Abstract
Direct solutions of the optimal control problem are considered. Two discretization schemes are proposed which are based on the parameterization of the control functions and on the parameterization of the control and the state functions, leading to direct shooting and direct collocation algorithms, respectively. The former is advantageous for problems with unspecified final state, the latter for prescribed final state and especially for stiff problems. The sparsity of the Jacobian matrix of the constraints and the Hessian matrix of the Lagrangian must be exploited in the direct collocation method in order to be efficient. The great advantage of the collocation approach lies in the availability of analytical gradients.
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References
Bertsekas, D.P., Constrained Optimization and Lagrange Multiplier Methods, Academic Press, New York, 1982.
Hestenes, M.R., Calculus of Variations and Optimal Control Theory, Wiley, New York, 1966.
Cesari, L., Optimization — Theory and Applications, Springer, New York, 1983.
Childs, B., Scott, M., Daniel, J.W., Denman, E., Nelson, P. (eds.), Codes for Boundary-Value Problems in Ordinary Differential Equations, Springer, Berlin, 1979.
Bryson, A.E., Ho, Y.C., Applied Optimal Control, Ginn & Company, Waltham, 1969.
Canon, M.D., Cullum, CD., Polak, E., Theory of Optimal Control and Mathematical Programming, McGraw-Hill, New York, 1970.
Tabak, D., Kuo, B.C., Optimal Control by Mathematical Programming, Prentice-Hall, Englewood Cliffs, 1971.
Brusch, R.G., Schappelle, R.H., “Solution of Highly Constrained Optimal Control Problems Using Nonlinear Programming”, AIAA Journal 11 (1973) 135-136.
Kamm, J.L., Johnson, I.L., “Nonlinear Programming Approach for Optimizing Two-Stage Lifting Vehicle Ascent to Orbit”, Automatica 9 (1973) 713-720.
Neuman, C.P., Sen, A., “A Suboptimal Control Algorithm for Constrained Problems Using Cubic Splines”, Automatica 9 (1973) 601-613.
Gill, P.E., Murray, W., “The Numerical Solution of a Problem in the Calculus of Variations”, in: Bell, D.J. (ed.), Recent Mathematical Developments in Control, Academic Press, London, 1973.
Rader, J.E., Hull, D.G., “Computation of Optimal Aircraft Trajectories Using Parameter Optimization Methods”, J. Aircraft 12 (1975) 864-866.
Hager, W.W., “Rates of Convergence for Discrete Approximations to Unconstrained Control Problems”, SIAM J. Numer. Anal. 13 (1976) 449-472.
Kraft, D., “Optimierung von Flugbahnen mit Zustandsbeschränkungen durch Mathematische Programmierung”, DGLR Jahrbuch, München, 1976.
Mantell, J.B., Lasdon, L.S., “A GRG Algorithm for Econometric Control Problems”, Annals of Economic and Social Measurement 6 (1978) 581-597.
Sargent, R.W.H., Sullivan, G.R., “The Development of an Efficient Optimal Control Package”, in: Stoer, J. (ed.), Optimization Techniques, Springer, Berlin, 1978.
Jonson, H., “A Newton Method for Solving Non-Linear Optimal Control Problems with General Constraints”, Linköping Studies in Science and Technologie 104, Linköping, 1983.
Di Pillo, G., Grippo, L., Lampariello, F., “A Newton-Type Computing Technique for Optimal Control Problems”, Optimal Control Applications & Methods 5 (1984) 149-166.
Powell, M.J.D., “Variable Metric Methods for Constrained Optimization”, in: Bachern, A., Grötschel, M., Korte, B. (eds.), Mathematical Programming — The State of the Art, Springer, Berlin, 1983.
Schittkowski, K., Nonlinear Programming Codes, Springer, Berlin, 1980.
Stoer, J., “Foundation of Recursive Quadratic Programming Methods for Solving Nonlinear Programs”, this volume.
Neustadt, L.W., Optimization, Princeton University Press, Princeton, 1976.
Ioffe, A.D., Tichomirov, V.M., Theorie der Extremalenaufgaben, Deutscher Verlag der Wissenschaften, Berlin, 1979. Also: Theory of Extremal Problems, North Holland, Amsterdam, 1979.
Kraft, D., “Finite-Difference Gradients versus Error-Quadrature Gradients in the Solution of Parameterized Optimal Control Problems”, Optimal Control Applications & Methods 2 (1981) 191-199.
Kraft, D., “Optimal Control of a Cryogenic Windtunnel”, to appear as DFVLR Research Report, 1985.
de Boor, C., A Practical Guide to Splines, Springer, New York, 1978.
Powell, M.J.D., Approximation Theory and Methods, Cambridge University Press, Cambridge, 1981.
Hall, C.A., Meyer, W.W., “Optimal Error Bounds for Cubic Spline Interpolation”, J. Approx. Theory 16 (1976) 105-122.
Shampine, L.F., Gordon, M.K., Computer Solution of Ordinary Differential Equations, W. H. Freeman & Company, San Francisco, 1975.
Shampine, L.F., Watts, H.A., “Practical Solution of Ordinary Differential Equations by Runge-Kutta Methods”, Sandia Laboratories, Report SAND 76-0585, Albuquerque, 1976.
Gill, P.E., Murray, W., Saunders, M.A., Wright, M.H., “Computing Forward-Difference Intervals for Numerical Optimization”, SIAM J. Sci. Stat. Comput. 4 (1983) 310-321.
Kraft, D., “Comparing Mathematical Programming Algorithms Based on Lagrangian Functions for Solving Optimal Control Problems”, in: Rauch, H.E. (ed.), Control Applications of Nonlinear Programming, Pergamon Press, New York, 1980.
Maier, M., “Die numerische Lösung von Halbleitermodellen mit Hilfe des Kollokationsverfahrens PITOHP unter Verwendung einer automatischen Schrittweitenkontrolle”, Thesis, TU München, 1979.
Dickmanns, E.D., Well, K.H., “Approximate Solution of Optimal Control Problems Using Third Order Hermite Polynomial Functions”, in: Marchuk, G.I. (ed.), Optimization Techniques, Springer, Berlin, 1975.
Russel, R.D., Christiansen, J., “Adaptive Mesh Selection Strategies for Solving Boundary Value Problems”, SIAM J. Numer. Anal. 15_ (1978) 59-80.
Fehlberg, E., “Klassische Runge-Kutta-Formeln fünfter und siebenter Ordnung mit Schrittweiten-Kontrolle”, Computing 4 (1969) 93-106.
Stoer, J., Bulirsch, R., Introduction to Numerical Analysis, Springer, New York, 1980.
Reinsch, C., Einführung in die numerische Mathematik, Mimeographed Lecture Notes, TU München, Abt. Mathematik, 3 1977.
Kraft, D., “FORTRAN-Programme zur numerischen Lösung optimaler Steuerungsprobleme”, DFVLR-Mitt. 80-03, Köln, 1980.
Powell, M.J.D., “A Fast Algorithm for Nonlinearly Constrained Optimization Calculations”, in: Watson, G. (ed.), Numerical Analysis, Springer, Berlin, 1978.
Schittkowski, K., “Powell with an Augmented Lagrangian Type Line Search Function”, Numer. Math. 38 (1981) 83–127.
Lawson, C.L., Hanson, R.J., Solving Least Squares Problems, Prentice Hall, Englewood Cliffs, 1974.
Gill, P.E., Murray, W., Saunders, M.A., Wright, M.H., “User’s Guide for SOL/QPSOL: A Fortran Package for Quadratic Programming”, TR SOL 83-7, Stanford University, Stanford, 1983.
Powell, M.J.D., “ZQPCVX A Fortran Subroutine for Convex Quadratic Programming”, DAMTP /1983/NA17, University of Cambridge, Cambridge, 1983.
Goldfarb, D., Idnani, A., “A Numerically Stable Dual Method for Solving Strictly Convex Quadratic Programs”, Math. Progr. 27 (1983) 1-33.
Fourer, R., “Staircase Matrices and Systems”, SIAM Review 26 (1984) 1-70.
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Kraft, D. (1985). On Converting Optimal Control Problems into Nonlinear Programming Problems. In: Schittkowski, K. (eds) Computational Mathematical Programming. NATO ASI Series, vol 15. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-82450-0_9
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DOI: https://doi.org/10.1007/978-3-642-82450-0_9
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