A finite element approximation for the initial-value problem for nonlinear second-order differential equations. II. Systems

Gregory, John; Zeman, Marvin; Badiey, Mohsen

doi:10.1016/0022-247x(86)90028-4

Journal of Mathematical Analysis and Applications

1986

DOI: 10.1016/0022-247x(86)90028-4

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A finite element approximation for the initial-value problem for nonlinear second-order differential equations. II. Systems

John Gregory

Marvin Zeman

Mohsen Badiey

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Cited by 3 publications

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“…x {h 3 Q + 0{h b )) implies that ||£*||oo < Ch 2 . We note that this last result is a significant generalization of the classical result of Henrici [4] for J -diag(-l,2,-1) when m = 1, where the elements of J -1 may be computed explicitly.…”

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Numerical methods for extremal problems in the calculus of variations and optimal control theory

Gregory¹

1988

Bull. Amer. Math. Soc.

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Introduction. New, general methods are given to find numerical solutions for extremal problems in the calculus of variations and optimal control theory. Theoretical methods are derived and used to establish pointwise a priori error estimates with maximum error at the node point, ||e||oo, equal to 0(h 2 ) and a Richardson error of 0(h 4 ). This is done under the weak assumption that there are no conjugate points on the interval and not under the usual convexity assumptions.Of practical interest is that these methods (i) are very easy to implement, (ii) hold for well-defined mixtures of initial value and boundary value problems, (iii) use multipliers, and not ill-conditioned penalty methods, for both equality and inequality constraints, in a natural, efficient manner, and (iv) are applicable to transversality, type-minimal time problems.The heart of these methods is the algorithm (4) and the a priori estimates in Theorem 2 for the m-dependent variable problem in the calculus of variations given below. Once this is established we quote Hestenes [5] and show that very general optimal control problems can be easily reformulated and solved as calculus of variations problems.

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Numerical methods for extremal problems in the calculus of variations and optimal control theory

Gregory¹

1988

Bull. Amer. Math. Soc.

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Numerical methods for nonlinear optimal control problems: Application to abnormal problems

Dapatadu

Gibson

Gregory

1991

Journal of Mathematical Analysis and Applications

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Discrete Variable Methods for the m-Dependent Variable, Nonlinear Extremal Problem in the Calculus of Variations

Gregory¹,

Wang²

1990

SIAM J. Numer. Anal.

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A finite element approximation for the initial-value problem for nonlinear second-order differential equations. II. Systems

Cited by 3 publications

References 5 publications

Numerical methods for extremal problems in the calculus of variations and optimal control theory

Numerical methods for extremal problems in the calculus of variations and optimal control theory

Numerical methods for nonlinear optimal control problems: Application to abnormal problems

Discrete Variable Methods for the m-Dependent Variable, Nonlinear Extremal Problem in the Calculus of Variations

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