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A note on junction conditions for partially singular trajectories

Authors

David Bell
Type: Article
Publication Date: 1978-07-01
Citations: 10
DOI: https://doi.org/10.1080/00207177808922436

Abstract

Abstract Certain necessary conditions for joining optimal singular and non-singular subarcs are well established in the control literature. This paper shows that different necessary conditions at junctions may be applicable under certain circumstances.

Locations

  • International Journal of Control
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Necessary conditions at the junction of singular and non-singular control subarcs

1979-06-01
David Bell , M. BOISSARD
Abstract It has been known for some time that modifications may be necessary to existing conditions at junctions on partially singular trajectories. Such modifications would be governed by the role … Abstract It has been known for some time that modifications may be necessary to existing conditions at junctions on partially singular trajectories. Such modifications would be governed by the role played by certain time derivatives which exhibit discontinuities at the junction. This paper investigates the simplest case in which such discontinuities can appear and demonstrates that either the modifications are not necessary or that the usual analysis is not sufficient to establish a result. An example is given to illustrate one of the existing conditions necessary at a junction.

Necessary conditions for joining optimal singular and nonsingular subarcs

2000-01-01
J. P. Mcdanell , William F. Powers
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Minimum Time Optimality of a Partially Singular Arc: Second Order Conditions

2007-10-05
Laura Poggiolini , Gianna Stefani

A Shooting Algorithm for Optimal Control Problems with Singular Arcs

2013-01-17
M. Soledad Aronna , J. Frédéric Bonnans , Pierre Martinon
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Singular Optimal Control - The State of the Art

1996-01-01
Volker Michel
The purpose of this paper is to present the state of the art in singular optimal control. If the Hamiltonian in an interval \([t_1,t_2]\) is independent of the control we … The purpose of this paper is to present the state of the art in singular optimal control. If the Hamiltonian in an interval \([t_1,t_2]\) is independent of the control we call the control in this interval singular. Singular optimal controls appear in many applications so that research has been motivated since the 1950s. Often optimal controls consist of nonsingular and singular parts where the junctions between these parts are mostly very difficult to find. One section of this work shows the actual knowledge about the location of the junctions and the behaviour of the control at the junctions. The definition and the properties of the orders (problem order and arc order), which are important in this context, are given, too. Another chapter considers multidimensional controls and how they can be treated. An alternate definition of the orders in the multidimensional case is proposed and a counterexample, which confirms a remark given in the 1960s, is given. A voluminous list of optimality conditions, which can be found in several publications, is added. A strategy for solving optimal control problems numerically is given, and the existing algorithms are compared with each other. Finally conclusions and an outlook on the future research is given.

Necessary optimality conditions for different phase portraits in a neighborhood of a singular arc

2006-12-01
A.A. Melikyan

A note on a singular optimal control problem

1969-11-01
J.B. Moore

On singular arcs in nonsmooth optimal control

2008-01-01
Hans Joachim Oberle , Ricki Rosendahl

Necessary integral optimality conditions for quasi-singular controls in one control problem

2008-06-01
E. E. Gafarov

Junction times in singular optimal control

1995-07-01
David J. W. Ruxton , David Bell

Necessary conditions for optimal control problems with discontinuous trajectories

1986-06-01
Фернандо Лобо Перейра , Richard Vinter

Integral and multipoint necessary optimality conditions of quasi-singular controls in one optimal control problem

2020-06-01
Shahla Majid gyzy Rasulzade
Ш.М Ш.М
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Singular Trajectories in Optimal Control

2021-01-01
Bernard Bonnard , Monique Chyba
Singular trajectories arise in optimal control as singularities of the end-point mapping. Their importance has long been recognized, at first in the Lagrange problem in the calculus of variations where … Singular trajectories arise in optimal control as singularities of the end-point mapping. Their importance has long been recognized, at first in the Lagrange problem in the calculus of variations where they are lifted into abnormal extremals. Singular trajectories are candidates as minimizers for the time-optimal control problem, and they are parameterized by the maximum principle via a pseudo-Hamiltonian function. Moreover, besides their importance in optimal control theory, these trajectories play an important role in the classification of systems for the action of the feedback group.

Singular Trajectories in Optimal Control

2015-01-01
Bernard Bonnard , Monique Chyba
Singular trajectories arise in optimal control as singularities of the end-point mapping. Their importance has long been recognized, at first in the Lagrange problem in the calculus of variations where … Singular trajectories arise in optimal control as singularities of the end-point mapping. Their importance has long been recognized, at first in the Lagrange problem in the calculus of variations where they are lifted into abnormal extremals. Singular trajectories are candidates as minimizers for the time-optimal control problem, and they are parameterized by the maximum principle via a pseudo-Hamiltonian function. Moreover, besides their importance in optimal control theory, these trajectories play an important role in the classification of systems for the action of the feedback group.

Near optimal closed-loop solutions to differential games with singular arcs

1972-12-01
G. M. Anderson

A planar intercept problem with a chattering junction of non-singular and singular subarcs

2006-02-06
Klaus Schnepper

6. Optimal Control

1999-01-01
Previous chapter Next chapter Classics in Applied Mathematics Singular Perturbation Methods in Control: Analysis and Design6. Optimal Controlpp.249 - 287Chapter DOI:https://doi.org/10.1137/1.9781611971118.ch6PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutExcerpt 6.1 Introduction In … Previous chapter Next chapter Classics in Applied Mathematics Singular Perturbation Methods in Control: Analysis and Design6. Optimal Controlpp.249 - 287Chapter DOI:https://doi.org/10.1137/1.9781611971118.ch6PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutExcerpt 6.1 Introduction In optimal control problems on sufficiently long intervals, boundary layer phenomena occur even if the system is not singularly perturbed. In this case, the time scales are due to the different nature of the control tasks near the ends of the interval and over the rest of the interval. In a minimum energy problem, for example, the requirement to satisfy the initial and end conditions dominates at the ends of the interval, while the energy minimization requirement is dominant over the rest of the interval. Optimal trajectories of this kind have a familiar "take-off", "cruise" and "landing" pattern, with the boundary layers being the "take-off" and "landing" parts. For fast subsystems of singularly perturbed systems the intervals are always long compared with their dynamics. Their boundary layers are superimposed on the solution of the slow subsystem. In this chapter, optimal control problems for singularly perturbed systems are approached by a preliminary study in Section 6.2 of boundary layers appearing in systems that are not singularly perturbed. This section is crucial for the understanding of the rest of the chapter. After a definition of the reduced problem in Section 6.3, we apply the results of Section 6.2 to near-optimal linear and nonlinear control in Sections 6.4 and 6.5, and to "cheap control" and singular arc analysis in Section 6.6. Previous chapter Next chapter RelatedDetails Published:1999ISBN:978-0-89871-444-9eISBN:978-1-61197-111-8 https://doi.org/10.1137/1.9781611971118Book Series Name:Classics in Applied MathematicsBook Code:CL25Book Pages:xv + 366Key words:stability, optimal control, eigenvalue, near-optimality, approximation, boundary-layer

Singular trajectories of control-affine systems

2006-07-19
Yacine Chitour , Frédéric Jean , Emmanuel Trélat
When applying methods of optimal control to motion planning or stabilization problems, some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, singular minimizing trajectories … When applying methods of optimal control to motion planning or stabilization problems, some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, singular minimizing trajectories of the underlying optimal control problem. In this article, we provide characterizations for singular trajectories of control-affine systems. We prove that, under generic assumptions, such trajectories share nice properties, related to computational aspects; more precisely, we show that, for a generic system -- with respect to the Whitney topology --, all nontrivial singular trajectories are of minimal order and of corank one. These results, established both for driftless and for control-affine systems, extend previous results. As a consequence, for generic systems having more than two vector fields, and for a fixed cost, there do not exist minimizing singular trajectories. We also prove that, given a control system satisfying the LARC, singular trajectories are strictly abnormal, generically with respect to the cost. We then show how these results can be used to derive regularity results for the value function and in the theory of Hamilton-Jacobi equations, which in turn have applications for stabilization and motion planning, both from the theoretical and implementation issues.

Singular trajectories of control-affine systems

2006-01-01
Yacine Chitour , Frédéric Jean , Emmanuel Trélat
When applying methods of optimal control to motion planning or stabilization problems, some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, singular minimizing trajectories … When applying methods of optimal control to motion planning or stabilization problems, some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, singular minimizing trajectories of the underlying optimal control problem. In this article, we provide characterizations for singular trajectories of control-affine systems. We prove that, under generic assumptions, such trajectories share nice properties, related to computational aspects; more precisely, we show that, for a generic system -- with respect to the Whitney topology --, all nontrivial singular trajectories are of minimal order and of corank one. These results, established both for driftless and for control-affine systems, extend previous results. As a consequence, for generic systems having more than two vector fields, and for a fixed cost, there do not exist minimizing singular trajectories. We also prove that, given a control system satisfying the LARC, singular trajectories are strictly abnormal, generically with respect to the cost. We then show how these results can be used to derive regularity results for the value function and in the theory of Hamilton-Jacobi equations, which in turn have applications for stabilization and motion planning, both from the theoretical and implementation issues.
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Discontinuous control problems for non-convex dynamics and near viability for singularly perturbed control systems

2010-07-05
Dan Goreac , Oana‐Silvia Serea

Necessary conditions at the junction of singular and non-singular control subarcs

1979-06-01
David Bell , M. BOISSARD
Abstract It has been known for some time that modifications may be necessary to existing conditions at junctions on partially singular trajectories. Such modifications would be governed by the role … Abstract It has been known for some time that modifications may be necessary to existing conditions at junctions on partially singular trajectories. Such modifications would be governed by the role played by certain time derivatives which exhibit discontinuities at the junction. This paper investigates the simplest case in which such discontinuities can appear and demonstrates that either the modifications are not necessary or that the usual analysis is not sufficient to establish a result. An example is given to illustrate one of the existing conditions necessary at a junction.

A numerical algorithm for singular optimal control synthesis using continuation methods

1994-10-01
Yaobin Chen , Jian Huang
Abstract This paper presents an efficient computational method for the synthesis of singular optimal control problems. The proposed numerical procedure consists of two phases. In the first phase the original … Abstract This paper presents an efficient computational method for the synthesis of singular optimal control problems. The proposed numerical procedure consists of two phases. In the first phase the original singular optimal control problem is converted into a non‐singular one by adding to the performance index a perturbed (or weighted) energy term. The resultant boundary value problem can easily be solved for an appropriately large value of the perturbation parameter. In the second phase the solution obtained from the first phase is refined in a systematic manner based on continuation methods (imbedding methods or homotopy methods) until the optimal (or suboptimal) solution to the original problem is achieved. One of the major advantages of the proposed algorithm is that the resultant two‐point boundary value problem need be solved just once for a properly large perturbation parameter and the refinement of the solution is accomplished by solving a set of initial value problems sequentially and/or in parallel as the perturbation parameter goes to zero. The proposed algorithm is therefore computationally efficient and applicable to a large class of optimal control problems with various boundary conditions (e.g. fixed and free terminal time). The practicability of the method is demonstrated by computer simulations on an example problem.

A Continuation Method for Singular Optimal Control Synthesis

1993-06-01
Yaobin Chen , Jian Huang
An efficient computational method is presented for the synthesis of singular optimal control in this paper. The proposed numerical procedure consists of two phases: initialization and refinement. In the initialization … An efficient computational method is presented for the synthesis of singular optimal control in this paper. The proposed numerical procedure consists of two phases: initialization and refinement. In the initialization phase, the original singular optimal contrl (SOC) problem is converted into nonsingular one by adding to the performance index a perturbed (or weighted) energy term. The resultant boundary value problem can easily be solved for an appropriately large value of the perturbation parameter. In the refinement phase, the solution obtained from the initialization phase is refined in a systematic manner based on continuation methods until the optimal (or sub-optimal) solution to the original SOC problem is achieved. One of the major advantages of the proposed algorithm is that the resultant two-point boundary value problem needs to be solved just one time and the refinement of the solution is accomplished by solving a set of initial value problems sequentially and/or in parallel as the perturbation parameter goes to zero. The proposed algorithm is, therefore, computationally efficient and applicable to a large class of optimal control problems with various boundary conditions (e.g., fixed and free terminal time). The practicability of the method is demonstrated by computer simulations on several example problems.

On Mcdanell's conjecture

2025-03-06
Mirhan Urkmez , Carsten Skovmose Kallesøe , Jan Dimon Bendtsen +1 more