Professor Stephen P Banks

Professor S P BanksAddress:
Professor S P Banks,
Department of Automatic Control and Systems Engineering
University of Sheffield
S1 3JD
Tel: (+44) (0)114 222 5137
Fax: (+44) (0)114 222 5683
Room: B16, Amy Johnson Building

Research Interests
Research Grants
Invited Lectures
Professional Activities
Selected Publications


Professor S P Banks has been Professor of Mathematical Control Theory since 1992 and has contributed to many areas of mathematical systems theory, including the solution of Lyapunov's problem in stability theory, the introduction of global nonlinear optimal control theories, knots, links and chaos in nonlinear systems and nonlinear partial differential equations.

He is an Editor of an Institute of Mathematics and Its Applications International Journal, is an Associate Editor of several other journals and regularly contributes to international activities through conference organisation, plenary lectures and invited talks.

He has been an external examiner for MSc in Control Systems, Imperial College (1990-1993) and for numerous PhDs throughout the world and has served on Habilitation juries in France and Poland.

Back to top

Research Interests

Nonlinear control theory, optimisation, global theory of dynamical systems, knots, chaos and cellular automata.

His main contributions are:

  • Determination of general Lyapunov functions for nonlinear systems (Lyapunov's problem)
  • The introduction of a global technique for solving nonlinear problems by reducing them to sequences of linear, time-varying ones.
  • The proof of the existence of higher order chaotic systems by the use of Lie theory.
  • The theory of global nonlinear dynamical systems on hyperbolic manifolds by the use of automorphic forms.

He has published over 200 refereed papers and 5 books including:

  • 'State-Space and Frequency Domain Methods in the Control of Distributed Parameter Systems', (Peter Peregrinus,1983)
  • 'Control Systems Engineering: Modelling and Simulation, Control Theory and Microprocessor Implementation', (Prentice-Hall ,1986)
  • 'Mathematical Theories of Nonlinear Systems', (Prentice-Hall, 1988)
  • 'Signal Processing, Image Processing and Pattern Recognition', (Prentice-Hall, 1990)

Back to top

Research Grants include:

SERC Award for 'Stategic Planning and Operations Control for Compex Mixed-Mode Systems', (with D.A.Linkens, M.J.Holcombe, S.Bennett, P.R.Croll, P.J.Fleming and N.Mort),1992-1995, £170,000
SERC Award for 'Frequency Domain Theory of Nonlinear Systems' (Principal Investigator, with D.J.Bell, UMIST), 1992-1995, £90,900
SERC Award for 'Optimal Control of Nonlinear Systems' (Principal Investigator, with D.J.Bell, UMIST, Prof. J.B.Edwards), 1994-1997, £127,518
SERC Award for 'Twin Satellite Studies of Waves and Turbulence in Space Plasmas', (with H.Alleyne), 1998-2001, £119,056
Royal Society for invited visit to Estonian Academy of Sciences, £1200
Royal Society for invited visit to Texas, USA, £1100

Back to top

Professional Activities

  • (1990-Present) Editor of IMA Journal of Information and Control.
  • (1989-Present) Member of Editorial Board of Journal of Computational and Applied Mathematics.
  • (1988-1995) Member of Editorial Board of IASTED International Journal of Modelling and Simulation.
  • (2003-Present) Member of Editorial Board of AGH University of Science and Tech Quarterly 'Mechankia'.
  • (2005-Present) Member of Honorary Advisory board of International Journal of Innovative Computing, Information and Control.
  • (2006-Present) Member of Editorial Board of Journal of Industrial and Management.

Back to top

Invited Lectures

  • 'Lie Algebras, Modal Control and Nonlinear Systems', Univ. of Mining and Minerals, Krakov, April, 1999.
  • 'Nonlinear Dynamical Systems, Lie Algebras and Knots', IEEE/CSS special seminar in METU and Bilkent Universities, Turkey, funded by TUBITAK-NATO, 2001.
  • 'Global nonlinear control, the HJB equation and the Maslov index', Dept. Electrical Eng., Melbourne University, April, 2002.
  • 'Global approximation methods in nonlinear systems', Dept. Electrical Eng., Melbourne University, April, 2002.
  • 'Nonlinear systems, Linear Approximations and Control', Dept. of Mathematics and Engineering, Anatal University, Eskisihir, Turkey, Sept. 2002.
  • 'Hamilton-Jacobi Equations, Viscosity Solutions and Optimal Control', METU (Middle East Technical University), Ankara, Turkey, Sept, 2002.
  • 'Robust Control of Uncertain Nonlinear Dynamical Systems via Linear Time-Varying Approximations', (WCNA), Orlando, Florida, July 7, 2004.
  • 'Dynamical systems on three manifolds', Ostrava, Czech Republic, May, 2006.

Back to top

Selected Publications since 2001

  Details of Publication Brief description of contribution
1 Banks, SP: `Exact boundary controllability and optimal control for a generalised Korteweg-de Vries equation´, J. Nonlinear Anal. – Apps and Methods, 47, 5537-5546, 2001. New technique for controlling nonlinear pdes. It applies the technique developed by the author for general nonlinear systems which is now being used in many places throughout the world. This method is applied in the papers 3 and 5. In this paper it is shown that one can control solitons (large nonlinear water waves, for example) from the boundary.
2 Banks, SP.: `Three-dimensional stratifications, knots and bifurcations of two-dimensional dynamical systems´,Int. J. Bifurcation & Chaos, 12, No.1, 1-21, 2002. Proof of the existence of topologically distinct knots in dynamical systems on surfaces. This has applications to global periodic behaviour in nonlinear communication systems, control of chaos, etc.
3 McCaffrey, D and Banks, SP: `Lagrangian manifolds, viscosity solutions and the Maslov index´, J. Convex Analysis, 9, 185-224, 2002. Shows deep theoretical connections between the topology of Lagrangian manifolds and the continuity of the Hamilton-Jacobi equation for optimisation. It explains the difficulty in obtaining global optimal solutions to general nonlinear optimisation problems in terms of global crossing of characteristics.
4 Cimen, T and Banks, S.P.: `Nonlinear Optimal Tracking Control with Application to Super-Tankers for Autopilot Design´. Automatica, 40, no. 11, 1845-1863, 2004. Fundamental results on optimal control of nonlinear systems with application to ship positioning. This paper uses the method of paper1 which requires very mild conditions (unlike all other techniques) and so is applicable to virtually all nonlinear problems.
5 Banks, S.P. and Diaz, D.: `Almost Radially-Invariant Systems Containing Arbitrary Knots and Links´, Int. J. Bifurcation & Chaos., 14, 1267-1276, 2004. Construction of nonlinear dynamical systems containing any knot and link. This proves for the first time that not only chaotic systems can contain knots and links of any type and shows that `regular´ systems on compact manifolds can exhibit `strange´ behaviour.
6 Cimen, T and Banks, S.P.: `Global Optimal Feedback Control for General Nonlinear Systems with Nonquadratic Performance Criteria´. Sys. Cont. Letts., 53, No. 5, 327-346, 2004. Fundamental global results on nonlinear control and viscosity solutions of HJB equations. This paper gives a theoretical justification of the work in papers 1 and 3 and shows when an optimal solution can be found. It also indicates that the method can be developed into a comprehensive technique for tackling many global problems in nonlinear systems, however complex.
7 Zheng, J ,Banks, S.P. and Alleyne, H: `Optimal Attitude Control for Three-Axis Stabilized Flexible Spacecraft´, Acta Astronautica, 56, 519-528, 2005. The global (as opposed to local, small angle) control for rotating spacecraft.
8 McCaffrey, D. and Banks, SP: `Geometric Existence Theory for the Control-Affine Nonlinear Optimal Regulator´, J. Math. Anal. App., 305, Issue 1, 1 May 2005, 380-390. Global theory of optimal control. Shows how the geometric properties of the space interact with the analytic ones.

Back to top

Other publications:

Banks, S.P. and Song, Y. `Elliptic and Automorphic Dynamical Systems on Surfaces´, to appear in Int. J. Bifur. Chaos, 2005.

Banks, S.P., Gurkan, E and Erkmen, I. `Stable Controller Design for T-S Fuzzy Systems based on Lie Algebras´, Fuzzy Sets and Systems, 156, 226-249, 2005.

Banks, S.P. and Navarro-Hernandez, C. `A New Proof of McCann´s Theorem and the Generalization of Lyapunov´s Equation to Nonlinear Systems´, Int. J. Inn. Comp., Inf. and Control, 1, 1-17, 2005.

Fahmy, S.F.F, and Banks, S.P.. `Robust Control of Uncertain Nonlinear Dynamical Systems via Linear Time-Varying Approximations´, Nonlinear Analysis, 2005.

Tomas-Rodriguez, M, Banks, S.P., Hartonen, J and Alleyne,H. `A New Method of Parameter Estimation in Nonlinear Systems and Application to Plasmas´. Amer. J. Phys – Phys of Plasmas, 11, No. 4, 1300-1307, 2004.

Yu, W, Palmiere, E.J., Banks, S.P and Han, J. `Cellular automata modelling of austenite grain coarsening during reheating  I. Normal grain coarsening, J. Univ. Sci. Tech. Beijing, 11, No. 6, 517-523, 2004.

Yu, W, Palmiere, E.J., Banks, S.P and Han, J. `Cellular automata modelling of austenite grain coarsening during reheating  II. Abnormal grain growth, J. Univ. Sci. Tech. Beijing, 12, No. 1, 26-32, 2004.

Banks, SP: `The Lie Algebra of a Dynamical System and its Application to Control´, Int. J. Sys. Sci., 32, 220-238, 2001.

McCaffrey, D and Banks, SP: `Lagrangian manifolds and asymptotically optimal stabilising feedback control´, Sys. Cont. Letts., 43, 219-224, 2001.

Banks, S.P. and Tomas-Rodriguez, M: `Frequency Analysis of Nonlinear Dynamical Systems and the Control of Vibration´, Arch. Control Sci., 13, 115-125, 2003.

Navarro Hernandez. C., Banks, S.P. and Aldeen, M.: `Observer Design for Nonlinear Systems using Linear Approximations´, IMA J. Math. Cont & Inf., 20, 359-370, 2003.

Lu, X-Y and Banks, S.P.: `Some New Results in Differential Algebraic Control Theory´, IMA J. Math. Cont & Inf., 20, 277-304, 2003.

Banks, SP.: `Nonlinear delay systems, Lie algebras and Lyapunov transformations´, IMA J. Math. Cont & Inf., 19, 59-72, 2002.

Tomas-Rodriguez, M. and Banks, S.P.: `Linear Approximations to Nonlinear Dynamical Systems with Applications to Stability and Spectral Theory´, IMA J. Math. Cont & Inf., 20, 89-103, 2003.

Song, Y, Banks, S.P. and Diaz,D.. `Dynamical Systems on Three Manifolds. Part 1:Knots, Links and Chaos´, to appear in Int. J. Bifur. Chaos, 2007.

Song, Y and Banks, S.P.. `Dynamical Systems on Three Manifolds. Part 2: 3-Manifolds and Heegaard Splittings´, to appear in Int. J. Bifur. Chaos, 2007

Back to top