The variational maximum principle and second-order optimality conditions for impulse processes and singular processes

Dykhta, V. A.

doi:10.1007/bf02104948

Cited by 21 publications

(18 citation statements)

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“…In fact, as already stated in the Introduction, the notion of limit solution coincides with a characterization of the concept of pointwise defined solution formerly proved for the particular case of commutative systems [7,10]. Let us also refer to [8,9,28] for other references dealing with systems where the Lie algebra generated by {g 1 , . .…”

Section: The Commutative Casementioning

confidence: 63%

“…, m (see e.g. [7,8,9,10]). Actually, under such hypothesis, a notion of solution to (E)-(IC) has been established for controls u ∈ L 1 (here we use L 1 to denote the set of pointwise defined Lebesgue integrable functions, while L 1 refers to the usual quotient with respect to the Lebesgue measure), possibly with unbounded variation.…”

mentioning

confidence: 99%

“…. , g m commute (see [7,23,10], and also [9,8]). We observe that, unlike the noncommutative case, uniqueness and continuous dependence is obtained also for discontinuous, L 1 controls u.…”

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confidence: 99%

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L1limit solutions for control systems

Aronna

Rampazzo

2015

Journal of Differential Equations

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Abstract. For a control Cauchy probleṁgα(x)uα, x(a) =x, on an interval [a, b], we propose a notion of limit solution x, verifying the following properties: i) x is defined for L 1 (impulsive) inputs u and for standard, bounded measurable, controls v; ii) in the commutative case (i.e. when [gα, g β ] ≡ 0, for all α, β = 1, . . . , m), x coincides with the solution one can obtain via the change of coordinates that makes the gα simultaneously constant; iii) x subsumes former concepts of solution valid for the generic, noncommutative case. In particular, when u has bounded variation, we investigate the relation between limit solutions and (single-valued) graph completion solutions. Furthermore, we prove consistency with the classical Carathéodory solution when u and x are absolutely continuous.Even though some specific problems are better addressed by means of special representations of the solutions, we believe that various theoretical issues call for a unified notion of trajectory. For instance, this is the case of optimal control problems, possibly with state and endpoint constraints, for which no extra assumptions (like e.g. coercivity, bounded variation, commutativity) are made in advance.

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Section: The Commutative Casementioning

confidence: 63%

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confidence: 99%

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L1limit solutions for control systems

Aronna

Rampazzo

2015

Journal of Differential Equations

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“…The first relation in (62) coincides with (18), while the last one immediately implies (19). Finally, observe that B ∈ B 0 if and only if −B ∈ B 0 , so that (63) yields (25).…”

Section: Conclusion Of the Proofmentioning

confidence: 87%

A Higher-Order Maximum Principle for Impulsive Optimal Control Problems

Aronna¹,

Motta²,

Rampazzo³

2020

SIAM J. Control Optim.

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We consider a nonlinear system, affine with respect to an unbounded control u which is allowed to range in a closed cone. To this system we associate a Bolza type minimum problem, with a Lagrangian having sublinear growth with respect to u. This lack of coercivity gives the problem an impulsive character, meaning that minimizing sequences of trajectories happen to converge towards discontinuous paths. As is known, a distributional approach does not make sense in such a nonlinear setting, where, instead, a suitable embedding in the graph-space is needed. We provide higher order necessary optimality conditions for properly defined impulsive minima, in the form of equalities and inequalities involving iterated Lie brackets of the dynamical vector fields. These conditions are derived under very weak regularity assumptions and without any constant rank conditions. Date: July 11, 2019.

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“…, m, and up to the second order (see e.g. [4], [5], [10]). Instead, our Higher Order Maximum Principle is established for the generic, non-commutative, case, and involves iterated brackets of any order.…”

Section: Introductionmentioning

confidence: 99%

Necessary conditions involving Lie brackets for impulsive optimal control problems

Aronna

Motta

Rampazzo

2019

2019 IEEE 58th Conference on Decision and Control (CDC)

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We obtain higher order necessary conditions for a minimum of a Mayer optimal control problem connected with a nonlinear, control-affine system, where the controls range on an m-dimensional Euclidean space. Since the allowed velocities are unbounded and the absence of coercivity assumptions makes big speeds quite likely, minimizing sequences happen to converge toward impulsive, namely discontinuous, trajectories. As is known, a distributional approach does not make sense in such a nonlinear setting, where instead a suitable embedding in the graph space is needed. We will illustrate how the chance of using impulse perturbations makes it possible to derive a Higher Order Maximum Principle which includes both the usual needle variations (in space-time) and conditions involving iterated Lie brackets. An example, where a third order necessary condition rules out the optimality of a given extremal, concludes the paper.

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The variational maximum principle and second-order optimality conditions for impulse processes and singular processes

Cited by 21 publications

References 0 publications

L1limit solutions for control systems

L1limit solutions for control systems

A Higher-Order Maximum Principle for Impulsive Optimal Control Problems

Necessary conditions involving Lie brackets for impulsive optimal control problems

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