State-local optimality of a bang–bang trajectory: a Hamiltonian approach

Poggiolini, Laura; Stefani, Gianna

doi:10.1016/j.sysconle.2004.05.005

Cited by 33 publications

(47 citation statements)

References 9 publications

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“…• the use of the Miele clock form (see [143,144]), widely generalized with the framework of symplectic geometry and methods, the latter being used to provide sufficient conditions for optimality either in terms of conjugate points, Maslov index, envelope theory, extremal field theory, or of optimal synthesis (see [60,68,19,17,69,66]);…”

Section: Geometric Optimal Control Results and Application To The Promentioning

confidence: 99%

“…Second order necessary and/or sufficient conditions of optimal control problems with nonlinear control systems and discontinuous controls have been developed in [62,63,64,65,66,67] and references therein. A different point of view is provided in [68,69] (see also [62,66]) in terms of extremal fields, and consists of embedding the reference trajectory into a local field of broken extremals (corresponding to piecewise continuous controls). The occurrence of a conjugate point is then related with an overlap of the flow near the switching surface.…”

Section: Generalizations Open Problems and Challengesmentioning

confidence: 99%

“…Hence, the methods e.g. of [68,62,69,65,66,77,67] that are applicable to bang-bang situations in any dimension should permit to derive local optimal syntheses in larger dimension under additional assumptions, and as well for control-affine systems with more than one control (although it can be expected that the situation is much more complicated). Note however that, according to the results of [45,46,47], generic (in the Whitney sense) controlaffine systems do not admit any minimizing singular trajectory whenever the number of controls is more than two (more precisely it is shown in these references that such generic control-affine systems do not admit any trajectories satisfying the Goh necessary condition derived in [160]).…”

Section: Open Challengesmentioning

confidence: 99%

See 2 more Smart Citations

Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

186

179

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This article surveys the classical techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the classical tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 80's and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric re-entry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem towards a simpler one, and then of solving a series of parametrized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low cost interplanetary space missions. The article ends with open problems and perspectives.

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Section: Geometric Optimal Control Results and Application To The Promentioning

confidence: 99%

Section: Generalizations Open Problems and Challengesmentioning

confidence: 99%

Section: Open Challengesmentioning

confidence: 99%

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Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

186

179

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“…This is motivated by the idea of adding Dirac variations at the switching points, although such variations cannot be realized or even approximated using bounded controls. Agrachev, Stefani, and Pezza [16] derived a second-order sufficient condition for optimality of bang-bang controls using a Hamiltonian approach (see also [17]). Miliutin and Osmolovskii [18] and Maurer and Osmolovskii [19] developed a rather different second-order optimality condition using an approach that is closely related to the calculus of variations [18,19].…”

Section: Sufficient Conditions For Local Optimalitymentioning

confidence: 99%

“…To make this more concrete, we cite here the sufficient optimality condition derived in [17], specialized for the case of a single-input system. The problem considered is minimumtime control for (1).…”

Section: Sufficient Conditions For Local Optimalitymentioning

confidence: 99%

A simplification of the Agrachev–Gamkrelidze second-order variation for bang–bang controls

Ratmansky

Margaliot

2010

Systems & Control Letters

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We consider an expression for the second-order variation (SOV) of bang-bang controls derived by Agrachev and Gamkrelidze. The SOV plays an important role in both necessary and sufficient second-order optimality conditions for bang-bang controls. These conditions are stronger than the one provided by the first-order Pontryagin maximum principle (PMP). For a bang-bang control with k switching points, the SOV contains k(k + 1)/2 Lie-algebraic terms. We derive a simplification of the SOV by relating k of these terms to the derivative of the switching function, defined in the PMP, evaluated at the switching points. We prove that this simplification can be used to reduce the computational burden associated with applying the SOV to analyze optimal controls. We demonstrate this by using the simplified expression for the SOV to show that the chattering control in Fuller's problem satisfies a second-order sufficient condition for optimality.

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