Ralph Chill
email: chill@univ-metz.fr

Wiktor Radzki
email: wiktorradzki@yahoo.com

STABILIZATION OF SOLUTIONS OF DISSIPATIVE HAMILTONIAN SYSTEMS

We study the stabilization of solutions of damped Hamiltonian systems. We give sufficient conditions for convergence of these solutions, decay estimate and examples of applications.

Introduction

Let (M, g) be a smooth Riemannian manifold, H ∈ C 2 (M) and let J and S be two smooth sections of skew-adjoint and selfadjoint linear mappings on the tangent space of M, respectively.

We consider the ordinary differential equation [START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF] ẋ + (J(x) + S(x))∇H(x) = 0, t ≥ 0, where ∇H is the gradient of H with respect to the Riemannian metric g.

The above equation includes two prominent examples of ordinary differential equations. First, if J vanishes and S is positive definite, then after an appropriate change of the Riemannian metric g we may assume that S ≡ I, and equation ( 1) is a gradient system. In particular, the function H is nonincreasing along solutions and strictly decreasing along nonstationary solutions. Second, if S vanishes, then, since J is skew-adjoint, equation [START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF] shares properties of a Hamiltonian system; in particular, the energy H is preserved along solutions. If J is nonsingular (and S still vanishes), then equation [START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF] actually is a Hamiltonian system for the induced symplectic structure ω given by ω(X, Y) = g(JX, Y). For other nontrivial choices of S and J, equation (1) also includes coupled systems.

In this article, we mainly consider Hamiltonian systems, but we always assume that some damping is present in the sense that S is nonnegative definite and not identically zero. In fact, we assume slightly more, and we address the question of stabilization/convergence to equilibrium of global solutions having relatively compact range in M. We note that for general gradient systems stabilization of such solutions need not necessarily take place [START_REF] Palis | Geometric Theory of Dynamical Systems. An Introduction[END_REF] (a counterexample was already described in [START_REF] Haskell | The method of steepest descent for non-linear minimization problems[END_REF]). This and a corresponding counterexample for infinite dimensional Hamiltonian systems [START_REF] Jendoubi | Nonstabilizing solutions of semilinear hyperbolic and elliptic equations with damping[END_REF] suggest, that one may expect also nonstabilization for general damped Hamiltonian systems. Thus, an additional assumption on the system is required. This assumption is here analyticity of the Hamiltonian H (that is, merely an assumption of regularity), or, more generally, the so-called Łojasiewicz-Simon gradient inequality for H.

We apply a version of general convergence theorems studied in [START_REF] Ch | Pointwise convergence of gradient-like systems[END_REF][START_REF] Chill | Applications of the Lojasiewicz-Simon gradient inequality to gradient-like evolution equations[END_REF] (see also [START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF]), as explained in Section 2.

The rest of this paper is organized as follows. Section 2 concerns general stabilization results for functions with values in manifolds. They are used in Section 3 to prove stabilization results for damped Hamiltonian systems (Theorem 3). Section 4 contains examples of application of the results from Section 3.

In Sections 3 and 4, stabilization of a solution means its convergence to an equilibrium point. However, the general results from Section 2 allow us to understand the stabilization of solutions in some more general sense. Namely, the function which converges to a point in Theorem 1 can be a superposition of a solution of given system with an appropriate mapping. This can mean, for example, a kind of directional or partial stabilization of solutions.

General stabilization results

This section is devoted to stabilization results for functions with values in manifolds, which will be applied in Section 3 to damped Hamiltonian systems.

In order to formulate the stabilization theorems, we recall that a function H ∈ C 1 (M) satisfies the Łojasiewicz-Simon gradient inequality near some element ϕ ∈ M if there exists θ ∈ (0, 1 2 ], C ≥ 0 and a neighbourhood U of ϕ such that

(2) |H(x) -H(ϕ)| 1-θ ≤ C ∇H(x) for every x ∈ U.
The ω-limit set of a function x : R + → M is, by definition, the set

ω(x) := {ϕ ∈ M : there exists (t n ) ր ∞ such that x(t n ) → ϕ}.
In order to prove the stabilization theorem for damped Hamiltonian systems, we use the following convergence result which has been formulated in [START_REF] Ch | Pointwise convergence of gradient-like systems[END_REF]Theorem 1.2] in a slightly different form on manifolds (see also [START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF]Theorem 2.2], [START_REF] Chill | Applications of the Lojasiewicz-Simon gradient inequality to gradient-like evolution equations[END_REF]Theorem 1]). The decay estimate is from [4, Theorem 2], at least in the case of euclidean space M = R n and in general Hilbert spaces. The generalization to manifolds is straightforward from the proof in [START_REF] Chill | Applications of the Lojasiewicz-Simon gradient inequality to gradient-like evolution equations[END_REF], but we give the complete proof for the convenience of the reader.

Theorem 1 (Convergence result). Let x ∈ C 1 (R + ; M) be a function having relatively compact range. Assume that there exists a control function E ∈ C 1 (M) and α > 0 such that E • x is differentiable almost everywhere,

(3) - d dt E(x(t)) ≥ α E ′ (x(t)) ẋ(t)
for almost every t ∈ R + , and

(4) if E(x(t)) is constant for t ≥ t 0 , then x(t) is constant for t ≥ t 0 .
Assume that there exists ϕ ∈ ω(x) such that E satisfies the Łojasiewicz-Simon gradient inequality near ϕ with exponent θ ∈ (0, 1 2 ]. Then x has finite length, and in particular lim

t→∞ x(t) = ϕ in M.
If, in addition, 2 for some β > 0 and almost every t ∈ R + ,

(5) - d dt E(x(t)) ≥ β E ′ (x(t))
then d(x(t), ϕ) =        O(e -ct ) for some c > 0 if θ = 1 2 , O(t -θ 1-2θ ) if θ < 1 2 .
Proof. It follows from condition (3) that the function E(x) is nonincreasing. Since x has relatively compact range and E is continuous, lim t→∞ E(x(t)) exists and is finite. Clearly, by continuity of the function E and by definition of the ω-limit set ω(x), this limit equals E(ϕ) for every ϕ ∈ ω(x). By changing E by an additive constant if necessary, we may assume without loss of generality that E(ϕ) = 0, so that E(x(t)) ≥ 0 and lim t→∞ E(x(t)) = 0.

If E(x(t 0 )) = 0 for some t 0 ≥ 0, then E(x(t)) = 0 for every t ≥ t 0 , and therefore, by condition (4), the function x is constant for t ≥ t 0 . In this case, there remains nothing to prove.

Hence, we may assume that E(x(t)) > 0 for every t ≥ 0. By assumption, there exists ϕ ∈ ω(x) such that E satisfies the Łojasiewicz-Simon gradient inequality near ϕ, that is, there exists θ ∈ (0, 1 2 ], C ≥ 0 and a neighbourhood U of ϕ such that E(x ′ ) 1-θ ≤ C ∇E(x ′ ) for every x ′ ∈ U. Let σ > 0 be such that the closed ball B(ϕ, σ) (with respect to the distance d induced by g) is contained in U.

Let t 0 ≥ 0 be such that d(x(t 0 ), ϕ) ≤ σ 3 and C αθ E(x(t 0 )) θ ≤ σ 3 ,
where α > 0 is as in condition (3), and let

t 1 := inf{t ≥ t 0 : d(x(t), ϕ) = σ}.
By continuity of the function x, we have t 1 > t 0 . Moreover, for every t ∈ [t 0 , t 1 ), by condition (3) and by the Łojasiewicz-Simon gradient inequality,

- d dt E(x(t)) θ = θ E(x(t)) θ-1 - d dt E(x(t)) (6) ≥ αθ E(x(t)) θ-1 E ′ (x(t)) ẋ(t) ≥ αθ C ẋ(t) .
Hence, for every t ∈ [t 0 , t 1 ), by definition of the distance d and by the choice of t 0 ,

d(x(t), ϕ) ≤ d(x(t), x(t 0 )) + d(x(t 0 ), ϕ) (7) ≤ t t 0 ẋ(s) ds + d(x(t 0 ), ϕ) ≤ C αθ E(x(t 0 )) θ + d(x(t 0 ), ϕ) ≤ 2 3 σ.
This inequality implies that t 1 = ∞. But then ẋ ∈ L 1 (R + ), by the estimate [START_REF] Gallot | Riemannian Geometry[END_REF]. This means that x has finite length. The existence of lim t→∞ x(t) follows from Cauchy's criterion and the fact that ϕ ∈ ω(x). Now, if in addition the condition (5) holds, then we can proceed differently in the second line of the estimate [START_REF] Gallot | Riemannian Geometry[END_REF] above in order to obtain, for all t ≥ t 0 ,

- d dt E(x(t)) θ = θ E(x(t)) θ-1 - d dt E(x(t)) ≥ βθ E(x(t)) θ-1 E ′ (x(t)) 2 ≥ βθ C 2 E(x(t)) θ 1-θ θ .
Solving this differential inequality for the function E(x) θ and inserting the resulting estimate into the inequality

d(x(t), ϕ) ≤ ∞ t ẋ(s) ds ≤ C αθ E(x(t)) θ ,
which follows from integrating the inequality (6), we obtain the desired decay estimate.

Stabilization results for damped Hamiltonian systems

In this section we prove a stabilization theorem for system [START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF]. To this aim we exploit Theorem 1.

The precise assumptions on the functions J, S and H are as follows. We denote by Π R the orthogonal projection of TM onto range S and we assume that (H1) the mappings J(x) and S(x) are skew-adjoint and selfadjoint on T x M, respectively, and there exists a continuous function

α 1 : M → (0, ∞) such that SX, X ≥ α 1 Π R X 2 for every tangent vector field X, (H2)
there exists a continuous function

c 1 : M → (0, ∞) such that X ≤ c 1 ( Π R X + Π R JX ) for every tangent vector field X, (H3)
there exists a continuous function

c 2 : M → (0, ∞) such that (I -Π R )JX ≤ c 2 Π R X for every tangent vector field X, (H4)
there exists a continuous function 2 for every tangent vector field X.

α 2 : M → (0, ∞) such that ∇ Π R X (Π R ∇H), Π R X ≥ α 2 Π R X
Remarks 2. (a) Observe that the hypotheses (H1)-(H4) imply that the differential equation ( 1) is gradient-like in the sense that the ω-limit set of every global solution with relatively compact range is contained in the set of stationary solutions, that is, in the set of all ϕ ∈ M such that ∇H(ϕ) = 0; see Lemma 6 below. (b) For the gradient-likeness and the following convergence result for global solutions of (1) it is in fact only important that the functions in hypotheses (H1) and (H4) have a sign; it is moreover not necessary that they have the same sign. In fact, the proofs below can be easily adapted by changing in an appropriate way the signs of the two terms in the perturbed energy functional defined in equation ( 8) below.

Our stabilization theorem for system (1) can be formulated as follows.

Theorem 3. Assume hypotheses (H1)-(H4) and let x ∈ C 1 (R + ; M) be a global solution of (1) having relatively compact range. Assume that there exists ϕ ∈ ω(x) such that H satisfies the Łojasiewicz-Simon gradient inequality near ϕ. Then x has finite length, and in particular lim t→∞ x(t) = ϕ.

In addition, if θ is the Łojasiewicz exponent from the Łojasiewicz-Simon inequality, then one has the following decay estimate:

d(x(t), ϕ) =        O(e -ct ) for some c > 0 if θ = 1 2 , O(t -θ 1-2θ ) if θ < 1 2 .
In order to prove Theorem 3, we apply Theorem 1 by using the control function E ∈ C 1 (M) given by ( 8)

E(x) = H(x) + ε F (x), x ∈ M,
where ε > 0 depends on the solution x and various constants from hypotheses (H1)-(H4) and the Łojasiewicz-Simon gradient inequality, and

F ∈ C 1 (M) is defined as (9) F (x) = J∇H(x), Π R ∇H(x) = Π R J∇H(x), Π R ∇H(x) , x ∈ M.
Here, as before, ∇ denotes the gradient with respect to the Riemannian metric g, and we write from now on for simplicity J, S and Π R instead of J(x), S(x) and Π R (x). Note that below ∇ also denotes the associated Riemannian connection acting on vector fields and tensors.

We shall check conditions (3), ( 4) and ( 5) from Theorem 1. We obviously have [START_REF] Ch | Pointwise convergence of gradient-like systems[END_REF] -

d dt E(x) = -∇H(x), ẋ -ε ∇F (x), ẋ .
The following lemma provides an estimate of the first term on the righthand side of the equality (10) above. Lemma 4. Assume hypothesis (H1), and let x ∈ C 1 (R + ; M) be a global solution of (1) having relatively compact range. Then there exists a constant α 1 > 0 such that, for every t ≥ 0, [START_REF] Łojasiewicz | Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles[END_REF] -

d dt H(x) = ∇H(x), ẋ ≥ α 1 Π R ∇H(x) 2 .
In particular, the function H(x) is nonincreasing.

Proof. Since J is skew-adjoint, we have ∇H(x), J∇H(x) = 0. Moreover, by hypothesis (H1) and since the closure of the range of x is compact, there exists a constant α 1 > 0 such that S∇H(x), ∇H(x) ≥ α 1 Π R ∇H(x) 2 for every t ≥ 0. Hence, for every t ≥ 0 we have the estimate

-∇H(x), ẋ = ∇H(x), (J(x) + S(x)) ∇H(x) = S(x)∇H(x), ∇H(x) ≥ α 1 Π R ∇H(x) 2 .
This is the claim.

The following lemma provides the estimate of the second term on the right-hand side of [START_REF] Ch | Pointwise convergence of gradient-like systems[END_REF]. Lemma 5. Assume hypotheses (H2)-(H4) and let x ∈ C 1 (R + ; M) be a global solution of (1) having relatively compact range. Then there exist constants α 2 > 0 and c > 0 such that for every λ > 0 and every t ≥ 0 one has

- d dt F (x) = ∇F (x), ẋ ≥ ≥ α 2 - cλ 2 Π R J∇H(x) 2 -c + c 2λ Π R ∇H(x) 2 . ( 12
)
Proof. In view of the definition of F and of the differential equation ( 1), we have

- d dt F (x) = -∇ ẋ Π R J∇H(x), Π R ∇H(x) = Π R J∇H(x), ∇ J∇H(x) (Π R ∇H(x)) + (13) + Π R J∇H(x), ∇ S∇H(x) (Π R ∇H(x)) + + ∇ J∇H(x) (Π R J∇H(x)), Π R ∇H(x) + + ∇ S∇H(x) (Π R J∇H(x)), Π R ∇H(x) .
Let us estimate the terms on the right-hand side of [START_REF] Maschke | Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation[END_REF]. We note that in the following, c denotes a constant which may change from line to line, which depends on various estimates of continuous functions on the (compact) closure of the range of x, but which is always finite and does not depend on t.

Note that the functions Π R ∇H and Π R J∇H are continuously differentiable and that their derivatives are therefore bounded on the range of x. Moreover, the function α 2 from hypothesis (H4) is positive and continuous and therefore bounded from below by a constant α 2 > 0 on the range of x which is assumed to be relatively compact. Hence, by hypothesis (H4), for every λ > 0 and every t ≥ 0 we have

Π R J∇H(x), ∇ J∇H(x) (Π R ∇H(x)) = = Π R J∇H(x), ∇ Π R J∇H(x) (Π R ∇H(x)) + + Π R J∇H(x), ∇ (I-Π R )J∇H(x) (Π R ∇H(x)) ≥ α 2 Π R J∇H(x) 2 -c Π R J∇H(x) (I -Π R )J∇H(x) (14) ≥ (α 2 - c λ 2 ) Π R J∇H(x) 2 - c 2λ Π R ∇H(x) 2 .
In the last inequality we have also used hypothesis (H3), and again the relative compactness of the range of x.

In order to estimate the second term, it suffices to note that S = SΠ R . This implies that for any λ > 0 and any t ≥ 0 we have

Π R J∇H(x), ∇ S∇H(x) (Π R ∇H(x)) = ≥ -c Π R J∇H(x) SΠ R ∇H(x) ≥ - c λ 2 Π R J∇H(x) 2 - c 2λ Π R ∇H(x) 2 . (15)
The third term is estimated in a similar way, using also hypothesis (H3). For every λ > 0 and every t ≥ 0 we have

∇ J∇H(x) (Π R J∇H(x)), Π R ∇H(x) = (16) ≥ -c Π R J∇H(x) + (I -Π R )J∇H(x) Π R ∇H(x) ≥ - cλ 2 Π R J∇H(x) 2 -( c 2λ + c) Π R ∇H(x) 2 .
Using again the equality S = SΠ R , the estimate of the fourth term is straightforward:

∇ S∇H(x) (Π R J∇H(x)), Π R ∇H(x) ≥ -c Π R ∇H(x) 2 . ( 17
)
The equality [START_REF] Maschke | Energy-based Lyapunov functions for forced Hamiltonian systems with dissipation[END_REF] and the estimates ( 14)-( 17) imply the estimate (12) which was the claim. Lemma 6. Assume the hypotheses of Theorem 3. Then, for every ε > 0 small enough, there exists α > 0 such that the function E defined in (8) satisfies 2 for every t ≥ 0.

(18) - d dt E(x) ≥ α ∇H(x)
In particular, the differential equation (1) is gradient-like in the sense that the ω-limit set ω(x) is contained in the set of stationary solutions.

Proof. By Lemmas 4 and 5, there exist constants α 1 , α 2 > 0 and c ≥ 0 such that, for every λ > 0 and every t ≥ 0,

- d dt E(x) ≥ α 1 -εc -ε c 2λ Π R ∇H(x) 2 + ε α 2 - cλ 2 Π R J∇H(x) 2 .
By choosing first λ > 0 small enough and then ε > 0 small enough, and by using hypothesis (H2), we thus obtain (18). The inequality (18) implies that E(x) is nonincreasing. Since x has relatively compact range, lim t→∞ E(x(t)) exists. As a consequence, the function ∇H(x) 2 is integrable on R + . Since this function is also uniformly continuous, we obtain lim t→∞ ∇H(x) = 0. Now, if ϕ ∈ ω(x), then there exists (t n ) ր ∞ such that x(t n ) → ϕ. By continuity of ∇H, we obtain ∇H(ϕ) = lim n→∞ ∇H(x(t n )) = 0. Since ϕ ∈ ω(x) was arbitrary, this means that (1) is gradient-like.

Lemma 7. Let H ∈ C 2 (M).

If H satisfies the Łojasiewicz-Simon inequality near some ϕ ∈ M and for some Łojasiewicz exponent θ ∈ (0, 1 2 ], then, for every ε > 0 small enough, the function E given by (8) satisfies the Łojasiewicz-Simon inequality near ϕ with the same Łojasiewicz exponent θ.

Proof. If ∇H(ϕ) 0 then, in view of the definition of E, ∇E(ϕ) 0 for sufficiently small ε > 0. The Łojasiewicz-Simon inequality is satisfied in this case.

So assume that ∇H(ϕ) = 0. Then also ∇E(ϕ) = 0, as one easily verifies. By assumption there exists θ ∈ (0, 1 2 ], a neighbourhood U of ϕ, and C > 0 such that H(x) -H(ϕ)

1-θ ≤ C ∇H(x) for every x ∈ U. By continuity of ∇H and the condition ∇H(ϕ) = 0 one can choose the neighbourhood U such that ∇H(x) ≤ 1 for every x ∈ U. Then, in view of ( 8), ( 9), and the condition 2(1θ) ≥ 1, there exist C 1 , C 2 > 0 such that for every x ∈ U one has

|E(x) -E(ϕ)| 1-θ ≤ C 1 |H(x) -H(ϕ)| 1-θ + ∇H(x) 2(1-θ) (19) ≤ C 1 C ∇H(x) + ∇H(x) 2(1-θ) ≤ C 2 ∇H(x) .
Next, for some C 3 > 0, sufficiently small ε > 0, and every x ∈ U the following inequality holds:

(20) ∇E(x) ≥ 1 2 ∇H(x) - 1 2 C 3 ε ∇H(x) ≥ 1 3 ∇H(x) .
The inequalities ( 19) and (20) give the Łojasiewicz-Simon inequality for E near ϕ.

Proof of Theorem 3. Let the function E be defined as in [START_REF] Jendoubi | Nonstabilizing solutions of semilinear hyperbolic and elliptic equations with damping[END_REF] with F as given by [START_REF] Kaufman | Dissipative Hamiltonian systems: a unifying principle[END_REF]. By Lemma 6, for every ε > 0 small enough there exists a constant α > 0 such that, for all t ≥ 0,

- d dt E(x) ≥ α ∇H(x) 2 .
On the other hand, from the differential equation ( 1) and the definition of E one easily obtains the estimate

ẋ + E(x) ≤ α ′ ∇H(x)
for some α ′ > 0 and every t ≥ 0. The preceding two estimates imply that conditions (3), ( 4) and ( 5) of Theorem 1 are satisfied. Moreover, by assumption, there exists ϕ ∈ ω(x) such that H satisfies the Łojasiewicz-Simon gradient inequality near ϕ. By Lemma 7, and if ε > 0 was chosen small enough (which we may always assume), we find that E satisfies the Łojasiewicz-Simon gradient inequality near ϕ, too. The claim therefore follows from Theorem 1.

Examples

4.1. Classical and relativistic mechanics. Let (N, g) be a Riemannian manifold (the state space), and let M := TN be the tangent bundle (the phase space), equipped with the natural (Sasaki) metric [START_REF] Gallot | Riemannian Geometry[END_REF] and the natural symplectic structure [2, Chapter 1, 3.3] (the natural symplectic structure on the cotangent bundle carries over to a symplectic structure on the tangent bundle by identification via the natural metric). Elements in M are written in the form of an ordered pair x = (p, q) with q ∈ N and p = p q ∈ T q N. Consider the Hamiltonian H ∈ C 2 (M) given by

H(x) = H(p, q) := T(|p|) + V(q), x = (p, q) ∈ M,
where V ∈ C 2 (N) is a potential energy, T is the kinetic energy, and |p| = |p| T q N denotes the norm of the generalized momentum p in T q N with respect to the Riemannian metric g. Usually, T(|p|) = 1 2 |p| 2 in classical mechanics, and

T(|p|) = |p| 2 + 1 in relativistic mechanics.
Let π : M → N, (p, q) → q be the canonical projection which is a Riemannian submersion. We have TM = ker π ′ ⊕ (ker π ′ ) ⊥ , where ker π ′ is the vertical bundle and (ker π ′ ) ⊥ is the horizontal bundle. Let J be the smooth section of skew-adjoint mappings on the tangent bundle TM giving the natural symplectic structure. Then J maps the vertical bundle into the horizontal bundle and vice versa. We assume that S is a smooth section of selfadjoint mappings on the tangent bundle TM such that range S equals the vertical bundle ker π ′ . Then Π R is the orthogonal projection onto the vertical bundle along the horizontal bundle. It is now an exercise to show that hypotheses (H1)-(H3) are satisfied in this case, and it is another exercise to show that hypothesis (H4) is satisfied as soon as the kinetic energy is a strictly convex function of |p|. Note that the two examples of kinetic energies belong to this case.

Hence, as a corollary to Theorem 3 we obtain Corollary 8. Let N, M, H, J and S be as above and assume that T is a strictly convex function of |p|. If N and H are real analytic, then every global solution x of the damped Hamiltonian system (1) having relatively compact range has finite length.

Proof. If N and H are real analytic then H satisfies the Łojasiewicz-Simon gradient inequality near every point x ∈ M by Łojasiewicz' classical result in [START_REF] Łojasiewicz | Colloques internationaux du C.N.R.S.: Les équations aux dérivées partielles[END_REF], [START_REF] Łojasiewicz | Ensembles semi-analytiques[END_REF]. The claim is therefore a direct consequence of Theorem 3 and the exercises above.

4.2.

A Hamiltonian system coupled with a gradient system. More generally, let (N 1 , g 1 ) and (N 2 , g 2 ) be two Riemannian manifolds (the state spaces) and let M = TN 1 × N 2 be the product manifold of the tangent bundle TN 1 and of N 2 . Elements of M are denoted in the form x = (p 1 , q 1 , q 2 ) with (p 1 , q 1 ) ∈ TN 1 and q 2 ∈ N 2 . On TN 1 we consider the situation as in the previous example, with the natural symplectic structure given by J 1 , a damping S 1 satisfying range S 1 = ker π 1 , and a kinetic energy T = T(|p 1 |). We assume that the Hamiltonian H is given by

H(x) = H(p 1 , q 1 , q 2 ) = T(|p 1 |) + V(q 1 , q 2 ), x = (p 1 , q 1 , q 2 ) ∈ M,
that J = J 1 0 0 0 and S = S 1 0 0 I . The resulting differential equation [START_REF] Absil | Convergence of the iterates of descent methods for analytic cost functions[END_REF] for these choices of H, J and S corresponds to a coupled system consisting of a Hamiltonian system (on N 1 ) and a gradient system (on N 2 ); the coupling takes place through the potential energy V.

4.3.

Damped spherical pendulum in electric field in R n+1 . To illustrate the first example, we consider the damped spherical pendulum which can serve as a model of a spherical pendulum with an electric charge Q in an electric field and in a fluid (for example, in a liquid), possibly inhomogeneous and anisotropic (which can be caused by the presence of the electric field).

The phase space of this system is the tangent bundle M = TS n r to the ndimensional sphere S n r ⊂ R n+1 centred at the origin, with radius r. The space M is equipped with the usual symplectic structure.

Every point q ∈ S n r is regarded as an element of R n+1 and the tangent space T q S n r is regarded as a linear subspace of R n+1 . Denote by •, • n+1 and | • | n+1 the standard inner product and norm in R n+1 . Let Φ : R n+1 → R be a scalar potential of the electric field (that is, the electric field is equal to -gradΦ). Assume that Φ is analytic. The Hamiltonian H : M → R of the spherical pendulum in this case is given, for every x = (p, q) ∈ M, q ∈ S n r , p ∈ T q S n r , by the formula

H(x) = H(p, q) = 1 2m
|p| 2 n+1 + QΦ(q) + mg( q, e n+1 + r), where e = (0, . . . , 0, 1) ∈ R n+1 , and m and g are positive constants which represent, for n = 3, the mass of the pendulum and the acceleration of gravity, respectively.

The damping S for this system satisfying conditions (H1)-(H4) and representing the friction caused by the fluid (for example, for n = 3) can be defined as follows. Let s : R n+1 × R n+1 → (0, +∞) be a continuous function such that for given x = (p, q), q ∈ R n+1 , p ∈ R n+1 , the vector -s(x)p ∈ R n+1 is the force of the fluid friction acting on given body moving in R n+1 at the point q with the momentum p. Then, in the case of damped spherical pendulum, one can assume S(x)(v, w) = (s(x)v, 0) for every x = (p, q), q ∈ S n r , p ∈ T q S n r , v, w ∈ T q S n r . More generally, one can consider the damping given by S(x)(v, w) = (S 1 (x)v, 0), where S 1 (x) is a selfadjoint positive operator on T q S n r and the mapping x → S 1 (x) is continuous. Theorem 10. Under the above assumptions, every solution x of (1) is global, has finite length and converges to a stationary point ϕ, that is, lim t→∞ x(t) = ϕ.

Proof. By compactness of the sphere S n r , the Hamiltonian H in this example is bounded from below. In particular, the kinetic energy is bounded along each solution, which in turn implies that every solution x stays in a compact subset of M. As a consequence, every solution is global and has relatively compact range in M. Since the electric potential is analytic, the Hamiltonian H is analytic and satisfies the Łojasiewicz-Simon gradient inequality near every every point ϕ ∈ M. The claim follows from Theorem 3.

Note that the set of stationary points in this example is always non-empty since every limit point of every solution is a stationary point. It can be non-discrete and one can shape it by changing the electric field.

The decay estimate from Theorem 3 holds in this case for appropriate θ depending on the electric field. 4.4. Damped Hénon-Heiles system. The Hénon-Heiles Hamiltonian system [START_REF] Hénon | The applicability of the third integral of motion: Some numerical experiments[END_REF], being a model of the movement of a star within a galaxy, is described by the Hamiltonian H : R 4 → R given by (21) H(p 1 , p 2 , q 1 , q 2 ) = 1 2 (p 2 1 + p 2 2 + q 2 1 + q 2 2 ) + q 2 1 q 2 -1 3 q 3 2 . In the case of the damped system (1) and with S of the form (22) S = S 1 0 0 0 , for S 1 being a strictly positive (2 × 2)-matrix, the damping can be caused by the matter in the galaxy. We shall prove that every solution of (1) starting from the sublevel {H < 1 6 } (for example, from the point with the norm less than 1 2 ) converges to the origin with an exponential decay rate. We have (∇H) -1 ({0}) = {x 0 , x 1 , x 2 , x 3 } ,

Date:

  February 5, 2016. 2000 Mathematics Subject Classification. 35P05, 35J70, 35K65. † Partially supported by a research scholarship of the French Government realized from September 1 to October 31, 2006, at the Université Paul Verlaine -Metz, Laboratoire de Mathématiques et Applications de Metz, France.

  It is again an exercise to show that hypotheses (H1)-(H4) are satisfied if in addition T is a convex function of |p|. Using again Łojasiewicz' classical result, we obtain the following corollary to Theorem 3. Let N 1 , N 2 , M, H, J and S be as above and assume that T is a strictly convex function of |p 1 |. If N 1 , N 2 and H are real analytic, then every global solution x of the damped coupled system (1) having relatively compact range has finite length.

	Corollary 9.

Acknowledgements

Wiktor Radzki wishes to express his gratitude to the French Government for the research scholarship, and to the Université Paul Verlaine -Metz, Laboratoire de Mathématiques et Applications de Metz, France, for its hospitality from September 1 to October 31, 2006.

where x i = (0, 0, z i ) for every i ∈ {0, . . . , 3} and z 0 = (0, 0), z 1 = (0, 1),

The Hamiltonian H attains a local minimum at x 0 , whereas x 1 , x 2 , x 3 are saddle points of H. Let V : R 2 → R by defined by

We shall show that the connected component of {H < 1 6 } containing the origin is bounded. To this end it suffices to prove that the connected component of {V < 1 6 } containing the origin is bounded. Observe that

Thus the set {V = 1 6 } is a union of three straight lines:

which connect the critical points z 1 , z 2 , z 3 of V. Since V(z 0 ) = 0, we have z 0 ∈ {V < 1 6 }. The connected component of {V < 1 6 } containing z 0 is the interior of the triangle ∆(z 1 , z 2 , z 3 ), which is bounded. (This triangle ∆(z 1 , z 2 , z 3 ) is the maximal bounded sublevel component of V containing the origin, since the connected component of {V ≤ 1 6 } containing the origin is not bounded.) Consequently, the connected component of {H < 1 6 } containing the origin is bounded.

Furthermore, the open ball

Now we shall show that H satisfies the Łojasiewicz-Simon inequality near the origin with exponent θ = 1 2 . Indeed, since the mapping G : R 4 → R, defined by

has a local minimum (equal to 0) at the origin, for every x = (p 1 , p 2 , q 1 , q 2 ) with sufficiently small norm we have

Taking into account the above conclusions, we obtain from Theorem 3 the following result for the damped Hénon-Heiles system. Theorem 11. Let H : R 4 → R be given by (21). Assume that S is of the form (22) for S 1 being a strictly positive (2 × 2)-matrix. Then every solution x of (1) such that H(x(0)) < 1 6 (for example, x such that x(0) < 1 2 ) converges to the origin and x(t) = O(e -ct ) as t → ∞.

The Hénon-Heiles Hamiltonian, written in this example in its original form, has a discrete set of stationary points, and therefore the convergence of a trajectory is guaranteed by the LaSalle invariance principle and the decay estimate can be obtained by Lyapunov's linearized stability. However, one can easily make the set of stationary points non-discrete in a neighbourhood of the origin by adding a perturbation potential depending on the generalized coordinates q and representing the change of the gravitational potential caused by an inhomogeneous mass distribution in the galaxy. Then the conclusion that every solution x with H(x(0)) < 1 6 converges remains valid if the perturbing potential is analytic and sufficiently small, in view of the results of the present paper. The decay estimate from Theorem 3 holds for θ depending on the perturbation.