domain: info.info-au spi math

text:
Pierre Bernhard
email: pierre.bernhard@inria.fr

Robust Control and Dynamic Games

Keywords: Robust control, H ∞ -optimal control, nonlinear H ∞ control, robust control appraoach to option pricing

We describe several problems of "robust control" that have a solution using game theoretical tools. This is by no means a general overview of robust control theory beyond that specific purpose, nor a general account of system theory with set-description of uncertaineties.

Introduction

Control theory is concerned with the design of control mechanisms for dynamic systems that compensate for (a priori) unknown disturbances acting on the system to be controlled. While early servomechanism techniques did not make use of much modeling, neither of the plant to be controlled nor of the disturbances, the advent of multi-input multi-output systems and the drive to more stringent specifications led researchers to use mathematical models of both. In that process, and most prominently with the famous LQG design technique, disturbances were described as unknown inputs to a known plant, and usually high frequency inputs.

"Robust control" refers to control mechanisms of dynamic systems that are designed to counter unknowns in the system equations describing the plant rather than in the inputs. This is also a very old topic. It might be said that a PID controller is a "robust" controller, since it makes little assumption on the controlled plant. Such techniques as gain margin and phase margin guarantees and loop shaping also belong in that class. However, the gap between "classic" and "robust" control designs became larger with the advent of "modern" control designs.

As soon as the mid seventies, some control design techniques appeared, often based upon some sort of Lyapunov function, to address that concern. A most striking feature is that they made use of bounded uncertainties rather than the prevailing Gaussian-Markovian model. One may cite [START_REF] Gutman | Stabilizing feedback control for dynamical systems with bounded uncertainty[END_REF], [START_REF] Gutman | Uncertain dynamical systems: -a Lyapunov min-max approach[END_REF], [START_REF] Corless | Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems[END_REF]. This was also true with the introduction of the so-called H ∞ control design technique in 1981 [START_REF] Zames | Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses[END_REF], and will become such a systematic feature that "robust control" became in many cases synonymous to control against bounded uncertainties, usually with no probabilistic structure on the set of possible disturbances.

We stress that this is not the topic of this chapter. It is not an account of system theory for systems with set description of the uncertainties. Such comprehensive theories have been developed using set-valued analysis and/or elliptic approximations. See e.g. [START_REF] Aubin | Viability Theory[END_REF] and [START_REF] Kurzhanski | Dynamics and Control of Trajectory Tubes[END_REF]. We restrict our scope to control problems, thus ignoring pure state estimation such as developed in the above references, or in [7, chap. 7] or [START_REF] Rapaport | Parallelotopic and practical observers for nonlinear uncertain systems[END_REF] for instance, and to game theoretic methods in control design, thus ignoring other approaches of robust control such as the gap metric [START_REF] Zames | Unstable ststems and feedback: the gap metric[END_REF][START_REF] Georgiou | Robustness analysis of nonlinear feedback systems: an input-output approach[END_REF][START_REF] Qiu | Robust control in gap metric[END_REF], an input-output approach in the spirit of the original H ∞ approach of [START_REF] Zames | Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses[END_REF], using transfer functions and algebraic methods in the ring of rational matrices, or Linear Matrix Inequality (LMI) approaches [START_REF] Kang-Zhi | Lmi approach to robust control[END_REF], and many other approaches found, e.g., in the same Encyclopaedia of Systems and Control [START_REF] Samad | Encyclopedia of Systems and Control[END_REF].

We will first investigate in section 2 the issue of modeling of the disturbances, show how non-probabilistic set descriptions naturally arise, and describe some specific tools we will use to deal with them. Subsection 2.1 is an elementary introduction to these questions, while subsection 2.2 is more technical, introducing typical control theoretic issues and examples in the discussion. From there on, a good control of linear systems theory, optimal control -specifically the Hamilton-Jacobi-Caratheodory-Isaacs-Bellman (HJCIB) theory-and dynamic games theory is required. Then, in section 3, we will give a rather complete account of the linear H ∞ -optimal control design for both continuous time and discrete time systems. Most of the material of these two sections can be found in more detail in [START_REF] Bas | H ∞ -Optimal Control and Related Minimax Design Problems : a Differential Games approach[END_REF]. In section 4, we will cover in less detail the so called "nonlinear H ∞ " control design, mainly addressing engineering problems, and a nonlinear example in the very different domain of mathematical finance.

2 Min-max problems in robust control 2.1 Decision theoretic formulation

Worst case design

We start this section with a very general set-up in terms of decision theory. Let U be a decision space, and W a disturbance space. We may think of W as being a bounded set in some ad hoc norm, but this is not necessary at this level of generality. Let J : U × W → R be a performance index, depending on the decision u ∈ U and on an unknown disturbance w ∈ W . The decision maker, choosing u, wants to make J as small as possible in spite of the a priori unknown disturbance w ∈ W . (One may think of J as a measure, such as the L 2 norm of an error signal.)

The so called worst case design method is as follows. Let us first emphasize that there is no "malicious adversary" manipulating the disturbance in the following description, contrary to many accounts of this method.

The basic concept is that of guaranteed performance of a given decision u ∈ U : any number g such that ∀w ∈ W , J(u, w) ≤ g.

Clearly, the best (smallest) guaranteed performance for a given decision u is

G(u) = sup w∈W J(u, w) .
Hence the phrase "worst case", which has to do with a guarantee, not any malicious adversary.

Now, the problem of finding the best possible decision in this context is to find the smallest guaranteed performance, or

inf u∈U G(u) = inf u∈U sup w∈W J(u, w) . (1)
If the infimum in u is reached, then the minimizing decision u deserves the name of optimal decision in that context.

Robust disturbance rejection or gain control

The following approach is mostly justified in the following linear set-up, but can be extended to a nonlinear one (see [START_REF] Van Der Schaft | L 2 -Gain and Passivity Techniques in Nonlinear Control[END_REF]). We let U and W be normed vector spaces, Z be an auxiliary normed vector space, z ∈ Z the output whose norm is to be kept small in spite of the disturbances w. We assume that for each decision u ∈ U , z depends linearly on w. Therefore, one has a (possibly nonlinear) application P : U → L(W → Z) and z = P (u)w .

Clearly, z cannot be kept bounded if w is not. A natural formalization of the problem of keeping it small is to try and make the operator norm of P (u) as small as possible. This may be expressed as follows: let B(W ) be the unit ball of W , one seeks inf u∈U sup w∈B(W )

P (u)w . (2)
But we will often prefer another formulation, again in terms of guaranteed performance, and which can easily be extended to nonlinear systems. Start from the problem of ensuring that P (u) ≤ γ for a given positive attenuation level γ. This is equivalent to ∀w ∈ W , z ≤ γ w , or equivalently (but leading to a smoother problem)

∀w ∈ W , z 2 ≤ γ 2 w 2 , (3)
or equivalently again

sup w∈W [ P (u)w 2 -γ 2 w 2 ] ≤ 0 .
Now, given a number γ, this has a solution (there exists a decision u ∈ U satisfying that inequality) if the infimum hereafter is reached or is negative, and only if

inf u∈U sup w∈W [ P (u)w 2 -γ 2 w 2 ] ≤ 0 . (4)
This is the method of H ∞ -optimal control. Notice, however, that equation (3) has a meaning even for a nonlinear operator P (u), so that the problem (4) is used in the so-called "nonlinear H ∞ " control. We will see a particular use of the control of the operator norm in feedback control when using the "small gain theorem". Although this could be set in the abstract context of nonlinear decision theory (see [START_REF] Vidyasagar | Nonlinear Systems Analysis[END_REF]), we will rather show it in the more explicit context of control theory.

Remark 2.1 The three problems outlined above as equations (1), [START_REF] Aubin | Viability Theory[END_REF], and (4) are all of the form inf u sup w , and therefore, in a dynamic context, are amenable to dynamic game machinery. They involve no probabilistic description of the disturbances.

Set description vs probabilistic

Two main reasons have driven robust control towards set theoretic, rather than probabilistic, descriptions of disturbances. On the one hand, the traditional Gauss-Markov representation is geared towards high frequency disturbances. When the main disturbance is, say, a misevaluated parameter in a dynamic equation, it is the exact opposite of high frequency: zero-frequency. And this is typically the type of unknowns that robust control is meant to deal with.

On the other hand, concerning the coefficients of a differential equation, using a probabilistic description requires that one deals with differential equations with random coefficients, a much more difficult theory than that of differential equations driven by a stochastic process.

Thus, "robust control approaches" to decision problems involve a description of the unknown disturbances in terms of a set containing all possible disturbances, often a bounded set, and hence hard bounds on the possible disturbances. As mentioned in the introduction, systematic theories have been developed to deal with set descriptions of uncertainties under the name of Viability theory [START_REF] Aubin | Viability Theory[END_REF] -particularly well suited to deal with non-smooth problems in engineering, environment, economy, finance [12, part V]-, or Trajectory Tubes which, with the use of Hamiltonian formalism and elliptic approximations, recover more regularity, and also leads to a theory of robust control in the presence of corrupted information [26, chap. 10]. We emphasize here somewhat different, and more classical, methods, making explicit use of dynamic game theory.

To practitioners accustomed to Gauss-Markov processes (such as a filtered "white noise"), hard bounds may seem a rather severe limitation, and providing the mathematical model with too much information. Yet, a probability law is an extremely rich information itself; in some sense, much richer than a simple set description. As an example, note that if we model a disturbance function as an ergodic stochastic process, then we tell the mathematical model that the long time average of the disturbance is exactly known. In the case of a Brownian motion, we say that the total quadratic variation of (almost) all realizations is exactly known. And these are only instances of the rich information we provide, that the mathematical machinery will use, may be far beyond what was meant by the modeler.

Control formulation

We want to apply the above ideas to control problems, i.e. problems where the decision to be taken is a control of a dynamic system. This raises the issues of causality, and open-loop versus closed-loop control. No control theoretician would want to rely on open-loop control to overcome uncertainties and disturbances in a system. But then, the issue of the available information to form one's control becomes crucial.

Closed-loop control

Let a dynamic system be represented by a differential equation in R n (we will therefore ignore the existing extensions of these results to infinite-dimensional systems, see [START_REF] Bensoussan | On the standard problem of H ∞optimal control for infinite dimensional systems[END_REF] for a game theoretic approach):

ẋ = f (t, x, u, w) (5)
and two outputs; y the observed output and z to be controlled:

z(t) = g(t, x, u) , (6)
y(t) = h(t, x, w) , (7)
Here, as in the sequel, we have

x ∈ R n , u ∈ U ⊂ R m , w ∈ W ⊂ R , y ∈ R p , z ∈ R q
, and u(•) ∈ U, w(•) ∈ W the sets of measurable time functions into U and W respectively. Likewise, we let y(•) ∈ Y and z(•) ∈ Z. Moreover, we assume that f enjoys regularity and growth properties that insure existence and uniqueness of the solution to (5) for any initial condition and any pair of controls

(u(•), w(•)) ∈ U × W.
Since we are concerned with causality, we introduce the following notation: For any time function v(•), let v t denote its restriction [v(s)|s ≤ t]. And if V is the set of functions v(•), let V t be the set of their restrictions v t :

V = {v(•) : [t 0 , t 1 ] → V : s → v(s)} =⇒ ∀t ∈ (t 0 , t 1 ) , V t = {v t : [t 0 , t] → V : s → v(s)} . (8)
The control u(t) will be synthesized as a closed-loop control

u(t) = φ(t, y t ) , (9)
typically by a dynamic compensator driven by the observed output y. Therefore, the operator inf u∈U in equations ( 1), (2), and (4) must be replaced by inf φ∈Φ . But then, we must specify the class Φ of admissible controllers. However, we postpone that discussion until after we deal with the disturbance w.

The question arises as to whether one should allow closed-loop "disturbance laws", some w(t) = ψ(t, u t ), or be content with open-loop disturbances w(•) ∈ W. This question is answered by the following lemma, occasionally attributed to Leonard Berkowitz [START_REF] Berkowitz | Lectures on differential games[END_REF]:

G w u z y φ - Figure 1: Disturbance rejection problem Lemma 2.1 Let a criterion J(u(•), w(•)) = K(x(•), u(•), w(•)
) be given. Let a control function φ be given, and Ψ be a class of closed-loop disturbance strategies compatible with φ (i.e. such that the differential equations of the dynamic system have a solution when u is generated by φ and w by any ψ ∈ Ψ). Then, (with a transparent abuse of notation)

sup w(•)∈W J(φ, w(•)) ≥ sup ψ∈Ψ J(φ, ψ) .
Therefore, it is never necessary to consider closed-loop disturbance laws. This greatly simplifies the discussion of the class Φ of admissible controllers. It only has to be such that solutions to the differential equations exist against all open-loop measurable disturbances. This is an important difference with "true" (dynamic) game theory, arising from the fact that we are not concerned with the existence of a Value, but only with inf sup operators.

The control form of the disturbance rejection problem of paragraph 2.1.2 may therefore be stated, for the system G given by ( 5), ( 6), ( 7) and ( 9) over a time span T , as follows: let Φ be the class of all causal controllers ensuring the existence in L 2 (T ) and uniqueness of the solution of the system equations.

Standard problem of H ∞ -optimal control with attenuation level γ: Does the following inequality hold:

inf φ∈Φ sup w∈L 2 (T ) z 2 L 2 -γ 2 w 2 L 2 ≤ 0 ? (10)
If the infimum is reached or is negative, find an admissible controller φ ensuring the inequality (3) in L 2 norms.

Minimax certainty equivalence principle

The standard problem as formulated above amounts to a min-sup differential game with a real time minimizer's information both partial and corrupted by the maximizer. A non-classical problem. In some favorable cases, including that of linear systems, the information problem can be solved via the following certainty equivalence theorem [START_REF] Bas | H ∞ -Optimal Control and Related Minimax Design Problems : a Differential Games approach[END_REF][START_REF] Bernhard | Min-max cetainty equivalence principle and differential games[END_REF]. Let a differential game be specified by ( 5) and a criterion:

J(x 0 ; u(•), w(•)) = M (x(t 1 )) + t 1 t 0 L(t, x(t), u(t), w(t)) dt + N (x 0 ) . ( 11
)
Assume that the full state information min-sup problem with N ≡ 0 has a state feedback solution

u(t) = ϕ (t, x(t)) , (12)
leading to a Value function

V (t, x) of class C 1 .
We now define a partial information mechanism. Since we assume that the minimizer does not know x(t), it is consistent to assume that he does not know x 0 either. We therefore allow, in the partial information problem, the added cost N (x 0 ). Let ω ∈ Ω = R n × W be the complete disturbance, i.e. the pair (x 0 , w(•)). Recall notation [START_REF] Bas | Dynamic Noncooperative Game Theory[END_REF]. An observation process is a device that, at each time instant t, defines a subset Ω t ⊂ Ω function of u(•) and ω, which enjoys the following three properties:

1. It is consistent: ∀u ∈ U , ∀ω ∈ Ω , ∀t , ω ∈ Ω t . 2. It is perfect recall: ∀u ∈ U , ∀ω ∈ Ω , ∀t ≥ t , Ω t ⊂ Ω t . 3. It is nonanticipative: ω t ∈ Ω t t ⇒ ω ∈ Ω t .
In the case of a system ( 5)- [START_REF] Bas | H ∞ -Optimal Control and Related Minimax Design Problems : a Differential Games approach[END_REF],

Ω t (u(•), ω) = {(x 0 , w(•)) | ∀s ≤ t , h(s, x(s), w(s)) = y(s)}.
We seek a controller

u(t) = φ(t, Ω t ) . (13)
Define the auxiliary criterion

G t (u t , ω t ) = V (t, x(t)) + t t 0 L(s, x(s), u(s), w(s)) ds + N (x 0 ) ,
and the auxiliary problem: max

ω t ∈Ω t t G t (u t , ω t ) .
Note that in this problem, u t is known, as our own past control.

Theorem 2.1 If the auxiliary problem admits one or several solutions leading to a unique state x(t) at time t, then, if the controller

u(t) = ϕ (t, x(t))
is admissible, it is a min-sup controller in partial information. Conversely, if there exists a time t such that sup ω t ∈Ω t t G t = ∞, then the criterion (11) has an infinite supremum in ω for any admissible feedback controller (13).

Remark 2.2 The state x(t) may be considered as the worst possible state given the available information.

One way to proceed in the case of an information such as ( 7) is to solve by forward dynamic programming (forward Hamilton-Jacobi-Caratheodory-Bellman equation) the constrained problem:

max ω t t 0 L(s, x(s), u(s), w(s)) ds + N (x 0 )
subject to the control constraint w(s) ∈ {w | h(s, x(s), w) = y(s)}, and the terminal constraint x(t) = ξ. It is convenient to call -W (t, ξ) the corresponding Value (or Bellman function). Assuming that, under the strategy ϕ , for every t the whole space is reachable by some ω ∈ Ω, x(t) is obtained via

max x∈R n [V (t, x) -W (t, x)] = V (t, x(t)) -W (t, x(t)) . ( 14
)
If this max is reached at a unique x = x(t), then the uniqueness condition of the theorem is satisfied.

Remark 2.3

In the duality between probability and optimization (see [START_REF] Franc ¸ois Baccelli | Synchronization and Linearity[END_REF][START_REF] Akian | Duality between probability and optimization[END_REF]) the function -W (t, •) is the conditional cost measure of the state for the measure t t 0 Lds + N (x 0 ) knowing y(•). The left-hand-side of formula [START_REF] Corless | Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems[END_REF] is the dual of a mathematical expectation.

Small gain theorem

We aim to show classical linear control problems that can be cast into a standard problem of H ∞ -optimal control. We develop some preliminary tools.

Linear operators and norms In the case of linear systems, operator norms have concrete forms.

Matrix: A p × m matrix M represents a linear operator M : R m → R p whose norm M is the maximum singular value σ max (M ) of the matrix.

Dynamic system: A stable stationary linear dynamic system with input u(t) ∈ R m and output y(t) ∈ R p will be considered as a linear operator, say G, from

L 2 (R → R m ) to L 2 (R → R p ).
It may be represented by its transfer function G(s), always a rational proper matrix since we confine ourselves to finite-dimensional state systems. Moreover, the system being stable (otherwise it would not map L 2 into L 2 ), the transfer function has all its poles in the left half complex plane. Thus it belongs to the Hardy space H ∞ of functions holomorphic in an open set of the complex plane containing the half-plane Re(s) ≥ 0. In that case, it follows from Parseval's equality that

G = sup ω∈R σ max G(ω) =: G(•) ∞ . The norm G(•) ∞ (or G ∞ ) is the norm of the transfer function in H ∞ .
Block operator: If the input and output spaces of a linear operator G are represented as product of two spaces each, the operator takes a block form

G = G 11 G 12 G 21 G 22 .
We will always consider the norm of product spaces as the Euclidean combination of the norms in the component spaces. In that case, it holds that G ij ≤ G . Furthermore, whenever the following operators are defined

G 1 G 2 ≤ G 1 2 + G 2 2 , ( G 1 G 2 ) ≤ G 1 2 + G 2 2 .
Hence

G 11 G 12 G 21 G 22 ≤ G 11 2 + G 12 2 + G 21 2 + G 22 2 .
We finally notice that for a block operator G = (G ij ), its norm is related to the matrix norm of its matrix of norms

( G ij ) by G ≤ ( G ij ) . G 0 u w y z ∆ - Figure 2:
The perturbed system G ∆ Small gain theorem Let a two input-two output linear system G 0 be given by

z y = G 11 G 10 G 01 G 00 w u . (15)
Assume that it is connected in feedback by a linear operator ∆ according to w = ∆z, leading to a composite system y = G ∆ u. (See figure 2.) Then, an easy consequence of Banach's fixed point theorem is:

Theorem 2.2 (Small gain theorem) If ∆G 11 = α < 1 or G 11 ∆ = α < 1, then the combined system G ∆ = G 01 (I -∆G 11 ) -1 ∆G 10 + G 00 or G ∆ = G 01 ∆(I -G 11 ∆) -1 G 10 + G 00
is well defined and is stable. Moreover, it holds that

G ∆ ≤ G 00 + G 01 G 10 ∆ 1 -α . Corollary 2.1 If G 11 < 1, then G ∆ is stable for any ∆ with ∆ ≤ 1.
This will be a motivation to try and solve the problem of making the norm of an operator smaller than 1, or smaller than a given number γ.

Robust servo-mechanism problem 2.3.1 Model uncertainty

Assume we deal with a linear system ẋ = Ax + Bu , y = Cx + Du .

But the system matrices are not precisely known. We have an estimate or "nominal system" (A 0 , B 0 , C 0 , D 0 ), and we set

A = A 0 +∆ A , B = B 0 +∆ B , C = C 0 +∆ C , D = D 0 + ∆ D .
The four matrices ∆ A , ∆ B , ∆ C , and ∆ D are unknown, but we know bounds δ A , δ B , δ C , and δ D on their respective norms. We rewrite the same system as

ẋ = A 0 x + B 0 u + w 1 , y = C 0 x + D 0 y + w 2 , z = x u ,
and obviously

w = w 1 w 2 = ∆ A ∆ B ∆ C ∆ D x u = ∆z .
We now have a known system G 0 connected in feedback by the unknown system ∆ for which we have a norm bound ∆ ≤ (δ

2 A + δ 2 B + δ 2 C + δ 2 D ) 1/2
, to which we can apply the small gain theorem.

Robust stabilization

Indeed, that representation of a partially unknown system can be extended to a dynamic unknown perturbation ∆. We rename w and z as v and ζ respectively, and we add a disturbance input w, an output z to be controlled, and a dynamic compensator u = Ky. From now on, to make things simpler, we assume that the signals v and ζ are scalar (See below a short discussion of vector signals). Assume that a frequency dependent bound of the modulus of ∆'s transfer function is known as |∆(ω)| ≤ δ(ω). Then, in a classical "loop shaping" fashion, we devise a dynamic filter W 1 (s) such that |W 1 (ω)| ≥ δ(ω), and consider the fictitious control system with output ζ = W 1 ζ. Stabilization of the system will be guaranteed if we can keep the transfer function norm from v to ζ less than one. We may also specify the disturbance rejection objective as keeping a fictitious output z = W 0 z small

G 0 v w u ζ z y ∆ - K - Figure 3:
The perturbed system and its compensator for some prescribed dynamic filter W 0 . The control objective of the compensator K will now be to keep the modulus of the transfer function from v to ζ less than one at all frequencies, while holding the norm of the transfer function from w to z as small as possible. This may be cast into a standard problem for instance by choosing the pair ((1/γ)v, w) as the input, ( ζ, β z) as the output, and finding β and γ as small as possible while holding inequality [START_REF] Berkowitz | Lectures on differential games[END_REF] true. But more clever weightings may also be tried. Some remarks are in order.

Remark 2.4

1. We might as well have placed the shaping filter W 1 on the input channel v. The two resulting control problems are not equivalent. One may have a solution and the other one none. Furthermore, the weight δ(ω) might be divided between two filters, one on the input and one on the output. The same remark applies to the filter W 0 and to the added weights β and γ. There is no known simple way to arbitrate these possibilities.

2. In case of vector signals v and ζ, one can use diagonal filters with the weight W 1 on each channel. But this is often inefficient. There exists a more elaborate way to handle that problem, called "µ-synthesis". (See [START_REF] Doyle | Analysis of feedback systems with structured uncertainties[END_REF][START_REF] Zhou | Robust and Optimal Control[END_REF].)

Robust servomechanism

We describe a particular case of the above problem which was at the inception of the H ∞ -optimal control problem in [START_REF] Zames | Feedback and optimal sensitivity: model reference transformations, multiplicative seminorms, and approximate inverses[END_REF]. All signals are scalar.

An uncertain system's transfer function G is described by a multiplicative uncertainty: G = (I + ∆)G 0 . We know G 0 and a bound |∆(ω)| ≤ δ(ω). However, it holds that as ω → ∞, G 0 (ω) → 0, and because of a possible fixed offset in the true plant, δ(ω) → ∞. The aim of the control system is to have the output y follow an unpredictable, but low frequency, reference signal w thanks to a dynamic compensator u = K(w -y). We name z the error signal w -y, ζ the (fictitious) output of the (fictitious) system G 0 , and v = ∆ζ. (See figure 4.) We can cast

+ w f z K u G 0 + f ζ ∆ ? + v y 6 - Figure 4:
The servomechanism this problem as a standard problem with input (v, w) and output (ζ, z). However, adding frequency dependent weights is now unavoidable. We have

z = -G 0 u -v + w , ζ = G 0 u , u = Kz , hence z = S(w -v) , S = (I + G 0 K) -1 , ζ = T (w -v) , T = G 0 K(I + G 0 K) -1 .
The control aim is to keep the sensitivity transfer function S small, while robust stability requires that the complementary sensitivity transfer function T be small. However, it holds that S + T = I, and hence, both cannot be kept small simultaneously. We need that

∀ω ∈ R , δ(ω)|T (ω)| ≤ 1 .
And if δ(ω) → ∞ as ω → ∞, this imposes that |T (ω)| → 0, and therefore |S(ω)| → 1. Therefore we cannot follow a high frequency reference input w. However, it may be possible to keep |S(ω)| small at low frequencies, where δ(ω) is small. The solution is to work with dynamic filters and fictitious outputs z = W 0 z,

ζ = W 1 ζ, with |W 1 (ω)| ≥ δ(ω) (a generalized differentiator).
If we can keep the transfer function from v to ζ less than one at all frequencies, we ensure stability of the control system. And we choose a weight W 0 large at low frequencies (a generalized integrator), and investigate the standard problem with the output (z, ζ).

If W 0 G 0 is strictly proper, as it will usually be, there is no throughput from u to the output z. In that case, it is necessary to add a third component ũ = W 3 u to the regulated output, with W 3 proper but not strictly (may be a multiplicative constant R). This is to satisfy the condition R > 0 of the next section.

H ∞ -optimal control

Given a linear system with inputs u and w and outputs y and z, and a desired attenuation level γ, we want to know whether there exist causal control laws u(t) = φ(t, y t ) guaranteeing inequality (3), and if yes, find one. This is the standard problem of H ∞ -optimal control. We propose here an approach of this problem based upon dynamic game theory, following [START_REF] Bas | H ∞ -Optimal Control and Related Minimax Design Problems : a Differential Games approach[END_REF]. Others exist. See [START_REF] Doyle | State-space solutions to the standard H 2 and H ∞ control problems[END_REF][START_REF] Stoorvogel | The H-infinity control problem[END_REF][START_REF] Glover | H-infinity control[END_REF].

We denote by an accent the transposition operation.

Continuous time

We start with a state space description of the system. Here

x ∈ R n , u ∈ R m , w ∈ R , y ∈ R p , z ∈ R q .
This prescribes the dimensions of the various matrices in the system equations. In the finite horizon problem, they may all be time dependent, say piecewise continuously. These equations are:

ẋ = Ax + Bu + Dw , x(t 0 ) = x 0 , (16) y
= Cx + Ew , (17) z
= Hx + Gu . ( 18
)
We will use the system matrix

S =   A B D C 0 E H G 0   . ( 19
)
Remark 3.1

1. The fact that "the same" input w drives the dynamics (16) and corrupts the output y in (17) is not a restriction. Indeed different components of w may enter the different equations. (Then DE = 0.)

2. The fact that we allowed no throughput from u to y is not restrictive. y is the measured output used to control the system. If there were a term +F u on the r.h.s. of (17), we could always use ỹ = y -F u as measured output.

3. A term +F w in (18) would create cross terms in wx and wu in the linear quadratic differential game to be solved. The problem would remain feasible, but the equations would be more complicated, which we would rather avoid.

We further let

H G ( H G ) = Q P P R and D E ( D E ) = M L L N . (20)
All the sequel makes use of the following hypotheses:

Assumption 3.1 1.
G is one to one or, equivalently, R > 0,

2. E is onto or, equivalently, N > 0.

Duality Notice that changing S into its transpose S swaps the two block matrices in (20) and also the above two hypotheses. This operation (together with reversal of time), that we will encounter again, is called duality.

Finite horizon

As in subsection 2.2.2, we include initial and terminal costs. For any symmetric nonnegative matrix Z, we write x 2 Z := x Zx and likewise for other vectors. Let Q 0 be a symmetric positive definite matrix, and Q 1 a nonnegative definite one, and

J γ (x 0 , u(•), w(•)) = x(t 1 ) 2 Q 1 + t 1 t 0 ( x (t) u (t) ) Q P P R x(t) u(t) -γ 2 w(t) 2 dt -γ 2 x 0 2 Q 0 . (21)
We introduce two Riccati equations for the symmetric matrices S and Σ, where we use the feedback gain F and the Kalman gain K appearing in the LQG control problem:

F = -R -1 (B S + P ) , K = (ΣC + L )N -1 , (22)
-Ṡ = SA + A S -F RF + γ -2 SM S + Q , S(t 1 ) = Q 1 , (23)
Σ = ΣA + AΣ -KN K + γ -2 ΣQΣ + M , Σ(t 0 ) = Q -1 0 . ( 24
)
Theorem 3.1 For a given γ, if both equations ( 23) and ( 24) have solutions over

[t 0 , t 1 ] satisfying ∀t ∈ [t 0 , t 1 ] , ρ(Σ(t)S(t)) ≤ γ 2 , ( 25
)
the finite horizon standard problem has a solution for that value of γ and any larger one. In that case, a (nonunique) min-sup controller is given by u = u as defined by either of the following two systems:

ẋ = (A+γ -2 M S)x + Bu + (I -γ -2 ΣS) -1 K[y-(C +γ -2 LS)x], x(t 0 ) = 0 , u = F x , or ẋ = (A + γ -2 ΣQ)x + (B + γ -2 ΣP )u + K(y -C x) , x(t 0 ) = 0 , u = F (I -γ -2 ΣS) -1 x .
If any one the of above conditions fails to hold, then, for any smaller γ, the criterion (21) has an infinite supremum for any causal controller u(t) = φ(t, y t ).

Remark 3.2

1. The notation ρ(X) stands for the spectral radius of the matrix X. If condition (25) holds, then indeed, the matrix (I -γ -2 ΣS) is invertible.

2. The two Riccati equations are dual of each other as in the LQG control problem.

3. The above formulas coincide with the LQG formulas in the limit as γ → ∞.

4. The first system above is the "certainty equivalent" form. The second one follows from placing x = (I -γ -2 ΣS)x. The symmetry between these two forms seems interesting.

5. The solution of the H ∞ -optimal control problem is highly nonunique. The above one is called the central controller. But use of Bas ¸ar's representation theorem [START_REF] Bas | Dynamic Noncooperative Game Theory[END_REF] yields a wide family of admissible controllers. See [START_REF] Bas | H ∞ -Optimal Control and Related Minimax Design Problems : a Differential Games approach[END_REF].

Infinite horizon stationary problem

In this section, we assume that the matrix S is a constant. The system is then said to be stationary. The spaces L 2 considered are to be understood as L 2 (R → R d ) with the appropriate dimension d. The dynamic equation ( 16) is to be understood with zero initial condition at -∞, and we will be interested in asymptotically stable solutions. There is no room for the terms in Q 0 and Q 1 of the finite time criterion [START_REF] Georgiou | Robustness analysis of nonlinear feedback systems: an input-output approach[END_REF], and its integral is to be taken from -∞ to ∞.

We further invoke a last pair of dual hypotheses:

Assumption 3.2
The pair (A, D) is stabilizable and the pair (A, H) is detectable.

The Riccati equations are replaced by their stationary variants, still using [START_REF] Glover | H-infinity control[END_REF]:

SA + A S -F RF + γ -2 SM S + Q = 0 , ( 26
)
ΣA + AΣ -KN K + γ -2 ΣQΣ + M = 0 . ( 27
)
Theorem 3.2 Under condition 3.2, if the two algebraic Riccati equations ( 26) and ( 27) have positive definite solutions, the minimal such solutions S and Σ can be obtained as the limit of the Riccati equations (23) when integrating from S(0) = 0 backward and, respectively, (24) when integrating from Σ(0) = 0 forward. If these solutions satisfy the condition ρ(Σ S ) < γ 2 , then the same formula as in Theorem 3.1 replacing S and Σ by S and Σ respectively provide a solution to the stationary standard problem. If moreover the pair (A, B) is stabilizable and the pair (A, C) is detectable, there exists such a solution for sufficiently small γ.

If the existence or the spectral radius condition fail to hold, there is no solution to the stationary standard problem for any smaller γ.

The condition of positive definiteness of S and Σ * can be slightly weakened, leading to a slightly more precise theorem (see [START_REF] Bas | H ∞ -Optimal Control and Related Minimax Design Problems : a Differential Games approach[END_REF]). But this does not seem to be very useful in practice.

Discrete time

We consider now the discrete-time system where t ∈ N:

x t+1 = A t x t + B t u t + D t w t , x t 0 = x 0 (28) y t = C t x t + E t w t , (29)
z t = H t x t + G t u t . ( 30
)
where the system matrices may depend on the time t. We use notation [START_REF] Scorletti | Control of rational systems using linear-fractional representations and linear matrix inequalities[END_REF], still with assumption 3.1 for all t, and

Āt = A t -B t R -1 t P t , A t = A t -L t N -1 t C t . (31)
We also invoke all along the following two dual hypotheses:

Assumption 3.3 ∀t , rank A t H t = n , rank ( A t D t ) = n .
We want to control the system with a strictly causal controller u t = φ t (t, y t-1 ) .

(See [START_REF] Bas | H ∞ -Optimal Control and Related Minimax Design Problems : a Differential Games approach[END_REF] for a nonstrictly causal controller or delayed information controllers.)

Finite horizon

We introduce two positive definite symmetric n × n matrices X and Y . The augmented criterion is now

J = x t 1 2 X + t 1 -1 t=t 0 ( z t 2 -γ 2 w t 2 ) -γ 2 x 0 2 Y . (32)
We will not attempt to describe here the (nonlinear) discrete-time certainty equivalence theorem used to solve this problem. (See [START_REF] Bernhard | A min-max certainty equivalence principle for nonlinear discrete time control problems[END_REF][START_REF] Bas | H ∞ -Optimal Control and Related Minimax Design Problems : a Differential Games approach[END_REF].) We go directly to the solution of the standard problem.

The various equations needed may take quite different forms. We choose one. We need the following notation (note that Γ t and St involve S t+1 ):

Γ t = (S -1 t+1 + B t R -1 t B t -γ -2 M t ) -1 , St = Ā t (S t+1 -γ -2 M t ) -1 Āt + Q t -P t R -1 t P t , ∆ t = (Σ -1 t + C t N -1 t C t -γ -2 Q t ) -1 . Σ t+1 = A t (Σ -1 t -γ -2 Q t ) -1 A t + M t -L t N -1 t L t . (33)
The two discrete Riccati equations may be written as

S t = Ā t Γ t Āt + Q t -P t R -1 t P t , S t 1 = X , (34)
Σ t+1 = A t ∆ t A t + M t -L t N -1 t L t , Σ t 0 = Y -1 . (35)
Theorem 3.3 Under the hypothesis 3.3, if both discrete Riccati equations (34) and (35) have solutions satisfying either ρ(M t S t+1 ) < γ 2 and ρ( Σ t+1 S t+1 ) < γ 2 or ρ(Σ t Q t ) < γ 2 and ρ(Σ t St ) < γ 2 , then the standard problem has a solution for that value of γ and any larger one, given by

u t = -R -1 t (B t Γ t Āt + P t )(I -γ -2 Σ t S t ) -1 xt , (36)
xt+1 = A t xt + B t u t + γ -2 A t ∆ t (Q t xt +P t u t ) + ( A t ∆ t C t +L t )N -1 t (y t -C t xt ), (37)
xt 0 = 0 . (38)
If any one of the above conditions fails to hold, then for any smaller γ, the criterion (32) has an infinite supremum for any strictly causal controller.

Remark 3.3

1. Equation (37) is a one-step predictor allowing one to get the worst possible state xt = (I -γ -2 Σ t S t ) -1 xt as a function of past y s up to s ≤ t -1.

2. Controller (36) is therefore a strictly causal, certainty-equivalent controller.

Infinite horizon stationary problem

We consider the same system as above, with all system matrices constant and with the strengthened hypothesis (as compared to assumption 3.2):

Assumption 3.4 The pair (A, D) is controllable and the pair (A, H) is observable.

The dynamics is to be understood with zero initial condition at -∞ and we seek an asymptotically stable and stabilizing controller. The criterion ( 32) is replaced by a sum from -∞ to ∞, thus with no initial and terminal terms.

The stationary versions of all the equations in the previous subsection are obtained by removing the index t or t + 1 to all matrix-valued symbols.

Theorem 3.4 Under assumption 3.4, if the stationary versions of (34) and ( 35) have positive definite solutions, the smallest such solutions S and Σ are obtained as, respectively, the limit of S t when integrating (34) backward from S 0 = 0, and the limit of Σ t when integrating (35) forward from Σ 0 = 0. If these limit values satisfy either of the two spectral conditions in Theorem 3.3, the standard problem has a solution for that value of γ and any larger one, given by the stationary versions of equations (36) and (37).

If any one of these conditions fails to hold, the supremum of J is infinite for all strictly causal controllers.

Nonlinear problems 4.1 Nonlinear H ∞ control

There are many different ways to extend H ∞ -optimal control theory to nonlinear systems. A driving factor is how much we are willing to parametrize the system and the controller. If one restricts its scope to a parametrized class of system equations (see below) and/or to a more or less restricted class of compensators (e.g. finite dimensional), then some more explicit results may be obtained, e.g. through special Lyapunov functions and LMI approaches (see [START_REF] Coutinho | Nonlinear H-infinity control: an LMI approach[END_REF][START_REF] Scorletti | Control of rational systems using linear-fractional representations and linear matrix inequalities[END_REF]), or passivity techniques (see [START_REF] Ball | H ∞ Control for Nonlinear Systems with Output Feedback[END_REF]).

In this section, partial derivatives are denoted by indices.

A finite horizon problem

As an example, we specialize [START_REF] Didinsky | Structural properties of minimax policies for a class of differential games arising in nonlinear H ∞optimal control[END_REF] for a problem slightly more natural than the standard problem [START_REF] Berkowitz | Lectures on differential games[END_REF] in a nonlinear setup (see [START_REF] Van Der Schaft | L 2 -Gain and Passivity Techniques in Nonlinear Control[END_REF]), i.e. finding a control law such that the controlled system has finite L 2 -gain: Definition 4.1 A system ω → z is said to have L 2 -gain less than or equal to γ if there exists a number β such that,

∀ω ∈ Ω , z 2 ≤ γ 2 ω 2 + β 2 . ( 39
)
We consider a nonlinear system defined by

ẋ = a(t, x, u) + b(t, x, u)w , x(0) = x 0 , z(t) = g(t, x, u) , y(t) = h(t, x) + v .
We work over the time interval [0, T ]. We use three symmetric matrices, R positive definite, P and Q nonnegative definite, and take as the norm of the disturbance:

ω 2 = T 0 w(t) 2 R + v(t) 2 P dt + x 0 2 Q .
Let also

B(t, x, u) := 1 4 b(t, x, u)R -1 b (t, x, u) .
We use the following two HJCIB equations, where φ : [0, T ] × R n → U is a state feedback and û(t) stands for the control actually used:

V t (t, x) + V x (t, x)a(t, x, φ(t, x)) + γ -2 V x (t, x)B(t, x, φ(t, x))V x (t, x) + g(t, x, φ(t, x)) 2 = -p(t, x) , V (T, x) = 0. (40)
ẋ = a(t, x, φ(t, x)) + 2γ -2 B(t, x, φ(t, x))V x (t, x) -[V xx (t, x)-W xx (t, x)] -1 2γ 2 h x (t, x) P (y-h(t, x)) -p x (t, x) , x(0) = Arg min x V (0, x) -γ 2 x 2 Q .
Remark 4.1 Although x is given by this ordinary differential equation, yet this is not a finite-dimensional controller since the partial differential equation (41) must be integrated in real time.

Stationary problem

We choose to just show a slight extension of the result of subsection 2.2.2 applied to the investigation of the standard problem [START_REF] Berkowitz | Lectures on differential games[END_REF], for a more general system, and requiring less regularity, than in the previous subsection. Let therefore a system be defined as in [START_REF] Bardi | Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations[END_REF] with x(0) = x 0 , (6), and ( 7), but with f , g and h time-independent. Let also

L(x, u, w) = g(x, u) 2 -γ 2 w 2 . ( 42
)
We want to find a nonanticipative control law u(•) = φ(y(•)) that would guarantee that there exists a number β such that the control û(•) and the state trajectory x(•) generated always satisfy

∀(x 0 , w(•)) ∈ Ω , J(û(•), w(•)) = ∞ 0 L(x(t), û(t), w(t)) dt ≤ β 2 . ( 43
)
We leave it to the reader to specialize the result below to the case (42), and further to such system equations as used, e.g. in [START_REF] Ball | H ∞ Control for Nonlinear Systems with Output Feedback[END_REF][START_REF] Coutinho | Nonlinear H-infinity control: an LMI approach[END_REF]. We need the following two HJCIB equations, using a control feedback φ(x) and the control û(t) actually used:

∀x ∈ R n , inf w∈W [-V x (x)f (x, φ(x), w) -L(x, φ(x), w)] = 0 , V (0) = 0 , (44) and, ∀x ∈ R n , inf w|y(t)
[-W t (t, x) -W x (t, x)f (t, û(t), w) -L(x, û(t), w)] = 0 , W (0, x) = 0 , (45) (by inf w|y(t) we mean inf w|h(t,x,w)=y(t) ). Let X(t) be the set of reachable states at time t from any x 0 with φ and a minimizing w in (45), and x(t) defined as 2. Equation (45), to be integrated in real time, behaves as an observer. More precisely, it is the counterpart for the conditional cost measure of Kushner's equation of nonlinear filtering (see [START_REF] Daum | Nonlinear filters[END_REF]) for the conditional probability measure.

V (x(t)) -W (t, x(t)) = max x∈X(t) [V (x) -W (t, x)] . (46
If equation ( 44) has no finite solution, then the full state information H ∞ control problem has no solution, and, not surprisingly, nor does the standard problem with partial corrupted information considered here. (See [START_REF] Van Der Schaft | L 2 -Gain and Passivity Techniques in Nonlinear Control[END_REF].) But in sufficiently extreme cases, we may also detect problems where the available information is the reason why there is no solution:

Theorem 4.3 If equation (44) has no finite solution, or, if V is a viscosity solution of (44) and, for all (û(•), y(•)), either (45) has no finite solution, or V -W is unbounded by above, then there is no causal controller that can keep the criterion bounded against all w(•).

It is due to its special structure that in the linear quadratic problem, either of the above two theorems applies.

Option pricing in Finance

As an example of the use of the ideas of subsection 2.1.1 in a very different dynamic context, we sketch here an application to the emblematic problem of mathematical finance, the problem of option pricing. The present theory is therefore an alternative to Black and Scholes' famous one, using a set description of the disturbances instead of a stochastic one. 2 As compared to the latter, the former allows us to include in a natural fashion transaction costs , and also discrete time trading. Many types of options are traded on various markets. As an example, we will emphasize here European "vanilla" buy options, or Call, and sell options, or Put, with closure in kind. Many other types are covered by several similar or related methods in [START_REF] Bernhard | The Interval Market Model in Mathematical Finance: a Game Theoretic Approach[END_REF].

The treatment here follows [START_REF] Bernhard | The Interval Market Model in Mathematical Finance: a Game Theoretic Approach[END_REF]Part 3], but it is quite symptomatic that independent works using similar ideas emerged around the year 2000, although they generally appeared in print much later. See the introduction of [START_REF] Bernhard | The Interval Market Model in Mathematical Finance: a Game Theoretic Approach[END_REF].

The problem

Portfolio For the sake of simplicity, we assume an economy without interest rate nor inflation. The contract is signed at time 0, and ends at the exercise time T . It bears upon a single financial security whose price u(t) varies with time in an unpredictable manner. We take as the disturbance in this problem its relative rate of growth u/u = τ . In the Black and Scholes theory, τ is modelled as a stochastic process, the sum of a deterministic drift and of a "white noise". In keeping with the topic of this chapter, we will not make it a stochastic process. Instead, we assume that two numbers are known: τ -< 0 and τ + > 0, and all we assume is boundedness and measurability:

∀t ∈ [0, T ] , τ (t) ∈ [τ -, τ + ] and τ (•) ∈ Ω = M([0, T ] → [τ -, τ + ]) . (47)
A portfolio is made of two components: shares of the security for a monetary amount v(t) and an amount y(t) of currency, for a total worth v + y = w. Both v and y may be positive, of course, but also negative through futures for v, a "short" portfolio, and borrowing for y. All variables may vary continuously.

A trader manages the portfolio in a self-financed fashion, meaning that he buys or sells shares of the security, withdrawing the money to buy from the currency part y of the portfolio, or adding to it the proceeds of the sales. In Merton's "continuous trading" fiction, the trader may trade at a continuous rate ξ(t), taken positive for a buy and negative for a sale. But he can also trade a finite block of shares instantly, resulting in jumps in v(•), represented by impulses in ξ(•). We will therefore allow a finite sum

ξ(t) = ξ c (t) + k ξ k δ(t -t k ) ⇔ ξ(•) ∈ Ξ
with ξ c (•) a measurable real function and {t k } ⊂ [0, T ] and {ξ k } ⊂ R two finite sequences, all chosen by the trader. We call Ξ the set of such distributions. There are transaction costs incurred in any transaction, that we will assume to be proportional to the amount of the transaction, with proportionality coefficients C -< 0 for a sale of shares of the security, and C + > 0 for a buy. We will write these transaction costs as C ε ξ with the convention that this means that ε = sign(ξ). The same convention holds for such notation as τ ε X , or later q ε X .

This results in the following control system, where the disturbance is τ and the have restricted the available trader's choices, thus we will end up with a larger premium.

We need now the parameters τ ε h = e hτ ε -1 , ε ∈ {-, +} .

We write u k , v k , w k for u(t k ), v(t k ), w(t k ). An exact discretization of our system is therefore as follows, with τ k ∈ [τ - h , τ + h ]:

u k+1 = (1 + τ k )u k , ( 59
) v k+1 = (1 + τ k )(v k + ξ k ) , (60)
w k+1 = w k + τ k (v k + ξ k ) -C ε ξ k . ( 61
)
The Value function of the restricted game is denoted by V h k (u k , v k ).

Theorem 4.6 The Value function {V h k } satisfies the Isaacs recurrence equation

V h k (u, v) = min ξ max τ ∈[τ - h ,τ + h ]
V h k+1 ((1+τ )u, (1+τ )(v+ξ)) -τ (v+ξ) + C ε ξ ,

∀(u, v) ∈ R + × R , V h N (u, v) = M (u, v) . (62)
Moreover, if one defines V h (t, u, v) as the Value of the game where the trader (maximizer) is allowed to make one jump at initial time, and then only at times t k as above, we have:

Theorem 4.7 The function V h interpolates the sequence {V h k } in the sense that, for all (u, v), V h (t k , u, v) = V h k (u, v). As the step size is subdivided and goes to zero (e.g. h = T /2 d , d → ∞), the function V h (t, u, v) decreases and converges to the function V (t, u, v) uniformly on any compact in (u, v).

Finally, one may extend the representation formula (58), just replacing v and ŵ by vh k and ŵh k given collectively by a carefully chosen finite difference approximation of equation (57) (but the representation formula is then exact) as follows:

Let q ε k = q ε (t k ) be alternatively given by q ε N = c ε and the recursion

q ε k+ 1 2 = (1 + τ ε h )q ε k+1 + τ ε h , q - k = max{q - k+ 1 2 , C -} , q + k = min{q + k+ 1 2 , C + } .
Also, let

Q ε k = ( q ε k 1 ) , W h k (u) = vh k (u) ŵh k (u)
.

The following algorithm is derived from a detailed analysis of equation (62):

W h k (u) = 1 q + k+ 1 2 -q - k+ 1 2 1 -1 -q - k+ 1 2 q + k+ 1 2 Q + k+1 W h k+1 ((1 + τ + h )u) Q - k+1 W h k+1 ((1 + τ - h )u) , W h N (u) = W (T, u) .
And as previously,

V h k (u, v) = ŵh k (u) + q ε k vh k (u) -v ,
and for all practical purposes, P (u 0 ) = ŵh 0 (u 0 ) + C ε vh 0 (u 0 ) .

Conclusion

As stated in the introduction, game theoretic methods are only one part, may be a prominent one, of modern robust control. They typically cover a wide spectrum of potential applications, their limitations being in the difficulty to solve a nonlinear dynamic game, often with imperfect information. Any advances in this area would instantly translate into advances in robust control.

The powerful theory of linear-quadratic games coupled with the min-max certainty equivalence principle makes it possible to efficiently solve the linear H ∞optimal control problem, while the last example above shows that some very nonlinear problems may also receive a rather explicit solution via these game theoretic methods.

Index

  If there exists an admissible state feedback φ, a nonnegative C 1 function p and a C 1 function V satisfying equation (40), then the strategy u(t) = φ(t, x(t)) solves the state feedback finite gain problem (39).2. If furthermore, V is C 2 , and there exists for all pairs (û(•), y(•)) a C 2 function W solution of equation (41), and x(t) in (14) is always unique, then the strategy u(t) = φ(t, x(t)) solves the output feedback finite gain problem (39). Moreover, x(t) may be computed recursively according to the equation

	W t (t, x) + W x (t, x)a(t, x, û(t)) + γ -2 W x (t, x)B(t, x, û(t))W x (t, x) + g(t, x, û(t)) 2 = γ 2 y(t) -h(t, x) 2 , W (0, x) = γ 2 x 2 Q .	(41)
	Theorem 4.1
	1.

  )

	2. If furthermore, there exists for every pair (û(•), y(•)) ∈ U × Y, a BUC vis-
	cosity solution W (t, x) of equation (45) in [0, ∞) × R n , and if furthermore,
	there is for all t ≥ 0 a unique x(t) satisfying equation (46), then the con-
	troller u(t) = φ(x(t)) solves the output feedback problem (43).
	Remark 4.2
	1. A supersolution of equation (44) may make the left hand side strictly positive.
	Theorem 4.2
	1. The state feedback problem (43) has a solution if and only if there exists an
	admissible stabilizing state feedback φ(x) and a BUC viscosity supersolu-
	tion 1 V (x) of equation (44). Then, u(t) = φ(x(t)) is a solution.

See[START_REF] Barles | Solutions de viscosité des équations de Hamilton-Jacobi[END_REF][START_REF] Bardi | Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman Equations[END_REF] for a definition and fundamental properties.

See, however, in[12, Chapter 2] a probability-free derivation of Black and Scholes' formula.

control ξ:

which the trader controls through a nonanticipative strategy ξ(•) = φ(u(•)), which may in practice take the form of a state feedback ξ(t) = ϕ(t, u(t), v(t)) with an additional rule saying when to make impulses, i.e. jumps in v, and by what amount.

Hedging A terminal payment by the trader is defined in the contract, in reference to an exercise price K. Adding to it the closure transactions costs with rates c -∈ [C -, 0] and c + ∈ [0, C + ], the total terminal payment can be formulated with the help of two auxiliary functions v(T, u) and ŵ(T, u) depending on the type of option considered according to the following table:

And the total terminal payment is M (u(T ), v(T )), with

Definition 4.2 An initial portfolio (v(0), w(0)) and a trading strategy φ constitute a hedge at u(0) if they ensure

meaning that the final worth of the portfolio is enough to cover the payment owed by the trader according to the contract signed. Il follows from (50) that ( 53) is equivalent to

a typical guaranteed value according to subsection 2.1.1. Moreover, the trader wishes to construct the cheapest possible hedge, and hence solve the problem

Let V (t, u, v) be the Value function associated with the differential game defined by ( 48), ( 49) and (54). The premium to be charged to the buyer of the contract, if

This problem defines the so-called "robust control approach to option pricing". An extensive use of differential game theory yields the following results.

The solution

Because of the impulses allowed in the control, Isaacs' equation is replaced by the following differential quasi-variational inequality (DQVI): Solving the DQVI (55) may be done with the help of the following auxiliary functions. We define

Note that, for ε ∈ {-, +}, q ε = C ε for t ≤ t ε and increases (ε = +) or decreases (ε = -) towards c ε as t → T , with

We also introduce the constant matrix S and the variable matrix T (t) defined by

Finally, we name collectively two functions:

involved in the pair of coupled linear partial differential equations

with the boundary conditions (51).

Theorem 4.5

• The partial differential equation (57) with boundary conditions (51) has a unique solution.

• The Value function of the game (48), ( 49), (54) (i.e. the unique Lipshitz continuous viscosity solution of (55)) is given by

• The optimal hedging strategy, starting with an initial wealth w(0) = P (u(0)), is to make an initial jump to v = v(0, u(0)) and keep v(t) = v(t, u(t)) as long as t < t ε as given by (56), and do nothing for

Remark 4.3

• In practice, for classic options, T -t ε is very small (typically less than one day) so that a simplified trading strategy is obtained by choosing q ε = C ε , and if T is not extremely small, P (u 0 ) = ŵ(0, u 0 ) + C ε v(0, u 0 ) .

• The curve P (u) for realistic [τ -, τ + ] is qualitatively similar to that of the Black and Scholes theory, usually larger because of the trading costs not accounted for in the classic theory.

• The larger the interval [τ -, τ + ] chosen, the larger P (u). Hence the choice of these bounds is a critical step in applying this theory. The hedge has been found to be very robust against occasional violations of the bounds in (47).

Discrete time trading

One of the advantages to relinquish the traditional "geometric diffusion" stochastic model for the disturbance τ (•) is to allow for a coherent theory of discrete-time trading. Let therefore h = T /N be a time step, with N ∈ N. Assume the trader is allowed to do some trading only at instants t k = kh, k ∈ N. This means that we keep only the impulsive part of ξ(•), and fix the impulse instants t k . We therefore


Paolo Scala

Miguel Mujica

Daniel Delahaye
email: delahaye@recherhe.enac.fr.

A DOWN TO EARTH SOLUTION: APPLYING A ROBUST SIMULATION-OPTIMIZATION APPROACH TO RESOLVE AVIATION PROBLEMS

This paper deals with the improvement of the robustness of heuristic solutions for aviation systems affected by uncertainty when the resolution of conflicts is implemented. A framework that includes the use of optimization and simulation is described which in turn generates pseudo-optimal schedules. The initial solution is progressively improved by iteratively evaluating the uncertainty in the generated solutions and calibrating in accordance with the objective function. Simulation is used for testing the feasibility of a solution generated by an optimization algorithm in an environment characterized by uncertainty. The results show that the methodology is able to improve solutions for the scenarios with uncertainty, thus making them excellent candidates for being implemented in real environments.

INTRODUCTION

Airport Systems are considered as critical infrastructures that promote connectivity, value to the region and key for fostering the global interaction of different countries, industries and people. For that reason the expected growth is a threat for the current facilities and it should not be desirable to limit the growth by the technical limitations of the facilities. Furthermore, the airport is a connection of elements and processes that require proper tools for efficiently manage them. Airport operations can be distinguished in landside operations and airside operations, where landside operations are the ones that occur inside the terminals, and airside operations are the ones that occur outside the terminal. Regarding landside operations, it is possible to identify the passengers as entities and check-in desks, gates, or even recreational places like shops and cafe as resources. Regarding airside operations, aircraft can be identified as entities, and ground handling, runways, taxiways, airspace sectors, just to cite some of them, as resources. So from this classification it can be noticed that, in the airport system, two subsystems are identified, very different from each other and governed by completely different dynamics and logic. Moreover, these two different subsystems interact with each other and affect each other performance. So, the characteristics of an airport system, as previously described, are in line with the definition of complex systems given at the beginning of this section and therefore, airport systems should be considered as complex systems when they are analyzed. Another element that characterize aviation systems is the stochasticity, aviation systems are by nature stochastic and for that reason simulation is a key technique for getting the proper required insight in order to increase predictability. Furthermore, aviation is highly dependent on schedules, this also makes their problems prone to be improved by the use of optimization techniques. However, practitioners have recognized that optimization techniques alone fall short when they are used for solving capacity problems [START_REF] Scala | Implementation of an Optimization and Simulation Based Approach for Detecting and resolving Conficts at Airports[END_REF]. In this context a fundamental concept that needs to be considered is the robustness of solution coming from optimization models. Due to the variability of the system, a solution provided by an optimization model can cover several scenarios with uncertainty, so a solution is defined robust if it can be applied to more uncertain scenarios without major changes in performance.

In the literature there are many works that are related to the robustness of solution for aviation problems, the outcome of these works is the development of many tools and methodologies. [START_REF] Chandran | A Dynamic Programming Algorithm for Robust Runway Scheduling[END_REF] proposed a dynamic programming algorithm for robust runway scheduling. In their work they considered uncertainties in the aircraft arrival times, coming up with a trade-off between runway throughput and probability of violating separation minimum on the runway as objective. [START_REF] Kim | Robust Gate Assignment[END_REF] tackled the robust gate assignment problem when stochastic delays are introduced. They defined robustness as the interval time between two aircraft that park at the same gate, so the bigger is the interval the more robust is the solution. They computed the number and level of disturbed assignments and they solved the problem using a linear mixed 0-1 optimization model and the tabu search meta heuristic showing improvement, when compared to the original schedule, in the number and level of disturbed assignments of the 92% and 95%, respectively. In the work of [START_REF] Arias | A Methodology Combining Optimization and Simulation for Real Applications of the Stochastic Aircraft Recovery Problem[END_REF] the stochastic aircraft recovery problem is solved by employing constraint programming paradigm in combination with simulation techniques, the robustness of the solutions is assessed comparing the standard deviation obtained from simulated results with the variation of the probabilistic distribution used for generating stochastic delays, and its expected propagation due to the problem size. In the PhD thesis of [START_REF] Lee | Airport Surface Traffic Optimization and Simulation in the Presence of Uncertainties[END_REF] airport surface congestion problem was tackled, in his work he employed optimization for scheduling runways and taxiways operations and then stochastic simulation to deal with uncertainties. [START_REF] Narciso | Robust gate assignment procedures from an airport management perspective[END_REF] considered the robust gate assignment problem proposing a causal model employing coloured Petri nets (CPN) formalism obtaining the minimum number of gates necessary to absorb the traffic.

The focus of this paper is the improvement of robustness, as it was defined previously, for solutions coming from optimization models solved by meta heuristics. The methodology is iterative, and it aims at finding solutions which fit best in a system affected by uncertainties. In this paper it is presented an application of this methodology to the airport conflict detection and resolution problem. Results confirm that after two iterations, by changing parameters of the algorithm implemented for solving this problem, it is possible to get an improvement in the initial solution. The paper is organized as follows, in section 2 the problem tackled in this paper is described, then in section 3 the methodology is presented together with the description of the optimization and the simulation model. Section 4 shows the implementation of the methodology applied to the airport conflict detection and resolution problem and in the same section results are shown and discussed. In the final section, conclusions and future work is presented.

PROBLEM DESCRIPTION

Nowadays, many European airports are on the edge of their capacity limitations and this can be attributed to the fast growing of the air traffic in the recent years. Moreover, a growth in the air traffic has been forecasted [START_REF] Airbus | Global Market Forecast[END_REF] for the upcoming years, therefore, in a future scenario of potential congestion, solutions need to be given to overcome this issue. Airport congestion is identified especially inside the airspace terminal manoeuvring area (TMA) and on the ground in the proximity of the runway entry. The TMA is a portion of the airspace that surrounds airports, here the flow of landing aircraft converge to the airport airside. A good management of arrivals and departures rate made by the air traffic controllers can relieve the airport, airspace and ground side, from potential congestion situations and at the same time increase the runway throughput [START_REF] Frankovich | Air traffic flow management at airports: A unified optimization approach[END_REF]. The airport conflict detection and resolution problem, which has been considered in this paper, refers to the sequencing of the aircraft landing flow and ground capacity management. In this specific problem, conflicts are detected in the airspace due to separation minima restrictions, and on the ground due to capacity overloads. In the next sections conflicts for airspace, runways and ground components are described in more detail. The objective of solving this problem is to ensure safety for the aircraft flow in the airspace reducing the airspace conflicts to zero and reduce capacity overloads on the ground. For further and more detailed information about this problem please refer to Ma, Delahaye, and Sbihi (2016).

Scala, Mujica and Delahaye

Airspace and Runway Conflicts

Concerning the airspace, given some landing routes merging at one final point (initial approach fix-IAF) (see Figure 1), and a set of landing aircraft, the main objective is to make the sequence of the aircraft so they will merge at the IAF without incurring in separation minima infringements. Regarding the runways, conflicts are detected when the separation between two consecutive aircraft that are crossing the runway is less than the required separation minima. Separation minima values are set according to the different runway operations, which can be arrivals (runway in) or departures (runway out), and they are based on the wake turbulence category of the leading and the trailing aircraft. Separation minima for the airspace are provided by ICAO standards (ICAO 2007) and separation minima for the runway are taken from [START_REF] Frankovich | Air traffic flow management at airports: A unified optimization approach[END_REF].

Ground Conflicts

Concerning ground operations, the developed model is a high-level one, meaning that ground operations are not modeled in detail. So first the main components of the ground are identified and then a capacity is attributed to each of them. The main ground components identified are taxiways and terminals. Capacity is defined as number of aircraft that are simultaneously using one of the components. Once the aircraft have landed and exited the runway, they cross the taxiway until they park at the gate. As mentioned before the model of the ground is a high-level one, which means that taxiway routes were not considered in detail, therefore, taxiway operations were modeled using average travel time. Turnaround times were based on a real schedule, and the travel time on the taxiway from the gate until the departing runway entry was considered using average travel time. The objective is to do not exceed the capacity of these components, conflicts in this case are represented by the sum of maximum capacity overload and the average capacity overload. Table 1 shows the capacity of the main ground components related to the example considered in this work. The framework which will be introduced in this section will be the combination of a sliding window-based optimization model together with a discrete event simulation based model to achieve the aforementioned scope. In the rest of this section the structure of the methodology will be introduced and then both the optimization process and the simulation environment will be described in more detail.

Methodology Description

The methodology proposed in this paper, can be represented by two main steps as depicted in Figure 2. In the first step, the system under study is subject to an optimization process with the aim of getting optimal or suboptimal solutions according to a specific objective. In this paper, this methodology is applied to the airport conflict detection and resolution problem, where the objective of the study is to achieve a scenario with zero conflicts. Bearing in mind that the optimization process leads to an exact solution, where most of the operational parameters are deterministic, it is likely that this solution, when tested in a real environment, will be sensitive to slight changes due to system variability. In the second step of the methodology, simulation is employed to introduce the variability of the system and to test the robustness of the solution provided by the optimization process. In the simulation environment, some relevant parameters that affect the solution in reality, are allowed to have uncertain values. This methodology is iterative, which means that these two steps will be applied as long as the robustness will be improved according to a certain level of satisfaction, which represents the stopping condition of the iterative process. In this work robustness is measured as the deviation in the performance of the simulated solution from the ideal scenario, where the performance is evaluated as number of conflicts and the ideal scenario is a solution that gives zero conflicts. In this context, the robustness is improved when this deviation is minimized.

Optimization Process

The optimization process that has been applied in this methodology, and specifically for the airport conflict detection and resolution problem, is based on a sliding window approach. This approach divides the entire problem in smaller subproblems according to a window size, and then shifts the window along the time horizon. The main advantages of this method are that the subproblems are easier to solve and require less computational time, moreover, it tackles the problem in a dynamical fashion by updating the decision variables while the window slides along the time horizon, resulting in an improvement of the quality of the solution. This approach has already been employed in similar problems giving good results ( In this approach, each entity (aircraft) is associated with a certain status according to its position relatively to the current window. As a result, the status of an entity can be complete if it is outside the current window; on-going if the entity is between previous and current window; active if the entity is inside the current window; planned if the entity is in the next window. In the current window, decisions will be made only for the entities with on-going and active status, while for the completed entities they have already been made. Concerning the planned entities, the decisions will be made in the next window (see Figure 3).

Scala, Mujica and Delahaye

The main parameters of the sliding window approach are the window size and the shift size, they affect the performance of the optimization process in terms of computational time and quality of the solution, so it is crucial to set these two parameters with proper values in order to get good solutions. A small window will lead to solve small subproblems but some flight info might not be considered. A big window will lead to solve bigger problems but at the same time some flight info might be considered too early. In both cases the effectiveness of the solution is affected. A small shift will lead to update many decision variables at every window shift, a big shift will lead to fix more decision variables and update less decision variables at every window shift affecting the performance of the algorithm.

Figure 3: Sliding window approach.

The solution space of this problem is considered NP-Hard [START_REF] Beasley | Scheduling Aircraft Landings -The Static Case[END_REF], which justifies the use of a meta heuristic in order to solve the problem and to obtain a (sub)optimal solution. To that purpose, each subproblem, defined by the sliding window approach, has been solved by means of the simulated annealing meta heuristic [START_REF] Kirkpatrick | Optimization by Simulated Annealing[END_REF]. This meta heuristic has been proven to be very efficient when dealing with similar problems [START_REF] Man | A novel approach to Automated Merge 4D Arrival Trajectories for Multi-parallel Runways[END_REF]Xiao-hao 2015, Ma, Delahaye, Sbihi, and[START_REF] Ma | Merging Flows in Terminal Maneuvring Area using Time Decomposition Approach[END_REF]. The definitions and detailed explanations of the sliding window approach and of the meta heuristic are out of the scope of this paper, for more detailed information and a better understanding of this optimization process applied to this specific type of problem please refer to Ma, Delahaye, and Sbihi (2016).

Simulation Environment

Discrete event simulation (DES) has been put in the loop for evaluating the effect of uncertainty in the solutions proposed by the optimization algorithm, in this section, the advantages of the use of simulation for this purpose are described. The use of DES allows to include the uncertainty in the scenarios which is not possible or highly complicated using other techniques. In this context, the introduction of uncertainty with simulation makes the situation closer to reality and with the definition of robustness it is possible to achieve a solution that might be not mathematically optimal but more resilient to disruption in the real system. DES has been widely used in many fields, from manufacturing to transportation, already since the last decades, and it has been proven to be useful for what-if analysis, scenario testing, and as a decision support tool [START_REF] Banks | Handbook of Simulation[END_REF][START_REF] Brunner | Toward increase use of simulation in transportation[END_REF][START_REF] Negahban | Simulation for manufacturing system design and operation: Literature review and analysis[END_REF]. In DES, time passes according to the occurrence of specific events and the sequence of actions that the entities make are already known in advance. The use of this modelling approach is better than other ones Scala, Mujica and Delahaye like agent-based or system dynamics due to the fact that the processes modelled follow sequential processes which can be properly modelled under this paradigm. The use of other ones like agent-based for instance is more suited for situations in which the modeled entities have more degrees of freedom and it is necessary to implement the logic behind the decision-makinig process that the entities undergo. In the system modelled in this work, this is not required, thus giving us a framework more flexible, easy to maintain and faster than the one that can be achieved with a different technique.

IMPLEMENTATION

The methodology presented in section 3.1 has been applied to the airport conflict detection and resolution problem. In the next sections the main parameters of the simulation and optimization model and two iterations of methodology implemented with the relative results are presented.

Parameters of the Methodology

The parameters involved in this methodology can be classified in three groups: controllable parameters, boundary parameters and stochastic parameters. The first ones refers to parameters that are related to the problem under study and therefore are problem specific. Boundary parameters are related to the optimization process itself, so they depend on the type of optimization that has been used. Controllable parameters are part of the optimization model, as they represent the decision variables of the problem. Stochastic parameters represent the sources of stochasticity of the system so they depend on the system under study. In the following sections these parameters are presented and discussed. The controllable and the boundary parameters are very important because they are the drivers of the methodology implementation. Each iteration of the methodology is based on a different setting of these parameters. Stochastic parameters are implemented by the simulation model resulting key factors for the goodness of the methodology. It is important to choose and calibrate them properly, in this work it has been made an initial estimation for them.

Controllable and Boundary Parameters

Controllable parameters of the airport conflict detection and resolution problem considered in this work are:

• Entry speed in the airspace: increase or decrease the speed of the aircraft entering in the airspace for the final approach. • Entry time in the airspace: delay or advance aircraft entering in the airspace for the final approach.

• Pushback time: delay aircraft at their gates before they leave for departing.

Boundary parameters refer to the sliding window parameters:

• Window size: define the size for each subproblem solved by the algorithm.

• Shift size: define the size of the shift for the window and therefore the amount of decision variables that are updated for each subproblem.

In Table 2 controllable and boundary parameters are summarized with their respective values used for the initial optimized solution. Note that values of the parameters entry time in the airspace and pushback time are discretized according to 5 second interval, while entry speed in the airspace is discretized according to 1 percent interval. Optimization parameters are continuous.

Stochastic Parameters

Stochastic parameters of the airport conflict detection and resolution problem considered in this work are:

Scala, Mujica and Delahaye

Experiments

The methodology proposed in this paper has been applied to the airport conflict detection and resolution problem, two iterations of the methodology are presented in this work. At the beginning a solution that represents the original schedule is given, then another solution, this time implementing the optimization process, is presented and the two solutions are compared to each other. After an optimized solution is found by the optimization algorithm, uncertainty is introduced with the simulation model, where 50 replications for each run were completed. The outcome from the simulation is then analyzed in order to test the robustness of the optimized solution. In the same way a second iteration was conducted by running the optimization model by changing some of the methodology parameters presented in 4.1.1, and then testing the new generated solution with the simulation model. The original schedule is based on one day real traffic data (7 February 2016), at Paris Charles de Gaulle airport, as both the optimization and simulation model have modelled the operations of this specific airport. In the following charts results from the initial schedule are illustrated, showing many conflicts for airspace, runways and a big capacity overload for taxiway network and terminals.

Regarding airspace and runway conflicts, they have been calculated on a hourly base, in this way it is possible to see in which part of the day they are concentrated. Figure 4(a) shows that the peak hours conflicts in the airspace happen around 8, 9, 12, 13 and 20 o'clock, ranging between 80 and 90 conflicts. Regarding conflicts on the runways (in and out), figures 4(b) and 4(c) show that there are many conflicts for them. In the runway in conflict trend there are peaks between 8 and 10 o'clock and also at 20 o'clock, in the runway out conflict trend there are peaks between 11 and 12 o'clock and also at 14 o'clock.

Regarding taxi out conflicts, in Figure 5(a) it can be seen that there is an overload in the capacity, which is set as 18, from 9 to 14 o'clock and from 19 to 21 o'clock, with a maximum peak of 25 around 21 o'clock. Figure 5(b) shows the occupancy for the three terminal considered, where especially for the terminal 1 (capacity 11) there is a constant overload from 6 to 21 o'clock.

First Iteration

According to the methodology procedure, in the first step of each iteration the optimization model is run and an optimized solution is obtained from it. In this case after running the optimization model a scenario with zero conflicts has been found for both airspace and ground components. Figures 6(a When uncertainty is included it can be seen that, despite the optimization solution gives a scenario without conflicts, new conflicts appear. Specifically for the airspace, there is a peak of 5 average hourly conflicts at 9 o'clock, and some other conflicts spread during the day (see figure 7(a)). Regarding the runway in, it can be seen that it is not much affected by the uncertainty since the average hourly conflict peak is almost 2 (see figure 7(b)). The runway out, instead, is more affected by the uncertainty, in this case there are many average hourly conflict peaks during the day, 10 between 11 and 12 o'clock, 12 between 14 and 15 o'clock and 9 at 22 o'clock (see figure 7(c)). Regarding ground component overload, terminals were not affected by the uncertainty since there were not capacity overload for them, taxiway average maximum capacity overload was 2.05.

Second Iteration

A second iteration is conducted since results from the first iteration showed the presence of conflicts after the inclusion of uncertainty. In this second iteration the optimization model has been run adjusting some of the controllable or boundary parameters so to have a new solution to be tested with the simulation model. In this case, running the optimization model, a new solution has been obtained by changing the window size of the sliding window optimization approach from a window of 3 hours to a window of 2 hours. This choice was justified by the fact that often when the window size is too long, the algorithm considers too early the info of some flights that are far ahead in the schedule, and this can lead to a loss of the dynamical 4. As can be noticed, this solution provides an almost conflict free scenario, since some conflicts appear for the runways, and for the terminals. So from the optimization point of view this solution is not optimal but suboptimal. In figures 8(a), 8(b) and 8(c) are illustrated the results obtained after testing this solution by means of the simulation model. In figure 8(a) it can be seen that three peaks of 4, 5 and 7 airspace average hourly conflicts are found at 6, 10 and 12 o'clock respectively. Regarding the runway in average hourly conflicts, as it can be seen from figure 8(b), there are two peaks of 1.6 and 2.1 at 8 and 12 o'clock respectively. Results for runway out records only one peak of 14 average hourly conflicts at 14 o'clock.

In table 5 are summarized the results from the two iterations. Looking at the second and the third column, relative to the first and second iteration, it can be noticed that the second iteration improves average runway out conflicts from 89.9 to 85.32, while does not improve average airspace conflicts. The values of other components, runway in, taxiway and terminals are stable between first and second iteration.

The expected result of the methodology should be the progressive decrease in the number conflicts after each iteration and, at the same time, obtaining a solution where the conflicts are balanced between all Scala, Mujica and Delahaye the components. In this case after two iterations it can be seen that some improvements have been reached for some parts of the systems, especially for the runway out, while for the airspace, the methodology was not able to make improvements. Concerning the ground components, the results confirmed that they were not much affected by variability meaning that they could be well predicted. However, the interpretation of the outcome after two iterations, when the overall airport system performance is considered, points out that the solution of the second iteration is more balanced than the solution of the first iteration, which is an improvement in terms of robustness. Implementing further iterations of this methodology, it is expected to obtain even more improvements in the robustness of the solutions.

CONCLUSION AND FUTURE STEPS

In this paper a framework for the improvement of the robustness of optimized solutions applied to the airport conflict detection and resolution problem was presented. The results demonstrate the feasibility of the concept of combining the two methodologies for providing more robust solutions. Uncertainty confirmed to be of high nuisance when applied to an optimized solution. Airspace and runway conflicts were affected the most after including variability in the system, while ground components were less sensitive to it. The methodology, iteratively implemented, was able to lower the conflicts for the runway out, however, it was not very effective for the conflicts in the airspace. In general after two iterations the methodology achieved a more balanced solution. The further discussion would be on how to implement a cyclic approach so that we are able to fine-tune the robustness of the solution obtained by the combination of the two approaches. Further studies can be conducted towards the choice of the parameters to adjust at each iteration in order to make it more effective and efficient. Another aspect that can be studied as a future development of the methodology is the choice of the values for the stochastic parameters that can be done, for example, by processing historical data.

Figure 1 :

 1 Figure 1: Aircraft landing routes.

Figure 2 :

 2 Figure 2: Methodology steps.

  [START_REF] Hu | Receding Horizon Control for Aircraft Arrival Sequencing and Scheduling[END_REF][START_REF] Scala | Receding Horizon Control for Aircraft Arrival Sequencing and Scheduling[END_REF][START_REF] Furini | Improved rolling horizon approaches to the aircraft sequencing problem[END_REF][START_REF] Toratani | Simultaneous Optimization Method for Trajectory and Sequence for receding Horizon Guidance in Terminal Area[END_REF][START_REF] Man | A novel approach to Automated Merge 4D Arrival Trajectories for Multi-parallel Runways[END_REF][START_REF] Ma | Merging Flows in Terminal Maneuvring Area using Time Decomposition Approach[END_REF].

  ) and 6(b) show

Figure 4 :

 4 Figure 4: Conflicts for airspace and runways, scenario with the original schedule.

Figure 5 :

 5 Figure 5: Ground occupancy, scenario with the original schedule.

Figure 6 :

 6 Figure 6: Ground occupancy after optimization, first iteration.

Figure 8 :

 8 Figure 8: Conflicts for airspace and runways, scenario with uncertainties, second iteration.

Table 1 :

 1 Capacity for the main ground component.

	Ground component Capacity
	Taxiway network	18
	Terminal 1	11
	Terminal 2	91
	Terminal 3	57

Table 2 :

 2 Controllable and boundary parameters.

	Entry speed in the airspace	[-10% , 10%]
	Entry time in the airspace [-5 min. , 30 min.]
	Pushback time	[0 , 15 min.]
	Window size	3 hrs.
	Shift size	30 min.
	• Entry time in the airspace deviation
	• Taxiway travel time deviation
	• Pushback time deviation
	In Table 3 stochastic parameters are summarized with their respective values used in the simulation model.
	Note that stochastic parameters are continuous.

Table 3 :

 3 Stochastic parameters.

	Entry time in the airspace deviation [-30 sec. , +30 sec.]
	Taxiway time deviation	[-10% , +10%]
	Pushback time deviation	[-60 sec. , +60 sec.]

Table 4 :

 4 Results from the optimization, second iteration.

	Airspace conflicts	0
	Runway (in and out) conflicts	1
	Taxiway network (maximum overload + average overload)	0
	Terminal 1 (maximum overload + average overload)	1.04
	Terminal 2 (maximum overload + average overload)	0
	Terminal 3 (maximum overload + average overload)	1.04

Table 5 :

 5 Comparison between the results from the two iterations.

		First iteration Second iteration
	Avg airspace conflicts	19.60	26.77
	Avg runway in conflicts	9.30	9.50
	Avg runway out conflicts	89.90	85.32
	Avg taxiway overload	2.05	2.06
	Avg terminals overload	0	0

ACKNOWLEDGMENTS

The authors would like to thank the Aviation Academy of the Amsterdam University of Applied Sciences and the Dutch Benelux Simulation Society and EUROSIM for disseminating the results of this work.

AUTHOR BIOGRAPHIES

PAOLO SCALA is a PhD student at ENAC (Ecole Nationale de l'Aviation Civile) (France) sponsored by the Aviation Academy of the Amsterdam University of Applied Sciences (the Netherlands). His interests are in modeling and simulation and optimization applied to the aviation field. His email address is p.m.scala@hva.nl.

MIGUEL MUJICA is an Associate Professor at the Aviation Academy of the Amsterdam University of Applied Sciences in the Netherlands. He is vice chair of the Dutch Benelux Simulation Society (www.dutchBSS.org) and executive board member in the EUROSIM federation of simulation societies of EUROPE (www.eurosim.info) besides being member of the Mexican National Research System. His research interests lie in simulation techniques and O.R. in industrial, logistics and aviation problems. His email address is m.mujica.mota@hva.nl and web page www.mmujicamota.com.