Singular Perturbations in Systems and Control pp 93-101 | Cite as

# On Nonlinear Optimal Control Problems

## Abstract

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)).

## Keywords

Optimal Control Problem Singular Perturbation Rensselaer Polytechnic Institute Singular Perturbation Theory Efficient Numerical Method## Preview

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## References

- 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
- 2.N. Fenichel, “Geometric singular perturbation theory for ordinary differential equations,” J. Differential Equations 31 (1979), 53–98.ADSCrossRefMATHMathSciNetGoogle 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
- 4.M. I. Freedman and B. Granoff, “Formal asymptotic solution of a singularly perturbed nonlinear optimal control problem,” J. Optim. Theory Appl. 19 (1976), 301–325.CrossRefMATHMathSciNetGoogle Scholar
- 5.M. I. Freedman and J. L. Kaplan, “Singular perturbations of two-point boundary value problems arising in optimal control,” SIAM J. Control Optim. 14 (1976), 189–215.CrossRefMATHMathSciNetGoogle Scholar
- 6.M. I. Freedman and J. L. Kaplan, “Perturbation analysis of an optimal control problem involving bang-bang controls,” J. Differential Equations 25 (1977), 11–29.ADSCrossRefMATHMathSciNetGoogle Scholar
- 7.V. Ja. Glizer and M. G. Dmitriev, “Singular perturbations in a linear optimal control problem with quadratic functional,” Soviet Math Dokl. 16 (1975), 1555–1558.MATHGoogle Scholar
- 8.V. Ja. Glizer and M. G. Dmitriev, “Singular perturbations and generalized functions,” Soviet Math. Dokl. 20 (1979), 1360–1364.MATHGoogle Scholar
- 9.C. R. Hadlock, “Existence and dependence on a parameter of solutions of a nonlinear two point boundary value problem,” J. Differential Equations 14 (1973), 498–517.ADSCrossRefMATHMathSciNetGoogle 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
- 11.F. Hoppensteadt, “Properties of solutions of ordinary differential equations with a small parameter,” Comm. Pure Appl. Math. 24 (1971), 807–840.CrossRefMATHMathSciNetGoogle Scholar
- 12.F. A. Howes, “Boundary-interior layer interactions in nonlinear singular perturbation theory,” Memoirs Amer. Math. Society 203, 1978.Google Scholar
- 13.: P. V. Kokotovic and P. Sannuti, “Singular perturbation method for reducing the model order in optimal control design,” IEEE Trans. Automatic Control 13 (1968), 377–384.CrossRefGoogle 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
- 15.J. J. Levin and N. Levinson, “Singular perturbations of nonlinear systems and an associated boundary layer equation,” J. Rational Mech. Anal. 3 (1954), 247–270.MATHMathSciNetGoogle 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
- 17.R. E. O’Malley, Jr., “Singular perturbation of a boundary value problem for a system of nonlinear differential equations,” J. Differential Equations 8 (1970), 431–447.ADSCrossRefMATHMathSciNetGoogle Scholar
- 18.R. E. O’Malley, Jr., Introduction to Singular Perturbations, Academic Press, New York, 1974.MATHGoogle 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
- 22.P. Sannuti, “Asymptotic expansions of singularly perturbed quasi-linear optimal systems,” SIAM J. Control 13 (1975), 572–592.CrossRefMATHMathSciNetGoogle Scholar
- 23.P. Sannuti and P. Kokotovic, “Singular perturbation method for near optimal design of high-order non-linear systems,” Automatica 5 (1969), 773–779.CrossRefMATHMathSciNetGoogle 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