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  • 1
    Electronic Resource
    Electronic Resource
    Symplectic phase flow approximation for the numerical integration of canonical systems (1992)
    Springer
    Numerische Mathematik 61 (1992), S. 501-521 
    Details
    ISSN: 0945-3245
    Keywords: 65L05 ; 65L07 ; 58F05 ; 70-08 ; 70F15 ; 70H15
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary New methods are presented for the numerical integration of ordinary differential equations of the important family of Hamiltonian dynamical systems. These methods preserve the Poincaré invariants and, therefore, mimic relevant qualitative properties of the exact solutions. The methods are based on a Runge-Kutta-type ansatz for the generating function to realize the integration steps by canonical transformations. A fourth-order method is given and its implementation is discussed. Numerical results are presented for the Hénon-Heiles system, which describes the motion of a star in an axisymmetric galaxy.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01385523
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  • 2
    Electronic Resource
    Electronic Resource
    A modified continuation method for the numerical solution of nonlinear two-point boundary value problems by shooting techniques (1976)
    Springer
    Numerische Mathematik 26 (1976), S. 327-343 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A modification of the well-known continuation (or homotopy) method for actual computation is worked out. Compared with the classical method, the modification seems to be a more reliable device for supplying useful initial data for shooting techniques. It is shown that computing time may be significantly reduced in the numerical solution of sensitive realistic two-point boundary value problems.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01395950
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  • 3
    Electronic Resource
    Electronic Resource
    Comparing routines for the numerical solution of initial value problems of ordinary differential equations in multiple shooting (1976)
    Springer
    Numerische Mathematik 27 (1976), S. 449-469 
    Details
    ISSN: 0945-3245
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary The numerical solution of two-point boundary value problems and problems of optimal control by shooting techniques requires integration routines. By solving 15 real-life problems four well-known intergrators are compared relative to reliability, fastness and precision. Hints are given, which routines could be used for a problem.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01399607
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  • 4
    Electronic Resource
    Electronic Resource
    Numerical treatment of delay differential equations by Hermite Interpolation (1981)
    Springer
    Numerische Mathematik 37 (1981), S. 235-255 
    Details
    ISSN: 0945-3245
    Keywords: AMS(MOS): 65L05 ; 65Q05 ; CR: 5.17
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Summary A class of numerical methods for the treatment of delay differential equations is developed. These methods are based on the wellknown Runge-Kutta-Fehlberg methods. The retarded argument is approximated by an appropriate multipoint Hermite Interpolation. The inherent jump discontinuities in the various derivatives of the solution are considered automatically. Problems with piecewise continuous right-hand side and initial function are treated too. Real-life problems are used for the numerical test and a comparison with other methods published in literature.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF01398255
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  • 5
    Electronic Resource
    Electronic Resource
    Direkte Cadmiumbestimmung in menschlichen Organen mit Zeeman-AAS (1984)
    Springer
    Fresenius' Zeitschrift für analytische Chemie 317 (1984), S. 461-461 
    Details
    ISSN: 1618-2650
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Notes: Abstract The analytical procedure is described, and the results and consequences are discussed.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00494516
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  • 6
    Electronic Resource
    Electronic Resource
    093 The mercury, cadmium and lead content of cigarette tobacco: Comparative analytical-statistical studies in 1987 and 1991 employing Zeeman-AAS (1992)
    Springer
    Fresenius' Zeitschrift für analytische Chemie 343 (1992), S. 152-153 
    Details
    ISSN: 1618-2650
    Source: Springer Online Journal Archives 1860-2000
    Topics: Chemistry and Pharmacology
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00332087
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  • 7
    Electronic Resource
    Electronic Resource
    The maximum principle, Bellman's equation, and Carathéodory's work (1994)
    Springer
    Journal of optimization theory and applications 80 (1994), S. 199-225 
    Details
    ISSN: 1573-2878
    Keywords: Maximum principle ; Bellman equation ; Carathéodory's work ; calculus of variations ; optimal control theory ; history of the calculus of variations ; history of optimal control theory
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract One of the most important and deep results in optimal control theory is the maximum principle attributed to Hestenes (1950) and in particular to Boltyanskii, Gamkrelidze, and Pontryagin (1956). Another prominent result is known as the Bellman equation, which is associated with Isaacs' and Bellman's work (later than 1951). However, precursors of both the maximum principle and the Bellman equation can already be found in Carathéodory's book of 1935 (Ref. 1a), the first even in his earlier work of 1926 which is given in Ref. 2. This is not a widely acknowledged fact. The present tutorial paper traces Carathéodory's approach to the calculus of variations, once called the “royal road in the calculus of variations,” and shows that famous results in optimal control theory, including the maximum principle and the Bellman equation, are consequences of Carathéodory's earlier results.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02192933
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  • 8
    Electronic Resource
    Electronic Resource
    Complex differential games of pursuit-evasion type with state constraints, part 2: Numerical computation of optimal open-loop strategies (1993)
    Springer
    Journal of optimization theory and applications 78 (1993), S. 443-463 
    Details
    ISSN: 1573-2878
    Keywords: Differential games ; pursuit-evasion games ; open-loop strategies ; multiple shooting ; multipoint boundary-value problems ; saddle-point trajectories ; barrier trajectories ; missile firing range
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In Part 1 of this paper (Ref. 1), necessary conditions for optimal open-loop strategies in differential games of pursuit-evasion type have been developed for problems which involve state variable inequality constraints and nonsmooth data. These necessary conditions lead to multipoint boundary-value problems with jump conditions. These problems can be solved very efficiently and accurately by the well-known multiple-shooting method. By this approach, optimal open-loop strategies and their associated saddle-point trajectories can be computed for the entire capture zone of the game. This also includes the computation of optimal open-loop strategies and saddle-point trajectories on the barrier of the pursuit-evasion game. The open-loop strategies provide an open-loop representation of the optimal feedback strategies. Numerical results are obtained for a special air combat scenario between one medium-range air-to-air missile and one high-performance aircraft in a vertical plane. A dynamic pressure limit for the aircraft imposes a state variable inequality constraint of the first order. Special emphasis is laid on realistic approximations of the lift, drag, and thrust of both vehicles and the atmospheric data. In particular, saddle-point trajectories on the barrier are computed and discussed. Submanifolds of the barrier which separate the initial values of the capture zone from those of the escape zone are computed for two representative launch positions of the missible. By this way, the firing range of the pursuing missile is determined and visualized.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939877
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  • 9
    Electronic Resource
    Electronic Resource
    Solution differentiability for parametric nonlinear control problems with control-state constraints (1995)
    Springer
    Journal of optimization theory and applications 86 (1995), S. 285-309 
    Details
    ISSN: 1573-2878
    Keywords: Parametric control problems ; mixed control-state constraints ; second-order sufficient conditions ; solution differentiability ; multipoint boundary-value problems ; shooting techniques ; Riccati equation
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract This paper considers parametric nonlinear control problems subject to mixed control-state constraints. The data perturbations are modeled by a parameterp of a Banach space. Using recent second-order sufficient conditions (SSC), it is shown that the optimal solution and the adjoint multipliers are differentiable functions of the parameter. The proof blends numerical shooting techniques for solving the associated boundary-value problem with theoretical methods for obtaining SSC. In a first step, a differentiable family of extremals for the underlying parameteric boundary-value problem is constructed by assuming the regularity of the shooting matrix. Optimality of this family of extremals can be established in a second step when SSC are imposed. This is achieved by building a bridge between the variational system corresponding to the boundary-value problem, solutions of the associated Riccati ODE, and SSC. Solution differentiability provides a theoretical basis for performing a numerical sensitivity analysis of first order. Two numerical examples are worked out in detail that aim at reducing the considerable deficit of numerical examples in this area of research.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF02192081
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  • 10
    Electronic Resource
    Electronic Resource
    Complex differential games of pursuit-evasion type with state constraints, part 1: Necessary conditions for optimal open-loop strategies (1993)
    Springer
    Journal of optimization theory and applications 78 (1993), S. 419-441 
    Details
    ISSN: 1573-2878
    Keywords: Differential games ; pursuit-evasion games ; singular surfaces ; barrier surface ; state constraints ; open-loop strategies ; multipoint boundary-value problems ; flight mechanics ; missile firing range
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract Complex pursuit-evasion games with state variable inequality constraints are investigated. Necessary conditions of the first and the second order for optimal trajectories are developed, which enable the calculation of optimal open-loop strategies. The necessary conditions on singular surfaces induced by state constraints and non-smooth data are discussed in detail. These conditions lead to multi-point boundary-value problems which can be solved very efficiently and very accurately by the multiple shooting method. A realistically modelled pursuit-evasion problem for one air-to-air missile versus one high performance aircraft in a vertical plane serves as an example. For this pursuit-evasion game, the barrier surface is investigated, which determines the firing range of the missile. The numerical method for solving this problem and extensive numerical results will be presented and discussed in Part 2 of this paper; see Ref. 1.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00939876
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