Analysis of the Optimal Relaxed Control to an Optimal Control Problem

Lou, Hongwei

doi:10.1007/s00245-008-9045-x

Cited by 10 publications

(8 citation statements)

References 20 publications

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“…With the aid of relaxed controls, Lou [12], [13] discussed the existence and nonexistence of optimal controls for linear elliptic equations. Later, motivated by the idea presented in [12], [13], the author [16] considered the case of semilinear elliptic controlled system. This present paper can be considered as a sequent work of [16] and there are three main differences between them.…”

Section: Shu Luanmentioning

confidence: 99%

“…The readers can refer to [7], [8], [9], [20]. When a convex condition is not assumed, relaxed controls have been proved to be an important tool for studying existence of optimal controls(see [12], [13], [14], [16], [17], for example). The concept of relaxed controls has evolved from generalized curves introduced by Young [24] and McShane [18] in the late 1930s.…”

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confidence: 99%

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On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions

Luan¹

2017

Mathematical Control &Amp; Related Fields

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Section: Shu Luanmentioning

confidence: 99%

mentioning

confidence: 99%

On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions

Luan¹

2017

Mathematical Control &Amp; Related Fields

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“…(8) and (10), respectively. For a givenν ∈ M , the function [13]), and since M is compact in the weak star topology, it attains a minimum in M . Therefore the following function, θ (μ) is well defined:…”

Section: Preliminary Version Of the Algorithmmentioning

confidence: 99%

“…( 8) and (10), respectively. For a given ν ∈ M , the function Θ( ν) := t f 0 H(x(t), ν(t), p(t))dt from M to R is continuous in the weak-star topology in M (see [13]), and since M is compact in the weak star topology, it attains a minimum in M . Therefore the following function, θ ( μ) is well defined:…”

Section: Preliminary Version Of the Algorithmmentioning

confidence: 99%

Hamiltonian-based algorithm for relaxed optimal control

Wardi

Egerstedt

Qureshi

2016

2016 IEEE 55th Conference on Decision and Control (CDC)

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This paper concerns a first-order algorithmic technique for a class of optimal control problems defined on switched-mode hybrid systems. The salient feature of the algorithm is that it avoids the computation of Fréchet or Gâteaux derivatives of the cost functional, which can be time consuming, but rather moves in a projected-gradient direction that is easily computable (for a class of problems) and does not require any explicit derivatives. The algorithm is applicable to a class of problems where a pointwise minimizer of the Hamiltonian is computable by a simple formula, and this includes many problems that arise in theory and applications. The natural setting for the algorithm is the space of continuoustime relaxed controls, whose special structure renders the analysis simpler than the setting of ordinary controls. While the space of relaxed controls has theoretical advantages, its elements are abstract entities that may not be amenable to computation. Therefore, a key feature of the algorithm is that it computes adequate approximations to relaxed controls without loosing its theoretical convergence properties. Simulation results, including cpu times, support the theoretical developments. arXiv:1609.03183v1 [math.OC]

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“…Comprehensive presentations of the theory of relaxed controls and their role in optimal control can be found in [10,11,35,36,37]; also see [38] for a re-cent survey. In the following paragraphs we summarize its main points that are relevant to our discussion.…”

Section: Relaxed Controlmentioning

confidence: 99%

Hamiltonian-Based Algorithm for Optimal Control

Hale¹,

Wardi²,

Jaleel³

et al. 2016

Preprint

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This paper proposes an algorithmic technique for a class of optimal control problems where it is easy to compute a pointwise minimizer of the Hamiltonian associated with every applied control. The algorithm operates in the space of relaxed controls and projects the final result into the space of ordinary controls. It is based on the descent direction from a given relaxed control towards a pointwise minimizer of the Hamiltonian. This direction comprises a form of gradient projection and for some systems, is argued to have computational advantages over direct gradient directions. The algorithm is shown to be applicable to a class of hybrid optimal control problems. The theoretical results, concerning convergence of the algorithm, are corroborated by simulation examples on switched-mode hybrid systems as well as on a problem of balancing transmission-and motion energy in a mobile robotic system.

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Analysis of the Optimal Relaxed Control to an Optimal Control Problem

Cited by 10 publications

References 20 publications

On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions

On the existence of optimal control for semilinear elliptic equations with nonlinear neumann boundary conditions

Hamiltonian-based algorithm for relaxed optimal control

Hamiltonian-Based Algorithm for Optimal Control

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