Jacobi fields in optimal control: Morse and Maslov indices

Agrachev, Andrei; Beschastnyi, Ivan

doi:10.1016/j.na.2021.112608

Cited by 5 publications

(14 citation statements)

References 49 publications

(77 reference statements)

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“…These cases were already studied by many authors (see for example [12,27,26,22,24,10]). These results can be recovered using the constructions from this paper and Morse-type theorems from our previous article [3].…”

Section: Introductionmentioning

confidence: 71%

“…which is essentially the linearization of a pull-back of the PMP Hamiltonian using the flow Φ t . In [3] we have proven the following characterization of the previously mentioned L-derivatives of optimal control problems.…”

Section: L-derivatives For Optimal Control Systemsmentioning

confidence: 89%

“…In the next section we will write down explicit expressions for the optimal control problem we are studying, and define Jacobi curves. In article [3] we gave a general algorithm for their approximation. In this article we focus on situations when an exact characterization is possible.…”

Section: L-derivativesmentioning

confidence: 99%

“…They can be used to obtain necessary [5] and sufficient [10] optimality conditions. We have discussed them in detail in [4] (see also [3]), so we just give a quick recap.…”

Section: L-derivatives For Optimal Control Systemsmentioning

confidence: 99%

“…We present this section for reader's convenience and in order to fix the basic notations. Then we recall all the necessary results from our article [3] about the L-derivatives and the construction of Jacobi curves for optimal control problems in Sections 2 and 3.…”

Section: Introductionmentioning

confidence: 99%

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Jacobi Fields in Optimal Control: One-dimensional Variations

Agrachev

Beschastnyi

2020

J Dyn Control Syst

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In this paper which is closely related to the previous paper [link] we specify general theory developed there. We study the structure of Jacobi fields in the case of an analytic system and piece-wise analytic control. Moreover, we consider only 1-dimensional control variations. Jacobi fields are piece-wise analytic in this case but may have jump discontinuities. We derive ODEs that these fields satisfy on the intervals of regularity and study behavior of the fields in a neighborhood of a singularity where the ODE becomes singular and the Jacobi fields may have jumps. arXiv:1811.09217v2 [math.OC] 3 Apr 2019 1 Linear symplectic geometry and Lagrangian GrassmanianIn this section we recall basic facts from symplectic geometry and fix notations that we use in the article. Given a symplectic space (R 2n , σ), where σ is a symplectic form we can always assume by the Darboux theorem thatwhere J is the standard complex structure J = 0 id n − id n 0 .

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Section: Introductionmentioning

confidence: 71%

Section: L-derivatives For Optimal Control Systemsmentioning

confidence: 89%

Section: L-derivativesmentioning

confidence: 99%

“…They can be used to obtain necessary [5] and sufficient [10] optimality conditions. We have discussed them in detail in [4] (see also [3]), so we just give a quick recap.…”

Section: L-derivatives For Optimal Control Systemsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Jacobi Fields in Optimal Control: One-dimensional Variations

Agrachev

Beschastnyi

2020

J Dyn Control Syst

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Operators Arising as Second Variation of Optimal Control Problems and Their Spectral Asymptotics

Baranzini

2022

J Dyn Control Syst

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Functional determinants for the second variation

Baranzini

2024

J. Fixed Point Theory Appl.

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We study the determinant of the second variation of an optimal control problem for general boundary conditions. Generically, these operators are not trace class and the determinant is defined as a principal value limit. We provide a formula to compute this determinant in terms of the linearisation of the extrenal flow. We illustrate the procedure in some special cases, proving some Hill-type formulas.

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Jacobi fields in optimal control: Morse and Maslov indices

Cited by 5 publications

References 49 publications

Jacobi Fields in Optimal Control: One-dimensional Variations

Jacobi Fields in Optimal Control: One-dimensional Variations

Operators Arising as Second Variation of Optimal Control Problems and Their Spectral Asymptotics

Functional determinants for the second variation

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