On the Generality of Variational Principles

Pedregal, Pablo

doi:10.1007/s00032-003-0023-0

Cited by 13 publications

(4 citation statements)

References 14 publications

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“…is the Langrangian function associated to problem ( 2)-( 3) (for more details see S. ATTAB -D. AFFANE -M. F. YAROU [10], [12], [24]) and…”

Section: An Optimal Control Problem With No Convexity Assumptionsmentioning

confidence: 99%

On a Non-Convex Lagrange Optimal Control Problem

Attab,

Affane,

Yarou

2024

Tatra Mountains Mathematical Publications

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“…is the Langrangian function associated to problem ( 2)-( 3) (for more details see S. ATTAB -D. AFFANE -M. F. YAROU [10], [12], [24]) and…”

Section: An Optimal Control Problem With No Convexity Assumptionsmentioning

confidence: 99%

On a Non-Convex Lagrange Optimal Control Problem

Attab,

Affane,

Yarou

2024

Tatra Mountains Mathematical Publications

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“…We start by recalling that under suitable conditions (see e.g., ,Prop. 3.1 for the details), OCP is equivalent to the variational problem

VP ({, \begin{matrix} falsenone nonefalsearray \\ arrayleft Minimize in x : & arrayleft \overset{true¯}{J} MathClass-open(x MathClass-close) = \int_{t_{0}}^{t_{f}} ϕ (t, x MathClass-open(t MathClass-close), x^{'} MathClass-open(t MathClass-close)) dt & arrayleft \\ arrayleft subject to & arrayleft x MathClass-open(t_{0} MathClass-close) = x^{0} & arrayleft \end{matrix})

here the density ϕ is given by

ϕ (t MathClass-punc, x MathClass-punc, ξ) MathClass-rel= \underset{normalmin}{msub} F (t MathClass-punc, x) MathClass-punc.

From now on, in this section, we assume that the mesh t 0 < t 1 < t 2 < ⋯ < t N + 1 = t f is given by

t_{j MathClass-bin+ 1} MathClass-rel= t_{j} MathClass-bin+ h MathClass-punc, h MathClass-rel= (t_{f} MathClass-bin- t_{0}) MathClass-bin∕ N MathClass-punc, N MathClass-rel\in double-struckN MathClass-punc.

…”

Section: Description Of the Numerical Schemementioning

confidence: 99%

“…That is why closed‐loop control systems are more appropriate in real‐world engineering problems as the one described previously. Nevertheless, as we shall show later in this work, from the usual variational reformulation of optimal control problems (see e.g., ), a numerical scheme can be derived to compute digital optimal controls in feedback form which can be successfully applied to solve the maneuvering problem for manned submarines indicated at the beginning. This is the main goal of the present work.…”

Section: Introduction – Problem Formulationmentioning

confidence: 99%

A numerical method for computing optimal controls in feedback and digital forms and its application to the blowing‐venting control system of manned submarines

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Pedregal

Periago

2012

Optim Control Appl Methods

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SUMMARYOn the basis of the classical variational reformulation of optimal control problems, we introduce a numerical scheme for solving those problems where the goal is the computation of optimal controls in feedback and digital forms defined on a discrete time mesh. The algorithm reduces the computation of such controls to solving a suitable nonlinear mathematical programming problem where the unknowns are the controls and slope of the state variable of the original problem. The motivation for this study comes from the real‐world engineering problem which consists of maneuvering a manned submarine by using the blowing‐venting control system of the ballast tanks of the vehicle. After checking the proposed algorithm in an academic example, we apply it to the maneuvering problem of submarines whose mathematical model includes a state law which is composed of a system of twenty‐four nonlinear ordinary differential equations. Numerical results illustrate the performance of the numerical scheme. Copyright © 2012 John Wiley & Sons, Ltd.

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“…The underlying idea is to use the differential expression (1) in order to define a new objective functional subject to a set of constraints which are easier to deal with. The construction of this equivalent problem is performed in an elementary way [13]:…”

Section: F Is Measurable In X and Continuous In (U λ ξ H)mentioning

confidence: 99%

A Review of an Optimal Design Problem for a Plate of Variable Thickness

̃oz¹,

Pedregal²

2007

SIAM J. Control Optim.

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Abstract. We revisit a classic design problem for a plate of variable thickness under the model of Kirchhoff. Our main contribution has two goals. One is to provide a rather general existence result under a main assumption on the structure of the tensor of material constants. The other focuses on providing a minimal number of additional design variables for a relaxation of the problem when that assumption on the tensor of elastic constants does not hold. In both situations, the cost functional can be pretty general.

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On the Generality of Variational Principles

Cited by 13 publications

References 14 publications

On a Non-Convex Lagrange Optimal Control Problem

On a Non-Convex Lagrange Optimal Control Problem

A numerical method for computing optimal controls in feedback and digital forms and its application to the blowing‐venting control system of manned submarines

A Review of an Optimal Design Problem for a Plate of Variable Thickness

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