Second Order Sufficient Conditions for Time-Optimal Bang-Bang Control

Maurer, Helmut; Osmolovskii, Nikolai P.

doi:10.1137/s0363012902402578

Cited by 104 publications

(78 citation statements)

References 30 publications

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“…The occurrence of a conjugate point is then related with an overlap of the flow near the switching surface. In [70,71] optimization methods are given to test second order sufficient optimality conditions for such bang-bang situations. The idea is to reduce the problem to the finitedimensional subproblem consisting of moving the switching times and a second variation is defined as a certain quadratic form associated to this subproblem.…”

Section: Generalizations Open Problems and Challengesmentioning

confidence: 99%

“…From the algorithmic point of view, note that, although the theory of conjugate times in the bang-bang case has been well developed, the computation of conjugate times in the bang-bang case is difficult in practice with the algorithms of the previously mentioned references (see in particular [70,71] and references therein). Besides, in the smooth case, as explained in the previous section efficient tools are available (see [23]).…”

Section: Generalizations Open Problems and Challengesmentioning

confidence: 99%

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Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

189

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: Generalizations Open Problems and Challengesmentioning

confidence: 99%

Section: Generalizations Open Problems and Challengesmentioning

confidence: 99%

Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

189

179

View full text Add to dashboard Cite

show abstract

“…In the literature on control problems governed by ordinary differential equations (ODEs) there are many contributions dealing with second-order conditions in the bang-bang case; see, e.g., [11,13,14,17,18,19,20]. In these contributions one typically assumes that the (differentiable) switching function σ : [0, T ] → R possesses only finitely many zeros and that |σ(t)| > 0 is satisfied for all zeros t of σ.…”

mentioning

confidence: 99%

Sufficient Second-Order Conditions for Bang-Bang Control Problems

Casas¹,

Wachsmuth²,

Wachsmuth³

2017

SIAM J. Control Optim.

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Abstract. We provide sufficient optimality conditions for optimal control problems with bangbang controls. Building on a structural assumption on the adjoint state, we additionally need a weak second-order condition. This second-order condition is formulated with functions from an extended critical cone, and it is equivalent to a formulation posed on measures supported on the set where the adjoint state vanishes. If our sufficient optimality condition is satisfied, we obtain a local quadratic growth condition in L 1 (Ω).

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“…The set Q n+1 can then be determined as follows. Equation (19) implies that there exists β ∈ R n \ {0} such that…”

Section: Necessary Condition For Optimalitymentioning

confidence: 99%

“…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]. The form of this condition is suitable for computational applications [20].…”

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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Second Order Sufficient Conditions for Time-Optimal Bang-Bang Control

Cited by 104 publications

References 30 publications

Optimal Control and Applications to Aerospace: Some Results and Challenges

Optimal Control and Applications to Aerospace: Some Results and Challenges

Sufficient Second-Order Conditions for Bang-Bang Control Problems

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

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