Application of multiple shooting to the numerical solution of optimal control problems with bounded state variables

Maurer, Helmut; Gillessen, W.

doi:10.1007/bf02252860

Cited by 61 publications

(25 citation statements)

References 12 publications

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“…On a boundary arc with x(t) = 2 0.05 it follows from u = -As, (12) and (15) that Xe = 0 and X, = 0. Then (14) and the sign condition (47) imply the desired…”

Section: Necessary Conditions Of Optimalitymentioning

confidence: 97%

The non‐linear beam via optimal control with bounded state variables

Maurer

Mittelmann

1991

Optim Control Appl Methods

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SUMMARYThe non-linear beam with bounded deflection is considered as an optimal control problem with bounded state variables. The theory of necessary optimality conditions leads to boundary value problems with jump conditions which are solved by multiple-shooting techniques. A branching analysis is performed which exhibits the different solution structures. In particular, the second bifurcation point is determined numerically. The bifurcation diagram reveals a hysteresis-like behaviour and explains the jumping to a different state at this bifurcation point.

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“…On a boundary arc with x(t) = 2 0.05 it follows from u = -As, (12) and (15) that Xe = 0 and X, = 0. Then (14) and the sign condition (47) imply the desired…”

Section: Necessary Conditions Of Optimalitymentioning

confidence: 97%

The non‐linear beam via optimal control with bounded state variables

Maurer

Mittelmann

1991

Optim Control Appl Methods

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“…Most of these methods successfully solve the unconstrained problem, but the presence of, for example, state variable inequality constraints (SVIC) often resulted in both analytical and computational difficulties. Theoretical aspects of the SVIC problem have been studied by Berkovitz (1961), Dreyfus (1962), Chang (1962), Berkovitz and Dreyfus (1965) and Speyer and Bryson (1968), while early contributions to the numerical computation were due to Dreyfus (1962), Kelley (1962), Bryson and Denham (1964), McGill (1965), Lasdon et al (1967), Jacobson and Lele (1969), Mehra and Davis (1972), Neuman and Sen (1973) and Maurer and Gillessen (1975). The approaches of Dreyfus (1962) and Bryson and Denham (1964) require guesses for both the number and location of the junction points between constrained and unconstrained arcs.…”

Section: Introductionmentioning

confidence: 99%

“…Such problems may turn out to be the last redoubt of the penalty-function technique (Kelley, 1962;McGill, 1965;Lasdon et al, 1967). For refinement of penaltyapproximation solutions one turns to the indirect variational method and multiple-shooting (Maurer and Gillessen, 1975). Jacobson and Lele (1969) used a slack variable to transform a SVIC problem into an unconstrained problem of higher dimension for which the optimal trajectory exhibits singular arcs.…”

Section: Introductionmentioning

confidence: 99%

A chebyshev polynomial method for optimal control with state constraints

Vlassenbroeck

1988

Automatica

107

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Abstrsct--This paper presents a numerical technique for solving non-linear constrained optimal control problems. The method extends previous contributions to non-linear unconstrained optimal control problems and is based upon a Chebyshev series expansion of state and control. The differential and integral expressions from the system dynamics and the performance index, the boundary conditions and other general conditions are converted into some algebraic equations. State inequality constraints are transformed into equality constraints through the use of slack variables. The technique may start from a feasible or non-feasible trajectory and avoids problems of singular arcs. The applicability is illustrated on two well-known state variable inequality constrained optimal control problems. Extensions of the approach to problems with other equality and inequality constraints on state and control are described but have not yet been tested on practical examples.

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“…The adjoint differential equations are 13) and the natural boundary conditions and the jump relations can be written as…”

Section: Necessary Conditions Derived By Optimal Control Theorymentioning

confidence: 99%

On the Construction of Optimal Monotone Cubic Spline Interpolations

Fredenhagen¹,

Oberle²,

Opfer³

1999

Journal of Approximation Theory

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In this paper we derive necessary optimality conditions for an interpolating spline function which minimizes the Holladay approximation of the energy functional and which stays monotone if the given interpolation data are monotone. To this end optimal control theory for state-restricted optimal control problems is applied. The necessary conditions yield a complete characterization of the optimal spline. In the case of two or three interpolation knots, which we call the local case, the optimality conditions are treated analytically. They reduce to polynomial equations which can very easily be solved numerically. These results are used for the construction of a numerical algorithm for the optimal monotone spline in the general (global) case via Newton's method. Here, the local optimal spline serves as a favourable initial estimation for the additional grid points of the optimal spline. Some numerical examples are presented which are constructed by FORTRAN and MATLAB programs. Academic Press

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Application of multiple shooting to the numerical solution of optimal control problems with bounded state variables

Cited by 61 publications

References 12 publications

The non‐linear beam via optimal control with bounded state variables

The non‐linear beam via optimal control with bounded state variables

A chebyshev polynomial method for optimal control with state constraints

On the Construction of Optimal Monotone Cubic Spline Interpolations

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