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Volumn 38, Issue 1, 2001, Pages 202-226

Second-order Runge-Kutta approximations in control constrained optimal control

Author keywords

Discretization; Numerical solution; Optimal control; Rate of convergence; Runge kutta scheme

Indexed keywords


EID: 0034453343     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142999351765     Document Type: Article
Times cited : (159)

References (53)
  • 3
    • 0015008071 scopus 로고
    • Error bounds of high order accuracy for the state regulator problem via piecewise polynomial approximations
    • W. E. BOSARGE, JR., AND O. G. JOHNSON, Error bounds of high order accuracy for the state regulator problem via piecewise polynomial approximations, SIAM J. Control, 9 (1971), pp. 15-28.
    • (1971) SIAM J. Control , vol.9 , pp. 15-28
    • Bosarge W.E., Jr.1    Johnson, O.G.2
  • 5
    • 0042467718 scopus 로고
    • Regular synthesis for the linear quadratic optimal control problem with, linear control constraints
    • P. BRUNOVSKÝ, Regular synthesis for the linear quadratic optimal control problem with, linear control constraints, J. Differential Equations, 38 (1980), pp. 344-360.
    • (1980) J. Differential Equations , vol.38 , pp. 344-360
    • Brunovský, P.1
  • 6
    • 0014469791 scopus 로고
    • Difference approximations in optimal control problems
    • B. M. BUDAK, E. M. BERKOVICH, AND E. N. SOLOV'EVA, Difference approximations in optimal control problems, SIAM J. Control, 7 (1969), pp. 18-31.
    • (1969) SIAM J. Control , vol.7 , pp. 18-31
    • Budak, B.M.1    Berkovich, E.M.2    Solov'eva, E.N.3
  • 8
    • 0040286883 scopus 로고
    • Discrete approximations to continuous optimal control problems
    • J. CULLUM, Discrete approximations to continuous optimal control problems, SIAM J. Control, 7 (1969), pp. 32-49.
    • (1969) SIAM J. Control , vol.7 , pp. 32-49
    • Cullum, J.1
  • 9
    • 0015097379 scopus 로고
    • An explicit procedure for discretizing continuous, optimal control problems
    • J. CULLUM, An explicit procedure for discretizing continuous, optimal control problems, J. Optim. Theory Appl., 8 (1971), pp. 15-34.
    • (1971) J. Optim. Theory Appl. , vol.8 , pp. 15-34
    • Cullum, J.1
  • 10
    • 0015432094 scopus 로고
    • Finite-dimensional approximatiosn of state-constrained continuous optimal control problems
    • J. CULLUM, Finite-dimensional approximatiosn of state-constrained continuous optimal control problems, SIAM J. Control, 10 (1972), pp. 649-670.
    • (1972) SIAM J. Control , vol.10 , pp. 649-670
    • Cullum, J.1
  • 11
    • 84968516204 scopus 로고
    • On the approximate minimization of functionals
    • J. W. DANIEL, On the approximate minimization of functionals, Math. Comp., 23 (1969), pp. 573-581.
    • (1969) Math. Comp. , vol.23 , pp. 573-581
    • Daniel, J.W.1
  • 12
    • 0014602475 scopus 로고
    • On the convergence of a numerical method in optimal control
    • J. W. DANIEL, On the convergence of a numerical method in optimal control, J. Optim. Theory Appl., 4 (1969), pp. 330-342.
    • (1969) J. Optim. Theory Appl. , vol.4 , pp. 330-342
    • Daniel, J.W.1
  • 13
    • 0039178857 scopus 로고
    • The Ritz-Galerkin method for abstract optimal control problems
    • J. W. DANIEL, The Ritz-Galerkin method for abstract optimal control problems, SIAM J. Control, 11 (1973), pp. 53-63.
    • (1973) SIAM J. Control , vol.11 , pp. 53-63
    • Daniel, J.W.1
  • 15
    • 0011918249 scopus 로고
    • Error estimates for a discrete approximation to constrained control problems
    • A. L. DONTCHEV, Error estimates for a discrete approximation to constrained control problems, SIAM J. Numer. Anal., 18 (1981), pp. 500-514.
    • (1981) SIAM J. Numer. Anal. , vol.18 , pp. 500-514
    • Dontchev, A.L.1
  • 16
    • 0003429641 scopus 로고
    • Perturbations, Approzimations and Sensitivity Analyse of Optimal Control Systems
    • Springer, New York
    • A. L. DONTCHEV, Perturbations, Approzimations and Sensitivity Analyse of Optimal Control Systems, Lecture Notes in Control and Inform. Sci. 52, Springer, New York, 1983.
    • (1983) Lecture Notes in Control and Inform. Sci. , vol.52
    • Dontchev, A.L.1
  • 17
    • 0030194741 scopus 로고    scopus 로고
    • An a priori estimate for discrete approximations in nonlinear optimal control
    • A. L. DONTCHEV, An a priori estimate for discrete approximations in nonlinear optimal control, SIAM J. Control Optim., 34 (1996), pp. 1315-1328.
    • (1996) SIAM J. Control Optim. , vol.34 , pp. 1315-1328
    • Dontchev, A.L.1
  • 19
    • 0027590079 scopus 로고
    • Lipschitzian stability in nonlinear control and optimization
    • A. L. DONTCHEV AND W. W. HAGER, Lipschitzian stability in nonlinear control and optimization SIAM J. Control Optim., 31 (1993), pp. 569-603.
    • (1993) SIAM J. Control Optim. , vol.31 , pp. 569-603
    • Dontchev, A.L.1    Hager, W.W.2
  • 20
    • 0032026016 scopus 로고    scopus 로고
    • Lipschitzian stability for state constrained nonlinear optimal control
    • A. L. DONTCHEV AND W. W. HAGER, Lipschitzian stability for state constrained nonlinear optimal control, SIAM J. Control Optim., 36 (1998), pp. 696-718.
    • (1998) SIAM J. Control Optim. , vol.36 , pp. 696-718
    • Dontchev, A.L.1    Hager, W.W.2
  • 21
    • 0001421413 scopus 로고    scopus 로고
    • A new approach to Lipschitz continuity in state constrained optimal control
    • A. L. DONTCHEV AND W. W. HAGER, A new approach to Lipschitz continuity in state constrained optimal control, Systems Control Lett., 35 (1998), pp. 137-143.
    • (1998) Systems Control Lett. , vol.35 , pp. 137-143
    • Dontchev, A.L.1    Hager, W.W.2
  • 22
    • 0042467717 scopus 로고    scopus 로고
    • The Euler Approximation in state constrained optimal control
    • to appear
    • A. L. DONTCHEV AND W. W. HAGER, The Euler Approximation in state constrained optimal control, Math. Comp., to appear.
    • Math. Comp.
    • Dontchev, A.L.1    Hager, W.W.2
  • 23
    • 0001496674 scopus 로고
    • Optimality, stability and convergence in nonlinear control
    • A. L. DONTCHEV W. W. HAGER, A. B. POORE, AND B. YANG, Optimality, stability and convergence in nonlinear control, Appl. Math. Optim., 31 (1995), pp. 297-326.
    • (1995) Appl. Math. Optim. , vol.31 , pp. 297-326
    • Dontchev, A.L.1    Hager, W.W.2    Poore, A.B.3    Yang, B.4
  • 24
    • 3543099411 scopus 로고
    • On regularity of optimal control
    • Recent Developments in Optimization, Springer, Berlin
    • A. L. DONTCHEV AND I. KOLMANOVSKY, On regularity of optimal control, in Recent Developments in Optimization, Lecture Notes in Econom. and Math. Systems 429, Springer, Berlin 1995, pp. 125-135.
    • (1995) Lecture Notes in Econom. and Math. Systems , vol.429 , pp. 125-135
    • Dontchev, A.L.1    Kolmanovsky, I.2
  • 25
    • 0007902573 scopus 로고    scopus 로고
    • A characterization of Lipschitzian stability in optimal control
    • Res. Notes Math. Chapman and Hall/CRC, Boca Raton, FL
    • A. L. DONTCHEV AND K. MALANOWSKI, A characterization of Lipschitzian stability in optimal control, in Calculus of Variations and Optimal Control, Res. Notes Math. 411, Chapman and Hall/CRC, Boca Raton, FL, 2000, pp. 62-76.
    • (2000) Calculus of Variations and Optimal Control , vol.411 , pp. 62-76
    • Dontchev, A.L.1    Malanowski, K.2
  • 26
    • 0030194742 scopus 로고    scopus 로고
    • 2 sufficient conditions and the gradient projection method for optimal control problems
    • 2 sufficient conditions and the gradient projection method for optimal control problems, SIAM J. Control Optim., 34 (1996), pp. 1270-1290.
    • (1996) SIAM J. Control Optim. , vol.34 , pp. 1270-1290
    • Dunn, J.C.1
  • 27
    • 0026955515 scopus 로고
    • Variants of the Kuhn-Tucker sufficient conditions in cones of nonnegative functions
    • J. C. DUNN AND T. TIAN, Variants of the Kuhn-Tucker sufficient conditions in cones of nonnegative functions, SIAM J. Control Optim., 30 (1992), pp. 1361-1384.
    • (1992) SIAM J. Control Optim. , vol.30 , pp. 1361-1384
    • Dunn, J.C.1    Tian, T.2
  • 28
    • 0039771277 scopus 로고
    • Runge-Kutta schemes applied to linear-quadratic optimal control problems
    • Bulgar. Akad. Nauk, Sofia
    • E. FARHI, Runge-Kutta schemes applied to linear-quadratic optimal control problems, in Mathematics and Mathematical Education, Bulgar. Akad. Nauk, Sofia, 1984, pp. 464-472.
    • (1984) Mathematics and Mathematical Education , pp. 464-472
    • Farhi, E.1
  • 29
    • 0016624279 scopus 로고
    • The Ritz-Trefftz method for state and control constrained optimal control problems
    • W. W. HAGER The Ritz-Trefftz method for state and control constrained optimal control problems, SIAM J. Numer. Anal., 12 (1975), pp. 854-867.
    • (1975) SIAM J. Numer. Anal. , vol.12 , pp. 854-867
    • Hager, W.W.1
  • 30
    • 0016994961 scopus 로고
    • Rate of convergence for discrete approximations to unconstrained control problems
    • W. W. HAGER Rate of convergence for discrete approximations to unconstrained control problems, SIAM J. Numer. Anal., 13 (1976), pp. 449-471.
    • (1976) SIAM J. Numer. Anal. , vol.13 , pp. 449-471
    • Hager, W.W.1
  • 31
    • 0039178856 scopus 로고
    • Convex control and dual approximations
    • W. W. HAGER, Convex control and dual approximations, Control Cybernet., 8 (1979), pp. 1-22, 73-86.
    • (1979) Control Cybernet , vol.8 , pp. 1-22
    • Hager, W.W.1
  • 32
    • 0018468242 scopus 로고
    • Lipschitz continuity for constrained processes
    • W. W. HAGER, Lipschitz continuity for constrained processes, SIAM J. Control Optim., 17 (1979), pp. 321-338.
    • (1979) SIAM J. Control Optim. , vol.17 , pp. 321-338
    • Hager, W.W.1
  • 33
    • 0025474996 scopus 로고
    • Multiplier methods for nonlinear optimal control
    • W. W. HAGER, Multiplier methods for nonlinear optimal control, SIAM J. Numer. Anal., 27 (1990), pp. 1061-1080.
    • (1990) SIAM J. Numer. Anal. , vol.27 , pp. 1061-1080
    • Hager, W.W.1
  • 34
    • 85046527030 scopus 로고    scopus 로고
    • Runge-Kutta methods in optimal control and the transformed adjoint system
    • to appear
    • W. W. HAGER, Runge-Kutta methods in optimal control and the transformed adjoint system, Numer. Math., to appear.
    • Numer. Math.
    • Hager, W.W.1
  • 35
    • 0021432726 scopus 로고
    • Dual approximations in optimal control
    • W. W. HAGER AND G. D. IANCULESCU, Dual approximations in optimal control, SIAM J. Control Optim., 22 (1984), pp. 423-465.
    • (1984) SIAM J. Control Optim. , vol.22 , pp. 423-465
    • Hager, W.W.1    Ianculescu, G.D.2
  • 36
    • 0023148667 scopus 로고
    • Optimal coatings, bang-bang controls, and gradient techniques
    • W. W. HAGER AND R. ROSTAMIAN, Optimal coatings, bang-bang controls, and gradient techniques, Optimal Control Appl. Methods, 8 (1987), pp. 1-20.
    • (1987) Optimal Control Appl. Methods , vol.8 , pp. 1-20
    • Hager, W.W.1    Rostamian, R.2
  • 39
    • 0003448429 scopus 로고    scopus 로고
    • Discrete approximations to differential inclusions
    • F. LEMPIO AND V. M. VELIOV, Discrete approximations to differential inclusions, GAMM Mitteilungen, 21 (1998), pp. 105-135.
    • (1998) GAMM Mitteilungen , vol.21 , pp. 105-135
    • Lempio, F.1    Veliov, V.M.2
  • 40
    • 0000463120 scopus 로고    scopus 로고
    • Convergence of approximations to nonlinear optimal control problems
    • Mathematical Programming with Data Perturbations, A. V. Fiacco, ed., Marcel Dekker, New York
    • K. MALANOWSKI, C. BÜSKENS, AND H. MAURER, Convergence of approximations to nonlinear optimal control problems, in Mathematical Programming with Data Perturbations, A. V. Fiacco, ed., Lecture Notes in Pure and Appl. Math 195, Marcel Dekker, New York, 1997, pp. 253-284.
    • (1997) Lecture Notes in Pure and Appl. Math , vol.195 , pp. 253-284
    • Malanowski, K.1    Büskens, C.2    Maurer, H.3
  • 41
    • 1842857981 scopus 로고
    • On difference approximations of optimal control systems
    • B. MORDUKHOVICH, On difference approximations of optimal control systems, J. Appl. Math. Mech., 42 (1978), pp. 452-461.
    • (1978) J. Appl. Math. Mech. , vol.42 , pp. 452-461
    • Mordukhovich, B.1
  • 42
    • 0001651781 scopus 로고
    • A historical survey of computations methods in optimal control
    • E. POLAK, A historical survey of computations methods in optimal control, SIAM Rev., 15 (1973), pp. 553-548.
    • (1973) SIAM Rev. , vol.15 , pp. 553-1548
    • Polak, E.1
  • 44
    • 0032380706 scopus 로고    scopus 로고
    • Runge-Kutta based procedure for the optimal control of differential-algebraic equations
    • R. PYTLAK, Runge-Kutta based procedure for the optimal control of differential-algebraic equations, J. Optim. Theory Appl., 97 (1998), pp. 675-705.
    • (1998) J. Optim. Theory Appl. , vol.97 , pp. 675-705
    • Pytlak, R.1
  • 45
    • 21444441848 scopus 로고    scopus 로고
    • A direct PRSQP method for path planning of satellite mounted robots
    • V. H. SCHULZ, A direct PRSQP method for path planning of satellite mounted robots, Z. Angew. Math. Mech., 76 (1996), pp. 17-20.
    • (1996) Z. Angew. Math. Mech. , vol.76 , pp. 17-20
    • Schulz, V.H.1
  • 46
    • 0030191875 scopus 로고    scopus 로고
    • Consistent approximations for optimal control problems based on Runge-Kutta integration
    • A. SCHWARTZ AND E. POLAK, Consistent approximations for optimal control problems based on Runge-Kutta integration, SIAM J. Control Optim., 34 (1996), pp. 1235-1269.
    • (1996) SIAM J. Control Optim. , vol.34 , pp. 1235-1269
    • Schwartz, A.1    Polak, E.2
  • 48
    • 21444445964 scopus 로고    scopus 로고
    • Stuctured interior point SQP methods in optimal control
    • M. STEINBACH, Stuctured interior point SQP methods in optimal control, Z. Angew. Math. Mech., 76 (1996), pp. 59-62.
    • (1996) Z. Angew. Math. Mech. , vol.76 , pp. 59-62
    • Steinbach, M.1
  • 50
    • 0041967200 scopus 로고
    • Bounds on the number of switchings for trajectories of piecewise analytic vector fields
    • H. SUSSMANN, Bounds on the number of switchings for trajectories of piecewise analytic vector fields, J. Differential Equations, 43 (1982), pp. 399-418.
    • (1982) J. Differential Equations , vol.43 , pp. 399-418
    • Sussmann, H.1
  • 51
    • 0026842736 scopus 로고
    • Second-order discrete approximation to linear differential inclusions
    • V. VELIOV, Second-order discrete approximation to linear differential inclusions, SIAM J. Numer. Anal., 29 (1992), pp. 439-451.
    • (1992) SIAM J. Numer. Anal. , vol.29 , pp. 439-451
    • Veliov, V.1
  • 52
    • 0031233819 scopus 로고    scopus 로고
    • On the time-discretization of control systems
    • V. M. VELIOV, On the time-discretization of control systems, SIAM J. Control Optim., 35 (1997), pp. 1470-1486.
    • (1997) SIAM J. Control Optim. , vol.35 , pp. 1470-1486
    • Veliov, V.M.1
  • 53
    • 0000423999 scopus 로고
    • Sufficient conditions for variational problems with variable endpoints: Coupled points
    • V. ZEIDAN, Sufficient conditions for variational problems with variable endpoints: Coupled points, Appl. Math. Optim., 27 (1993), pp. 191-209.
    • (1993) Appl. Math. Optim. , vol.27 , pp. 191-209
    • Zeidan, V.1


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