Formulas for the Variation of the Solution of a Differential Equation with Retarded Arguments and with a Continuous Initial Condition

Kharatishvili, G. L.; Tadumadze, Tamaz

doi:10.1023/b:dieq.0000035791.37236.08

Cited by 3 publications

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“…We thereby derive a modified transversality condition, which supplies additional information about the optimal end-time, expressed in terms of the 'essential value' of a maximized Hamiltonian-like function, introduced in [5]. Our free-time transversality condition, which is 'two-sided', is stronger than the 'one-sided' condition in [14].…”

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confidence: 99%

“…While integral forms have been proved in special cases ('additively-coupled' non-commensurate time delays in the control (see, e.g. [19]) or commensurate control delays [14], only pointwise forms (or weak 'differentiated' forms), of the Weierstrass condition are provided for general time delays in the control, elsewhere in the literature. An exception is the important, but apparently overlooked, work of Warga and Zhu [20].…”

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confidence: 99%

“…the sup norm) and D(t) is required to be closed for each t. The relevant component of the necessary conditions is a less informative 'weak' Weierstrass condition governing the initial data. [14] provides a 'strong' Weierstrass condition (in integrated form) for the initial data, but under stronger hypotheses: D(t) is must be a closed, convex product set and the control delays are assumed commensurate.…”

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confidence: 99%

“…They improve on earlier nonsmooth necessary conditions for time delay problems, which allow delays only in the state [6], [7], or require a separable structure for the control delay dependence [19]. In common with [20], they improve on available necessary conditions in [14] for smooth problems with delays in both state and control, because they do not require the control delays to be commensurate.…”

Section: Commentsmentioning

confidence: 99%

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Free time optimal control problems with time delays

Boccia

Falugi

Maurer

et al. 2013

52nd IEEE Conference on Decision and Control

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This paper provides necessary conditions of optimality for optimal control problems with time delays in both state and control variables. Different versions of the necessary conditions cover fixed end-time problems and, under additional hypotheses, free end-time problems. The conditions improve on previous available conditions in a number of respects. They can be regarded as the first generalized Pontryagin Maximum Principle for fully non-smooth optimal control problems, involving delays in state and control variables, only special cases of which have previously been derived. Even when the data is smooth, the conditions advance the existing theory. For example, we provide a new 'two-sided' generalized transversality condition, associated with the optimal end-time, which gives more information about the optimal end-time than the 'one-sided' condition in the earlier literature. But there are improvements in other respects, relating to the treatment of initial data, specifying past histories of the state and control, and to the unrestrictive nature of the hypotheses under which the necessary conditions are derived. Nonsmoothness:We provide the first set of necessary conditions, in the form of a generalized Pontryagin Maximum Principle, for 'fully' nonsmooth problems (i.e. problems in which the only regularity hypothesis on the data w.r.t. the state variable is 'Lipschitz continuity') involving delays in states and controls. They resemble the classical necessary conditions for 'smooth' problems except that, in the costate relation (1.3) classical derivatives are replaced by set-valued subdifferentials of nonsmooth analysis. Two earlier papers [6], [7] provide necessary conditions for nonsmooth optimal control problems with time delays in the state alone. The difference is that, in these papers, the dynamic constraint is modelled as a differential inclusion, and the relation for the costate arc (combined with the Weierstrass condition) is a generalization of Clarke's Hamiltonian inclusion condition with 'advanced' arguments. As observed in [7, Section 1,], necessary conditions expressed in terms of the Hamiltonian inclusion imply the nonsmooth Maximum Principle only for problems having special structure and not 'fully' nonsmooth problems, as in this paper. Furthermore, the methods of [6] and [7] cannot be adapted to cover problems with time delays in the control, because it is not possible to express a controlled delay differential equation (with delays in the control) as a delay differential inclusion (which can take account only of delays in the state).[7] allows both distributed and discrete delays (in the state variable), whereas we allow only discrete delays (in both state and control variables). Necessary conditions for optimal control problems involving differential inclusions are also provided in [3] and [15] for fixed time optimal control problems involving a single time delay in the state. We mention that Warga [19] showed that a broad class of optimal control problems involving delays and/or functional diff...

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Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: )mentioning

confidence: 99%

Section: Commentsmentioning

confidence: 99%

See 2 more Smart Citations

Free time optimal control problems with time delays

Boccia

Falugi

Maurer

et al. 2013

52nd IEEE Conference on Decision and Control

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“…If t 00 C 0 D t 10 ; then Theorem 1.1 is valid on the interval OEt 10 ; t 10 C ı 2 and Theorem 1.2 is valid on the interval OEt 10 ı 2 ; t 10 : Finally, we note that variation formulas for the solution of various classes of functional-differential equations without perturbations of delay can be found in [1][2][3][4][5][6][7][8].…”

Section: Some Commentsmentioning

confidence: 99%

Variation Formulas for the Solution of Delay Differential Equations with Mixed Initial Conditions and Delay Perturbations

Tadumadze

2015

J Math Sci

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Variation formulas are proved for the solution of nonlinear differential equations with constant delays. The essential novelty of the work is the effect of delay perturbations in the variation formulas. Mixed initial conditions mean that, at the initial time, some coordinates of the trajectory do not coincide with the corresponding coordinates of the initial function, whereas the other coordinates coincide. The variation formulas are used in the proof of necessary optimality conditions.

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Variation formulas of solution for a functional differential equation with delay function perturbation

Tadumadze

Gorgodze

2014

J. Contemp. Mathemat. Anal.

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Variation formulas of solution are proved for a controlled non-linear functionaldifferential equation with constant delay and the continuous initial condition. In this paper, the essential novelty is the effect of delay perturbation in the variation formulas.The continuity of the initial condition means that the values of the initial function and the trajectory always coincide at the initial moment.

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Formulas for the Variation of the Solution of a Differential Equation with Retarded Arguments and with a Continuous Initial Condition

Cited by 3 publications

References 0 publications

Free time optimal control problems with time delays

Free time optimal control problems with time delays

Variation Formulas for the Solution of Delay Differential Equations with Mixed Initial Conditions and Delay Perturbations

Variation formulas of solution for a functional differential equation with delay function perturbation

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