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A shooting algorithm for optimal control problems with singular Arcs. (English) Zbl 1275.49045

The authors investigate a shooting algorithm for the optimal control problem \(\varphi_0(x_0,x_T) \to \min\), \(\dot{x}_t=\sum_{i=0}^m u_{i,t} f_i(x_t)\) a.e. on \([0,T]\), \(\eta_j(x_0,x_T)=0\), \(j=1,\dots,d\), where the final time \(T\) is fixed, \(f_i: R^n \to R^n\) for \(i=0, \dots,m\) and \(\eta_j:R^{2n} \to R\) for \(j=1,\dots,d\). The functions \(\varphi_0\), \(f_i\), and \(\eta_j\) have Lipschitz-continuous second derivatives. In general, the shooting system has more equations than unknowns, and the Gauss-Newton method is used to compute a zero of the shooting system. 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 the paper asserts that the latter holds whenever a sufficient condition for weak optimality is satisfied. Numerical tests that validate the proposed method are included.

MSC:

49M15 Newton-type methods
49K40 Sensitivity, stability, well-posedness
49K30 Optimality conditions for solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)

Software:

minpack; Cotcot; Bocop; Shoot
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References:

[1] Goodman, T.R.; Lance, G.N., The numerical integration of two-point boundary value problems, Math. Tables Other Aids Comput., 10, 82-86, (1956) · Zbl 0071.34006
[2] Morrison, D.D.; Riley, J.D.; Zancanaro, J.F., Multiple shooting method for two-point boundary value problems, Commun. ACM, 5, 613-614, (1962) · Zbl 0106.31903
[3] Keller, H.B.: Numerical Methods for Two-Point Boundary-Value Problems. Ginn-Blaisdell, Waltham (1968) · Zbl 0172.19503
[4] Bulirsch, R.: Die Mehrzielmethode zur numerischen Lösung von nichtlinearen Randwertproblemen und Aufgaben der optimalen Steuerung. Technical report, Carl-Cranz-Gesellschaft, Deutsches Zentrum für Luft- und Raumfahrt (DLR), Oberpfaffenhofen, Germany (1971)
[5] Maurer, H., Numerical solution of singular control problems using multiple shooting techniques, J. Optim. Theory Appl., 18, 235-257, (1976) · Zbl 0302.65063
[6] Oberle, H.J.: Numerische Behandlung singulärer Steuerungen mit der Mehrzielmethode am Beispiel der Klimatisierung von Sonnenhäusern. Ph.D. thesis, Technische Universität München (1977) · Zbl 0396.49021
[7] Oberle, H.J., Numerical computation of singular control problems with application to optimal heating and cooling by solar energy, Appl. Math. Optim., 5, 297-314, (1979) · Zbl 0428.49007
[8] Fraser-Andrews, G., Finding candidate singular optimal controls: a state-of-the-art survey, J. Optim. Theory Appl., 60, 173-190, (1989) · Zbl 0633.49015
[9] Martinon, P.: Numerical resolution of optimal control problems by a piecewise linear continuation method. Ph.D. thesis, Institut National Polytechnique de Toulouse (2005). Online: http://www.cmap.polytechnique.fr/ martinon/docs/Martinon-Thesis.pdf
[10] Vossen, G., Switching time optimization for bang-bang and singular controls, J. Optim. Theory Appl., 144, 409-429, (2010) · Zbl 1185.49022
[11] Aronna, M.S.: Singular solutions in optimal control: second order conditions and a shooting algorithm. Research report Nr. 7764, INRIA (2011)
[12] Bonnard, B., Kupka, I.: Théorie des singularités de l’application entrée/sortie et optimalité des trajectoires singulières dans le problème du temps minimal. Forum Math. 5(2), 111-159 (1993) · Zbl 0779.49025
[13] Bonnard, B.; Caillau, J.B.; Trélat, E., Second order optimality conditions in the smooth case and applications in optimal control, ESAIM Control Optim. Calc. Var., 13, 207-236, (2007) · Zbl 1123.49014
[14] Bonnard, B., Chyba, M.: Singular Trajectories and Their Role in Control Theory. Mathematics & Applications, vol. 40. Springer, Berlin (2003) · Zbl 1022.93003
[15] Malanowski, K.; Maurer, H., Sensitivity analysis for parametric control problems with control-state constraints, Comput. Optim. Appl., 5, 253-283, (1996) · Zbl 0864.49020
[16] Bonnans, J.F.; Hermant, A., Second-order analysis for optimal control problems with pure state constraints and mixed control-state constraints. annals of I.H.P, Nonlinear Anal., 26, 561-598, (2009) · Zbl 1158.49023
[17] Dennis, J.E.; Jacobs, D. (ed.), Nonlinear least-squares, 269-312, (1977), London
[18] Fletcher, R., Practical methods of optimization, No. 1, (1980), Chichester · Zbl 0439.93001
[19] Dennis, J.E.; Gay, D.M.; Welsch, R.E., An adaptive nonlinear least-squares algorithm, ACM Trans. Math. Softw., 7, 348-368, (1981) · Zbl 0464.65040
[20] Dmitruk, A.V., Quadratic weak minimum conditions for special situations in optimal control problems, Dokl. Akad. Nauk SSSR, 233, 523-526, (1977)
[21] Dmitruk, A.V., Quadratic order conditions for a Pontryagin minimum in an optimal control problem linear in the control, Math. USSR, Izv., 28, 275-303, (1987) · Zbl 0682.49020
[22] Felgenhauer, U., Structural stability investigation of bang-singular-bang optimal controls, J. Optim. Theory Appl., 152, 605-631, (2012) · Zbl 1237.49033
[23] Felgenhauer, U.: Controllability and stability for problems with bang-singular-bang optimal control (2011, submitted)
[24] Pontryagin, L., Boltyanski, V., Gamkrelidze, R., Michtchenko, E.: The Mathematical Theory of Optimal Processes. Wiley-Interscience, New York (1962) · Zbl 0102.32001
[25] Kelley, H.J., A second variation test for singular extremals, AIAA J., 2, 1380-1382, (1964) · Zbl 0136.10101
[26] Goh, B.S., The second variation for the singular Bolza problem, SIAM J. Control, 4, 309-325, (1966) · Zbl 0146.11906
[27] Goh, B.S., Necessary conditions for singular extremals involving multiple control variables, SIAM J. Control, 4, 716-731, (1966) · Zbl 0161.29004
[28] Goh, B.S.: Necessary conditions for the singular extremals in the calculus of variations. Ph.D. thesis, University of Canterbury, New Zealand (1966)
[29] Kelley, H.J.; Kopp, R.E.; Moyer, H.G., Singular extremals, 63-101, (1967), New York
[30] Robbins, H.M., A generalized Legendre-Clebsch condition for the singular case of optimal control, IBM J. Res. Dev., 11, 361-372, (1967) · Zbl 0153.41202
[31] Bonnans, J.F.: Optimisation Continue. Dunod (2006)
[32] Levitin, E.S.; Milyutin, A.A.; Osmolovskiĭ, N.P., Higher order conditions for local minima in problems with constraints, Usp. Mat. Nauk, 33, 85-148, (1978) · Zbl 0409.49016
[33] Aronna, M.S.; Bonnans, J.F.; Dmitruk, A.V.; Lotito, P.A., Quadratic order conditions for bang-singular extremals, Numer. Algebra Control Optim., 2, 511-546, (2012) · Zbl 1252.49028
[34] Bell, D.J., Jacobson, D.H.: Singular Optimal Control Problems. Academic Press, New York (1975) · Zbl 0338.49006
[35] Zeidan, V., Sufficiency criteria via focal points and via coupled points, SIAM J. Control Optim., 30, 82-98, (1992) · Zbl 0780.49018
[36] Bonnard, B., Caillau, J.-B., Trélat, E.: Cotcot: Short reference manual. Technical report RT/APO/05/1, ENSEEIHT-IRIT, (2005) · Zbl 1185.49022
[37] Fuller, A.T., Study of an optimum non-linear control system, J. Electron. Control, 15, 63-71, (1963)
[38] Betts, J.T., Survey of numerical methods for trajectory optimization, AIAA J. Guid. Control Dyn., 21, 193-207, (1998) · Zbl 1158.49303
[39] Biegler, L.T., Nonlinear programming: concepts, algorithms, and applications to chemical processes, (2010), Philadelphia · Zbl 1207.90004
[40] Gergaud, J.; Martinon, P.; Seeger, A. (ed.), An application of PL continuation methods to singular arcs problems, No. 563, 163-186, (2006), Berlin · Zbl 1108.49026
[41] Bonnans, J.F.; Martinon, P.; Trélat, E., Singular arcs in the generalized goddard’s problem, J. Optim. Theory Appl., 139, 439-461, (2008) · Zbl 1159.49027
[42] Clark, C.W.: Mathematical Bioeconomics. Wiley, New York (1976) · Zbl 0364.90002
[43] Aly, G.M., The computation of optimal singular control, Int. J. Control, 28, 681-688, (1978) · Zbl 0396.49021
[44] Goddard, R.H., A method of reaching extreme altitudes, No. 71, (1919), Washington
[45] Seywald, H.; Cliff, E.M., Goddard problem in presence of a dynamic pressure limit, J. Guid. Control Dyn., 16, 776-781, (1993) · Zbl 0779.70020
[46] Martinon, P., Gergaud, J.: Shoot2.0: An indirect grid shooting package for optimal control problems, with switching handling and embedded continuation. Research report Nr. 7380, INRIA, (2011)
[47] Garbow, B.S., Hillstrom, K.E., More, J.J.: User Guide for Minpack-1. National Argonne Laboratory, Illinois (1980)
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