On Nonlinear Optimal Control Problems

  • Robert E. O’MalleyJr.
Part of the International Centre for Mechanical Sciences book series (CISM, volume 280)


Some of the first papers which consciously used a singular perturbations approach in optimal control were about nonlinear regulators (cf. Kokotovic and Sannuti (1968) and Sannuti and Kokotovic (1969)). They showed how such problems can be reduced to nonlinear two-point singularly perturbed boundary value problems for the states and costates, constrained by an optimality condition. Even without such constraints, however, a general theory for such nonlinear systems of ordinary differential equations is not available (cf. O’Malley (1980)). Limited success has been achieved, but largely for quasilinear problems (cf.,e.g.,Hadlock (1973), O’Malley (1974), Sannuti (1975), Freedman and Granoff (1976), and Freedman and Kaplan (1976) and (1977)). Efficient numerical methods are now becoming available for such boundary value problems (cf., e.g., Flaherty and O’Malley (1982), Weiss (1982), and Ascher and Weiss (1982)). For more nonlinear problems, interior shocks and transition layers can be expected in addition to endpoint boundary layers (cf. Howes (1978) for a treatment of second order scalar equations and O’Malley (1982) concerning some special vector systems). One must anticipate that such theories will become important in various control contexts. “In addition, the associated mathematics may become increasingly geometrical (cf. Lavin and Levinson (1954), Fenichel (1979), Sastry, Desoer, and Varaiya (1980), and Kurland (1981)). One would hope that the underlying Hamiltonian structure of control problems might ultimately allow a special development of the necessary asymptotic analysis. Our approach will largely follow McIntyre (1977) (see O’Malley (1978)).


Optimal Control Problem Singular Perturbation Rensselaer Polytechnic Institute Singular Perturbation Theory Efficient Numerical Method 
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Copyright information

© Springer-Verlag Wien 1983

Authors and Affiliations

  • Robert E. O’MalleyJr.
    • 1
  1. 1.Department of Mathematical SciencesRensselaer Polytechnic InstituteTroyUSA

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