On Nonlinear Optimal Control Problems
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)).
KeywordsOptimal Control Problem Singular Perturbation Rensselaer Polytechnic Institute Singular Perturbation Theory Efficient Numerical Method
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- 1.U. Ascher and R. Weiss, “Collocation for singular perturbation problems II: Linear first order systems without turning points,” Technical Report 82–4, Department of Computer Science, University of British Columbia, Vancouver, 1982.Google Scholar
- 3.J. E. Flaherty and R. E. O’Malley, Jr., “Asymptotic and numerical methods for vector systems of singularly perturbed boundary value problems,” Proceedings, Army Numerical Analysis and Computer Conference, Vicksburg, 1982.Google Scholar
- 10.W. A. Harris, Jr., “Singularly perturbed boundary value problems revisited,” Lecture Notes in Math. 312, Springer-Verlag, Berlin, 1973, 54–64.Google Scholar
- 12.F. A. Howes, “Boundary-interior layer interactions in nonlinear singular perturbation theory,” Memoirs Amer. Math. Society 203, 1978.Google Scholar
- 14.H. L. Kurland,. “Solutions to boundary value problems of fast-slow systems by continuing homology in the Morse index along a path of isolated invariant sets of the fast systems,” preprint, Boston University, 1981.Google Scholar
- 16.H. D. McIntyre, “The formal asymptotic solution of a nonlinear singularly perturbed state regulator,” pre-thesis monograph, University of Arizona, 1977.Google Scholar
- 19.R. E. O’Malley, Jr., “Singular perturbations and optimal control,” Lecture Notes in Math. 680, Springer-Verlag, Berlin, 1978, 170–218.Google Scholar
- 20.R. E. O’Malley, Jr., “On multiple solutions of singularly perturbed systems in the conditionally stable case,” Singular Perturbations and Asymptotics, R. E. Meyer and S. V. Parter, editors, Academic Press, New York, 1980, 87–108.Google Scholar
- 21.R. E. O’Malley, Jr., “Slow/fast decoupling — Analytical and numerical aspects,” preprint, Rensselaer Polytechnic Institute, 1982.Google Scholar
- 24.S. S. Sastry, C. A. Desoer and P. P. Varaiya, “Jump behavior of circuits and systems,” Memorandum UCB/ERL M80/44, University of California, Berkeley, 1980.Google Scholar
- 25.R. Weiss, “An analysis of the box and trapezoidal schemes for linear singularly perturbed boundary value problems,” preprint, Technische Universität Wein, 1982.Google Scholar