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  • 1
    Electronic Resource
    Electronic Resource
    Remarks on some existence theorems for optimal control (1969)
    Springer
    Journal of optimization theory and applications 3 (1969), S. 296-305 
    Details
    ISSN: 1573-2878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract First, a remark is made that a growth condition contained in previous papers by Cesari concerning existence theorems for optimal controls can be replaced by a slightly more general condition. In this more general condition, a constantM ɛ⩾0 is replaced by any functionM ɛ(t)⩾0 which is assumed to beL-integrable in every finite interval. Then, the remark is made that the same condition, which is usually required to be satisfied by the functionsf 0(t, x, u),f(t, x, u) characterizing the control, can be required to be satisfied only by the admissible pairsx(t),u(t) of the class Ω in which the optimum is being sought. This generalization requires a subtle argument. The new condition parallels now the usual conditions of the type ∝|x′| p dt⩽M, which are required to be satisfied by the admissible pairs of the class Ω.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00931369
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  • 2
    Electronic Resource
    Electronic Resource
    Remarks on certain growth conditions in problems of optimal control (1969)
    Springer
    Journal of optimization theory and applications 4 (1969), S. 321-329 
    Details
    ISSN: 1573-2878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In the Tonelli-Nagumo existence theorem for free problems of the calculus of variations, a uniform growth condition (T 1) was employed to prove the equiabsolute continuity of the trajectories of a minimizing sequence. This same growth condition was shown to be equivalent to a pointwise growth condition (T 2). Cesari utilized a generalization, condition (α), in order to extend the Tonelli-Nagumo existence theorem to optimal control problems. In the present paper, we show that, for problems of optimal control, condition (α) is equivalent to a uniform growth condition (β) and that both conditions (α) and (β) are stronger than a pointwise growth condition (γ), a generalization of condition (T 2). Two situations are given where conditions (α), (β), and (γ) are equivalent, and the result of Tonelli and Turner concerning the equivalence of conditions (T 1) and (T 2) is obtained as a particular case of one of these situations.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00927674
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  • 3
    Electronic Resource
    Electronic Resource
    A recent significant growth condition in optimal control theory (1969)
    Springer
    Journal of optimization theory and applications 4 (1969), S. 378-385 
    Details
    ISSN: 1573-2878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Notes: Abstract In 1934, Tonelli proved, for free problems withn=1, that the trajectories of a minimizing sequence are equiabsolutely continuous provided a pointwise growth condition is satisfied everywhere except at the points of an exceptional set where an additional mild hypothesis is required, condition (T). The author extended this result to optimal control problems by the use of a uniform growth condition, called (α), which is equivalent to Tonelli's pointwise growth condition for free problems with convex integrand (Refs. 4–5, 10). More recently, condition (α), or condition (Φ), which is equivalent to (α) for optimal control problems, was replaced by the much weaker condition (β) in existence theorems for optimal control problems (Refs. 1–2). In the present paper, we show that this same combination of growth condition (α) and condition (T) at the points of the exceptional set is actually a special case of the growth condition (β), which does not use the concept of the exceptional set.
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00927690
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  • 4
    Electronic Resource
    Electronic Resource
    Book reviews (1971)
    Springer
    Journal of optimization theory and applications 7 (1971), S. 217-220 
    Details
    ISSN: 1573-2878
    Source: Springer Online Journal Archives 1860-2000
    Topics: Mathematics
    Type of Medium: Electronic Resource
    URL: http://dx.doi.org/10.1007/BF00932481
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    Paper (German National Licenses)
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