Convergence Results for Smooth Regularizations of Hybrid Nonlinear Optimal Control Problems

Haberkorn, Thomas; Trélat, Emmanuel

doi:10.1137/100809209

Cited by 47 publications

(61 citation statements)

References 22 publications

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“…Ignoring the coordinates (L, l, χ), the induced optimal control problem is to steer the above three-dimensional control system, from a given x(0) to a final target (γ(t f ) is free but the two first coordinates are fixed), with a control satisfying the constraint |u| 1, and moreover, under the three state constraints on the thermal flux, normal acceleration and dynamic pressure (which depend only on x), by minimizing the cost (21).…”

Section: Geometric Optimal Control Results and Application To The Promentioning

confidence: 99%

“…Indeed, those two conditions are local, whereas the maximization condition (9) is global. In the proof of the general version of the Pontryagin Maximum Principle, needle-like variations of the control are the main tool in order to derive the strong condition (9) (note that that a short proof of the Pontryagin Maximum Principle is provided in the general case, with needle-like variations and with a conic implicit function theorem, in [21]). …”

Section: Pontryagin Maximum Principlementioning

confidence: 99%

“…The Pontryagin Maximum Principle can be generalized to wider classes of functionals and boundary conditions; for instance, periodic boundary conditions (see [17]), systems involving some delays (see [14,26]), intermediate conditions (see [9,27,28]), or, more generally, hybrid systems where the dynamics may change along the trajectory, accordingly to time and/or state conditions (see [29,30,31,32,33,21]). In particular in this last case, when the system crosses a given boundary then a jump condition must hold on the adjoint vector, which means that the adjoint vector is no more continuous (but is however piecewise absolutely continuous).…”

Section: Generalizationsmentioning

confidence: 99%

“…The advantage of this method is that along the boundary arcs the control can be easily expressed in closed-loop (feedback), which is greatly convenient for stabilization issues and for real-time embarked implementation. Anyway this strategy is not optimal for the minimization criterion (21), and it was the aim of [42,41,97,117] to solve this optimal control problem with geometric considerations.…”

Section: The Atmospheric Re-entry Problemmentioning

confidence: 99%

“…To eliminate the first possibility, since a limit of optimal controls is optimal as well (see e.g. [25,21]), we can make the assumption of the absence of minimizing singular trajectory and of conjugate point over all the domain under consideration (not only along the continuation path), and for every λ ∈ [0, 1]. As said before, the absence of singular minimizing trajectory over the whole space is generic for large classes of systems, hence this is a reasonable assumption; however the global absence of conjugate point is a strong assumption.…”

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confidence: 99%

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Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

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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: Geometric Optimal Control Results and Application To The Promentioning

confidence: 99%

Section: Pontryagin Maximum Principlementioning

confidence: 99%

Section: Generalizationsmentioning

confidence: 99%

Section: The Atmospheric Re-entry Problemmentioning

confidence: 99%

mentioning

confidence: 99%

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Optimal Control and Applications to Aerospace: Some Results and Challenges

Trélat

2012

J Optim Theory Appl

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189

179

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An interior penalty method for optimal control problems with state and input constraints of nonlinear systems

Malisani

Chaplais

Petit

2014

Optim Control Appl Methods

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SUMMARYThis paper exposes a methodology to solve state and input constrained optimal control problems for nonlinear systems. In the presented 'interior penalty' approach, constraints are penalized in a way that guarantees the strict interiority of the approaching solutions. This property allows one to invoke simple (without constraints) stationarity conditions to characterize the unknowns. A constructive choice for the penalty functions is exhibited. The property of interiority is established, and practical guidelines for implementation are given. A numerical benchmark example is given for illustration.

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Minimal time crisis versus minimum time to reach a viability kernel: A case study in the prey‐predator model

Bayen

Rapaport

2018

Optim Control Appl Methods

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In this work, we provide a reformulation of the minimal time crisis problem associated with a constraint set as a free terminal time control problem. The proof requires the existence of a nonempty viability kernel that is reachable from the state space. In addition, we suppose a uniform lower bound between two consecutive crossing times of the constraint set. Thanks to this result, we compute an optimal synthesis for the minimal time crisis problem governed by the prey-predator dynamics, with a controlled mortality on the predators. Finally, we compare the time spent in the crisis set by optimal trajectories of the minimal time crisis problem with the minimum time problem to reach the viability kernel. This is made possible by means of an exact characterization of the viability kernel. KEYWORDSoptimal control, Pontryagin's maximum principle, prey-predator model, viability theory 330

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Convergence Results for Smooth Regularizations of Hybrid Nonlinear Optimal Control Problems

Cited by 47 publications

References 22 publications

Optimal Control and Applications to Aerospace: Some Results and Challenges

Optimal Control and Applications to Aerospace: Some Results and Challenges

An interior penalty method for optimal control problems with state and input constraints of nonlinear systems

Minimal time crisis versus minimum time to reach a viability kernel: A case study in the prey‐predator model

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