Mohammad Hosein Houshmandrad
email: hoseinhoushmandrad@gmail.com

Keywords: developing countries, project Delivery system, contracting companies, infrastructure plans, technical cooperation

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Identifying the necessary indicators to confirm the qualification of the executive contract of infrastructure plans

Mohammad Hosein Houshmandrad, Mohammad Beizapour, Rahim Aminzadeh

To cite this version:

Mohammad Hosein Houshmandrad, Mohammad Beizapour, Rahim Aminzadeh. Identifying the necessary indicators to confirm the qualification of the executive contract of infrastructure plans. 5th national conference on applied research in civil engineering, architecture and urban management, Dec 2017, Tehran, Iran. hal-04110772

Identifying the necessary indicators to confirm the qualification of the executive contract of infrastructure plans , Mohammad Beizapour, Rahim Aminzadeh

1.Introduction

The advancements resulting from the sustainable economic and social development of societies have caused vast changes in project management. The construction industry as well as engineering projects in the oil, gas and petrochemical industries, hydropower and hydropower plants, dams, etc., need to be developed due to the increasing needs of communities and the greater impact of time, cost and quality in evaluations and decisions. On the other hand, achieving the goals of the project through the explanation of procedures, actions, sequence of events, contractual relationships and the limits of responsibilities and obligations of each of the involved factors, in order to successfully achieve the goals of the project. One of the main indicators of development and sustainable development in today's world is that countries have reliable infrastructures and rely on infrastructure facilities and parent industries, which is one of the main conditions for achieving this. Technical knowledge and financing are mentioned as two basic sources and main bottlenecks for the implementation of infrastructure projects and parent industries, which always cause challenges for the employer: 1 -Financing the construction of infrastructure projects sometimes puts the employer under such pressure that employers are forced to borrow from outside their organization and sometimes, if the borrowing problem is not fulfilled, it leads to the non-implementation or incompleteness of the infrastructure project.

-

The next influential source is the necessary technical knowledge to carry out and develop infrastructure facilities, which is transferred to the contractor if the necessary knowledge exists in the employer's system. However, in case of lack of knowledge of technical knowledge, he is forced to acquire or transfer it from outside the organization, and if this transfer or acquisition of knowledge is not realized, the employer is bound to terminate the project. Therefore, in order to get rid of this challenge in infrastructure projects, the employer entrusts the provision of these two items to the contracting company implementing infrastructure projects by using new contract methods, therefore the role of the contractor in these projects is of particular importance. that the ability or weakness of the contractor implementing these projects has a significant impact on achieving goals in the development path. Therefore, according to the aforementioned propositions and the importance of the role of the contracting company or group of contracting companies (consortium), we are looking for indicators to determine the ability of the contracting company (domestic or foreign) before referring the work [START_REF] Fuomshi | Drafting and writing of all types of contracts[END_REF][START_REF] Arjmand | Meters and estimates and the basic principles of contracting[END_REF][START_REF] Parisi | Research method in science[END_REF][START_REF] Azar | Applied decision making: MADM approach[END_REF][START_REF] Azar | Fuzzy management science[END_REF][START_REF] Maleki | Management of contractual affairs of construction projects[END_REF][START_REF] Houshmandrad | Choosing the most optimal type of contract in infrastructure projects from the employer's point of view in the field of technical participation and getting rid of the pressures of liquidity provision[END_REF][START_REF] Houshmand | employer in the field of technical assistance and relief from the pressures of funding[END_REF].

PROMETHEE technique

In this part, PROMETHEE technique is introduced based on J.P. Burns and B. Marshall's article .PROMETHEE-1 (partial ranking) and PROMETHEE-2 (full ranking) by J. P. Burns presented for the first time in the conference organized by R. Nadeo and M. Landry was presented at the University of Lowell, Quebec, Canada. In the same year, many applications of this method in the field of health care by J. Davignon was used. A few years later, J. P. Burns and B. Marshall presented PROMETHEE-3 (ranking based on distances) and PROMETHEE-4 (continuous mode). In 1988, the same authors proposed the interactive visual module Gaia, which is a stunning graphical representation to support the PROMETHEE methodology. In 1992 and 1994, J. P. Burns and B. In addition to the previous methods, Marshall proposed two interesting appendices called PROMETHEE-5 (MCDA including partial constraints) and PROMETHEE-6 (representation of the human mind). So far, a significant number of successful applications of PROMETHEE methodology in various fields such as banking, industrial location, human resource planning, water resources management, investment, medicine, chemistry, health care [START_REF] Mobasseri | Impact of driving style on fuel consumption[END_REF][START_REF] Mobasseri | Traffic noise and it's measurement methods[END_REF][START_REF] Mobasseri | Intelligent Circuit Application for Detecting the amount of Air in Automobile Tires[END_REF][START_REF] Mobasseri | A Comparative Study Between ABS and Disc Brake System Using Finite Element Method[END_REF], tourism, dynamic management, etc. have been discussed. The success of this method is mainly due to its mathematical properties and especially the ease of working with them. Consider the following multi-criteria problem: max So that A is a set of possible options and It is a set of evaluation criteria. Here, the maximum (Max) and minimization (Min) of some criteria are not considered, what is expected from the decision maker is to identify an option that optimizes all the criteria. Below is the basic information of the multi-criteria problem, including 1 evaluation [START_REF] Maleki | Management of contractual affairs of construction projects[END_REF][START_REF] Bordogna | A fuzzy linguistic approach generalizing boolean information retrieval: A model and its evaluation[END_REF][START_REF] Collier K | Managing construction contract[END_REF][START_REF] Bucklet | fuzzy hierarchical analysis. fuzzy sets and systems[END_REF].

Table1.Evaluation

Providing a solution for a multi-criteria problem does not only depend on the basic information such as the evaluation table, but also depends on the decision maker himself. There is no absolute optimal solution! The best consensus solution (middle) depends on the personal preferences of each decision maker, which are present in the "mind" of each decision maker.

Consequently, peripheral information is needed to represent these subjective preferences to guide the decision maker toward a useful decision goal.

Consequently, peripheral information is needed to represent these subjective preferences to guide the decision maker toward a useful decision goal. The usual relation related to multi-criteria problems of type ( 1) is defined as follows.

For every So that P, I, and R are respectively abbreviations for preference, indifference, and incomparability, which are very clear definitions. An option is better than other options if it is better or at least equal to the assumed option in all criteria. If one option is better than others in criterion s and another option is better in criterion r. It is not possible to choose the best option without using additional information. Therefore, in this case, the two options are not comparable.

Options that are not defeated by other options are called efficient solutions. In a given evaluation table for a specific multi-criteria problem, usually most of the options (and often all of them) are efficient. Often there is a weak correlation of type from I and P. When one option is better from the point of view of one criterion, other options are better on other criteria. As a result, incomparability has happened in most of the 2 pairwise comparisons, in which case it is not possible to make a decision without additional information. For example, this information can include:

1-interactions between criteria; 2-The value function that combines all the criteria in a single function to obtain a single criteria problem for which there is an optimal solution; 3-Weights that show the relative importance of the criterion; 4-the preferences related to each pairwise comparison within each criterion; 5-Thresholds that establish the preferred limits.

So far, a large number of multi-criteria decision support methods have been proposed, all of these methods are derived from the same evaluation table, but they are different in terms of additional information needed. PROMETHEE methods require clear supplementary information and this information is easily obtained and understandable by decision makers and analysts.

The goal of all multi-criteria methods is to enrich the dominance graph, for example; Reducing the number of inconsistencies. (R) When the utility function is built, the multicriteria problem becomes a single-criteria problem for which there is an optimal solution. This seems like an overstatement because it is based on very strong assumptions (are all our decisions really based on a utility function defined somewhere in our brains?) and completely changes the structure of a decision problem. For this reason, B. Roy suggested that non-ranking relationships be created that only include realistic enrichments of dominance relationships. PEOMETHEE methods are among non-ranked methods.

In order to make a suitable multi-criteria method, some necessary items are listed below: Prerequisite 1: The range of deviations (differences) between the evaluations of the options for each criterion should be considered: This information can be easily calculated, but it is not used in efficiency theory. When these deviations are imperceptible, the communication of mastery can be enriched.

Necessary condition 2: Since the evaluations of gj(a) of each criterion are expressed according to its own unit, scale effects should be completely eliminated. Obtaining results based on the scales provided by the evaluations is not acceptable. Unfortunately, not all multi-criteria guidelines meet this condition.

Necessary condition 3: In the case of pairwise comparisons, a suitable multi-criteria method should provide the necessary information, which are: b is preferable to a; b and a are without difference; b and a are not comparable; Of course, the goal of this work is to reduce at least the number of incommensurability, but not when they are not realistic. So this instruction may be considered as a fair instruction. When all incommensurability for a particular instruction has been systematically retrieved by the information provided, this instruction can be further discussed.

Prerequisite 4: Different multi-criteria methods require different additional information and perform different calculation instructions, so that their proposed solutions are different. Therefore, it is important to develop methods that are understandable for decision makers. Prerequisite 5: An appropriate guideline should not include technical parameters that are not important to the decision maker. Requirement 6: An appropriate method should provide information about the conflicting nature of criteria.

Requirement 7: Most multi-criteria methods allocate relative importance weights for each criterion. These weights represent an important part of the decision maker's mentality. Assigning these weights to criteria is not an easy task and usually decision makers strongly dislike this practice. A suitable method should include sensitivity tools (sensitivity analysis) to easily test different sets of weights. PROMETHEE methods and the visual interactive module related to Gaia take into account all these conditions. On the other hand, some mathematical properties that multi-criteria problems may have can also be considered. PROMETHEE methods are designed to handle multi-criteria problems as well as related evaluation tables. Additional information required to implement PROMETHEE is clear and understandable for both analysts and decision makers, this information includes information between criteria and information within each criterion: According to the understanding of the set {wj, j=1,2,...,k}, the weights of the relative importance of different criteria should be completed. These weights are non-negative numbers and are independent of the measurement unit of each criterion. There is nothing wrong with considering the weights as normal because [START_REF] Azar | Fuzzy management science[END_REF][START_REF] Maleki | Management of contractual affairs of construction projects[END_REF][START_REF] Bordogna | A fuzzy linguistic approach generalizing boolean information retrieval: A model and its evaluation[END_REF][START_REF] Collier K | Managing construction contract[END_REF][START_REF] Bucklet | fuzzy hierarchical analysis. fuzzy sets and systems[END_REF][START_REF] Dalkey | An experimental application of the Delphi method to the use of experts[END_REF]:

Results

Data analysis in this research is done by the following three techniques, which are:

• Receiving feedback and statistical analysis of answers to questions in a group and finalizing the effective criteria in evaluating and ranking contractors implementing infrastructure projects, using the fuzzy Delphi technique [START_REF] Dalkey | An experimental application of the Delphi method to the use of experts[END_REF];

• Carrying out paired comparisons and weighting the selected criteria using the fuzzy hierarchical analysis technique;

• Final ranking of indicators with the help of Prometheus technique.

Using library studies and the use of experts' opinions, about 50 related indicators were identified to identify the competence of contracting companies implementing infrastructure projects, and finally 32 of them were approved, as described in the following table: The results of index ranking with the Prometheus method state that the indexes that have more out-of-ranking flow are more prioritized. Therefore, the definition of the acceptable limit for confirming the indicators is as described in the table below. In the following, the rejection or acceptance status of the criteria is as described in the following table.

Conclusion

As a result, after collecting the opinions of the elites and summarizing the indicators and performing the analysis, 12 indicators were finally accepted. Therefore, the criterion for identifying and distinguishing the contracting company qualified to carry out the country's infrastructure plans (projects) is as follows:

1-Technical dimension

• Having a comprehensive project planning and control system

• How to comply with standards and technical specifications in previous projects

• How to implement the previous projects in terms of quality, cost and expected schedule

• 3 -

 3 Condition and usability of equipment and machines 2-Experimental dimension • Continuous communication and coordination with the contractor and the monitoring system The dimension of organization and management • Level of education, field of study and executive experience of expert staff and key elements 4-Good record and credibility • Acquiring various qualifications from official and reliable organizations and bodies • Credit and goodwill of the company and key personnel 5-Dimension of financial qualification • Financial ability and support 6-After the suggested price •Price

Table 2 .

 2 Final relative importance weight … … ... … The highest weight is the most important criterion.

Table No . 3-Contractor rating criteria

 No

	3		How to implement the previous projects in terms of
			quality, cost and schedule
	4		Compliance with relevant current laws such as
			environmental, labor and social security laws
	5		Compliance with the safety and protection instructions of
			the workshop
	6		Condition and usability of equipment and machines
	7		Complete and timely equipment of the workshop
	8	experimental	Executive experience in the desired field and field of
			work
	9		Being a local contractor or having experience at the
			project site
	10		Creativity and innovation in previous projects
	11		Application of appropriate executive methods and
			organization and order in workshop affairs
	12		Classification of workshop documents and documentation
			of previous works
	13		Continuous communication and coordination with the
			contractor and the monitoring system [17]
	14		The scale of completed projects
	15		Volume and type of works referred to the subcontractor
	16		Use of new technologies
	17	Organization and	Efficient management and suitable management system
		management and	for work execution
	18	planning of the country	Stability of board members and specialized staff
	19		The level of education, field of study and executive
			experience of the expert staff and key elements
	20		Presentation of articles in conventions, conferences and
			specialized magazines
	21		Works, compositions and scientific and technical research
	22		Continuous training of employees
	23		Organization size
	24		Organizational behavior of the contractor
	25	Good history and	Acquiring various qualifications from official and reliable
		credibility	organizations and bodies
	26		Receiving official awards and recognitions
	27		Good record in previous jobs
	28		Credit and reputation of the company and key personnel
	29	Financial qualification	Financial ability and support
	30		Timely payment of wages for employees, workshop
			workers and subcontractors
	31		Insuring all facilities, equipment and personnel against
			possible accidents
	32		How to analyze the price
	33	Proposed price	price
	Row	aspect	index
	1	technical	Having a comprehensive project planning and control
			system
	2		How to comply with standards and technical
			specifications in previous projects

Table No . 4-Definition of acceptable limit

 No

	Determining	domain	Criterion	Scoring	The score	Score the
	the	share	selection	range of	of the	highest
	acceptable		factor	each	lowest	criterion
	score limit			criterion	criterion	value
					value

Table No . 5 -The final status of the criteria

 No

			subcontractor
	16		Use of new technologies	5	5 /	rejection
	17	Organization and	Efficient management and suitable management system for work execution	9	25 /	confirmation
	18	management and planning	Stability of board members and specialized staff	3	66 /	rejection
	19	of the country	The level of education, field of study and
			executive experience of the expert staff and key elements	9	25 /	confirmation
	20		Presentation of articles in conventions, conferences and specialized magazines	4	66 /	rejection
	21		Works, compositions and scientific and technical research	3	25 /	rejection
	22		Continuous training of employees	5	5 /	rejection
	23		Organization size	5	5 /	rejection
	Row 24 1 25 2 26 3 27 4 28 5 29 30 6 31 32	aspect technical Good history and credibility Financial qualification	index Organizational behavior of the contractor Acquiring various qualifications from Having a comprehensive project planning and control system official and reliable organizations and bodies How to comply with standards and technical specifications in previous projects Receiving official awards and recognitions Good record in previous jobs How to implement the previous projects in terms of quality, cost and schedule Credit and reputation of the company and key personnel Compliance with relevant current laws such as environmental, labor and social security laws Compliance with the safety and protection instructions of the workshop Financial ability and support Timely payment of wages for employees, workshop workers and subcontractors Condition and usability of equipment and machines Insuring all facilities, equipment and personnel against possible accidents How to analyze the price	Final 25 / 3 average 25 / 9 25 / 9 25 / 48 / 5 9 14 / 7 25 / 9 5 / 5 5 / 5 5 / 5 25 / 9 25 / 3 25 / 9 25 / 3 25 / 3	Condition rejection confirmation confirmation confirmation rejection rejection confirmation rejection rejection confirmation rejection rejection rejection confirmation rejection
	7 33 Proposed price price Complete and timely equipment of the workshop	5 / 93 / 5 8	rejection confirmation
	8	experimental Executive experience in the desired field and field of work	9	25 /	confirmation
	9		Being a local contractor or having experience at the project site	9	25 /	confirmation
	10		Creativity and innovation in previous projects	5	5 /	rejection
	11		Application of appropriate executive
			methods and organization and order in workshop affairs	5	5 /	rejection
	12		Classification of workshop documents and documentation of previous works	5	5 /	rejection
	13		Continuous communication and
			coordination with the contractor and the monitoring system	9	25 /	confirmation
	14		The scale of completed projects	5	5 /	rejection
	15		Volume and type of works referred to the	5	5 /	rejection


Térence Bayen
email: terence.bayen@univ-avignon.fr

F Mairet

Approximation of chattering arcs in optimal control problems governed by mono-input affine control systems

Keywords: optimal control, singular arcs, chattering, numerical methods

In this paper, we consider a general Mayer optimal control problem governed by a mono-input affine control system whose optimal solution involves a second-order singular arc (leading to chattering). The objective of the paper is to present a numerical scheme to approach the chattering control by controls with a simpler structure (concatenation of bang-bang controls with a finite number of switching times and first-order singular arcs). Doing so, we consider a sequence of vector fields converging to the drift such that the associated optimal control problems involve only first-order singular arcs (and thus, optimal controls necessarily have a finite number of bang arcs). Up to a subsequence, we prove convergence of the sequence of extremals to an extremal of the original optimal control problem as well as convergence of the value functions. Next, we consider several examples of problems involving chattering. For each of them, we give an explicit family of approximated optimal control problems whose solutions involve bang arcs and first-order singular arcs. This allows us to approximate numerically solutions (with chattering) to these original optimal control problems.

Introduction

In numerous optimal control problems arising in various areas such as aerospace, biology, mechanics, the optimal solution involves what is called Fuller's phenomenon. In that case, we also say that chattering occurs. This property means that the optimal control possesses an infinite number of extremal bang arcs over a finite time interval. This particularly interesting behavior of optimal controls was originally discovered by A.T. Fuller in the seminal works [START_REF] Fuller | Study of an optimum non-linear control system[END_REF][START_REF] Fuller | Further study of an optimum non-linear control system[END_REF][START_REF] Fuller | Relay control systems optimized for various performance criteria[END_REF]. Since then, many properties related to chattering were studied by several authors (see, e.g., [START_REF] Ryan | Singular optimal controls for second-order saturating systems[END_REF] and the monograph [START_REF] Zelikin | Theory of chattering control, Systems & Control: Foundations & Applications[END_REF] giving a large panel on the chattering phenomenon).

Fuller's phenomenon arises whenever an optimal control problem presents a second order singular arc. Recall that in geometric optimal control theory, a singular singular arc is a time interval on which the optimal control takes values within the interior of the admissible control set. The occurrence of such arcs is of utmost importance to compute an optimal solution since it completely determines the optimal synthesis. This situation is encountered typically when computing minimum time syntheses in the two-dimensional setting in presence of a turnpike singular arc (see [START_REF] Boscain | Optimal syntheses for control systems on 2-D manifolds[END_REF] or [START_REF] Bayen | Optimal synthesis for the minimum time control problems of fedbatch bioprocesses for growth functions with two maxima[END_REF]). In the setting of Mayer optimal control problems governed by an affine control system, the application of the Pontryagin maximum Principle (in short, PMP) shows that several cases may occur (see, e.g., [START_REF] Chitour | Singular trajectories of control-affine systems[END_REF]). First, geometric control theory predicts that the control input is present explicitly in the expression defining the derivative of order 2n, n ≥ 1, of the switching function. So, if the control input is present in the second-order derivative of the switching function (allowing to compute the singular control as a function of the state and co-state), then, we say that a singular arc of first order occurs. But, it may happen that the control is present explicitly only in the fourth-order derivative of the switching function (and not in the second-order derivative) leading that way to a second-order singular arc and to chattering, see, e.g., [START_REF] Zelikin | Theory of chattering control, Systems & Control: Foundations & Applications[END_REF][START_REF] Schättler | Geometric optimal control[END_REF]. More generally, it is possible to encounter optimal control problems involving singular arcs of order greater than three. But, we shall restrict our attention here to the case of first and second order singular arcs that are the most common ones in the application models.

A striking point related to second order singular arcs is that they arise in simple examples from mathematical modeling such as in aerospace [START_REF] Zhu | Minimum time control of the rocket attitude reorientation associated with orbit dynamics[END_REF][START_REF] Zhu | Geometric optimal control and applications to aerospace[END_REF], in biomedicine [START_REF] Ledzewicz | Singular controls and chattering arcs in optimal control problems arising in biomedicine[END_REF][START_REF] Grigorieva | Chattering and its approximation in control of psoriasis treatment[END_REF][START_REF] Schättler | Optimal control for mathematical models of cancer therapies. An application of geometric methods[END_REF], in biology [START_REF] Mairet | The promise of dawn: microalgae photoacclimation as an optimal control problem of resource allocation[END_REF][START_REF] Mairet | Parameter estimation for dynamic resource allocation in microorganisms: a bi-level optimization problem[END_REF], or in physics [START_REF] Robin | Chattering phenomenon in quantum optimal control[END_REF]. The simplest problem involving chattering is known as Fuller's problem for which the underlying system is the two-dimensional double integrator associated with a quadratic cost. It is worth noticing that chattering renders the analysis of optimal solutions more delicate than for instance in the case of a first-order singular arc for which the optimal synthesis involves in general a low number of switching times. In particular, the occurrence of chattering raises the question of implementation of an optimal solution from a practical viewpoint. The question is how to bypass non-implementable solutions and how synthesizing an adequate sub-optimal control (typically based on a finite number of bang-bang arcs). This question was studied by several authors for instance in [START_REF] Caponigro | Regularization of chattering phenomena via bounded variation controls[END_REF] by modifying the objective function via bounded variation controls.

In the present paper, we consider another approach to tackle chattering based on a perturbation technique. More precisely, we consider a general optimal control problem governed by a mono-input affine control system whose optimal solution involves a second-order singular arc. We suppose that there is a sequence of dynamics converging to the original one such that the associated optimal solution involves only first-order singular arcs. This sequence is constructed by approximating the drift term. From a practical viewpoint, we thus approach the original optimal control by a perturbed one for which the optimal solution does not involve chattering. We first prove that, up to a sub-sequence, an extremal of the perturbed optimal control problems converges to an extremal to the original optimal control problem. As well, we prove the convergence of the value function associated to the perturbed optimal control problem to the value function associated with the original optimal control problem. Convergence of optimal controls (up to a sub-sequence) is also guaranteed on time intervals for which the limit trajectory is non-singular. Interestingly, the approximated optimal control is a concatenation of a finite number of bang and singular arcs. Next, we show how to construct such approximating sequence on three examples with chattering : a variation on Fuller's problem, a resource-allocation model [START_REF] Giordano | Dynamical allocation of cellular resources as an optimal control problem: novel insights into microbial growth strategies[END_REF], and a quantum control system [START_REF] Robin | Chattering phenomenon in quantum optimal control[END_REF].

The paper is organized as follows. In Section 2, we introduce our methodology and we prove convergence results between the sequence of approximated optimal control problems and the original optimal control problem. Next, in Section 3, we address three examples for which we provide an explicit sequence of approximated optimal control problems. For each of them, we verify that the limit problem possesses chattering whereas perturbed optimal control problems involve only first-order singular arcs. In addition, for each example, we provide numerical solutions (with an increasing number of arcs as the sequence approaches the original dynamics) to the approximated optimal control problems via a direct method [START_REF]BOCOP: an open source toolbox for optimal control[END_REF].

Approximation and convergence result 2.1 Second order singular arcs

Hereafter, we denote by A ⊤ the transpose of a matrix A and the inner product in the Euclidean space is written a • b where a, b are two vectors. Let n ∈ N * and consider two vector fields f, g : R n → R n of class C ∞ , a terminal pay-off ψ : R n → R of class C 1 , and let C be a non-empty closed convex subset of R n . Given T > 0 and x 0 ∈ R n , we consider the Mayer optimal control problem inf

u∈U ψ(x(T )) s.t.    ẋ(t) = f (x(t)) + u(t)g(x(t)) a.e. t ∈ [0, T ], x(0) = x 0 , x(T ) ∈ C, (OCP)
where U denotes the set of admissible controls u : [0, T ] → [-1, 1] that are measurable. The aim of this paper is to provide alternative strategies in order to study (OCP) whenever f and g satisfy the condition:

[g, [f, g]] = 0 over R n , (2.1)
or whenever the covector is orthogonal to [g, [f, g]] along a singular arc, where [f, g](x)

:= D x g(x)f (x) - D x f (x)g(x)
denotes the Lie bracket of f and g and where D x f (x), D x g(x) stand respectively for the Jacobian matrix of f and g. Under one of these two conditions, we know that optimal controls may involve an infinite number of switching times over [0, T ]. This phenomenon renders the analysis of optimal trajectories quite involved (e.g., the determination of the switching locus in this context). Thus, from a numerical point of view, it is desirable to deal with controls having a finite number of switching times.

First, we would like to remind why condition (2.1) implies the so-called Fuller's phenomenon (or chattering). From Pontryagin's Principle [START_REF] Pontryagin | Mathematical theory of optimal processes[END_REF][START_REF] Vinter | [END_REF], if u is an optimal control and x denotes the associated solution of the system defined over [0, T ], there exist p 0 ≤ 0 and an absolutely continuous function p : [0, T ] → R n (the covector) satisfying the adjoint equation

ṗ(t) = -D x f (x(t)) ⊤ p(t) -u(t)D x g(x(t)) ⊤ p(t) a.e. t ∈ [0, T ], (2.2)
together with the transversality condition

p(T ) + p 0 ∇ψ(x(T )) ∈ -N C (x(T )), (2.3)
where N C (x) denotes the (convex) normal cone to the set C at some point x ∈ C. The pair (p 0 , p(•)) also fulfills the non-triviality condition (p 0 , p(•)) ̸ = 0. Finally, the Hamiltonian maximization condition gives us the following control law:

u(t) =
φ(t) = p(t) • [f, g](x(t)), t ∈ [0, T ],
and that φ

(t) = p(t) • [f, [f, g]](x(t)) + u(t)p(t) • [g, [f, g]](x(t)) a.e. t ∈ [0, T ]. (2.5)
Thus, under (2.1), if a singular arc occurs, the (corresponding) singular control u cannot be expressed thanks to (2.5), but further derivations of φ are needed to compute the singular control (at least two according to the geometric control theory). This leads to the singular arcs of order at least two. In this direction, let us remind the following result about the occurrence of the chattering phenomenon for second-order singular arcs (see [START_REF] Schättler | Geometric optimal control[END_REF]). This means that the only possibility for an optimal control to connect the trajectory to the singular arc is an infinite sequence of bang arcs since the proposition excludes the concatenation of the singular arc to the optimal trajectory by a finite number of bang arcs. Also, singular controls may exceed the largest admissible value leading to saturation [START_REF] Bayen | Tangency property and prior-saturation points in minimal time problems in the plane[END_REF], that is why, the singular control is supposed to be admissible in Proposition 2.1. In addition to this proposition, let us also recall Legendre-Clebsch's condition [START_REF] Schättler | Geometric optimal control[END_REF] in the above context (with a single input). If a singular arc of order2 q ∈ N * is optimal over some time interval [t 1 , t 2 ], then it must satisfy the condition

- ∂ ∂u d 2q dt 2q ϕ ≤ 0.
If this inequality is strict, we then speak of the strict Legendre-Clebsch condition. Typically, if q = 1, then, ϕ must satisfy φ|u > 0 along a singular arc whereas if q = 2, then, it must satisfy φ|u = 0 and .... ϕ |u < 0.

Approximation of the optimal control problem

In order to avoid Fuller's phenomenon, we propose the following approach. We suppose that there is a sequence

f k : R n → R n (of class C ∞ for each k ∈ N) such that ∥f k -f ∥ L ∞ → 0, (2.7)
together with

∥D x f k -D x f ∥ L ∞ → 0, (2.8)
as k → +∞ (where

L ∞ := L ∞ (R n ) is the set of bounded functions) and such that [g, [f k , g]] ̸ = 0 over R n , (2.9)
for every k ∈ N. This hypothesis will allow us to prevent chattering (see the discussion in Section 2.4). Consider now the approximated Mayer optimal control problem:

inf u∈U ψ(x(T )) s.t.    ẋ(t) = f k (x(t)) + u(t)g(x(t)) a.e. t ∈ [0, T ], x(0) = x 0 , x(T ) ∈ C, (OCP k )
and let us introduce additional hypotheses in order to ensure existence of solutions to (OCP) and (OCP k ).

Hereafter | • | denotes the Euclidean norm in R n .

Assumption 2.1. We suppose that for every k ∈ N, the target set C is reachable from x 0 for the controlled dynamics

ẋ(t) = f k (x(t)) + u(t)g(x(t)) a.e. t ∈ [0, T ], (2.10)
where u ∈ U. Similarly, we assume that C is also reachable from x 0 for the controlled dynamics

ẋ(t) = f (x(t)) + u(t)g(x(t)) a.e. t ∈ [0, T ], (2.11)
where u ∈ U. In addition, we suppose that there is c ≥ 0 such that

∀k ∈ N, |f k (x)| ≤ c(|x| + 1) ; |f (x)| ≤ c(|x| + 1) ; |g(x)| ≤ c(|x| + 1) for all x ∈ R n .
(2.12)

Note that for simplicity, we supposed that the sequence (f k ) satisfies a linear growth condition, but this hypothesis could be removed in view of (2.7).

Lemma 2.1. If Assumption 2.1 is satisfied, then for every k ∈ N, (OCP k ) has a solution as well as (OCP).

Proof. Since the velocity sets associated to (2.10) and (2.11) are convex w.r.t. u, we can apply Filipov's existence Theorem (see [START_REF] Cesari | Optimization-Theory and applications. Problems with ordinary differential equations[END_REF]), whence the result.

Convergence results

For every k ∈ N, let u k be an optimal solution of (OCP k ). According to the PMP, there exist p 0 k ≤ 0 and an absolutely continuous function

p k : [0, T ] → R n such that (p 0 k , p k (•)) ̸ = 0 and such that ṗk (t) = -D x f k (x k (t)) ⊤ p k (t) -u k (t)D x g(x k (t)) ⊤ p k (t) a.e. t ∈ [0, T ], (2.13)
together with the transversality condition

p k (T ) + p 0 k ∇ψ(x k (T )) ∈ -N C (x k (T )). (2.14)
In addition, the Hamiltonian maximization condition implies that

u k (t) = sign(ϕ k (t)) a.e. t ∈ [0, T ] s.t. ϕ k (t) ̸ = 0, (2.15)
where ϕ k (t) := p k (t) • g(x k (t)). Finally, since the problem is autonomous the Hamiltonian of the problem is constant almost everywhere, that is, for all k ∈ N, there is

H k ∈ R such that H k := p k (t) • f k (x k (t)) + u k (t)p k (t) • g(x k (t)) a.e. t ∈ [0, T ].
The transversality condition (2.14) implies that for every k ∈ N, there exists

ξ k ∈ N C (x k (T )) such that p k (T ) + p 0 k ∇ψ(x k (T )) = -ξ k , so, the scalar r k := (p 0 k ) 2 + |ξ k | 2 is with positive value for all k ∈ N since the pair (p 0 k , p k (•)) is non-trivial. Hence, we can set q k := p k /r k . (2.16)
Hereafter, we say that a sequence (x k ) of absolutely continuous functions over [0, T ] strongly-weakly converges to a function

x ⋆ : [0, T ] → R n if and only if ∥x k -x ⋆ ∥ L ∞ ([0,T ] ; R n ) → 0 and ẋk ⇀ ẋ⋆ (weak convergence in L 2 ([0, T ] ; R n )).
Proposition 2.2. Suppose that Assumption 2.1 is satisfied. For every k ∈ N, denote by u k an optimal solution of (OCP k ) and let q k be defined by (2.13)-(2.14), and (2.16). Then, there exists an extremal (x ⋆ , p, , u ⋆ ) associated with (OCP) such that, up to a sub-sequence, (x k , q k ) strongly-weakly converges to (x ⋆ , p) over [0, T ]. In addition, one has the pointwise convergence

u k (t) → u ⋆ (t) on every time interval [t 1 , t 2 ] such that ϕ(t) ̸ = 0 where ϕ(t) := p(t) • g(x ⋆ (t)), t ∈ [0, T ].
In addition, u ⋆ is an optimal solution to (OCP).

Proof. From the linear growth assumption on the sequence of dynamics (f k ) and f (Assumption 2.1), we easily obtain that the sequence (x k ) is uniformly bounded over [0, T ] since the set of admissible controls is bounded. So, the sequence (x k (T )) is also bounded, hence, up to a subsequence, we may assume that it converges to some value w ∈ R n . Now, let us set α k := p 0 k /r k and β k := -ξ k /r k so that the transversality condition reads as follows:

q k (T ) + α k ∇ψ(x k (T )) = β k .
By compacteness of the unit ball in R 1+n , we may assume (extracting a sub-sequence if necessary) that the sequence (α k , β k ) k converges to a limit (α, β) ∈ R 1+n which is non-zero and such that α ≤ 0. By closedness property of the normal cone, one also obtains that -β ∈ N C (w). Next, observe that the Hamiltonian of the problem and that the adjoint equation are linear w.r.t. the covector, so the triple (x k , q k , u k ) satisfies:

Hk := H k r k = q k (t) • f k (x k (t)) + u k (t)q k (t) • g(x k (t)) a.e. t ∈ [0, T ], (2.17)
together with the state-adjoint system

ẋk (t) = f (x k (t)) + u k (t)g(x k (t)) + ηk (t), qk (t) = -D x f (x k (t)) ⊤ q k (t) -u k (t)D x g(x k (t)) ⊤ q k (t) + ηk (t), (2.18)
where for a.e. t ∈ [0, T ], ηk (t) and ηk (t) are defined as:

ηk (t) := f k (x k (t)) -f (x k (t)) ; ηk (t) := D x f (x k (t)) ⊤ q k (t) -D x f k (x k (t)) ⊤ q k (t).
Setting z k := (x k , q k ) and using that (x k ) is uniformly bounded over [0, T ], we obtain that there exists c ′ ≥ 0 such that for every k ∈ N one has

|z k (t)| ≤ c ′ (|z k (t)| + 1) a.e. t ∈ [0, T ], (2.19)
where | • | denotes the Euclidean norm in R 2n . From the preceding inequality and the fact that (x k (T ), q k (T )) converges, we obtain after application of Gronwall's Lemma that (

x k (•), q k (•)) is uniformly bounded over [0, T ]. Now the multi-function F (x, q) := {(f (x) + ug(x), -D x f (x) ⊤ q -uD x g k (x) ⊤ q) ; u ∈ [-1, 1]},
is with compact convex values for every (x, q) ∈ R 2n . Remind that z k (T ) converges and note also (using the uniform boundedness of (x k , q k ) in L ∞ ([0, T ] ; R n )) that the sequences ηk (•) and ηk (•) uniformly converge to zero. Hence, we can apply [START_REF] Clarke | Nonsmooth analysis and control theory[END_REF]Theorem 1.11] which shows the existence of an absolutely continuous pair (x ⋆ , p) as well as a control function u ⋆ ∈ U such that

ẋ⋆ (t) = f (x ⋆ (t)) + u ⋆ (t)g(x ⋆ (t)), ṗ(t) = -D x f (x ⋆ (t)) ⊤ p(t) -u ⋆ (t)D x g(x ⋆ (t)) ⊤ p(t), (2.20)
for a.e. t ∈ [0, T ]. In addition, (x k , q k ) strongly-weakly converges to (x ⋆ , p) over [0, T ]. Also, by passing to the limit as k → +∞, we obtain that x ⋆ (0) = x 0 and that p(T ) + α∇ψ(x ⋆ (T )) = β ∈ -N C (x ⋆ (T )) (using the closedness property of the normal cone) where (α, β) is non-zero and α ≤ 0. This proves the first property of Proposition 2.2.

Let us now address convergence of the sequence of controls (u k ) to u ⋆ . Suppose that there is t 0 ∈ [0, T ] such that ϕ(t 0 ) > 0. By uniform convergence of ϕ k to ϕ, there exist k 0 ∈ N, ε > 0, and κ > 0 such that for every k ≥ k 0 and every t ∈ [t 0 -ε, t 0 ], one has ϕ k (t) ≥ κ. Thus, by the Hamiltonian maximization condition, we obtain that u k (t) = +1 for every t ∈ [t 0 -ε, t 0 ] and every k ≥ k 0 . Hence, for all η ∈ (0, ε) and for all k ≥ k 0 , one has 1

η t0 t0-η ẋk (t) dt = 1 η t0 t0-η [f k (x k (t)) + g(x k (t))] dt.
Thus, if we let k → +∞ (utilizing uniform convergence of the sequences (f k ) and (x k ) and also weak convergence of ( ẋk )) and next η ↓ 0, one obtains using that almost every

t 0 ∈ [0, T ] is a Lebesgue point of u ⋆ (since u ⋆ is bounded): ẋ⋆ (t 0 ) = f (x ⋆ (t 0 )) + g(x ⋆ (t 0 )).
It follows that u ⋆ (t 0 ) = +1 = sign(ϕ(x ⋆ (t 0 ))). By repeating this argumentation in a neighborhood of every point t 0 ∈ [0, T ] such that ϕ(x ⋆ (t 0 )) ̸ = 0, we obtain that u ⋆ satisfies the Hamiltonian maximization condition (2.4) as desired as well as the pointwise convergence of (u k ) to u ⋆ at every point t 0 ∈ [0, T ] such that ϕ(t 0 ) ̸ = 0. This concludes the proof of the convergence of the sequence of controls (u k ) to u ⋆ . Finally, let us check that u ⋆ is an optimal solution to (OCP). Doing so, observe that

ψ(x k (T )) ≤ ψ(x k (T )),
for every admissible pair (x k , ũ) of (OCP). Note also that ũ is a fixed control in U. So, again using Gronwall's Lemma, we can check that (x k ) strongly-weakly converges to x over [0, T ] where x is the unique solution to ẋ(t) = f (x(t)) + u(t)g(x(t)), a.e. t ∈ [0, T ] together with x(0) = x 0 . Hence, passing to the limit as k → +∞ (ψ being continuous) implies that

ψ(x ⋆ (T )) ≤ ψ(x(T )),
for every admissible pair (x, ũ) to (OCP). This shows that u ⋆ is an optimal solution to (OCP) which ends the proof of the proposition.

Remark 2.1. As a byproduct of the previous proposition, we obtain that the value function associated to (OCP k ) converges to the value function associated to (OCP). In addition, the control u k uniformly converges to u ⋆ on every segment [t 1 , t 2 ] such that ϕ(t) ̸ = 0 over [t 1 , t 2 ].

Discussion about the employed methodology

Recall that in the present setting, chattering may occur whenever (2.1) is verified or whenever the covector is orthogonal to [g, [f, g]] along a singular arc. So, under one of these two conditions, we propose to introduce a sequence of functions (f k ) approaching f (see (2.7) and (2.8)) and such that (2.9) is verified for every k ∈ N.

The main advantage of considering (OCP k ) in place of (OCP) is that the behavior of an optimal solution to (OCP k ) might be simpler to handle. Indeed, as shows Proposition 2.1, one can expect Fuller's phenomenon for an optimal solution to (OCP) involving a singular arc. In contrast, if (x k , p k , u k ) is an extremal of (OCP k ), then, according to the PMP, the corresponding switching function ϕ k fulfills:

φk (t) = p k (t) • [f k , [f k , g]](x k (t)) + u k (t)p k (t) • [g, [f k , g]](x k (t)).
We can thus expect that along a singular arc

p k (t) • [g, [f k , g]](x k (t)
) ̸ = 0, so Fuller's phenomenon may not arise. In addition, if Legendre-Clebsch's condition is verified for every k ∈ N, then, we can also expect the occurrence of a singular arc of turnpike type (thus a first-order singular arc without involving an infinite number of switching times, see, e.g., [START_REF] Bayen | Optimal synthesis for the minimum time control problems of fedbatch bioprocesses for growth functions with two maxima[END_REF]). Hence, for the approximated problem, Fuller's phenomenon may not arise even if Legendre-Clebsch's condition is not verified (in that case, we speak of anti-turnpike [START_REF] Bonnard | Singular trajectories and their role in control theory[END_REF][START_REF] Boscain | Optimal syntheses for control systems on 2-D manifolds[END_REF]).

Note that we supposed the terminal time T > 0 fixed, but it can also be free (in that case, we can adapt the previous properties by using classical techniques considering an augmented state, see, e.g., [START_REF] Bressan | Introduction to the mathematical theory of control[END_REF]). In the next section, we provide three examples for which an optimal solution contains a second-order singular arc. Doing so, we introduce an explicit sequence of functions f k approaching the dynamics such that the associated optimal control problem does not possesses chattering.

Application to three examples involving chattering

Throughout this section, we are given a sequence (ε k ) of positive numbers such that ε k ↓ 0 as k → +∞. By limit problem, we mean the optimal control problem for which ε k = 0 (i.e. without perturbation).

3.1 A variant of Fuller's problem

   ẋ1 = x 2 , ẋ2 = ρ(x 1 ) + u(t), ẋ3 = ℓ(x 1 ) + ε k x 2 2 , ;    x 1 (0) = x 0 1 ∈ R x 2 (0) = x 0 2 ∈ R x 3 (0) = 0 ; x(T ) ∈ C, (3.1)
where C ⊂ R 3 is a non-empty closed convex subset. Suppose that ℓ : R → R has a unique minimum x m 1 over R such that ℓ ′ (x m 1 ) = 0 together with ℓ ′′ (x m 1 ) > 0. Without any loss of generality, we may also assume that ℓ(x m 1 ) = 0 (by translation). Furthermore, assume that -ρ(x m 1 ) ∈ [-1, 1] (this is required for admissibility of singular arcs).

Remark 3.1. Fuller's Problem corresponds to the "limit" case ε k = 0 together with ρ(x 1 ) = 0, ℓ(x 1 ) = x 2 1 , and C = {(0, 0)} × R.

The previous framework encompasses this example setting

f (x) :=   x 2 ρ(x 1 ) ℓ(x 1 )   ; g(x) :=   0 1 0   ; f k (x) :=   x 2 ρ(x 1 ) ℓ(x 1 ) + ε k x 2 2   , since we can easily check that [g, [f, g]] = 0 and [g, [f k , g]] = (0, 0, -2ε k ) ⊤ ̸ = 0 (for ε k > 0)
. We now provide more details on optimal solutions of (3.1) for ε k > 0 and in the limit case.

1. Study of the optimal control problem in the case where ε k > 0. By application of the PMP considering only normal lifts, the Hamiltonian of the problem becomes:

H k := p 1 x 2 + p 2 ρ(x 1 ) + p 2 u -ℓ(x 1 ) -ε k x 2 2 .
Indeed, using the transversality condition, we obtain that p 3 (T ) = -1 and because H k does not depend on x 3 , the function p 3 must be constant. It follows that the adjoint equation can be written as:

ṗ1 = -p 2 ρ ′ (x 1 ) + ℓ ′ (x 1 ), ṗ2 = -p 1 + 2ε k x 2 .
The switching function is p 2 which satisfies p2 = p 2 ρ ′ (x 1 ) -ℓ ′ (x 1 ) + 2ε k ρ(x 1 ) + 2ε k u.

We are now in a position to characterize singular arcs for every ε k > 0. Using that the Hamiltonian is zero along any extremal (since the terminal time is free), every singular arc is such that:

ε k x 2 2 = ℓ(x 1 ) ; u ⋆ k = ℓ ′ (x 1 ) 2ε k -ρ(x 1 ) ; ẍ1 = ℓ ′ (x 1 ) 2ε k ,
where u ⋆ k denotes the singular control. The set of points (x 1 , x 2 ) ∈ R 2 such that ε k x 2 2 = ℓ(x 1 ) is the singular locus. In addition, for every k ∈ N, we find that p2 |u = 2ε k > 0 thus Legendre-Clebsch's condition is verified along a singular arc. In particular, since ε k ̸ = 0, every singular arc is of first order.

2. Study of the optimal control problem in the limit case. Using similar notation as when ε k > 0, we find that the switching function associated with the optimal control problem with ε k = 0 now satisfies p2 = p 2 ρ ′ (x 1 ) -ℓ ′ (x 1 ), thus, we need to differentiate p 2 at least four times to check the order of the singular control. Note also that we must have -ℓ ′ (x 1 ) = 0 along a singular arc, hence, x 1 = x m 1 . By differentiating p2 w.r.t. t, we find that ... p 2 = -ℓ ′′ (x 1 )x 2 , hence x 2 = 0 along the singular arc (because ℓ ′′ (x m 1 ) > 0), and finally .... p 2 = -ℓ ′′ (x m 1 )u. In that case, a singular arc is then characterized by

x ⋆ 1 = x m 1 ; x ⋆ 2 = 0 ; u ⋆ = -ρ(x m 1 )
, where u ⋆ is the singular control. We conclude that the limit case has a second-order singular arc which is such that .... p 2 |u = -ℓ ′′ (x m 1 ) < 0, thus Legendre-Clebsch's condition is also verified. In addition, |u ⋆ | = | -ρ(x m 1 )| ≤ 1, thus the singular arc is admissible. Remark 3.2. (i) When ε k > 0, the singular arc is of turnpike type, however, a saturation phenomenon may occur since for t > 0, one has |u ⋆ k (t)| ≤ 1 if and only if

ℓ ′ (x 1 (t)) 2ε k -ρ(x 1 (t)) ≤ 1.
Since ε k is small, the singular arc is admissible only if x 1 is close to x m 1 . This defines a "small" subset of the singular locus in the plane (x 1 , x 2 ) for the admissibility of the singular arc. We also see that the singular locus as ε k ↓ 0 is a small deformation of the origin (the singular locus for the problem with ε k = 0). (ii) Uniform convergence of the sequence (f k ) and its derivative (D x f k ) to f and D x f respectively is straightforward over compact subsets of R 3 . But, since for a given initial condition (x 0 1 , x 0 2 , 0) ∈ R 3 , optimal trajectories of (3.1) are uniformly bounded, the uniform convergence of f k to f on compact subsets is enough to apply the results of Proposition 2.2.

In the next section, we depict numerically optimal solutions when ε k > 0 and also in the limit case.

Numerical simulations on the perturbed optimal control problem

For all the examples presented in this article, simulations were carried out with the direct method, using the Bocop solver [START_REF]BOCOP: an open source toolbox for optimal control[END_REF]. For sake of simplicity, the final time was fixed. The Gauss II method 3 was used for discretization, with 1000 time steps. Simulations of the perturbed Fuller's problem (i.e., Problem (3.1) with ρ(x 1 ) = 0 and ℓ(x 1 ) = x 2 1 ) are depicted on Fig. 1 and2 for different values of ε k . In line with our previous results, the trajectories are of type Bang-Singular. In particular, Fig. 2 highlights the fact that optimal paths (for ε k > 0) lie in the singular locus ε k x 2 2 = x 2 1 before approaching the target point (see Remark 3.2). As ε k decreases, we can see that the number of bang arcs increase and the trajectories approximate that of the original problem. Optimal controls are a succession of bang arcs together with a "small" terminal first-order singular arc such that the optimal path lies in the singular locus and then reaches the target point.

Also, we have performed a numerical simulation of optimal paths when ε k < 0, see Fig. 3. In that case, Legendre-Clesbsch's condition has the opposite sign leading that way to an anti-turnpike behavior. It follows that approximated optimal controls are a succession of bang arcs (see, e.g., [START_REF] Bonnard | Singular trajectories and their role in control theory[END_REF]). In that case, no singular arcs occurs.

Application to a resource allocation model

Presentation of the model

We consider a control system describing a self-replicator model of bacterial growth where x 1 is the concentration of precursors within a cell and x 2 denotes the concentration of gene expression machinery. The control u with values in [0, 1] will denote the resource allocation parameter defining the proportion of precursors used for making gene expression machinery. The dynamical resource allocation model amounts then to maximize biomass, in order to understand cellular regulations acquired through evolution. For more details on the modeling, we refer to [START_REF] Giordano | Dynamical allocation of cellular resources as an optimal control problem: novel insights into microbial growth strategies[END_REF]. This yields the three-dimensional Mayer optimal control problem inf

u(•)∈[0,1] x 3 (T ) s.t.    ẋ1 = e(1 -x 2 ) -(1 + x 1 )h(x 1 , x 2 ), ẋ2 = h(x 1 , x 2 )(u(t) -x 2 ), ẋ3 = -h(x 1 , x 2 ), ;    x 1 (0) = x 0 1 , x 2 (0) = x 0 2 , x 3 (0) = 0, (3.2)
3 It is an implicit method of fourth order in two stages to solve numerically an ordinary differential equation, [START_REF]BOCOP: an open source toolbox for optimal control[END_REF]. where h(x 1 , x 2 ) := x1x2 K+x1 , e, K are positive constants, (x 0 1 , x 0 2 ) ∈ R + × [0, 1], and T > 0 is fixed. The objective function in (3.2) represents the biomass to be maximized. We can observe that x 2 is always bounded between 0 and 1 provided that x 0 2 ∈ [0, 1], but, this property does not hold for x 1 (for more details, see [START_REF] Giordano | Dynamical allocation of cellular resources as an optimal control problem: novel insights into microbial growth strategies[END_REF]). The expression of h will lead to chattering (see below), making the use of this model and its variants (see Note that in that case, Legendre-Clebsch's condition has the opposite sign (i.e., p2 |u = 2ε k < 0), so, we have an anti-turnpike behavior for the solution to the approximated optimal control problem leading to an increasing number of bang arcs. e.g. [START_REF] Yegorov | Optimal feedback strategies for bacterial growth with degradation, recycling, and effect of temperature[END_REF][START_REF] Yabo | Dynamical analysis and optimization of a generalized resource allocation model of microbial growth[END_REF]) more complex. A way to avoid this behavior is to make a slight perturbation of the system replacing h(x 1 , x 2 ) by

h ε (x 1 , x 2 ) := x1 K+x1 x2
1+εx2 (actually, we replace x 2 → x 2 by x 2 → x2 1+εx2 ). Hereafter, we keep ε in place of ε k for convenience. Setting f (x 1 ) := x1 K+x1 and g ε (x 2 ) := x2 1+εx2 , we can rewrite the above control system as follows:

   ẋ1 = e(1 -x 2 ) -(1 + x 1 ) f (x 1 )g ε (x 2 ), ẋ2 = f (x 1 )g ε (x 2 )(u(t) -x 2 ), ẋ3 = -f (x 1 )g ε (x 2 ). (3.3)
So, for ε = 0, we are in the limit case with chattering (see below) whereas for every ε > 0, the optimal synthesis may exhibit singular arcs of order at most 1 (see below). In addition, it is easily seen that

|g ε (x 2 ) -x 2 | = εx 2 2 1 + εx 2 ≤ ε ; |g ′ ε (x 2 ) -1| ≤ 3ε,
for every ε ∈ (0, 1] and every x 2 ∈ [0, 1]. Hence, the convergence results of the previous section can be applied.

We shall now verify the properties related to the occurrence of a singular arc by application of the PMP.

1. Study of the optimal control problem in the cases where ε > 0. We apply the PMP with the maximum convention. Observe that the cost to be minimized is Φ(x(T )) := x 3 (T ), so the terminal adjoint vector is p(T ) := -∇Φ(x(T )) = (0, 0, -1). It follows that H can be written

H = p1e(1 -x2) -p1(1 + x1) f (x1)gε(x2) + p2 f (x1)gε(x2)(u -x2) + f (x1)gε(x2) → max u∈[0,1]
so u(t) = sign(p 2 (t)) except along (possible) singular arcs. Note that the switching function ϕ is ϕ := p 2 f (x 1 )g ε (x 2 ), but since f (x 1 )g ε (x 2 ) > 0, we will compute the derivatives w.r.t. t of p 2 in place of ϕ for simplification. Now, the covector fulfills over [0, T ] the adjoint equation:

ṗ1 = p 1 g ε (x 2 )( f (x 1 ) + (1 + x 1 ) f ′ (x 1 )) -p 2 f ′ (x 1 )g ε (x 2 )(u -x 2 ) -f ′ (x 1 )g ε (x 2 ), ṗ2 = p 1 e + p 1 (1 + x 1 ) f (x 1 )g ′ ε (x 2 ) -p 2 f (x 1 )(g ′ ε (x 2 )(u -x 2 ) -g ε (x 2 )) -f (x 1 )g ′ ε (x 2 ).
Let us now address the occurrence of a singular arc. Doing so, suppose that p 2 ≡ 0 over [t 1 , t 2 ] where [t 1 , t 2 ] is a sub-interval of [0, T ]. We get ṗ2 = 0 thus

p 1 (e + (1 + x 1 ) f (x 1 )g ′ ε (x 2 )) = f (x 1 )g ′ ε (x 2 ).
We now turn to the computation of p2 (along p 2 = ṗ2 = 0 over [t 1 , t 2 ]). We get:

p2 = ṗ1[e + (1 + x1) f (x1)g ′ ε (x2)] + p1 d dt (e + (1 + x1) f (x1)g ′ ε (x2)) - d dt ( f (x1)g ′ ε (x2)) = [p1gε(x2)( f (x1) + (1 + x1) f ′ (x1)) -f ′ (x1)gε(x2)][e + (1 + x1) f (x1)g ′ ε (x2)] + p1[ ẋ1( f (x1)g ′ ε (x2) + (1 + x1) f ′ (x1)g ′ ε (x2)) + (1 + x1) f (x1)g ′′ ε (x2) ẋ2] -f ′ (x1)g ′ ε (x2) ẋ1 -f (x1)g ′′ ε (x2) ẋ2
To compute p2 |u there are two contributing terms (from ẋ2 ). We get replacing p 1 by its value in the second equality:

p2 |u = (1 + x1) f 2 (x1)g ′′ ε (x2)gε(x2)p1 -f 2 (x1)g ′′ ε (x2)gε(x2) = (1 + x1) f 2 (x1)g ′′ ε (x2)gε(x2) f (x1)g ′ ε (x2) e + (1 + x1) f (x1)g ′ ε (x2) -f 2 (x1)g ′′ ε (x2)gε(x2) = -e f 2 (x1)gε(x2)g ′′ (x2) e + (1 + x1) f (x1)g ′ ε (x2)
.

Proposition 3.1. Let ε > 0 and consider an extremal of the optimal control problem. If a singular arc occurs, then, the Legendre-Clebsch condition is fulfilled.

Proof. From the preceding expression of p2 |u , we deduce that along a singular arc, the switching function

satisfies φ|u = -ef 3 (x 1 )g ε (x 2 ) 2 g ′′ ε (x 2 ) e + (1 + x 1 ) f (x 1 )g ′ ε (x 2 )
> 0, since g ε is strictly concave, whence the result.

2. Study of the optimal control problem in the limit case. We can rewrite the system as follows

   ẋ1 = e(1 -x 2 ) -(1 + x 1 ) f (x 1 )x 2 , ẋ2 = f (x 1 )x 2 (u(t) -x 2 ), ẋ3 = -f (x 1 )x 2 .
Again, we apply the PMP with the maximum convention and similarly, the terminal adjoint vector is p(T ) = (0, 0, -1). It follows that the Hamiltonian H can be written

H = p1e(1 -x2) -p1(1 + x1) f (x1)x2 + p2 f (x1)x2(u -x2) + f (x1)x2 → max u∈[0,1]
so u(t) = sign(p 2 (t)) except along (possible) singular arcs. As previously, we shall compute the derivatives of p 2 w.r.t. t in place of the switching function ϕ := p 2 f (x 1 )x 2 (since f (x 1 )x 2 > 0). The covector fulfills the adjoint equation over [0, T ]:

ṗ1 = p 1 x 2 ( f (x 1 ) + (1 + x 1 ) f ′ (x 1 )) -p 2 f ′ (x 1 )x 2 (u -x 2 ) -f ′ (x 1 )x 2 , ṗ2 = p 1 (e + (1 + x 1 ) f (x 1 )) + p 2 f (x 1 )(2x 2 -u) -f (x 1 ).
Let us now address the occurrence of a singular arc. Doing so, suppose that p 2 ≡ 0 over [t 1 , t 2 ] where [t 1 , t 2 ] is a sub-interval of [0, T ]. We get ṗ2 = 0 thus p 1 (e + (1 + x 1 ) f (x 1 )) = f (x 1 ).

We now turn to the computation of p (k)

2 , k = 2, 3, 4 (along p 2 = ṗ2 = 0 over [t 1 , t 2 ]). We get replacing p 1 , ṗ1 , and ẋ1 by their respective value:

p2 = ṗ1[e + (1 + x1) f (x1)] + p1[ f (x1) + (1 + x1) f ′ (x1)] ẋ1 -f ′ (x1) ẋ1 = [p1x2( f (x1) + (1 + x1) f ′ (x1)) -f ′ (x1)x2][e + (1 + x1) f (x1)] + p1[ f (x1) + (1 + x1) f ′ (x1)][e(1 -x2) -(1 + x1) f (x1)x2] -f (x1)[e(1 -x2) -(1 + x1) f (x1)x2] = p1[ f (x1) + (1 + x1) f ′ (x1)] -f ′ (x1)e = Ψ(x1), where Ψ(x 1 ) := e[ f 2 (x 1 ) -e f ′ (x 1 )] e + (1 + x 1 ) f (x 1
) .

We deduce that if a singular arc occurs, then it is at least of second order since φ|u = f (x 1 )x 2 p2 |u = 0, i.e., p(t) • [g, [f, g]](x(t)) = 0 along the singular arc. In addition, we obtain that if a singular arc occurs, then, we have p 2 = ṗ2 = p2 = 0, so Ψ(x 1 ) = 0 which gives

x 1 = x ⋆ 1 := √ eK.
We also obtain ... p 2 = Ψ ′ (x 1 ) ẋ1 = 0. A computation shows that

Ψ ′ (x ⋆ 1 ) = 2 √ Kee (2Ke + e √ Ke + √ Ke)(K + √ Ke) ,
which is obviously non-zero, hence, x 1 remains constant along the singular arc and, so, the singular arc is at least of second order. Finally, differentiating a last time w.r.t. t yields ....

p 2 = Ψ(x 1 )ẍ 1 = Ψ ′ (x 1 )(-ẋ2 (e + (1 + x 1 ) f (x 1 ))).
This gives ẋ2 = 0, hence x 2 = 0 or x 2 = u. The case x 2 = 0 implies that ẋ1 = e > 0 in contradiction with the fact that ẋ1 = 0. So, we can conclude that u = x 2 . Hence, the singular arc can be characterized as:

x ⋆ 1 = √ eK ; x ⋆ 2 := e e + (1 + x ⋆ 1 )f (x ⋆ 1 ) = √ e( √ e + √ K) 1 + e + 2 √ e √ K ; u ⋆ = x ⋆ 2 .
Finally, we get that ....

p 2 |u = -Ψ ′ (x ⋆ 1 )(e + (1 + x ⋆ 1 )f (x ⋆ 1
))x ⋆ 2 (x ⋆ 1 ) < 0, hence, Legendre-Clebsch's condition is satisfied.

Numerical simulations on the perturbed optimal control problem

Fig. 4 depicts trajectories obtained for different values of ε k > 0. Following [START_REF] Giordano | Dynamical allocation of cellular resources as an optimal control problem: novel insights into microbial growth strategies[END_REF], simulations are carried out with a fixed final time, and only the transients until reaching the optimal steady-state are depicted (bang arcs also appear at the end of the simulations, but they are not relevant for the biological problem). The results are similar to those of the Fuller's problem. For ε k > 0, optimal controls are a concatenation of bang arcs together with a terminal singular arc approaching the singular arc in the limit case. A value of ε k = 0.01 is enough to obtain an accurate approximation of the optimal trajectory to the original problem.

Application to a quantum control system

Presentation of the problem

Here, we consider an optimal control problem introduced in [START_REF] Robin | Chattering phenomenon in quantum optimal control[END_REF] in which chattering occurs. The model describes the control of a three-level quantum system to be steered from a given initial condition on the unit sphere of R 3 (denoted by S 1 ) to the north pole (0, 0, 1). This yields the following optimal control problem (written in Mayer form):

min T ≥0, u∈U x 4 (T ) s.t.        ẋ1 = -x 2 , ẋ2 = x 1 -u(t)x 3 , ẋ3 = u(t)x 2 , ẋ4 = x 2 1 2 ,

and

(x 1 (0), x 2 (0), x 3 (0), x 4 (0)) ∈ S 1 × {0}, (x 1 (T ), x 2 (T ), x 3 (T ), x 4 (T )) ∈ {(0, 0, 1)} × R. As it can be verified, the dynamics of (x 1 , x 2 , x 3 ) is with values in S 1 . We assume for sake of simplicity that x 3 (t) ≥ 0, ∀t ∈ [0, T ] (by choosing initial conditions) so that one can reduce the dimension of the system. Thus, we can introduce the following control system:

   ẋ1 = -x 2 , ẋ2 = x 1 -u(t) 1 -x 2 1 -x 2 2 , ẋ3 = 1 2 x 2 1 + ε 2 x 2 2 ,
proceeding as previously, that is, by adding a small perturbation term in the state x 3 . The optimal control problem then amounts to minimize x 3 (T ) w.r.t. the pair (T, u) such that x 1 (T ) = x 2 (T ) = 0 starting from some initial condition (x 0 1 , x 0 2 , 0) ∈ R 3 such that (x 0 1 ) 2 + (x 0 2 ) 2 < 1. 1. Study of the optimal control problem in the case where ε > 0. In that case, the Hamiltonian associated with the problem can be written (taking into account that p 3 = -1 is constant):

H = -p 1 x 2 + p 2 x 1 -up 2 1 -x 2 1 -x 2 2 - x 2 1 2 -ε x 2 2 2 → max u∈[-1,1]
and it is conserved over [0, T ] (since the problem is autonomous) and also equal to zero (because the terminal time is free). The adjoint equation reads as follows:

   ṗ1 = -p 2 -ux1p2 √ 1-x 2 1 -x 2 2 + x 1 , ṗ2 = p 1 -ux2p2 √ 1-x 2 1 -x 2 2 + εx 2 Now, the switching function is ϕ := -p 2 1 -x 2 1 -x 2 2
, so, it is enough to study the behavior of p 2 . Along a singular arc, we have p 2 = ṗ2 = 0, thus p 1 = -εx 2 . Replacing into the Hamiltonian yields the singular locus εx 2 2 = x 2 1 in the state space. By differentiating this expression, we obtain the value of the singular control

u ⋆ ε := ± (1 + ε)|x 1 | ε 1 -x 2 1 -x 2 2 .
We note that the value of |u ⋆ ε | may exceed 1, so in that case, we have saturation (see [START_REF] Bayen | Tangency property and prior-saturation points in minimal time problems in the plane[END_REF]). However, numerical simulations indicate that it is admissible when the trajectory is close to the origin (what can be expected).

Finally, the second order derivative of p 2 satisfies: p2 |u = -ε 1 -x 2 1 -x 2 2 < 0. It follows that Legendre-Clesbsch's condition is satisfied (since ϕ = -p 2 ).

2. Study of the optimal control problem in the limit case. In this case, the switching function is also ϕ = -p 2 . By differentiating w.r.t. t, we find that φ = -p 1 + ux2p2 √ 1-x 2 1 -x 2

2

. Let us examine singular arcs in this case. Supposing that ϕ = 0 over some time interval [t 1 , t 2 ] gives us -p 1 x 2 -

x 2 1 2 = 0. The equation φ = 0 also implies that p 1 = 0, hence x 1 = 0. Along the singular arc, we can write φ = -ṗ1 = x 1 , thus ... ϕ = -ẋ1 = x 2 , and finally .... ϕ = ẋ2 = x 1 -u 1 -x 2 1 -x 2 2 . To conclude this case, we obtain that the singular arc satisfies:

x ⋆ 1 = x ⋆ 2 = u ⋆ = 0 ; φ|u = 0 ;

.... ϕ |u = -1 < 0.

We can thus conclude that the limit case is indeed a second-order singular arc for which Legendre-Clesbch's condition is verified.

Numerical simulations on the perturbed optimal control problem

The results obtained for the perturbed quantum control system are presented on Fig. 5. Here again, we can see that the number of bang arcs increases as ε k decreases. In addition, the trajectory ends up with a singular arc approaching the singular arc of the limit problem (corresponding to u = 0). This example, like the previous ones, confirms the applicability of the method to various control systems.

Conclusion and perspectives

In this work, we have studied how to slightly modify optimal control problems involving a second order singular arc in order to obtain a "simpler" optimal control problem whose solutions do not possess chattering. The advantage of these modified optimal control problems is that they are more tractable from a numerical viewpoint since optimal controls involve only a finite number of switching times. The examples we have studied show that we obtain that way a deformation of the singular locus (corresponding to the second-order singular arc) into a "small" singular locus (corresponding to a first-order singular arc). Since approximated optimal controls may involve both singular and bang arcs, our approach somehow differs from other approaches using only a sequence of bang-bang arcs to approach the chattering control. In particular, our methodology allows to apply the PMP on the approximated problem and to obtain that way qualitative properties of the approximated control (like the value of the singular control for the approximated problem). Other applications such as in [START_REF] Ledzewicz | Singular controls and chattering arcs in optimal control problems arising in biomedicine[END_REF] could be investigated in future works. As well, an interesting question could be to find out if every control system involving a second order singular arc (typically, in our setting, when [g, [f, g]] is identically zero) can be approximated by control systems with a first-order singular arc (for which this Lie bracket is nonzero), and how to determine an approximated system. Additionally, convergence of the singular control (for the approximated problem) to the singular control (for the second order singular arc) could also be addressed in future works.

3. 1 . 1

 11 Presentation of the problem Let ρ, ℓ : R → R be two smooth functions. Consider the three-dimensional optimal control problem inf u∈U , T ∈R+ x 3 (T ) s.t.

2 Figure 1 : 2 Figure 2 :

 2122 Figure 1: Simulations of the perturbed Fuller's problem with different values of ε k .

2 Figure 3 :

 23 Figure3: Simulations of the perturbed Fuller's problem with ε k < 0. Note that in that case, Legendre-Clebsch's condition has the opposite sign (i.e., p2 |u = 2ε k < 0), so, we have an anti-turnpike behavior for the solution to the approximated optimal control problem leading to an increasing number of bang arcs.

2 Figure 4 :

 24 Figure 4: Simulations of the perturbed resource allocation problem (3.2) with different values of ε k .

2 Figure 5 :

 25 Figure 5: Simulations of the perturbed quantum control system with different values of ε k .

  If this situation occurs, then, we say that the extremal 1 (x, p, u) is singular over [t 1 , t 2 ] (equivalently, we say that the trajectory has a singular arc over [t 1 , t 2 ]). The study of singular arcs relies on the expressions defining the derivatives of ϕ w.r.t. the time. A simple computation shows that one has

	sign(ϕ(t)) a.e. t ∈ [0, T ] s.t. ϕ(t) ̸ = 0,	(2.4)
	where ϕ(t) := p(t) • g(x(t)) is the switching function. On a sub-interval [t 1 , t 2 ] ⊂ [0, T ] on which ϕ vanishes,
	a different analysis should be carried out in order to find the expression of an optimal control (in particular
	(2.4) is useless).

An extremal is a triple (x, p, u) satisfying the state-adjoint system, the transversality condition, and the Hamiltonian maximization condition.

[START_REF] Bayen | Optimal synthesis for the minimum time control problems of fedbatch bioprocesses for growth functions with two maxima[END_REF] We mean that along a singular arc, the input is present explicitly in the expression of ϕ (2q) and not in ϕ (2r) , r = 1, ..., q -1.

Acknowledgments

We thank Anas Bouali and the Inria Biocore team for helpful discussions on the subject.

Declarations

Ethical Approval: not applicable.

Competing interests: not applicable.

Authors' contributions: All authors contributed equally in this manuscript.

Funding: not applicable.

Availability of data and materials: not applicable.