traditional techniques involve the solutions of Riccati differential or difference equations, the formulae used here to solve the finite-horizon LQ control problem only rely on solutions of the algebraic Riccati equations. In this entry, aspects of the framework are described within a discrete-time context.


Cyclic boundary conditions Discrete-time linear systems Fixed end-point Initial value Point-to-point boundary conditions Quadratic cost Riccati equations 
This is a preview of subscription content, log in to check access.


  1. Anderson BDO, Moore JB (1971) Linear optimal control. Prentice Hall International, Englewood CliffsCrossRefGoogle Scholar
  2. Balas G, Bokor J (2004) Detection filter design for LPV systems – a geometric approach. Automatica 40: 511–518MathSciNetCrossRefGoogle Scholar
  3. Bilardi G, Ferrante A (2007) The role of terminal cost/reward in finite-horizon discrete-time LQ optimal control. Linear Algebra Appl (Spec Issue honor Paul Fuhrmann) 425:323–344MathSciNetCrossRefGoogle Scholar
  4. Ferrante A (2004) On the structure of the solution of discrete-time algebraic Riccati equation with singular closed-loop matrix. IEEE Trans Autom Control AC-49(11):2049–2054MathSciNetCrossRefGoogle Scholar
  5. Ferrante A, Levy B (1998) Canonical form for symplectic matrix pencils. Linear Algebra Appl 274:259–300MathSciNetCrossRefGoogle Scholar
  6. Ferrante A, Ntogramatzidis L (2005) Employing the algebraic Riccati equation for a parametrization of the solutions of the finite-horizon LQ problem: the discrete-time case. Syst Control Lett 54(7):693–703MathSciNetCrossRefGoogle Scholar
  7. Ferrante A, Ntogramatzidis L (2007a) A unified approach to the finite-horizon linear quadratic optimal control problem. Eur J Control 13(5):473–488MathSciNetCrossRefGoogle Scholar
  8. Ferrante A, Ntogramatzidis L (2007b) A unified approach to finite-horizon generalized LQ optimal control problems for discrete-time systems. Linear Algebra Appl (Spec Issue honor Paul Fuhrmann) 425(2–3):242–260MathSciNetCrossRefGoogle Scholar
  9. Ferrante A, Ntogramatzidis L (2012) Comments on “structural invariant subspaces of singular Hamiltonian systems and nonrecursive solutions of finite-horizon optimal control problems”. IEEE Trans Autom Control 57(1):270–272CrossRefGoogle Scholar
  10. Ferrante A, Ntogramatzidis L (2013a) The extended symplectic pencil and the finite-horizon LQ problem with two-sided boundary conditions. IEEE Trans Autom Control 58(8):2102–2107MathSciNetCrossRefGoogle Scholar
  11. Ferrante A, Ntogramatzidis L (2013b) The role of the generalised continuous algebraic Riccati equation in impulse-free continuous-time singular LQ optimal control. In: Proceedings of the 52nd conference on decision and control (CDC 13), Florence, 10–13 Dec 2013bGoogle Scholar
  12. Ferrante A, Marro G, Ntogramatzidis L (2005) A parametrization of the solutions of the finite-horizon LQ problem with general cost and boundary conditions. Automatica 41:1359–1366MathSciNetCrossRefGoogle Scholar
  13. Kalman RE (1960) Contributions to the theory of optimal control. Bulletin de la Sociedad Matematica Mexicana 5:102–119MathSciNetGoogle Scholar
  14. Kwakernaak H, Sivan R (1972) Linear optimal control systems. Wiley, New YorkzbMATHGoogle Scholar
  15. Ntogramatzidis L, Ferrante A (2010) On the solution of the Riccati differential equation arising from the LQ optimal control problem. Syst Control Lett 59(2): 114–121MathSciNetCrossRefGoogle Scholar
  16. Ntogramatzidis L, Ferrante A (2013) The generalised discrete algebraic Riccati equation in linear-quadratic optimal control. Automatica 49:471–478. https://doi.org/10.1016/j.automatica.2012.11.006 MathSciNetCrossRefGoogle Scholar
  17. Ntogramatzidis L, Marro G (2005) A parametrization of the solutions of the Hamiltonian system for stabilizable pairs. Int J Control 78(7):530–533MathSciNetCrossRefGoogle Scholar
  18. Prattichizzo D, Ntogramatzidis L, Marro G (2008) A new approach to the cheap LQ regulator exploiting the geometric properties of the Hamiltonian system. Automatica 44:2834–2839MathSciNetCrossRefGoogle Scholar
  19. Zattoni E (2008) Structural invariant subspaces of singular Hamiltonian systems and nonrecursive solutions of finite-horizon optimal control problems. IEEE Trans Autom Control AC-53(5):1279–1284MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag London Ltd., part of Springer Nature 2020

Authors and Affiliations

  1. 1.Dipartimento di Ingegneria dell’InformazioneUniversità di PadovaPadovaItaly
  2. 2.Department of Mathematics and StatisticsCurtin UniversityPerthAustralia

Section editors and affiliations

  • Michael Cantoni
    • 1
  1. 1.Department of Electrical & Electronic EngineeringThe University of MelbourneParkvilleAustralia