A Shooting Algorithm for Optimal Control Problems with Singular Arcs

In this article, we propose a shooting algorithm for a class of optimal control problems for which all control variables appear linearly. The shooting system has, in the general case, more equations than unknowns and the Gauss–Newton method is used to compute a zero of the shooting function. This shooting algorithm is locally quadratically convergent, if the derivative of the shooting function is one-to-one at the solution. The main result of this paper is to show that the latter holds whenever a sufficient condition for weak optimality is satisfied. We note that this condition is very close to a second order necessary condition. For the case when the shooting system can be reduced to one having the same number of unknowns and equations (square system), we prove that the mentioned sufficient condition guarantees the stability of the optimal solution under small perturbations and the invertibility of the Jacobian matrix of the shooting function associated with the perturbed problem. We present numerical tests that validate our method.

[1]  T. Goodman,et al.  The numerical integration of two-point boundary value problems , 1956 .

[2]  Bernard Bonnard,et al.  Théorie des singularités de l'application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal , 1993 .

[3]  George Leitman,et al.  Topics in optimization , 1967 .

[4]  Pierre Martinon,et al.  An Application of PL Continuation Methods to Singular Arcs Problems , 2006 .

[5]  B.D.O. Anderson,et al.  Singular optimal control problems , 1975, Proceedings of the IEEE.

[6]  B. Goh.  Necessary Conditions for Singular Extremals Involving Multiple Control Variables , 1966 .

[7]  Emmanuel Trélat,et al.  Cotcot: short reference manual , 2005 .

[8]  G. Fraser-Andrews,et al.  Finding candidate singular optimal controls: A state of the art survey , 1989 .

[9]  Bean San Goh.  Necessary conditions for singular extremals in the calculus of variations , 1966 .

[10]  D. Jacobson,et al.  Computation of optimal singular controls , 1970 .

[11]  Andrei V. Dmitruk,et al.  Quadratic order conditions for bang-singular extremals , 2011, 1107.0161.

[12]  R. Snodgrass,et al.  SMITHSONIAN MISCELLANEOUS COLLECTIONS , 2003 .

[13]  L. S. Pontryagin,et al.  Mathematical Theory of Optimal Processes , 1962 .

[14]  H. Gardner Moyer,et al.  3 Singular Extremals , 1967 .

[15]  Joseph Frédéric Bonnans,et al.  Revisiting the analysis of optimal control problems with several state constraints , 2009 .

[16]  G. Aly.  The computation of optimal singular control , 1978 .

[17]  B. Goh.  The Second Variation for the Singular Bolza Problem , 1966 .

[18]  A. Fuller.  Study of an Optimum Non-linear Control System† , 1963 .

[19]  E. Trélat,et al.  Singular Arcs in the Generalized Goddard’s Problem , 2007, math/0703911.

[20]  R. Fletcher.  Practical Methods of Optimization , 1988 .

[21]  David G. Wilson,et al.  2. Constrained Optimization , 2005 .

[22]  H. Kelley.  A second variation test for singular extremals , 1964 .

[23]  Ursula Felgenhauer,et al.  Structural Stability Investigation of Bang-Singular-Bang Optimal Controls , 2012, J. Optim. Theory Appl..

[24]  Hans Joachim Oberle.  Numerical computation of singular control problems with application to optimal heating and cooling by solar energy , 1979 .

[25]  David A. H. Jacobs,et al.  The State of the Art in Numerical Analysis. , 1978 .

[26]  W. Marsden.  I and J , 2012 .

[27]  Eugene M. Cliff,et al.  GODDARD PROBLEM IN PRESENCE OF A DYNAMIC PRESSURE LIMIT , 1993 .

[28]  Helmut Maurer,et al.  Sensitivity analysis for parametric control problems with control-state constraints , 1996, Comput. Optim. Appl..

[29]  John E. Dennis,et al.  An Adaptive Nonlinear Least-Squares Algorithm , 1977, TOMS.

[30]  Peter Teunissen,et al.  Nonlinear least squares , 1990 .

[31]  R. Goddard.  A Method of Reaching Extreme Altitudes. , 1920, Nature.

[32]  M. Chyba,et al.  Singular Trajectories and Their Role in Control Theory , 2003, IEEE Transactions on Automatic Control.

[33]  Lorenz T. Biegler,et al.  Nonlinear Waves in Integrable and Nonintegrable Systems , 2018 .

[34]  M. L. Chambers.  The Mathematical Theory of Optimal Processes , 1965 .

[35]  M. S. Aronna,et al.  Partially affine control problems : second order conditions and a well-posed shooting algorithm , 2011 .

[36]  J. Frédéric Bonnans,et al.  Quadratic conditions for bang-singular extremals , 2011 .

[37]  G. Fairweather,et al.  Numerical Methods for Two – Point Boundary Value Problems , 2008 .

[38]  H. Robbins.  A generalized legendre-clebsch condition for the singular cases of optimal control , 1967 .

[39]  A. Dmitruk,et al.  QUADRATIC CONDITIONS FOR A PONTRYAGIN MINIMUM IN AN OPTIMUM CONTROL PROBLEM LINEAR IN THE CONTROL. I: A DECODING THEOREM , 1987 .

[40]  H. Maurer,et al.  Numerical solution of singular control problems using multiple shooting techniques , 1976 .

[41]  Vera Zeidan,et al.  Sufficiency criteria via focal points and via coupled points , 1992 .

[42]  Jorge J. Moré,et al.  User guide for MINPACK-1. [In FORTRAN] , 1980 .

[43]  M. Soledad Aronna,et al.  Singular solutions in optimal control: second order conditions and a shooting algorithm , 2011, 1210.7425.

[44]  A. Seeger.  Recent advances in optimization , 2006 .

[45]  Pierre Martinon,et al.  SHOOT2.0: An indirect grid shooting package for optimal control problems, with switching handling and embedded continuation , 2010 .

[46]  G. Vossen.  Switching Time Optimization for Bang-Bang and Singular Controls , 2010 .

[47]  Jean-Baptiste Caillau,et al.  Second order optimality conditions in the smooth case and applications in optimal control , 2007 .

[48]  J. Betts.  Survey of Numerical Methods for Trajectory Optimization , 1998 .

[49]  R. Kalaba,et al.  Nonlinear Least Squares , 1986 .

[50]  Jorge J. Moré,et al.  User Guide for Minpack-1 , 1980 .

[51]  J. Frédéric Bonnans,et al.  Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints , 2009 .

[52]  David D. Morrison,et al.  Multiple shooting method for two-point boundary value problems , 1962, CACM.