Samir Adly
email: samir.adly@unilim.fr

Abderrahim Hantoute
email: ahantoute@dim.uchile.cl

Michel Théra
email: michel.thera@unilim.fr

Jong-Shi Pang

Nonsmooth Lyapunov pairs for differential inclusions governed by operators with nonempty interior domain

Keywords: Evolution differential inclusions, Maximally monotone operators, Lower semicontinuous Lyapunov pairs and functions, Invariant sets, Generalized subdifferentials Mathematics Subject Classification 37B25, 47J35, 93B05

The general theory of Lyapunov stability of first-order differential inclusions in Hilbert spaces has been studied by the authors in the previous paper (Adly et al. in Nonlinear Anal 75(3): 985-1008, 2012). This new contribution focuses on the case when the interior of the domain of the maximally monotone operator governing the given differential inclusion is nonempty; this includes in a natural way the finitedimensional case. The current setting leads to simplified, more explicit criteria and permits some flexibility in the choice of the generalized subdifferentials. Some consequences of the viability of closed sets are given. Our analysis makes use of standard tools from convex and variational analysis.

Introduction

In 1964, Stampacchia [START_REF] Stampacchia | Formes bilinéaires coercitives sur les ensembles convexes[END_REF] extended the well known Lax-Milgram lemma (or variational equation) on coercive bilinear forms to convex and closed sets (representing in general some constraints). This important result applied by Fichera [START_REF] Fichera | Problemi elastostatici con vincoli unilaterali: Il problema di Signorini con ambigue condizioni al contorno[END_REF] to the Signorini problem on the elastic equilibrium of a body under unilateral constraints and by Stampacchia to the definition of the capacitary potential associated to a non symmetric bilinear form, is considered as the starting point of the theory of variational inequalities, see e.g., the excellent survey by Mazzone and Lions [START_REF] Lions | The work of G. Stampacchia in variational inequalities[END_REF][START_REF] Mazzone | Guido Stampacchia. In: Variational Analysis and Applications[END_REF]. This theory was considerably extended with the work by Hartman and Stampacchia [START_REF] Hartman | On some non-linear elliptic differential-functional equations[END_REF], in reflexive Banach spaces, for nonlinear partial differential operators arising in elasticity. Then, Lions and Stampacchia [START_REF] Lions | Variational inequalities[END_REF] extended Fichera's analysis to abstract variational inequalities associated to bilinear forms which are coercive or simply non negative in real Hilbert spaces as a tool for the study of partial differential elliptic and parabolic equations (see, also [START_REF] Ernst | A converse to the Lions-Stampacchia theorem[END_REF] and [START_REF] Duvaut | Les inéquations en mécanique et en physique[END_REF] for applications to the unilateral mechanics). In an abstract framework, if K is a closed and convex subset of an ambient space X and f a given element in the dual space, a variational inequality is the problem of finding u ∈ K such that Auf, vu ≥ 0 for each v ∈ K .

(

The evolution analogue of (1), i.e., the problem of finding a function t → u(t), where t is the time, modelizes evolution problems such as parabolic or hyperbolic equations (see [START_REF] Duvaut | Les inéquations en mécanique et en physique[END_REF]). In this context, evolution variational inequalities or more generally differential inclusions have been considered as a natural generalization of ordinary differential equations (ODE's) when dealing with unilateral constraints in mechanics, for example. On the other hand, studies were conducted in parallel in linear and nonlinear complementarity problems (a part of mathematical programming and optimization) in finite dimensional spaces with a large number of applications in economics, finance, transportation planning, Nash equilibrium and game theory. The two subjects of variational inequalities and complementarity are closely related.

In many applications, the models lead to an ordinary differential equation parametrized by a variational inequality or complementarity condition in the constraint. Such a combination appears naturally for instance in optimal control theory where the state x(t) and the control u(t) are related by a constraint. For illustration purpose, let us consider the following nonlinear input/output system

⎧ ⎪ ⎨ ⎪ ⎩ ẋ(t) = f (t, x(t)) + B T u(t), a.e. t ∈ [0, T ] (2)
y(t) = Bx(t), (3) 0 ≤ x(t) ⊥ y(t) ≥ 0. (4)
Here x(t) ∈ R n is the state variable, u(t) ∈ R m is the input variable, y(t) ∈ R m is the output variable, B ∈ R m×n is a given matrix and f :[0, T ] × R n → R n is a given vector field. This problem belongs to the class of piecewise smooth systems which consist of an ODE parametrized by an algebraic variable y that is required to satisfy a complementarity condition or more generally a variational inequality. This can be extended to the more general abstract framework of Differential Variational Inequalities (DVI's) studied thoroughly by Pang and Stewart [START_REF] Pang | Differential variational inequalities[END_REF]. Problem ( 2)-( 4) plays an important role in nonsmooth mechanics (multibody dynamics with unilateral contact), in nonregular electrical circuits (switching systems, relay, diodes and transistors) as well as in dynamic games. Using tools from convex analysis, we show that problem [START_REF] Adly | A stability theory for second-order nonsmooth dynamical systems with application to friction problems[END_REF] can be reduced to the study of a general differential inclusion governed by a maximally monotone operator which is our object problem to study in this paper. In fact, the complementarity condition (4) can be rewritten in terms of the normal cone. More precisely,

0 ≤ y(t) ⊥ u(t) ≥ 0 ⇐⇒ y(t) ∈ -N R n + (u(t))
, where N R n + stands for the normal cone to the positive orthant R n + . Hence, relation ( 2) is equivalent to

ẋ(t) ∈ f (t, x(t)) -B T N R n + (Bx(t)).
Using the chain rule formula and using the indicator function (defined in Sect. 2) I R n + , we obtain

ẋ(t) ∈ f (t, x(t)) -∂ϕ(x(t)), ( 5
)
where

ϕ(x) = (I R n + • B)(x)
, which is a convex, lower semicontinuous (lsc for short) and proper function. Let us notice that problem (5) is of the form (6) since A = ∂ϕ is a maximally monotone operator.

Equally important is the study of the stability in the sense of Lyapunov of dynamical systems due to its usefulness in system theory and engineering. This concept has been studied extensively in the literature in the smooth case. In various applications modeled by ordinary differential equations, one may be forced to work with systems that have nondifferentiable solutions. For example, Lyapunov functions (positive-definite functions, which are nonincreasing along the trajectories) used to establish a stability of a given system may be nondifferentiable. The need to extend the classical differentiable Lyapunov stability to the nonsmooth case is unavoidable when studying discontinuous systems. In practice, many systems arising in physics, engineering, biology, etc, exhibit generally nonsmooth energy functions, which are usually typical candidates for Lyapunov functions. Thus, the use of elements of nonsmooth analysis is essential [START_REF] Bacciotti | Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions[END_REF][START_REF] Camlibel | Lyapunov stability of complementarity and extended systems[END_REF][START_REF] Clarke | Nonsmooth analysis and control theory[END_REF][START_REF] Shevitz | Lyapunov stability theory of nonsmooth systems[END_REF].

Instead of considering inclusion [START_REF] Borwein | Fifty years of maximal monotonicity[END_REF], throughout this article we are interested in the general framework of infinite-dimensional dynamical systems, that is, systems of the form:

ẋ(t; x 0 ) ∈ f (x(•; x 0 )) -Ax(•; x 0 ), x 0 ∈ cl(Dom A) a.e. t ≥ 0. ( 6
)
In the sequel, cl(Dom A) is the closure of the domain of a maximally monotone operator A : H ⇒ H defined on a real Hilbert space H , possibly nonlinear and multivalued with domain Dom A, and f is a Lipschitz continuous mapping defined on cl(Dom A). A pair of proper lower semicontinuous functions V, W : H → R∪{+∞} is said to form an a-Lyapunov pair for [START_REF] Borwein | Recent progress on monotone operator theory[END_REF], with some a ≥ 0, if for all x 0 ∈ cl(Dom A) the solution x(•; x 0 ) of ( 6), in the sense that will be made precise in Sect. [START_REF] Bacciotti | Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions[END_REF], satisfies

e at V (x(t; x 0 )) + t s W (x(τ ; x 0 ))dτ ≤ e as V (x(s; x 0 )) for all t ≥ s ≥ 0. ( 7
)
Observe that when W ≡ 0 and a = 0 one recovers the classical notion of Lyapunov functions; see, e.g., [START_REF] Smirnov | Introduction to the theory of differential inclusions[END_REF]. The main motivation for this definition is that many stability concepts for the equilibrium sets of (6) (namely stability, asymptotic or finite-time stability, etc.) can be obtained just by choosing appropriate values for a and function W in [START_REF] Borwein | Techniques of variational analysis[END_REF]. The method of Lyapunov functions has a long history that has been described in several places. We refer the reader to Clarke [START_REF] Clarke | Lyapunov functions and feedback in nonlinear control[END_REF][START_REF] Clarke | Nonsmooth analysis in systems and control theory[END_REF] for an overview of the recent developments of the theory, where he pointed out that for nonlinear systems, the Lyapunov method turns out to be essential to consider nonsmooth Lyapunov functions, even if the underlying control dynamics are themselves smooth. Among the various contributions, Kocan and Soravia [START_REF] Kocan | Lyapunov functions for infinite-dimensional systems[END_REF] characterized Lyapunov pairs in terms of viscosity solutions to a related partial differential inequality. Another well-established approach consists of characterizing Lyapunov pairs by means of the contingent derivative of the maximally monotone operator A-see e.g. Cârjȃ and Motreanu [START_REF] Cârjȃ | Flow-invariance and Lyapunov pairs[END_REF] for the case of a linear maximally monotone operator and also when A is a multivalued m-accretive operator on an arbitrary Banach space [START_REF] Cârjȃ | Characterization of Lyapunov pairs in the nonlinear case and applications[END_REF]. In these approaches the authors used tangency and flow-invariance arguments combined with a priori estimates and approximation. We also refer to the paper by Adly and Goeleven [START_REF] Adly | A stability theory for second-order nonsmooth dynamical systems with application to friction problems[END_REF] in which smooth Lyapunov functions were used in the framework of the second order differential equations, that can be rewritten in the form of [START_REF] Borwein | Recent progress on monotone operator theory[END_REF].

In [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions[END_REF], we followed a different approach that did not make use of viscosity solutions or contingent derivatives associated to the operator A. We provided general criteria for nonsmooth Lyapunov pairs associated to [START_REF] Borwein | Recent progress on monotone operator theory[END_REF] in terms of proximal and horizon subgradients of the involved function V . Such conditions were written by considering limiting processes required by the fact that the initial condition in (6) was allowed to be any point in the closure of the domain A.

Our objective in this work is to refine the approach of [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions[END_REF] to the setting where the interior of the domain of the involved maximally monotone operator is nonempty. This setting subsumes the finite-dimensional case where the relative interior of the convex envelope of the domain of the operator is always nonempty. Moreover, as in [START_REF] Adly | A stability theory for second-order nonsmooth dynamical systems with application to friction problems[END_REF] which deals with the smooth case, the criteria for Lyapunov pairs are checked only in the interior of the domain (or the relative interior) instead of the closure of the whole domain. In contrast to [START_REF] Adly | A stability theory for second-order nonsmooth dynamical systems with application to friction problems[END_REF], this setting also ensures obtaining global Lyapunov pairs and allows us to control the whole trajectory of the solution to the given differential inclusion. This additional interiority assumption provides more explicit criteria for nonsmooth Lyapunov pairs than the ones given in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions[END_REF]. Indeed, on the one hand, the conditions we present here are given at the nominal point and do not involve limiting processes. On the other hand, the current analysis is flexible regarding the choice of the subdifferentials, which can be either the proximal, the Fréchet (regular), or the limiting subdifferentials. Moreover, a notable difference with the previous results of [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions[END_REF] is that it is not necessary to consider the horizon subgradient, a fact that leads us to sharper conditions for Lyapunov pairs. The structure of the paper is as follows. In Sect. 2 we introduce the main tools and basic results used in the paper. In Sect. 3 we give new criteria for lower semicontinuous Lyapunov pairs, achieved in Theorems 3.1, 3.2, and 3.3. Section 4 is devoted to the finite-dimensional setting.

Notation and main tools

Throughout the paper, H is a (real) Hilbert space endowed with the inner product

•, • and the associated norm • . Given a nonempty set S ⊂ H (or S ⊂ H × R), by co S, cone S, and aff S, we denote the convex hull, the conic hull, and the affine hull of the set S, respectively. Moreover, IntS is the interior of S, and clS and S are indistinctly used for the closure of S (with respect to the norm topology on H ). We note ri S the (topological) relative interior of S, i.e., the interior of S in the topology relative to cl(aff S). For x ∈ H (or x ∈ H × R) and ρ ≥ 0, B ρ (x) is the open ball with center x and radius ρ, and B ρ (x) is the closure of B ρ (x) (with

B := B 1 (0)).
Our notation is the standard one used in convex and variational analysis and in monotone operator theory; see, e.g., [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF][START_REF] Rockafellar | Variational analysis[END_REF]. The indicator function of S ⊂ H is the function defined as

I S (x) := 0 ifx ∈ S +∞ otherwise.
The distance function to S is denoted by

d(x, S) := inf{ x -y | y ∈ S},
and the orthogonal projection on S, π S , is defined as

π S (x) := {y ∈ S | x -y = d(x, S)}.
Given a function ϕ : H → R, its (effective) domain and epigraph are defined by

Dom ϕ := {x ∈ H | ϕ(x) < +∞}, epi ϕ := {(x, α) ∈ H × R | ϕ(x) ≤ α}. For λ ∈ R, the open upper level set of ϕ at λ is [ϕ > λ] := {x ∈ H | ϕ(x) > λ};
The sets [ϕ ≤ λ] and [ϕ < λ] are defined similarly. We say that ϕ is proper if Dom ϕ = ∅ and ϕ(x) > -∞ for all x ∈ H. We say that ϕ is convex if epi ϕ is convex, and (weakly) lower semicontinuous if epi ϕ is closed with respect to the (weak topology) norm-topology on H . We denote

F (H ) := {ϕ : H → R | ϕ is proper and lsc}, F w (H ) := {ϕ : H → R | ϕ is

proper and weakly lsc};

F (H ; R + ) and F w (H ; R + ) stand for the subsets of nonnegative functions of F (H ) and F w (H ), respectively.

As maximally monotone set-valued operators play an important role in this work, it is useful to recall some of basic definitions and some of their properties. More generally, they have frequently shown themselves to be a key class of objects in both modern Optimization and Analysis; see, e.g., [START_REF] Bauschke | Convex analysis and monotone operator theory in Hilbert spaces[END_REF][START_REF] Borwein | Fifty years of maximal monotonicity[END_REF][START_REF] Borwein | Recent progress on monotone operator theory[END_REF][START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF][START_REF] Rockafellar | Variational analysis[END_REF][START_REF] Simons | Minimax and monotonicity[END_REF].

For an operator A : H ⇒ H, the domain and the graph of A are given respectively by

Dom A := {z ∈ H | Az = ∅} and gph A := {(x, y) ∈ H × H | y ∈ Ax};
for notational simplicity we identify the operator A to its graph. The inver se operator of A, denoted by A -1 , is defined as

(y, x) ∈ A -1 ⇐⇒ (x, y) ∈ A.
We say that an operator A is monotone if

y 1 -y 2 , x 1 -x 2 ≥ 0 for all (x 1 , y 1 ), (x 2 , y 2 ) ∈ A,
and maximally monotone if A is monotone and has no proper monotone extension (in the sense of graph inclusion). If A is maximally monotone, it is well-known (e.g., [START_REF] Simons | Minimax and monotonicity[END_REF]) that Dom A is convex, and Ax is convex and closed for every x ∈ Dom A. Note that the domain or the range of a maximally monotone operator may fail to be convex, see, e.g., [28, page 555]. In particular, if A is the subdifferential ∂ϕ of some lower semicontinuous convex and proper function ϕ : H → R, then A is a classical example of a maximally monotone operator, as is a linear operator with a positive symmetric part. We know that

Dom A ⊂ Dom ϕ ⊂ Dom ϕ = Dom A.
For x ∈ Dom A, we shall use the notation (Ax) • to denote the principal section of A, i.e., the set of points of minimal norm in Ax. For λ > 0, the resolvent and the Yoshida approximation of A are given, respectively, by

J λ := (I + λ A) -1 , A λ := I -J λ λ ,
where I stands for the identity mapping on H .

We introduce next some basic concepts of nonsmooth and variational analysis (more details can be found for instance in [START_REF] Borwein | Techniques of variational analysis[END_REF][START_REF] Clarke | Optimization and nonsmooth analysis[END_REF][START_REF] Clarke | Nonsmooth analysis and control theory[END_REF][START_REF] Mordukhovich | Variational analysis and generalized differentiation[END_REF][START_REF] Rockafellar | Variational analysis[END_REF]). We assume that ϕ ∈ F (H ), and take x ∈ Dom ϕ. We say that a vector ξ ∈ H is a proximal subgradient of ϕ at x, and we write ξ ∈ ∂ P ϕ(x), if there are ρ > 0 and σ ≥ 0 such that

ϕ(y) ≥ ϕ(x) + ξ, y -x -σ y -x 2 for all y ∈ B ρ (x).
The set ∂ P ϕ(x) is convex, possibly empty and not necessarily closed. The set ∂ F ϕ(x) of Fréchet (regular) subgradient of ϕ at x is defined as the set of those ξ ∈ H satisfying

ϕ(y) ≥ ϕ(x) + ξ, y -x + o( y -x ).
Associated to proximal and Fréchet subdifferentials, limiting objects have been introduced:

-the limiting Mordukhovich subdifferential ∂ L ϕ(x) of ϕ at x, which is the set of those ξ ∈ H such that there exist sequences

(x k ) k∈N and (ξ k ) k∈N satisfying x k → ϕ x (that is, x k → x and ϕ(x k ) → ϕ(x)), ξ k ∈ ∂ P ϕ(x k ) and ξ k → ξ ; -the horizon (singular) subdifferential ∂ ∞ ϕ(x) of ϕ at x, which is the set of those ξ ∈ H such that there exist sequences (α k ) k∈N ⊂ R + , (x k ) k∈N and (ξ k ) k∈N satisfying α k → 0 + , x k → ϕ x, ξ k ∈ ∂ P ϕ(x k ) and α k ξ k → ξ .
Note that the use of strong convergence in the definition of the limiting and horizontal subdifferentials above is due to the current Hilbert setting (e.g. [24, Theorem 2.34]). The Clarke subdifferential of ϕ at x denoted by ∂ C ϕ(x) coincides with co w {∂ L ϕ(x) + ∂ ∞ ϕ(x)} (see, e.g., [START_REF] Mordukhovich | Variational analysis and generalized differentiation[END_REF] and [START_REF] Rockafellar | Variational analysis[END_REF]), where the superscript w refers to the weak topology in H . Then it follows from the definition that

∂ P ϕ(x) ⊂ ∂ F ϕ(x) ⊂ ∂ L ϕ(x) ⊂ ∂ C ϕ(x). ( 8
)
In particular, if ϕ is convex, then

∂ P ϕ(x) = ∂ C ϕ(x) = ∂ϕ(x),
where ∂ϕ(x) is the usual subdifferential of convex analysis:

∂ϕ(x) := {ξ ∈ H | ϕ(y) -ϕ(x) ≥ ξ, y -x for all y ∈ H }.
If ϕ is Gâteaux-differentiable at x ∈ Dom ϕ, then we have

∂ P ϕ(x) ⊂ {ϕ G (x)} ⊂ ∂ C ϕ(x).
If ϕ is C 1 , then

∂ P ϕ(x) ⊂ {ϕ (x)} = ∂ C ϕ(x) and ∂ ∞ ϕ(x) = {θ }. If ϕ is C 2 , then ∂ P ϕ(x) = ∂ C ϕ(x) = {ϕ (x)}.
From a geometrical point of view, if S ⊂ H is closed and x ∈ S, the proximal normal cone to S at x is

N P S (x) := ∂ P I S (x).
or, equivalently (e.g. [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF]),

N P S (x) = cone(π -1 S (x) -x) if π -1 S (x) = ∅, {θ } if π -1 S (x) = ∅, where π -1 S (x) := {y ∈ H \S | x ∈ π S (y)}. Similarly, N L S (x) := ∂ L I S (x) (= ∂ ∞ I S (x))
is the limiting normal cone to S at x, and N C S (x) := co w {N L S (x)} is the Clarke normal cone to S at x. In that way, the above subdifferentials of ϕ ∈ F (H ) satisfy

∂ P ϕ(x) = {ξ ∈ H | (ξ, -1) ∈ N P epi ϕ (x, ϕ(x))}, ∂ ∞ ϕ(x) ⊂ {ξ ∈ H | (ξ, 0) ∈ N P epi ϕ (x, ϕ(x))}. Conversely, if ξ ∈ H is such that (ξ, 0) ∈ N P epi ϕ (x, ϕ(x)), then (e.g. [24, Lemma 2.37]) there exist sequences (α k ) k∈N ⊂ R + , (x k ) k∈N and (ξ k ) k∈N such that α k → 0 + , x k → ϕ x, ξ k ∈ α k ∂ F ϕ(x k ) and ξ k → ξ .
Note that when ϕ is a proper extended-real valuedconvex function, we have [28, Proposition 8.12]

∂ ∞ ϕ(x) ⊂ N Dom ϕ (x). ( 9
)
We use the notation T S (x) to denote the contingent cone to S at x ∈ S (also called Bouligand tangent cone) defined by

T S (x) := {ξ ∈ H | x + τ k ξ k ∈ S for some ξ k → ξ and τ k → 0 + }. The Dini directional derivative of the function ϕ (∈ F (H )) at x ∈ Dom ϕ in the direction v ∈ H is given by ϕ (x, v) = lim inf t→0 + ,w→v ϕ(x + tw) -ϕ(x) t . The relation epi ϕ (x, •) = T epi ϕ (x, ϕ(x)) is verified. When ϕ := d(•, S)
is the distance function to a closed subset S of H , then we have

∂ C ϕ(x) = N C S (x) ∩ B, for all x ∈ S,
while for x / ∈ S with ∂ P ϕ(x) = ∅, we have that π S (x) is a singleton and (see, e.g., [START_REF] Clarke | Nonsmooth analysis and control theory[END_REF])

∂ P ϕ(x) = x -π S (x) d(x, S) .
Hence,

∂ L ϕ(x) = w-lim k x k -π S (x k ) ϕ(x) ; x k → x ,
where wlim stands for the weak limit. More generally, we have

N P S (x) = R + ∂ P d S (x) and N C S (x) = R + ∂ C d S (x)
w (with the convention that 0.∅ = {θ}).

Finally, we recall that ϕ ∈ F (R) is nonincreasing if and only if ξ ≤ 0 for every ξ ∈ ∂ P ϕ(x) and x ∈ R, (e.g., [START_REF] Clarke | Nonsmooth analysis and control theory[END_REF]). We shall use the following Lemma: Lemma 2.1 Given t 2 > t 1 ≥ 0, a = 0, and b ≥ 0, we assume that an absolutely continuous function ψ

: [t 1 , t 2 ] → R + satisfies ψ (t) ≤ aψ(t) + b a.e. t ∈ [t 1 , t 2 ].
Then, for all t ∈ [t 1 , t 2 ],

ψ(t) ≤ ψ(t 1 ) + b a e a(t-t 1 ) - b a .
Proof Just apply Gronwall's Lemma to the function θ := ψ + b a .

Characterization of Lyapunov pairs

In this section we provide the desired explicit criterion for lower semicontinuous (weighted-) Lyapunov pairs associated to the differential inclusion (6):

ẋ(t; x 0 ) ∈ f (x(•; x 0 )) -Ax(•; x 0 ), x 0 ∈ cl (Dom A) ,
where A : H ⇒ H is a maximally monotone operator and f : cl (Dom A) ⊂ H → H is a Lipschitz continuous mapping. Recall that for a fixed real T > 0 and for a given x 0 ∈ cl (Dom A) , we call strong solution of (6), the unique absolute continuous function x(•; x 0 ) : [0, T ] → H, which satisfies x(0; x 0 ) = x 0 together with (see, e.g., [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF])

ẋ(t; x 0 ) ∈ L ∞ loc ((0, T ], H ), ( 10
)
x(t; x 0 ) ∈ Dom A, for all t > 0, (11)
ẋ(t; x 0 ) ∈ f (x(t; x 0 )) -Ax(t; x 0 ), a.e. t ≥ 0. ( 12
)
Existence of strong solutions is known to occur if, for instance,

x 0 ∈ Dom A, Int (co (Dom A)) = ∅, dim H < ∞, or if A ≡ ∂ϕ where ϕ : H → R ∪ {+∞} is a lsc extended-real-valued convex proper function. Moreover, we have ẋ(•; x 0 ) ∈ L ∞ ([0, T ], H ) if and only if x 0 ∈ Dom A.
In this later case, x(•; x 0 ) is rightdifferentiable at each s ∈ [0, T ) and

d + x(•; x 0 ) t (s) = f (x(s; x 0 )) -π Ax(s;x 0 ) ( f (x(s; x 0 ))).
The strong solution also satisfies the so-called semi-group property,

x(s; x(t; x 0 )) = x(s + t; x 0 ) for all s, t ≥ 0, [START_REF] Clarke | Lyapunov functions and feedback in nonlinear control[END_REF] together with the relationship

x(t; x 0 ) -x(t; y 0 ) ≤ e L f t x 0 -y 0 ( 14
)
whenever t ≥ 0 and x 0 , y 0 ∈ cl(Dom A); hereafter, L f denotes the Lipschitz constant of the mapping f on cl(Dom A).

In the general case, it is well established that ( 6) admits a unique weak solution x(•; x 0 ) ∈ C(0, T ; H ) which satisfies x(t; x 0 ) ∈ cl(Dom A) for all t ≥ 0. More precisely, there exists a sequence (z k ) k∈N ⊂ Dom A converging to x 0 such that the strong solution x k (•; z k ) of the equation

ẋk (t; z k ) ∈ f (x(t; z k )) -Ax k (t; z k ), x k (0, z k ) = z k , ( 15
)
converges uniformly to x(•; x 0 ) on [0, T ]. Moreover, we know that ( 13) and ( 14) also hold in this case on cl(Dom A) .

The following condition on the interior of the operator A will play a crucial role in our analysis,

Int (co (Dom A)) = ∅. ( 16
)
Applying well known results from the theory of maximally monotone operators, the last assumption implies that Int

(Dom A) is convex, Int (Dom A) = Int (co (Dom A)) = Int (cl (Dom A)),
and A is locally bounded on Int (Dom A). Therefore, a (unique) strong solution of (6) always exists (see e.g. [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF]).

The following technical lemma, adds more information on the qualitative behavior of this solution.

Lemma 3.1 Let ȳ ∈ Dom A and ρ > 0 be such that B ρ ( ȳ) ⊂ Int (co (Dom A)) . Then, there exists ρ ∈ (0, ρ) such that

M := sup z∈B ρ ( ȳ) ( f (z) -Az) • < ∞
and, for all y ∈ B ρ ( ȳ) and t ≥ 0,

d + x(•; y) dt (t) ≤ e L f t M.
Proof By virtue of the semi-group property [START_REF] Clarke | Lyapunov functions and feedback in nonlinear control[END_REF], the following inequality holds for all y ∈ cl(Dom A) and 0 ≤ t < s (e.g., [8, Lemma 1.1]):

x(t + s; y)x(t; y) = x(t; x(s; y))x(t; y) ≤ e L f t x(s; y)y . [START_REF] Ernst | A converse to the Lions-Stampacchia theorem[END_REF] Hence, taking limits as s goes to 0,

d + x(•; y) dt (t) = lim s↓0 s -1 x(t + s; y) -x(t; y) ≤ e L f t lim s↓0 s -1 x(s; y) -y = e L f t d + x(•; y) dt (0) = e L f t ( f (y) -Ay) • .
Finally, by the monotonicity of A and the Lipschitz continuity of f , we can choose ρ ∈ (0, ρ) such that

( f (y) -Ay) • ≤ M
for some constant M ≥ 0. This concludes the proof of the lemma.

Definition 3.1 Given functions V ∈ F (H ), W ∈ F (H ; R + ), and a number a ∈ R + , we say that (V, W ) forms an a-Lyapunov pair for [START_REF] Borwein | Recent progress on monotone operator theory[END_REF] if for all y ∈ cl(Dom A)

we have

e at V (x(t; y)) + t 0 W (x(τ ; y))dτ ≤ V (y) for all t ≥ 0. ( 18
)
We note that if a = 0 and W = 0, then we recover the classical concept of Lyapunov functions.

The lack of regularity properties of a-Lyapunov pairs (V, W ) in Definition 3.1 is mainly due to the non-smoothness of the function V. Let us remind that inequality (18) also holds if instead of W one considers its Moreau-Yosida regularization, which is Lipschitz continuous on every bounded subset of H . This follows from the following lemma (e.g [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions[END_REF]). Lemma 3.2 For every W ∈ F (H ; R + ), there exists a sequence of functions (W k ) k∈N ⊂ F (H, R + ) converging to W (for instance, W k ↑ W ) such that each W k is Lipschitz continuous on every bounded subset of H, and satisfies W (y) > 0 if and only if W k (y) > 0.

The following proposition shows that, generally, Lyapunov pairs have to be checked only on the domain of the involved maximally monotone operator.

Proposition 3.1 Let V ∈ F (H ) and W ∈ F (H ; R + ) be given. Suppose that V satisfies lim inf Dom A z→y V (z) = V (y) for all y ∈ cl(Dom A) ∩ Dom V. ( 19
)
Then, given a ∈ R + , the following statements are equivalent:

(i) (V, W ) forms an a-Lyapunov pair with respect to Dom A; that is, [START_REF] Borwein | Techniques of variational analysis[END_REF] holds on Dom A; (ii) (V, W ) forms an a-Lyapunov pair with respect to cl(Dom A); that is, [START_REF] Borwein | Techniques of variational analysis[END_REF] holds on cl(Dom A);

Property [START_REF] Hartman | On some non-linear elliptic differential-functional equations[END_REF] has been already used in [START_REF] Kocan | Lyapunov functions for infinite-dimensional systems[END_REF], and implicitly in [START_REF] Pazy | The Lyapunov method for semigroups of nonlinear contractions in Banach spaces[END_REF][START_REF] Pazy | On the Lyapunov method for evolution equations governed by accretive operators[END_REF], among other works. It holds, if for instance, V (∈ F (H ) ) is convex and its effective domain has a nonempty interior such that Int(Dom V ) ⊂ Dom A.

Our starting point is the next result which characterizes a-Lyapunov pairs locally in Int(Dom A). This is a specification of the analysis of [2, Theorems 3.3 and 3.4] to the current setting where Int (co(Dom A)) = ∅. Here we give complete and general criteria by means of either the proximal, the Fréchet, or the limiting subdifferentials; this last one coincides with the viscosity subdifferential (see Borwein [START_REF] Borwein | Techniques of variational analysis[END_REF]). Moreover, there is no need for the horizontal subgradient.

Theorem 3.1 Assume that Int (co(Dom

A)) = ∅. Let V ∈ F w (H ), W ∈ F (H ; R + ), and a ∈ R + be given. Let ȳ ∈ H, λ ∈ [-∞, V ( ȳ)), and ρ ∈ (0, +∞] be such that Dom V ∩ B ρ ( ȳ) ∩ [V > λ] ⊂ Int(Dom A).
Then, the following statements are equivalent provided that ∂ is either the proximal, the Fréchet, or the limiting subdifferentials:

(i) ∀y ∈ Dom V ∩ B ρ ( ȳ) ∩ [V > λ] sup ξ ∈∂ V (y) min υ∈Ay ξ, f (y) -υ + aV (y) + W (y) ≤ 0; (ii) ∀y ∈ Dom V ∩ B ρ ( ȳ) ∩ [V > λ] sup ξ ∈∂ V (y) ξ, f (y) -π Ay ( f (y)) + aV (y) + W (y) ≤ 0; (iii) ∀y ∈ B ρ ( ȳ) ∩ [V > λ]
we have e at V (x(t; y))

+ t 0 W (x(τ ; y))dτ ≤ V (y) ∀t ∈ [0, ρ(y)] ,
where

ρ(y):= sup ⎧ ⎪ ⎨ ⎪ ⎩ ν >0 ∃ρ >0 s.t. B ρ (y) ⊂ B ρ ( ȳ)∩[V > λ]
, and for all t ∈[0, ν] 2 x(t; y)y < ρ 2 and (e -at -1)V (y) -

t 0 W (x(τ ; y))dτ < ρ 2 ⎫ ⎪ ⎬ ⎪ ⎭ . ( 20
)
Remark 3.1 The constant ρ(y) defined in ( 20) is positive whenever

y ∈ cl(Dom A) ∩ B ρ ( ȳ) ∩ [V > λ].
Proof of Theorem 3.1 For simplicity, we suppose that W ≡ 0 and a = 0 (the general case follows similarly by noting that the function W may be taken Lipschitz on bounded sets, according to Lemma 3.2).

(iii) ⇒ (ii). According to the sequence of inclusions [START_REF] Brézis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF], it is enough to give the proof for the limiting subdifferential. We will proceed into two steps.

First step Let us give the proof for the Fréchet subdifferential. Let us fix y

∈ B ρ ( ȳ)∩ [V > λ] and, first, take ξ in ∂ F V (y) so that y ∈ B ρ ( ȳ) ∩ [V > λ] ∩ Dom V ⊂ Dom A and there exists T ∈ (0, ρ(y)) such that ξ, x(t; y) -y ≤ V (x(t; y)) -V (y) + α x(t; y) -y 2 ≤ o( x(t; y) -y ) for all t ∈ [0, T ),
where o(.) is a function satisfying o(s) → 0 as s → 0. But y ∈ Dom A and so there exists a constant l ≥ 0 such that (taking a smaller T if necessary) ξ, t -1 (x(t; y)y) ≤ l x(t; y)y for all t ∈ [0, T );

hence, taking the limit as t → 0 + we obtain that

ξ, f (y) -π Ay ( f (y)) ≤ 0;
that is, (ii) follows in the case when ξ ∈ ∂ F V (y).

Second step Let us give the proof for the limiting subdifferential. Now, we suppose that ξ ∈ ∂ L V (y) and let the sequences (y k ) k∈N and (ξ k ) k∈N be such that

y k → V y, ξ k ∈ ∂ F V (y k ) and ξ k → ξ . Then, from the paragraph above, for each k we find v k ∈ Ay k such that ξ k , f (y k ) -v k ≤ 0.
But since y ∈ Int (Dom A) and y k → y, the maximal monotonicity of A allows us to suppose without loss of generality that the sequence (v k ) is bounded and, so, weakly convergent, up to a subsequence denoted in the same way, to some v ∈ Ay. Thus, because ξ k strongly converges to ξ , by taking the limits as n goes to ∞ in the last inequality above we obtain that

inf v∈Ay ξ, f (y) -v ≤ 0 ≤ ξ, f (y) -v ≤ 0. (i) ⇒ (iii).
Again by virtue of (8), it is sufficient to give the proof for the Fréchet subdifferential. We fix y ∈ Dom V ∩ B ρ ( ȳ) ∩ [V > λ] and let ρ > 0 and ν > 0 be such that

B ρ (y) ⊂ B ρ ( ȳ) ∩ [V > λ] (21)
and sup t∈[0,ν] 2 x(t; y)y < ρ; [START_REF] Lions | Variational inequalities[END_REF] the existence of such scalars ρ and ν is a consequence of the lower semicontinuity of V and the Lipschitz continuity of x(•; •) (see Lemma 3.1). Moreover, due to the maximal monotonicity of A, we may assume that A is bounded on B ρ (y). Let T < ν be fixed and define the functions

z(•) : [0, T ] ⊂ R + → H × R and η(•) : [0, T ] ⊂ R + → R + as z(t) := (x(t; y), V (y)), η(t) := 1 2 d 2 (z(t), epi V ); ( 23
)
observe that z(•) and η(•) are Lipschitz continuous on [0, T ). Now, using a standard chain rule (e.g. [START_REF] Clarke | Optimization and nonsmooth analysis[END_REF]), for a fixed t ∈ (0, T ) it holds that

∂ C η(t) = d(z(t), epi V )∂ C d(z(•), epi V )(t).
Hence, whenever z(t)

∈ epi V we get ∂ C η(t) = {θ }. Otherwise, if z(t) / ∈ epi V , then ∂ C d(z(•), epi V )(t) ⊂ co ⎡ ⎣ (u,μ)∈ epi V (z(t)), u∈B ρ (y) x(t; y) -u, -Ax(t; y) d(z(t), epi V ) ⎤ ⎦
and, consequently,

∂ C η(t) ⊂ co ⎡ ⎣ (u,μ)∈ epi V (z(t)), u∈B ρ (y) x(t; y) -u, -Ax(t; y) ⎤ ⎦ . ( 24
)
Note that the condition u ∈ B ρ (y) for (u, μ) ∈ epi V (z(t)) in this last formula is a consequence of the following inequalities:

u -y ≤ x(t; y) -u + x(t; y) -y ≤ (x(t; y), V (y)) -(u, μ) + x(t; y) -y ≤ (x(t; y), V (y)) -(y, V (y)) + x(t; y) -y ≤ 2 x(t; y) -y < ρ
(recall ( 22)). Take now ξ ∈ Ax(t; y) and (u, μ) ∈ epi V (z(t)) with u ∈ B ρ (y). Thus,

(x(t; y) -u, V (y) -μ) ∈ N P epi V (u, μ),
and therefore V (y)-μ ≤ 0 and

V (u)-μ ≤ 0. Hence, u ∈ Dom V ∩ B ρ ( ȳ)∩[V > λ]
(recall ( 21)).

If V (y) -μ < 0, we write (μ -V (y)) -1 (x(t; y)u) ∈ ∂ V P (u) and so, by the current assumption (i), we select υ ∈ Au such that

(μ -V (y)) -1 (x(t; y) -u), -υ ≤ 0.
Therefore, invoking the monotonicty of A we get

x(t; y)-u,-ξ = x(t; y)-u, -υ + x(t; y)-u, υ -ξ ≤ x(t; y) -u, -υ ≤0.
Consequently, since ξ ∈ Ax(t; y) is arbitrary, [START_REF] Mordukhovich | Variational analysis and generalized differentiation[END_REF]

leads us to ∂ C η(t) ⊂ R -.
It remains to investigate the other case corresponding to V (y) -μ = 0; that is, (x(t; y)u, 0)

∈ N P epi V (u, V (u)). Let us first observe that x(t; y) -u = θ. Next, we choose an ε > 0 such that B ε (u) ⊂ B ρ (y) ∩ Int(Dom A) (recall that u ∈ B ρ (y) ∩ Dom V ∩ B ρ ( ȳ) ∩ [V > λ] ⊂ Int(Dom A))
and, according to [24, Lemma 2.37 and formulas in Page 240], take

u ε ∈ B ε (u) ∩ Dom V with |V (u) -V (u ε )| ≤ ε, α ∈ (0, ε) and ξ ε ∈ B ε (x(t; y)-u) such that α -1 ξ ε ∈ ∂ V P (u ε ). Therefore, using the current assumption, select v ε ∈ Au ε such that ξ ε , -v ε ≤ 0. Hence, x(t; y) -u, -v ε ≤ ε v ε + ξ ε , -v ε ≤ ε v ε
and, consequently by the monotonicity of A,

x(t; y) -u, -ξ ≤ x(t; y) -u ε , -ξ + ε ξ ≤ x(t; y) -u ε , -v ε + ε ξ ≤ x(t; y) -u, -v ε + u ε -u v ε + ε ξ ≤ 2ε v ε + ε ξ .
Moreover, as (v ε ) ε≤1 ⊂ B ρ (y) and A is bounded on B ρ (y), by passing to the limit as ε goes to 0 we get

x(t; y) -u, -ξ ≤ 0.
This gives the desired inclusion

∂ C η(t) ⊂ R -(recall ( 24
)
) and so establishes the proof of (iii).

The next theorem adds more information on the uniform-like behavior of Lyapunov pairs for [START_REF] Borwein | Recent progress on monotone operator theory[END_REF].

Theorem 3.2 Under the condition Int (co{Dom A}) = ∅, let us consider functions V ∈ F w (H ) and W ∈ F (H ; R + ), and a nonnegative number a. Fix ȳ ∈ Dom V, λ ∈ (-∞, V ( ȳ)) and let ρ > 0 be such that

Dom V ∩ [V > λ] ∩ B ρ ( ȳ) ⊂ Int(Dom A).
Assume that for all y

∈ Dom V ∩ B ρ ( ȳ) ∩ [V > λ] we have sup ξ ∈∂ P V (y) min υ∈Ay ξ, f (y) -υ + aV (y) + W (y) ≤ 0. ( 25
)
Then, for every y ∈ Dom V ∩ B ρ ( ȳ) ∩ [V > λ] there exist ρ > 0 and T > 0 such that e at V (x(t; z))

+ t 0 W (x(τ ; z))dτ ≤ V (z)
, for all z ∈ B ρ (y) and t ≤ T.

Consequently, under condition [START_REF] Hartman | On some non-linear elliptic differential-functional equations[END_REF], the pair (V, W ) is an a-Lyapunov pair for [START_REF] Borwein | Recent progress on monotone operator theory[END_REF] provided that Dom A is open and ( 25) holds on Dom A.

Proof We shall suppose that W = 0 and a = 0 (the general case is similar). For this aim we pick A)) and we choose ρ > 0 such that A is bounded on B 2ρ ( ŷ). Taking into account the lower semicontinuity of V , we have

ŷ in Dom V ∩ B ρ ( ȳ) ∩ [V > λ] (⊂ Int(Dom
B 2ρ ( ŷ) ⊂ B ρ ( ȳ) ∩ [V > λ] ∩ Int(Dom A), V (z) ≥ V ( ŷ) -1 ∀z ∈ B 2ρ ( ŷ); (26)
moreover, by virtue of Lemma 3.1, we may assume that there exists a positive constant M satisfying, for all t ≥ 0 and all z ∈ B 2ρ ( ŷ),

d + x(•; z) dt (t) ≤ e L f t M. ( 27
)
Hence, x(t; z)z ≤ Mte L f t and therefore by [START_REF] Pazy | The Lyapunov method for semigroups of nonlinear contractions in Banach spaces[END_REF],

V (x(t; z)) ≥ V ( ŷ) -1 ≥ λ -1 (28)
for all z ∈ B ρ ( ŷ) and all t ≥ 0 such that te

L f t ≤ ρ M . Now writing, for all z ∈ B 2ρ ( ŷ) ∩ Dom V and 0 ≤ t ≤ 1, 2 x(t; z) -z ≤ 2Me L f t, there exists T > 0 such that for all z ∈ B 2ρ ( ŷ) we have sup t∈[0,T ] 2 x(t; z) -z < ρ 2 .
Thus, since for any given z ∈ B ρ ( ŷ) we have

B ρ (z) ⊂ B 2ρ ( ŷ) ∩ [V > λ] ⊂ B ρ ( ȳ) ∩ [V > λ]
the main conclusion of the theorem follows from Theorem 3.1. Finally, we assume that Dom A is open and (25) holds on Dom A. For fixed y ∈ Int Dom A we introduce the nonempty set E ⊂ R + given as

E := {λ ∈ R + | V (x(t; y)) ≤ V (y) ∀t ≤ λ }.
Then, from the first part of the proof, by taking into account the continuity of x(•; y) and the openness of Dom A it follows that E is open and closed so that E = R + . Then, the fact that the pair (V, W ) is an a-Lyapunov pair for (6) follows from Proposition 3.1.

Corollary 3.1 Under the assumption of Theorem 3.2 we also suppose that

Dom V ∩ B ρ ( ȳ) ∩ [V > λ] is compact. Then, there exists T > 0 such that e at V (x(t; y)) + t 0 W (x(τ ; y))dτ ≤ V (y), for every y ∈ Dom V ∩ B ρ ( ȳ) ∩ [V > λ] and t ∈ [0, T ]. Proof According to Theorem 3.2, for every y ∈ Dom V ∩ B ρ ( ȳ) ∩ [V > λ] there exist T y > 0 and ρ y > 0 such that e at V (x(t; z)) + t 0 W (x(τ ; z))dτ ≤ V (z), for all z ∈ B ρ y (y) and t ≤ T y . Then, since Dom V ∩ B ρ ( ȳ)∩[V > λ] is compact, we can find y 1 , . . . , y k ∈ Dom V ∩ B ρ ( ȳ) ∩ [V > λ], T 1 , . . . , T k > 0 and ρ 1 , . . . , ρ k > 0 such that Dom V ∩ B ρ ( ȳ) ∩ [V > λ] ⊂ i=1,...,k B ρ i (y i )
and, for each i = 1, . . . , k, e at V (x(t; z))

+ t 0 W (x(τ ; z))dτ ≤ V (z), for all z ∈ B ρ i (y i ) and t ≤ T y i .
Consequently, the conclusion follows by taking T = min i=1,...,k T y i .

In the following theorem we do not assume that Int (co{Dom A}) = ∅. Theorem 3. [START_REF] Bacciotti | Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions[END_REF] We consider functions V ∈ F w (H ) and W ∈ F (H ; R + ), and a nonnegative number a. Assume the existence of λ 0 > 0 such that for all y ∈ Dom V it holds sup

ξ ∈∂ P V (y) ξ, f (y) -A λ y + aV (y) + W (y) ≤ 0 for all λ ≤ λ 0 .
Then, for every z ∈ cl(Dom A) we have that e at V (x(t; z))

+ t 0 W (x(τ ; z))dτ ≤ V (z), for all t ≥ 0;
that is, (V, W ) is an a-Lyapunov pair for [START_REF] Borwein | Recent progress on monotone operator theory[END_REF].

Proof Since Dom A λ = H , by Theorem 3.2 for every λ ≤ λ 0 we have that

e at V (x λ (t; z)) + t 0 W (x λ (τ ; z))dτ ≤ V (z)
, for all t ≥ 0 and all z ∈ H, where x λ (.; z) is the (strong) solution of the differential equation

ẋλ (t; z) = -A λ x λ (t; z) + f (x λ (t; z)); x λ (0; z) = z.
Hence, by taking the limits as λ goes to 0 in the last inequality above, the conclusion follows due to the uniform convergence of x λ (.; z) to x(.; z).

Characterizations of finite-dimensional nonsmooth Lyapunov pairs

This section is devoted to the finite-dimensional setting. Assuming that dim H < ∞, we give multiple primal and dual characterizations for nonsmooth a-Lyapunov pairs for the differential inclusion [START_REF] Borwein | Recent progress on monotone operator theory[END_REF].

Theorem 4.1 Assume that dim H < ∞. Let V ∈ F (H ), W ∈ F (H ; R + ),
and a ∈ R + be given, and let ∂ be either the proximal, the Fréchet, or the limiting subdifferentials. Fix ȳ ∈ rint(cl(Dom A)) and let ρ > 0 be such that B 2ρ ( ȳ) ∩ aff(cl(Dom A)) ⊂ Dom A. Then, the following assertions (i)-(v) are equivalent:

(i) there exists T > 0 such that for every y

∈ Dom A ∩ Dom V ∩ B ρ ( ȳ)
e at V (x(t; y)) + t 0 W (x(τ ; y))dτ ≤ V (y) for all t ≤ T ;

(ii) for every y

∈ Dom A ∩ Dom V ∩ B ρ ( ȳ) sup ξ ∈∂ P V (y) ξ, f (y) -π Ay ( f (y)) + aV (y) + W (y) ≤ 0; (iii) for every y ∈ Dom A ∩ Dom V ∩ B ρ ( ȳ) sup ξ ∈∂ V (y) inf υ∈Ay ξ, f (y) -y * + aV (y) + W (y) ≤ 0; (iv) for every y ∈ Dom A ∩ Dom V ∩ B ρ ( ȳ) V (y; f (y) -π Ay ( f (y))) + aV (y) + W (y) ≤ 0; (v) for every y ∈ Dom A ∩ Dom V ∩ B ρ ( ȳ) inf υ∈Ay V (y; f (y) -υ) + aV (y) + W (y) ≤ 0.
If V is nonnegative, each one of the statements above is equivalent to

(vi) for every y ∈ Dom A ∩ Dom V ∩ B ρ ( ȳ) V (x(t; y)) + a t 0 V (x(τ ; y))dτ + t 0 W (x(τ ; y))dτ ≤ V (y) for all t ≥ 0.
Proof (iii with∂ ≡ ∂ P ) ⇒ (i): We may assume that 0 ∈ Dom A and denote H 0 := lin(cl(Dom A)). Let A 0 : H 0 ⇒ H 0 be the operator given by

A 0 y = Ay ∩ H 0 , ( 29
)
and define the Lipschitz continuous mapping f 0 : H 0 → H 0 as

f 0 (y) = π H 0 ( f (y)), ( 30
)
where π H 0 denotes the orthogonal projection onto H 0 . According to the Minty Theorem, it follows that A 0 is also a maximally monotone operator. Further, for every y ∈ Dom A we have Ay + N cl(Dom A) (y) = Ay, and therefore Ay + H ⊥ 0 = Ay. Hence,

Ay = (Ay ∩ H 0 ) + H ⊥ 0 = A 0 y + H ⊥ 0 . ( 31
)
From this inequality we deduce that Dom A 0 = Dom A and, so,

rint(cl(Dom A)) = Int(cl(Dom A 0 )) = Int(Dom A 0 );
(for the last equality see, e.g., [8, Remark 2.1-Page 33]). Further, since for y ∈ cl(Dom A) we have

f 0 (y) -A 0 y ⊂ f (y) -A 0 y + H ⊥ 0 = f (y) -Ay,
it follows that x(•; y) is the unique solution of the differential inclusion ẋ(t; y) ∈ f 0 (x(t; y)) -A 0 x(t; y), x(0, y) = y.

Next, we are going to show that condition (i) of Theorem 3.2 holds for the pair

(A 0 , f 0 ). Fix y ∈ Dom A ∩ Dom V ∩ B ρ ( ȳ) and ξ ∈ ∂ V (y) (if any).
For a fixed ε > 0, by assumption take υ ∈ Ay in such a way that

ξ, f (y) -υ + aV (y) + W (y) ≤ ε. Since f (y) ∈ f 0 (y) + H ⊥ 0 and υ + H ⊥ 0 ∈ Ay + H ⊥ 0 = A 0 y, we have inf υ∈A 0 y ξ, f 0 (y) -υ ≤ inf υ∈Ay ξ, f (y) -υ ≤ ε -aV (y) -W (y), (32)
and condition (i) of Theorem 3.2 follows as ε → 0. Consequently, by this Theorem 3.2, for every y ∈ Dom A ∩ Dom V ∩ B ρ ( ȳ), there exist ρ > 0, small enough, and T (y) > 0 such that e at V (x(t; z)) + t 0 W (x(τ ; z))dτ ≤ V (z) for all t ≤ T (y) and z ∈ B ρ (y).

Thus, as Dom A ∩ Dom V ∩ B ρ ( ȳ) is a precompact set, and cl(Dom

A ∩ Dom V ∩ B ρ ( ȳ)) ⊂ B 2ρ ( ȳ) ∩ aff(cl(Dom A)) ⊂ rint(cl(Dom A))
, instead of T (y) we can choose a uniform T which gives us statement (i).

(i) ⇒ (iv): Fix y ∈ Dom A ∩ Dom V ∩ B ρ ( ȳ). Then, as shown in the paragraph above, the solution x(t; y) of ( 6) is also the unique strong solution of the equation

ẋ(t; y) ∈ f 0 (x(t; y)) -A 0 x(t; y), x(0; y) = y ∈ cl (Dom A) ,
where A 0 and f 0 are defined in ( 29) and [START_REF] Simons | Minimax and monotonicity[END_REF], respectively. Let (t n ) n∈N ⊂ (0, T ) be a sequence such that t n → 0 + and set

w n := x(t n ; y) -y t n .
Because x(•; y) is derivable from the right at 0 (recall that y ∈ Dom A) and

d + x(•; y) dt (0) = ( f (y) -Ay) • = f (y) -π Ay ( f (y)),
we get

w n → f (y) -π Ay ( f (y)).
Therefore, using the current assumption (i),

V (y + t n w n ) -V (y) t n = V (x(t n , y)) -V (y) t n ≤ e -at n (1 -e at n ) t n V (y) - e -at n t n t n 0 W (x(s; y))ds,
and taking limits yields

V (y; f (y)-π Ay ( f (y))) ≤ lim inf n e -at n (1 -e at n ) t n V (y) - e -at n t n t n 0 W (x(s; y))ds = -aV (y) -W (y); this proves (iv). (iv) ⇒ (v) is trivial. (v) ⇒ [(iii) with ∂ ≡ ∂ L ]. Take y ∈ Dom A ∩ Dom V ∩ B ρ ( ȳ).
For fixed ε > 0, by (v) we let υ ∈ Ay be such that

V (y; f (y) -υ) ≤ ε -aV (y) -W (y); that is, ( f (y) -υ, ε -aV (y) -W (y)) ∈ epi V (y, •) = T epi V (y, V (y)) ⊂ N p epi V (y, V (y)) • . If ξ ∈ ∂ P V (y), since (ξ, -1) ∈ N p epi V (y, V (y)), the above inequality leads to ξ, f (y) -υ ≤ (ξ, -1), ( f (y) -υ, ε -aV (y) -W (y)) + ε -aV (y) -W (y) ≤ ε -aV (y) -W (y), so that (ii) follows when ε → 0. If ξ ∈ ∂ L V (y)
, then there are sequences (y n ) and (ξ n ) such that y n → y, ξ n → ξ, V (ξ n ) → V (ξ ) and ξ n ∈ V (y n ) (for n sufficiently large). As just shown above, given ε > 0, for each n there exists y * n ∈ Ay n such that

ξ n , f (y n ) -y * n ≤ ε -aV (y n ) -W (y n ).
Since (y n ) n converges to y, then we may suppose that y * n → υ ∈ Ay. Thus, passing to the limit in the above inequality, and taking into account the lower semicontinuity of V and the continuity of W, we obtain ξ, f (y) -υ ≤ ε -aV (y) -W (y). This shows that (iii) holds with ∂ ≡ ∂ L .

At this point we have proved that (i) ⇐⇒ (iii with ∂ ≡ ∂ L ) ⇐⇒ (iv) ⇐⇒ (v). To see that (ii) is also equivalent to the other statements we observe that (ii) ⇒ (iii) holds obviously. On the other hand, the implication (iv) ⇒ (ii) follows in a similar way as in the proof of the statement (v) ⇒ (iii). This proves the equivalences of (i) through (v).

Finally, if V is nonnegative, (vi) is nothing else but (i) with a and W replaced by θ and aV + W, respectively. Thus, (vi) is equivalent to (iii).

The following result is an immediate consequence of the previous theorem and Proposition 3.1.

Corollary 4.1 Assume that dim H < ∞. Let V ∈ F (H ), W ∈ F (H ; R + )
, and a ∈ R + be given, and let ∂ be either the proximal, the Fréchet, or the limiting subdifferentials. Assume that rint(cl(Dom A)) = Dom A. Then, under condition [START_REF] Hartman | On some non-linear elliptic differential-functional equations[END_REF], (V, W ) forms an a-Lyapunov pair for [START_REF] Borwein | Recent progress on monotone operator theory[END_REF] provided that one of the following assertions holds:

(i) for all y ∈ Dom A ∩ Dom V sup ξ ∈∂ P V (y) ξ, f (y) -π Ay ( f (y)) + aV (y) + W (y) ≤ 0; (ii) for all y ∈ Dom A ∩ Dom V sup ξ ∈∂ V (y) inf υ∈Ay ξ, f (y) -y * + aV (y) + W (y) ≤ 0; (iii) for all y ∈ Dom A ∩ Dom V V (y; f (y) -π Ay ( f (y))) + aV (y) + W (y) ≤ 0; (iv) for all y ∈ Dom A ∩ Dom V inf υ∈Ay V (y; f (y) -υ) + aV (y) + W (y) ≤ 0.
In contrast to the (analytic) Definition 3.1, Lyapunov stability can also be approached from a geometrical point of view using the concept of invariance: Definition 4.1 A non-empty closed set S ⊂ H is said to be invariant for ( 6) if for all y ∈ S ∩ cl(Dom A) one has that x(t; y) ∈ S for all t ≥ 0. Corollary 4.2 Assume that dim H < ∞ and rint(cl(Dom A)) = Dom A. Then, a nonempty closed set S ⊂ H is invariant for [START_REF] Borwein | Recent progress on monotone operator theory[END_REF] if and only if one of the following assertions is satisfied:

(i) for all y ∈ Dom A ∩ S sup ξ ∈N P S∩cl(Dom A) (y) ξ, f (y) -π Ay ( f (y)) ≤ 0; (ii) for all y ∈ Dom A ∩ S sup ξ ∈N P S∩cl(Dom A) (y) inf υ∈Ay ξ, f (y) -υ ≤ 0; (iii) for all y ∈ Dom A ∩ S f (y) -π Ay ( f (y)) ∈ T S∩cl(Dom A) (y); (iv) for all y ∈ Dom A ∩ S [ f (y) -Ay] ∩ T S∩cl(Dom A) (y) = ∅; (v) for all y ∈ Dom A ∩ S [ f (y) -Ay] ∩ co T S∩cl(Dom A) (y) = ∅.
T S∩cl(Dom A) (y) = {w ∈ H | d (•, S ∩ cl(Dom A)(w) = 0},
the current assertions (iii) and (iv) follow from statements (iii) and (iv) of Theorem 4.1, respectively. This shows that (i) ⇐⇒ (ii) ⇐⇒ (iii) ⇐⇒ (iv). It remains to show that (v) is equivalent to the other statements. We obviously have that (iv) ⇒ (v) and so (i) ⇒ (v). To prove the reverse implication it suffices to show that (v) ⇒ (ii). Indeed, fix y ∈ S ∩ Dom A and ξ ∈ N P S∩cl(Dom A) . Then, by (v) there exists υ ∈ Ay such that f (y) -υ ∈ co T S∩cl(Dom A) (y) ⊂ N P S∩cl(Dom A)

• . Therefore, ξ, f (y) -υ ≤ 0; that is (ii) follows.

The characterization of Gâteaux differentiable Lyapunov functions is given in the following corollary. Finally, we treat the simple case when A ≡ 0 so that our inclusion (6) becomes an ordinary differential equation which reads: for every y ∈ H there exists a unique trajectory x(•; y) ∈ C 1 (0, ∞; H ) such that x(0, y) = y and ẋ(t; y) = f (x(t; y)) for all t ≥ 0.

(33)

In this case, Theorem 3.1 gives in a simplified form the characterization of the associated a-Lyapunov pairs.

Concluding remarks

The main goal of this paper was to explore the existence of local nonsmooth Lyapunov pairs for a first-order evolution differential inclusion governed by a maximal monotone operator. From the mathematical point of view, our major contribution is establishing the fact that the variational criteria for the existence of Lyapunov pairs need to be verified on the interior of the domain of the operator A, while Lyapunov pairs are defined on the whole cl(Dom A). The first version of these ideas appeared in [START_REF] Adly | Nonsmooth Lyapunov pairs for infinite-dimensional first-order differential inclusions[END_REF]. The flow-invariance of a closed set has been also investigated as a particular case of the theory of Lyapunov functions. The characterization of invariance involves a "proximal aiming" condition, as well as the convex hull of the contingent cone. An important issue, left for the future work, is to go beyond maximally monotone operators, since in some applications the monotonicity assumption is not satisfied. This is the case e.g. when A coincides with the normal cone operator of a prox-regular set. It would be interesting to perform a stability analysis of differential variational inequalities involving locally-prox-regular sets. This is beyond the scope of this paper and will be a subject of a forthcoming research project. For its huge potential of applications, Lyapunov stability is used by other communities, particularly in nonlinear systems and control. As noticed by one of the referees, it should be interesting to find practical applications of the theory developed in this paper. Bridging the communities of applied mathematicians, controllers and engineers, is one of our future objectives.

Corollary 4 . 3

 43 Assume that dim H < ∞ and rint(cl(Dom A)) = Dom A. Let V ∈ F (H ), W ∈ F (H, R + ), and a ∈ R + be given. If V is Gâteaux differentiable, then the following statements are equivalent:(i) (V, W ) is an a-Lyapunov pair for (6); (iii) for every y ∈ Dom A ∩ Dom V V G (y)( f (y) -π Ay ( f (y))) + aV (y) + W (y) ≤ 0; (iv) for all y ∈ Dom A ∩ Dom V inf υ∈Ay V G (y)( f (y) -υ) + aV (y) + W (y) ≤ 0.

Corollary 4 . 4

 44 Assume that dim H < ∞. Let be given V ∈ F (H ), W ∈ F (H ; R + ), and a ∈ R + . The following statements are equivalent:(i) (V, W ) is an a-Lyapunov pair for (33); (ii) for every y ∈ Dom V V (y; f (y)) + aV (y) + W (y) ≤ 0; (iii) for all y ∈ Dom V sup ξ ∈∂ V (y) ξ, f (y) + aV (y) + W (y) ≤ 0,where ∂ V stands for either the proximal, the Fréchet, the limiting, or the Clarke subdifferentials.Proof According to Theorem 4.1 we only need to show that (iii) is also a characterization when ∂ ≡ ∂ C . For this aim, in view of the relationship∂ L ⊂ ∂ C , it suffices to show that [(iii) with ∂ ≡ ∂ L ] implies [(iii) with ∂ ≡ ∂ C ]. Indeed, fix y ∈ Dom V so that sup ξ ∈∂ ∞ V (y) ξ, f (y) ≤ 0.So, according to[START_REF] Rockafellar | Variational analysis[END_REF], (iii with∂ ≡ ∂ C ) follows since that sup ξ ∈∂ C V (y) ξ, f (y) + aV (y) + W (y) = sup ξ ∈co{∂ L V (y)+∂ ∞ V (y)}ξ, f (y) + aV (y) + W (y) ≤ 0.

  Proof It is an immediate fact that S is invariant if and only if I S∩cl(Dom A) is a Lyapunov function. Then, the current assertions (i) and (ii) come from statements (i) and (ii) of Theorem 4.1, respectively. Similarly, S is invariant if and only d(•, S ∩ cl(Dom A)) is a Lyapunov function. Thus, by virtue of the relationship

Acknowledgments Research partially supported by the Australian Research Council under Grant DP-110102011, by the Projects FONDECYT-CONICYT no. 1110019, ECOS-CONICYT no. C10E08, Math-Amsud no. 13MATH-01 2013, and Ministerio de Economía y Competitividad MTM2011-29064-C03(03). The authors thank two anonymous referees and also Alexander Kruger for constructive comments.


The Debate on Property during the First English Revolution 1647-1659 : A Historical Perspective

Myriam-Isabelle Ducrocq

The aim of this paper is to provide a historical background to the radical proposals 1 formulated by Thomas Spence concerning property, insofar as they are reminiscent of the debates that took place at the time of the First English Revolution. In Crusonia (1782) 2 , the traveller and narrator describes life in the island where Robinson was once shipwrecked. The island is now populated by the offspring of European sailors and native women, who at first had retained the law of male primogeniture as applied in England, but soon discarded it and opted for new institutions. Having declared every man's right to "Liberty, Subsistence (and consequently to Land) and to be of the Legislature, and other Offices", the island's inhabitants called "the Malcontents" had proclaimed the abolition of landownership so that every parish become a corporation where everyone is allocated a tract of land and pays to the public treasury a rent apportioned to the land's value. The narrator confesses that where he expected to find "Anarchy, Idleness, Poverty, and Meanness" he sees "nothing but Order, Industry, Wealth, and the most pleasing Magnificence !" Such are the unexpected effects of what he "narrowly thought of a ridiculous Levelling Scheme". The opening of Crusonia contains many echoes of English seventeenth-century revolutionary period. The reproduction of a proto-English society in a faraway country is evocative of Henry Neville's Isle of Pines (1668), except that Crusonia's inhabitants are wiser in departing from what soon appears as a flawed social organisation 3 . The denomination of "Malcontents" evokes the way the General Council referred to "the Agitators" in the New Model Army, whereas the term "Levelling" alludes to the kind of projects born in the previous century, with a view to promoting a greater equality among the members of the Commonwealth trough the reform of property structures and political rights.

I will take the well-known Army debates of 1647-49 as a starting point for this study not only because they are well documented 4 , but because beyond the pressing question of the king's fate they epitomize the conflicting views about property in revolutionary England and its relation to political rights. As Colonel Rich, one of the Putney protagonists, put it, the debates revealed two "extremes": on the one hand, the idea "[that] men that have no interest as to estate should have no interest as to election", on the other, the conviction "that they should have an equal interest" (63-64). However, as he pointed out, there may have been a middle way between on the one hand, connecting the voting rights to property rights, thus excluding the poor from civic representation, and on the other hand, disconnecting them altogether so as to allow manhood suffrage, which by the law of majority was bound to end in voting "equality of goods and estates" (63). This middle way consisted of a "distribution of election" so that "there may be a representative of the poor as well as the rich, and not to exclude all." (64) Here is an interpretative framework that helps us read various proposals for a better society that emerged in the wake of the civil wars. I will chronologically expose that of the Levellers, as expounded in the Agreements of the People and discussed at Putney, who stood for the extension of the franchise; that of the Diggers voiced by Gerrard Winstanley, who claimed collective property rights over the land; and that of James Harrington who derived political rights from economic power while proposing a more equalitarian distribution of property. It will be shown that all three positions engaged the understanding of such fundamental notions as liberty and equality.

"Property… the most fundamental constitution" : the Putney Debates

What took place at St Mary's Church in Putney (south London) in the Autumn of 1647 and again in the winter of 1648-49 was the debating of a declaration entitled An Agreement of the People, signed by sixteen regiments (nine of horsemen, seven of footmen) and presented to the General Council of Officers of the New Model Army. The authors vindicated what they called "the native rights" of "the freeborn people of England" which they hoped to see entrenched in law, at a time when they knew Cromwell and the Army grandees were negotiating the peace terms with King Charles I. They claimed a better representation in Parliament achieved though frequent elections every two years, a more equal distribution of seats throughout the country apportioned to the population and last but not least, popular sovereignty.

When A Case of the Army Stated claimed the payment of the arrears to the rank and file who had been fighting the royalist camp 5 , the Agreement called for a legal and political reward for the war effort. The demand for successive parliaments, which should meet regularly and be frequently renewed, as opposed to indefinite ones summoned at will by the sovereign, was of course not specific to the Levellers. It had been the battle cry of the Parliamentary camp and it had even been conceded by Charles I who had accepted to sign the Triennal Act in 1641, an act which required that Parliament should meet at least once in three years for a minimum period of fifty days. This was common ground throughout the Putney Debates. But there were other sources of disagreement.

After three days of intense debating (and praying) no consensus was reached. When an amended version of the Agreement was presented in the midst of the king's trial on January 20 th 1649, "Ireton and Cromwell saw an opportunity to let the Agreement die a quiet death while the nation's attention was fixed upon the fate of their monarch" (Vallance 2009, 170). By March 1649, all the leaders of the Leveller movement were imprisoned 6 . What led the Army leaders to "let the Agreement die a quiet death" then ? Ireton may have sympathized with some of the Levellers' claims in principle, especially when it came to the extension of the franchise (Morton 215,Vallance 164), and may have buried it because political realism commanded it ; but the discussions reveal some insuperable divisions.

Colonel Thomas Rainborough and other Levellers 7 advocated the extension of the franchise below the 40-shilling threshold and a better electoral distribution of seats throughout the country. This it should be noted did not amount to manhood suffrage but would have granted small proprietors and wage earners the right to vote, excluding servants, labourers and paupers to the extent that they were economically dependent on others. They grounded their claim in the existence of the native liberties Englishmen had been deprived of ever since the Norman invasion. They combined these liberties to natural liberties, so that "every Englishman that is an inhabitant of England should choose have a voice in the representatives" (Petty 61). In St Edward's Ghost : or, Anti-Normanisme (1646), Mark Overton had resorted to the topos of the Norman Yoke that has been well analysed by Christopher Hill and others 8 . As Joseph Franck noted :

A little more than a year later, during the debate at Putney, the myth of historical rights was more fully fused with the myth of inherent natural right, until, in combination, they became the most powerful weapon in the Leveller propaganda arsenal. (Franck 84) Army Grandees, through the voice of Oliver Cromwell and Henry Ireton, stood firmly against such proposals. They perceived the far-reaching implications of positing equal rights for every man living in the community, every resident. First of all, it could mean granting a voice to strangers, a principle which ran contrary to the fundamental constitution of the realm "that no person that hath not a local and permanent interest in the kingdom should have an equal dependence in election [with those that have]" (Ireton 63). Secondly, and perhaps more importantly, they were keen to show that the notion of a Law of Nature on which the Levellers based their argument severely undermined the institution of property, one of the pillars of English society :

All the main thing that I speak for, is because I would have an eye to property. Let everyman consider with himself that he do not go that way to take away all property. For here is the case of the most fundamental part of the constitution of the Kingdom, which if you take away, you take all away, by that […] if this be allowed, we are free, we are equal, one man must have as much voice as another […] Upon these grounds, if you do, paramount [to] all constitutions, hold up this law of Nature, I would fain have any man show me their bounds, where you will end, and [why you should not] take away all property (Ireton 57).

While the more moderate Levellers like Maximilian Petty or Wildman proclaimed to be set against "Kings and Lords and property" -meaning the overwhelming power of the landed aristocracy -while denying to attack property per se, the more radical Rainborough or Sexby insistently questioned the connection between property and franchise : "I would desire to know how this comes to be a property in some men and not in others." (60-61) Ireton's following declaration voices his ultimate trust in God's ways, however incomprehensible they may be, should He allow the Leveller principles to be adopted :

If God saw it good to destroy not only King and Lords, but all distinctions of degrees -nay if it go further to destroy all property, that there's no such thing left, that there be nothing at all of civil constitution left in the kingdom-if I see the hand of God in it I hope I shall with quietness acquiesce, and submit to it, and not resist it (Ireton 50).

He went on to express confidence that God will not let the Army "incur sin or bring scandal in the name of God and the name of the people of God" (Ireton 50). There is no doubt that with peace looming ahead, the Army victors and parliamentary Independents were returning to "the government policy as it affected the land, trading interests and foreign policy" (Morton 216) and that they became increasingly concerned with restoring public order. As Cromwell told them : "These things that you have now offered, they are new to us" ; he recognized that the Agreement contained in it very great alterations of the fundamental laws of the realm that the people may not be ready to receive (7-8). A few days later, the Leveller leaders were examined before the Council of State set up to act as a temporary executive body before being sent to the Tower. To Ireton who, though conceding that property was a human and not a divine institution, repeated he could not approve in his heart the destruction of property, a bitter Edward Sexby 9 , replied : I see that though liberty were our end, there is a degeneration from it […] There are many thousands of us soldiers that have ventured our lives ; we have had little propriety in the kingdom as to our estates, yet we have had a birthright. But it seems now, except a man hath a fixed estate in this kingdom, he hath no right in this kingdom. (69) Cromwell and the Army Grandees proclaimed to be the defenders of liberty though. What they found much more difficult to accept, or found too dangerous to accept for the sake of the country's unity, was the "utter confusion" and "absolute desolation" which would result from implementing a greater equality in the system of representation (7) 10 . Not only would it "make England like the Switzerland country, one canton of the Swiss against another" (7), but it would put master and servant on an equal footing by making them "equal electors", so that "those that have no interest in the kingdom will make it their interest to choose those that have no interest. It may happen, that the majority may by law, not in confusion, destroy property" (63). The idea of democracy brought about a train of fears -confusion of rank, rivalry between the regions, both conducive to a state of anarchy. Equality, more than liberty, appears to have been the stumbling-block of the Putney Debates.

"Once the earth becomes a common treasury again" 11 : Winstanley and the Digger experiment

In April 1649 a group of men set up a community on St George's Hill in the parish of Walton-on-Thames. Against the backdrop of severe poverty due to a series of bad harvests, they sought to put into practice the principles that Gerrard Winstanley had not ceased to defend in his writings, and started digging and planting the waste lands together. The land was conveniently located near the Windsor Great Forest, the resources of which the people could reclaim after the fall of the monarchy and the seizing of the Crown's lands. The group of Diggers amounted to twenty to thirty people who were offered meat, drink and clothes (Hill 1972, 110). The labourers were almost immediately rejected by locals, so in August Winstanley decided to move to the nearby community of Cobham where they were better accepted. For example, Gurney notes that John Coulton, "a yeoman farmer and a solid member of his local community" was one of those who joined Winstanley on St George's Hill and remained there until the end of their activity in April 1650 (Gurney 2007, 1-2). Overall the Digger experiment lasted one year 12 .

For Winstanley, tyrannical oppression by "kingly power" was irretrievably connected to the dispossession of the poor by the Crown and the landed gentry. This vision was anchored in a rational reading of the Bible : And hereupon the earth, which was made to be a common treasury of relief for all, both beasts and men, was hedged into enclosures by the teachers and rulers, and the others were made servants and slaves. And the earth that is within this creation made a common storehouse for all, is bought and sold and kept in the hands of a few […] (379).

He went on :

That we may work in righteousness, and lay the foundation for making the earth a common treasury for all, both rich and poor. That every one that is born in the land may be fed by the earth, his mother […] not enclosing any part into any particular hand, but all as one man working together, and feeding together as sons of one father, members of one family ; not one lording over another, but all looking upon each other as equals in the creation (380).

"Civil propriety is the curse", Winstanley declared. He saw a historical continuity between the Babylonian yoke over Israel and the Norman yoke over England, as in both cases "the freeholders or landlords must be the choosers." The "poor enslaved English Israelites" had been deprived of their means of subsistence by the conquest of the land and by the same token, had been deprived of their electoral voices. That is why Winstanley did not advocate any extension of the franchise, a system still based on private property. Tilling the common lands was the only means by which the people could retrieve and assert collective rights of property and be eventually freed from bondage (382-385).

As Christopher Hill described it, the Digger experiment "was a symbolic assumption of ownership of the common lands" (Hill 1972, 110). For local landowners and some observers, it amounted to illegal appropriation. Thus James Harrington famously referred to "robbers or levellers" (Pocock 1977, 292) 13 . But I would like to suggest that doing so, Harrington denounced the ill-consequences of Cromwell and the Army grandees' policy, and intervened in the ongoing debate about property in one of the directions pointed out by Colonel Rich at Putney.

Harrington and the balance of property Rich had declared :

There may be a more equitable division and distribution than that he that hath nothing, should have an equal voice ; and certainly there may be some other way thought of, that there may be a representative of the poor as well as the rich, and not to exclude all. (Woodhouse 63) James Harrington certainly never went as far as to advocate some form of agrarian communism. The institution of private property remained one of the bases of his model republic depicted in The Commonwealth of Oceana (1656). However, what relates him to the Levellers and the Diggers is the way in which he linked economic inequalities to political exclusion and believed the reduction of such inequalities was at the core of a wellordered commonwealth.

Harrington was not an actor of the Revolution ; he was an eye witness to the fall of the monarchy as he had served as gentleman of the bedchamber to Charles I in captivity between 1647 and 1649 . He was a theoretician who reflected on the historical mechanisms by which states maintained themselves or collapsed 14 . His conviction was that the ancient constitution had been overthrown because of imbalances in the distribution of property. To him, these imbalances could only be remedied by the entrenchment of an agrarian law inspired by the Roman Republic 15 .

Inequalities in the distribution of property primarily derived from the law of male primogeniture, which deprived the younger brothers in order to avoid the division of the estate. He humorously compared this process to the drowning of newborn puppies. A pillar of the monarchy and the aristocracy, where land and power rested within the hands of a few, such a law ran counter to the principles of a commonwealth : "If the monarchy could not bear such divisions [of property] because they tended to be a commonwealth, neither can a commonwealth connive at such accumulations, because they tend to a monarchy." (Pocock 237) 16 Hence the need for a voluntary redistribution of property, not along levelling lines, but in way that : every man who is at present possessed, or shall hereafter be possessed, of an estate in land exceeding the revenue of five thousand pounds a year, and having more than one son, shall leave his lands either equally divided among them, in case the lands amount unto above 2000 £ a year unto each, or so near equally, in case they come under, that the greater part or portion of the same remaining unto the eldest exceed not the value of two thousand pounds revenue (Pocock 231).

Harrington was taking part in a long-standing debate as to whether the agrarian law had accelerated the fall of the Roman Republic which had adopted it. In this particular respect, he firmly opposed his master Machiavelli . Conversely, he stood closer to the Leveller soldier Cowling who, during the Putney debates, had raised the following question : "whether the younger son have not as much right to the inheritance as the eldest" (64). In fact, several 1640s pamphlets had asked for a government intervention meant to restore a better distribution of land property. For instance, in August 1649, the Tyrannipocrit Discovered had attacked the new government of England for not establishing "an equality of goods and lands" (cited in Hill 1972, 116). Thus, it clearly appears that "The agrarian law was made famous by James Harrington's advocacy of it in Oceana (1656) from which many thinkers adapted the idea. But Harrington was only summing up a tradition." (Hill 1972, 115) As seen above, Harrington's agrarian was not about achieving "an equality of goods and lands" but it was about preventing blatant discrepancies between the citizens of his ideal republic, inequalities which would inevitably result in civil strife : "Take the bread out of the people's mouths, as did the Roman patricians, and you are sure enough of a war, in which case they may be levellers ; but our agrarian causeth their industry to flow with milk and honey." (Pocock 293) As the rest of the passage shows, this was a clear indictment of Cromwell's policy in the late 1640s which had deprived the common soldiers of their pay and thus captured the public revenue. The term "leveller" remained a derogatory one, but Harrington attenuated it by asserting that social circumstances, not any ingrained wickedness, made people so. Similarly, the persistence of immovable castes who lived to receive the dividends of their property was a disgrace :

The aristocracy is ravenous, and not the people. Your highwaymen are not such as have trades or have been brought up unto industry, but such whose education hath pretended unto that of gentlemen. ( 292) Therefore the utopian vein served him to convey the beneficial effects of restoring the balance, a notion which is at the core of his political theory.

Harrington was deeply concerned with establishing a well-ordered commonwealth on solid foundations. The agrarian law was the first fundamental as defined in the "Epitome of the whole commonwealth":

proportioned at two thousand pounds a year in land, lying and being within the proper territory of Oceana, and so stating property in land at such a balance that the power can never swerve out of the hands of the many. (Pocock 333) Harrington held the nobility was necessary to the Commonwealth, because of their education and "leisure for the public" (Pocock 257,259,261); yet, their wealth should never overbalance that of the people. The agrarian would even allow for a degree of social mobility by avoiding the concentration of property within the hands of a few and encouraging the most industrious to purchase greater estates (293). Just as the agrarian would secure the balance of dominion, the rotation of mandates in a bicameral parliament would secure the balance of power and would prevent it from being overruled by one class of citizens.

Harrington parted with the radical Levellers who would have granted the right to vote to any citizen. In Oceana citizens were categorized along strict property lines so that freeholders could become electors. But the smaller proprietors detained the sovereign power in a constitutional framework that maintained the balance of power and property between the people and the aristocracy. In a sense, the Levellers were not mistaken in asking for more democracy, but for not asking enough of it:

Government should be established upon a rock, not set upon a precipice; a representative consisting but of four hundred, though in the nature thereof it be popular, is not in itself a weapon that is fixed, but hath something of the broken bow, as still apt to start aside unto monarchy. (Pocock 657) What they had overlooked was the importance of the balance of power at work in a bicameral Parliament, an institution he ceaselessly defended until the last hours of the Protectorate. It consisted of a senate elected among the richer proprietors replacing the hereditary House of Lords (three hundred); and a large popular assembly or prerogative tribe, composed of one thousand and fifty small proprietors (four-hundred-and-fifty horsemen and six-hundred footmen) (Pocock 284). Contrary to the old system of government, each house would be ascribed a specific role, so that power would be equally distributed between the two: the Senate debating, the Prerogative tribe deciding. This, according to Harrington, would prevent a House from supporting a law that would supersede the other House's interest:

The over-balance in the suffrage remaining unto the foot by one hundred and fifty votes, you have, unto the support of a true and natural aristocracy, the deepest root of a democracy that hath been planted. (Pocock 284) In addition, the members of the prerogative chamber and of the Senate would be elected and both houses would be renewed by third during annual elections, according to the principle of rotation so as to prevent the corruption of the people's representatives.

Harrington staunchly believed in democracy as incarnated by a numerous popular assembly. On the eve of the Restoration, in The Art of Lawgiving (1659), he expressed impatience at the resistance he felt even among his fellow citizens, including those who had overthrown the monarchy 17 , against this notion of a popular assembly. Commenting on Thucydides (in Hobbes's translation) who recommended the combination of the few and the many to redress Athens:

But we in England are not apt to believe that to decree the sovereignty unto thousands were the way to make city or a nation recover of wounds or to raise her head. We have a loathing, we are sick of such thoughts. An assembly of the people sovereign! Nay, and an assembly of the people consisting in the major vote of the lower sort! (Pocock 676) Even when it voted bad laws he insisted, a popular assembly was the "touchstone" of an equalitarian commonwealth (Pocock 660,676).

In July 1659, following the death of his father, Richard Cromwell convened the dissolved Rump Parliament in the midst of a severe economic and constitutional crisis. The project of a constitution drafted along Harringtonian principles was submitted to the Rump thanks to the support of a few MPs. It was immediately rejected. If it retained the principles of an elected bicameral parliament and of the rotation of mandates, the constitutional principle of an agrarian had been carefully left out. As Harrington well knew, "agrarian laws of all others have ever been the greatest bugbears" (Pocock 231).

Conclusion

It is quite obvious that the radical thinkers of the next century kept memory of these discussions, vehiculed by seventeenth and eighteenth-century radicals and "commonwealthmen" 18 . Thomas Spence's proposal of a land plan was reminiscent of the agrarian law debated during the English Revolution. In some ways, it echoed Winstanley's views "in that it identified the private ownership of land as the source of inequality and evil in the country" (Vallance 231). On the other hand, Spence was highly defiant of "big government" and proposed that land should be owned by the parish and distributed among its inhabitants. This was quite different from the kind of centralizing system advocated by Winstanley or Harrington, for whom the agrarian was inscribed in the constitution of his ideal republic. However, the common thread between these thinkers was the way they connected the reform of society and the reform of property structures, a reform for which they deserve the name of "radicals".

Quite symptomatically, eighteenth-century designs involving a better distribution of property among men (and sometimes women) as the basis for a more equalitarian society, met with the same opposition as seventeenth-century ones. This is what one can read in this political tract circulated by the Association for preserving Liberty and Property against Republicans and Levellers in 1792 :

It appears to us, the tendency of these Opinions is, that we are voluntarily to surrender every thing we now possess ; our Religion and our Laws ; our civil Government and Civil Society ; and that we are to trust to the formation of something New, upon the principles of Equality, and under the auspices of speculative men, who have conceived ideas of perfection that never yet were known in the World.

Thus, when in the midst of another revolutionary turmoil Edmund Burke denounced the destruction of property (Burke 141) and "the fabrication of a new government" (117) in France, the petitioners lamented the planned destruction of English institutions and their substitution by the dangerous brainchild of utopian thinkers.

NOTES

1. Some authors have warned us against the indiscriminate use of the labels "radical" and "radicals" as that they tend to reify ideological divisions and run the risk of anachronism by overlooking the links between supposed radicals and tradition. Radicalism will be taken here as referring to "an attitude to the status quo" and "a demand for the structural transformation of an existing order". (Burgess 2007, 62-81) 2. Thomas Spence, A Supplement to the History of Robinson Crusoe being the History of Crusonia, Newcastle, printed and sold by T. Saint in 1782. In the following passage, I am referring to the text from the 1782 digitized by the Thomas Spence Society and available on their website: http:// thomas-spence-society.co.uk 3. For a recent analysis of Neville's Isle of Pines and republicanism, see Mahlberg 2009, 83-89. 4. The general secretary of the Army William Clarke took a verbatim record of them. For a discussion of the various editions of the Putney Debates, see Michael Mendle 2001, 1-18. Woodhouse based his edition on the Clarke Manuscript in Oxford (Clarke Ms 65, Worcester College). As the debates (and only the first three days) were taken in shorthand by Clarke, the editor took the liberty to reconstruct whole groups of words. These are the words between brackets. Whenever referring to the Debates I will refer to the Woodhouse edition.

5.

For a treatment of the "bread-and-butter grievances" at stake, see Morrill's "The Army Revolt" (1977) in Morrill 1993, 307-331. 6. These were John Lilburne, Richard Overton, William Walwyn and Thomas Prince. 8. Hill 1997[START_REF] Hill | The Intellectual Origins of the English Revolution[END_REF], [START_REF] Pocock | The Ancient Constitution and the Feudal Law[END_REF][START_REF] Woolrych | Soldiers and Statesmen[END_REF], Burgess 1992[START_REF] Lurbe | L'interprétation de la conquête normande dans la pensée politique anglaise (XVII-XVIIIe siècles)[END_REF]. 9. Edward Sexby is the author of the famous pamphlet in favour of tyrannicide Killing No Murder (1657) advocating Cromwell's assassination.

10. There seems to be a degree of nuance between Cromwell and Ireton on that point as Ireton proclaimed "I am agreed with you if you insist upon a more equal distribution of elections, I will agree with you, not only to dispute for it, but to fight for it and contend for it." (77) ' Standard Advanced (Woodhouse 380). Whenever quoting from this text, I will refer to the Woodhouse edition.

The True Levellers

12. There seems to have been other Digger communities, or at least sympathisers, in the southern part and central part of the country. Cf. Hill 1972, 124-128. 13. In the next passage, I will refer to the Pocock edition of Harrington's Political Works.

14. He described himself as a "political anatomist" whose complex description of the body politic was "understood by so few" (656).

15.

He became more politically active in the later years of the Protectorate when he and others ran constitutional debates within the Rota Club, which came up with a draft constitution presented to the Rump Parliament in 1659. Cf. [START_REF] Foxley | Democracy in 1659 : Harrington and the Good Old Cause[END_REF]Ducrocq 2014. 16. Insofar as the law of male primogeniture was the main foundation of hereditary monarchy and aristocracy, notably expounded by Sir Robert Filmer in Patriarcha or the Natural Power of Kings (posth.1679), most of radical and republican thinkers proceeded to a scathing critique of it, to the point that seventeenth-century republicanism can be seen as "anti-patriarchalism". This does not mean to say that all radicals or republicans asserted gender equality, quite the contrary. Cf. 18. This umbrella term came to designate a group of political thinkers and activists who from the First English Revolution onward defended such principles as limited government, republicanism, parliamentary reform, the rule of law, religious toleration in the name of natural rights and individual liberty. Cf. [START_REF] Robbins | The Eighteenth-Century Commonwealthman : Studies in the Transmission, Development and Circumstances of English Liberal Thought from the Restoration of Charles II until the War with the Thirteen Colonies[END_REF]Pocock 1975.


Rémi Douvenot
email: douvenot@recherche.enac.fr

Alexandre Chabory
email: chabory@recherche.enac.fr

Sébastien Rougerie
email: sebastien.rougerie@cnes.fr

Modelling the Radiowave Propagation with a Split-Step Wavelet Method for Radio Occultation

Keywords: split-step wavelet, radio occultation, GPS signal, propagation

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Modelling the radiowave propagation with a split-step wavelet method for radio occultation

I. INTRODUCTION

Radio occultation (RO) consists in inferring the atmospheric conditions from satellite-to-satellite radiowave measurements when the satellites are at the limit of the line-of-sight configuration. In this condition, the radiowave propagation is significantly altered by the variations of the atmosphere in the lowest layers. This alteration carries information about the propagation channel.

On Earth, RO is often applied with GNSS signals of opportunity since lots of transmitters are available [START_REF] Kursinski | Observing Earth's atmosphere with radio occultation measurements using the Global Positioning System[END_REF]. Therefore, we consider a geometry with a GPS transmitter and a dedicated low-elevation orbit (LEO) satellite as a receiver.

A simple one-ray model is often used to model the propagation and infer the atmospheric conditions. However, efforts are made by the community to develop more accurate models [START_REF] Hour | 2D modeling of the atmospheric refraction based on Gaussian beams[END_REF]. In this context, the aim of this paper is to demonstrate the usability of the recently developed split-step wavelet (SSW) algorithm [START_REF] Zhou | A fast split-step wavelet algorithm for the simulation of long-range propagation[END_REF] for RO. This method considers the atmosphere using phase screens. Compared to the one-ray model, SSW can deal with any atmosphere varying in both altitude and distance. Moreover, the ground relief and composition can be accounted. For this study, we consider the Earth as a sphere. The atmosphere is defined from an ITU model [START_REF] Itu | Reference standard atmospheres[END_REF]. In the end, the impacts of the atmospheric layers on the magnitude, group and phase delays, and Doppler shift received on the LEO satellite are expected to be accurately quantified.

In the first section, SSW and the atmospheric model are briefly introduced. Simulation results are presented in the second section. Finally, the relevancy to use accurate models as SSW is discussed in the conclusion.

II. THE ELECTROMAGNETIC MODELLING

A. Atmospheric model

In this paper, we consider the atmospheric model P835-6 proposed by ITU [START_REF] Itu | Reference standard atmospheres[END_REF]. This model allows for the calculation of the refractive index n with respect to standard values of the pressure, humidity, and temperature on Earth. This model is used here as an example. More complex atmospheric models could be used without modifying the propagation method itself.

We only model the troposphere. Ionosphere may be included in the global atmospheric model by using (for example) the ITU-R P531. However, as the refractive index is very close to one in the ionosphere, there is almost no refraction. Thus we decide to neglect it to reduce our computation domain.

B. Split-step wavelet propagation method

The radiowave is propagated using the split-step wavelet (SSW) method. This method is inspired by the split-step Fourier (SSF) method. The main difference with SSF is that the field is decomposed on wavelets instead of plane waves. The wavelet decomposition has the advantage of being fast and rendering a sparse representation of the signal [START_REF] Mallat | A Wavelet Tour of Signal Processing[END_REF].

The propagation method is described in details in [START_REF] Zhou | A fast split-step wavelet algorithm for the simulation of long-range propagation[END_REF] and briefly summarised here. The field is supposed known on a first vertical. The propagation is performed iteratively following these steps: the field is represented as a sparse wavelet vector by applying the fast wavelet transform (FWT) and compression. Then, the propagation on a step ∆x is considered using a precomputed wavelet-to-wavelet scattering matrix. A sparse wavelet vector after free-space propagation on ∆x is obtained. Then, an inverse FWT gives the corresponding field.

Finally, the refractivity and relief are accounted in the space domain like for SSF. The ground is accounted by means of a dedicated local image method. This method is performed iteratively in distance to obtain the complete field.

SSW is validated in [START_REF] Zhou | A fast split-step wavelet algorithm for the simulation of long-range propagation[END_REF] by comparisons with analytically propagated complex source points in free space and with SSF on complex scenarios. The additional RMS error due to the matrix compression follows E M = N x V M , where V M is the compression threshold for the precomputed wavelet-to-wavelet scattering matrix. The additional RMS error due to the fields compression during the propagation follows E s = N 0.3

x V s , where V s is the compression threshold applied on the field at each step. Finally, the maximal expected RMS error due to the wavelet compressions is given by E tot = E M + E s .

Therefore, the simulation methods fulfils the same limitations as SSF, with an acceptable and tunable error.

Here, the propagation domain is separated in 3 areas: the ground is accounted in the second one only.

III. NUMERICAL SIMULATIONS

A. Configuration

We consider a GPS satellite at frequency f = 1.575 GHz with a gain of 16 dBi and a power of 25 W. The receiver is positioned on a LEO satellite. The RO configuration is chosen such that the trajectory between the GPS and the LEO satellites is tangent to Earth. The total distance between the satellites is 29200 km.

Concerning the computation domain, the aim is to find the best trade-off between a domain large enough to model all the physical contributions of the RO process and an acceptable computation time. As we assume no ionosphere, we use a 2D plane of height 50 km. Moreover, the first vertical is set at x = 24647 km. The chosen steps are ∆z = 4 m ≈ 20 λ and ∆x = 1000 m. Finally, the size of the computation domain is 50 × 4553 km 2 , i.e., 12500 × 4553 points.

The compression thresholds are chosen as V M = 2 × 10 -7 and V s = 10 -4 . Therefore, the additional modelling error due to the wavelet compression is -53 dB, which is negligible.

The polarisation is TE. Circular polarisations could be obtained with 2 simulations in TE and TM. Note that the coarse discretisation along z implies that the calculation is only valid for propagation angles close to the main propagation direction. This is not an issue considering the geometry of RO.

The refractive index obtained with the ITU model is represented in figure 1. As expected, the variation of the refractive index is stronger close to the Earth surface. The position of the LEO satellite corresponds to the vertical distance 0 km at range 4553 km.

B. Results

The electric field obtained with respect to the horizontal and vertical distances is plotted in figure 2. The vertical white dotted lines show the separations between the 3 calculation zones. The ground in accounted in the second one only. Calculation time of SSW is approximately 10 min 30 s on a standard computer with a 2.7 GHz processor. One can see that the Earth surface is accounted. The bending due to atmosphere is also visible. Few numerical artefacts due to the coarse vertical meshing can be observed, without influence on the field calculated at the LEO. The field on the last vertical 500 m on each side of the LEO satellite is plotted in figure 3. The variability of the field with respect to the position is strong. Indeed, the reflection over Earth has here a significant effect.

IV. DISCUSSION

This paper introduces split-step wavelet (SSW) to calculate propagation for radio occultation (RO). We have shown that accurate electromagnetic modelling including atmosphere and ground can be used rather than a simplified one-ray method. This study has also shown the ability of SSW to deal with very large scenes (12500 vertical points).

The main interest of SSW is to simulate propagation in complex environments. Complex refractive index and relief can be used as input. Moreover, simulations can be performed in acceptable times -typically few minutes or tens of minutes for one scene on a standard computer.

The second interest of SSW is to consider the ground reflection. The relevance of considering the ground is still to be discussed. The reflected wave is mainly in cross-polarisation.

Fig. 1 .

 1 Fig. 1. N index along the propagation scene. Earth is in black.

Fig. 2 .

 2 Fig. 2. Electric field with respect to horizontal and vertical distances.

Fig. 3 .

 3 Fig. 3. Electric field with respect to the vertical distance on the last vertical.

Nevertheless, the antenna polarisations are not pure and a modelling considering both transmission and reception antennas should be lead to investigate this question.