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  • 1
    ISSN: 1573-2878
    Keywords: Discrete systems ; periodic processes ; maximum principle ; sufficient conditions ; calculus of variations
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The problem of improving the optimal steady state of a discrete system by means of a suitable cyclic operation is discussed. Two sufficient conditions are found by negating a necessary condition and by following a frequency-domain approach, respectively. The connection between the two conditions is also shown.
    Type of Medium: Electronic Resource
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  • 2
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Nonlinear systems ; calculus of variations ; necessary conditions ; maximum principle ; optimization theorems
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is the purpose of this paper to develop and present new approaches to optimal control problems for which the state evolution equation is nonlinear. For bilinear systems in which the evolution equation is right invariant, it is possible to use ideas from differential geometry and Lie theory to obtain explicit closed-form solutions.
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  • 3
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Optimal remodeling ; structural optimization ; calculus of variations ; optimality conditions ; structural mechanics
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract We present the problem of remodeling a given structure such as to improve structural performance optimally within a specified available resource. The development pertains to all types of problems where the mode of structural response is governed by an extremum principle. A variational formulation is used, and the idea is illustrated for maximum-stiffness remodeling of single-purpose structures.
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  • 4
    ISSN: 1573-2878
    Keywords: Optimal control ; junction conditions ; calculus of variations ; inequality constraints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract It is known that extremal arcs governed by inequality constraints of third order (constraint relations that must be differentiated three times to generate a control equation) cannot join an unconstrained arc, except in special cases. But a control problem is exhibited, for which every extremal includes a constrained arc of third order. The constrained arc joins the end of an infinite sequence of consecutive unconstrained arcs of finite total duration. Evidence (but not proof) is given that this phenomenon is typical, rather than exceptional. An analogous phenomenon is well known for optimal control problems with singular arcs of second order.
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  • 5
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Constrained minimum problems ; problem of Lagrange ; calculus of variations ; augmentability ; generalized Lagrangians
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The present paper is concerned with the concept of augmentability. One form of augmentability is based on the generalized Lagrangian and leads to the method of multipliers for solving constrained minimum problems. An assumption of augmentability can be used in place of regularity to establish first-order and second-order necessary conditions for a minimum for constrained minimum problems. Another type of augmentability leads to the theory of Mayer fields in variational theory.
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  • 6
    ISSN: 1573-2878
    Keywords: Optimal control problems ; current value Hamiltonian ; constant optimal controls ; analytic functions ; calculus of variations ; differential games
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The conditions under which an optimal control problem commonly employed in economics gives rise to a constant optimal control are characterized. The conditions are stated in terms of the properties of the functional form of the integrand of the objective function and of the state equation. These conditions can be checked prior to computation of the optimal control and thereby can simplify its calculation.
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  • 7
    ISSN: 1573-2878
    Keywords: Lagrange multiplier rule ; problem of Lagrange ; continuity ; calculus of variations ; fixed endpoints
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract A new simple proof of the Lagrange multiplier rule is presented in this paper. The approach used involves simple analytical techniques that are very easy to follow and does not involve theorems on imbeddability in a one-parameter family of curves or matrix-rank analysis as do most of the existing techniques. The proof is here developed for the fixed-endpoint problem in a three-dimensional space.
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  • 8
    Electronic Resource
    Electronic Resource
    Springer
    ISSN: 1573-2878
    Keywords: Optimal harvesting ; renewable resources ; calculus of variations ; bang-bang fishing
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract The aim of the paper is to investigate different types of calculus of variations solutions maximizing the present value of a renewable resource. It is found that there are two types of optimal stationary harvesting policies. For certain cost functions, bang-bang harvesting may be more profitable than either of these.
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  • 9
    ISSN: 1573-2878
    Keywords: Weak and strong local minima ; optimal control ; smooth constraints ; calculus of variations ; Jacobi condition
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In order to tighten the gap between necessary and sufficient conditions, new second-order sufficient conditions are developed for optimal control problems, where the control set is given by smooth functions. When the control set is polyhedral, our criterion generalizes prior results of the same kind, namely, the Jacobi criterion in Hamiltonian form and that in Lagrangian form (Refs. 1–3).
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  • 10
    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.
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