Asymptotic Stability of Dissipated Hamilton-Poisson Systems
aa r X i v : . [ m a t h . D S ] J u l ASYMPTOTIC STABILITY OF DISSIPATED HAMILTON-POISSONSYSTEMS
Petre Birtea and Dan Com˘anescu
Abstract
We will further develop the study of the dissipation for a Hamilton-Poisson system introducedin [3]. We will give a tensorial form of this dissipation and show that it preserves the Hamiltonianfunction but not the Poisson geometry of the initial Hamilton-Poisson system. We will give pre-cise results about asymptotic stabilizability of the stable equilibria of the initial Hamilton-Poissonsystem.
MSC : 37C10, 37C75, 70E50.
Keywords : dynamical systems, stability theory, rigid body.
Let ˙ x = Π ▽ H ( x )be a Hamilton-Poisson system. If C ∈ C ∞ ( R n , R ) is a Casimir function then we have the followingobvious equality Π ▽ C = 0.We will add to the Hamilton-Poisson system that we will denote by ξ Π a dissipation term of the form G ▽ C , where G is a certain symmetric matrix that will be discussed below and C is a Casimir function.The dissipated system ξ will be ˙ x = Π ▽ H ( x ) + G ▽ C ( x ) (1.1)The notion of dissipative bracket was introduced by A. Kaufman [10] in his study of dissipativeHamilton-Poisson systems. In [15], P.J. Morrison introduced the notion of metriplectic systems whichare Hamilton-Poisson systems with a dissipation of metric type. Metric type dissipation was alsointroduced in [15] and is given by a dissipative bracket constructed from a metric defined on the phasespace. Dissipative terms and their implications for dynamics have been studied in connection withvarious dynamical systems derived from mathematical physics, see [8], [9], [11],[12], [13], [14].The dissipation that we will study is the one introduced in [3]. Here we will write it in a tensorialform which enables us to better understand its geometry. It is a particular form of the metric dissi-pation introduced in [15]. More precisely, our system (1.1) is the same system with the one describedby equations (25) in [15] under the condition ( g ij ) ▽ H = 0. Like the dissipation from [7], our dissipa-tion conserves the energy. A dissipation form that conserves the symplectic leaves and dissipates theHamiltonian function have been studied in [5].We will also illustrate the improvement of the stability result obtained in [3]. This improvement isobtained by the observation that the dissipation conserves the energy. G = ▽ H ⊗ ▽ H − || ▽ H || I In this section we introduce the dissipation matrix G = ▽ H ⊗ ▽ H − || ▽ H || I . This is the tensorial form of the dissipation constructed in [3] where the matrix I is the identitymatrix on R n . We denote by ξ the dissipated system (1.1) in the case when the matrix G is as above.1 emma 2.1. We have the following properties for G :(i) G ▽ H = 0 ;(ii) ▽ C · G ▽ C ≤ , for any Casimir function C .The equality holds in a point x ∈ R n iff ▽ C ( x ) and ▽ H ( x ) are linear dependent.Proof. For (i) we have G ▽ H = ( ▽ H ⊗ ▽ H ) ▽ H − || ▽ H || I ▽ H = ( ▽ H · ▽ H ) ▽ H − || ▽ H || ▽ H = 0Analogous for (ii) we have ▽ C · G ▽ C = ▽ C · [( ▽ H ⊗ ▽ H ) ▽ C − || ▽ H || I ▽ C ]= ▽ C · [( ▽ C · ▽ H ) ▽ H − || ▽ H || ▽ C ]= ( ▽ C · ▽ H ) − || ▽ C || · || ▽ H || ≤ . by C-B-S inequality. Equality holds iff ▽ H and ▽ C are linear dependent.For the perturbed system ξ the initial Hamiltonian remains a conservation low but the Casimirfunction C does not. Lemma 2.2.
We have the following behavior:(i) H is conserved along the solutions of ξ .(ii) C decreases along the solutions of ξ .Proof. For (i) we have ddt H = ˙ x · ▽ H = (Π ▽ H + G ▽ C ) · ▽ H = Π ▽ H · ▽ H + [( ▽ H ⊗ ▽ H ) ▽ C − || ▽ H || ▽ C ] · ▽ H = − ▽ H · Π ▽ H = 0since Π is an antisymmetric matrix.We do the same type of computation for (ii) ddt C = ˙ x · ▽ C = (Π ▽ H + G ▽ C ) · ▽ C = − ▽ H · Π ▽ C + ▽ C · G ▽ C = ▽ C · G ▽ C ≤ E ξ Π for the unperturbed system ξ Π and the set of equilibria E ξ for the perturbed system ξ . Proposition 2.3.
We have:(i) If ▽ C ( x e ) = 0 for a point x e ∈ R n then x e ∈ E ξ ⇔ ▽ H ( x e ) and ▽ C ( x e ) are linear dependent.(ii) E ξ ⊂ E ξ Π . roof. By definition we have that x e ∈ E ξ Π ⇔ Π ▽ H ( x e ) = 0and x e ∈ E ξ ⇔ Π ▽ H ( x e ) + G ▽ C ( x e ) = 0(i) Let x e ∈ E ξ . If we multiply both sides of the above equality with ▽ C ( x e ) then ▽ C ( x e ) · Π ▽ H ( x e ) + ▽ C ( x e ) · G ▽ C ( x e ) = 0 ⇔ − Π ▽ C ( x e ) · ▽ H ( x e ) + ▽ C ( x e ) · G ▽ C ( x e ) = 0 ⇔ ▽ C ( x e ) · G ▽ C ( x e ) = 0which by Lemma 2.1 (ii) implies that ▽ H ( x e ) and ▽ C ( x e ) are linear dependent.For the converse, if ▽ H ( x e ) and ▽ C ( x e ) are linear dependent then there exists λ ∈ R such that ▽ H ( x e ) = λ ▽ C ( x e ). Consequently, Π ▽ H ( x e ) = λ Π ▽ C ( x e ) = 0 . In the case ▽ H ( x e ) = 0 we obtain that G ( x e ) = 0.In the case ▽ H ( x e ) = 0 we obtain that ▽ C ( x e ) = λ ▽ H ( x e ) and using Lemma 2.1 (i) we have G ▽ C ( x e ) = λ G ▽ H ( x e ) = 0. In both cases we conclude that x e ∈ E ξ .(ii) Let x e ∈ E ξ . If ▽ C ( x e ) = 0 then G ▽ C ( x e ) = 0 and we obtain that Π ▽ H ( x e ) = 0 which implies x e ∈ E ξ Π .If ▽ C ( x e ) = 0 then there exists λ ∈ R such that ▽ H ( x e ) = λ ▽ C ( x e ). We observe that Π ▽ H ( x e ) = λ Π ▽ C ( x e ) = 0 and consequently, x e ∈ E ξ Π . We will briefly recall some definitions of stability for a dynamical system on R n that will be used later˙ x = f ( x ) , (3.1)where f ∈ C ∞ ( R n , R n ). Definition 3.1.
An equilibrium point x e is stable if for any small neighborhood U of x e there is aneighborhood V of x e , V ⊂ U such that if initially x is in V , then φ ( t, x ) ∈ U for all t > . If inaddition we have lim t →∞ φ ( t, x ) = x e then x e is called asymptotically stable. For studying more complicated asymptotic behavior we need to introduce the notion of ω -limit set.Let φ t be the flow defined by equation (3.1). The ω -limit set of x is ω ( x ) := { y ∈ R n |∃ t , t ... → ∞ s.t. φ ( t k , x ) → y as k → ∞} . The ω -limit sets have the following properties that we will use later. For moredetails, see [16].(i) If φ ( t, x ) = y for some t ∈ R , then ω ( x ) = ω ( y ).(ii) ω ( x ) is a closed subset and both positively and negatively invariant (contains complete orbits).The above properties of ω -limit sets have been used in the proof of the following version of LaSalletheorem, see [3]. 3 heorem 3.2. Let x e be an equilibrium point of a dynamical system ˙ x = f ( x ) and U a small compact neighborhood of x e . Suppose there exists L : U → R a C differentiable functionwith L ( x ) > for all x ∈ U \ { x e } , L ( x e ) = 0 and ˙ L ( x ) ≤ . Let E be the set of all points in U where ˙ L ( x ) = 0 . Let M be the largest invariant set in E . Then there exists a small neighborhood V of x e with V ⊂ U such that ω ( x ) ⊂ M for all x ∈ V . The next result describes the asymptotic behavior for the solutions of the dissipated system ξ . Weintroduce the following set C ∗ = { x ∈ R n | ▽ C ( x ) = 0 } . Theorem 3.3.
Let x e ∈ E ξ be an equilibrium point for the dissipated system ξ . Suppose there existsa function ψ ( H, C ) ∈ C ∞ ( R , R ) such that ∂ψ∂C ( H ( x e ) , C ( x e )) > and x e is a strict relative minimumfor L ( x ) = ψ ( H ( x ) , C ( x )) − ψ ( H ( x e ) , C ( x e )) . Then there exists a small compact neighborhood U of x e and another neighborhood V of x e with V ⊂ U such that every solution of ξ starting in V approachesthe largest invariant set M in U T ( E ξ S C ∗ ) as t → ∞ ( M is an attracting set).Proof. We have L ( x e ) = 0 with x e being a strict local minimum for L . Then there exists a smallcompact neighborhood U of x e such that L ( x ) > ∂ψ∂C ( H ( x ) , C ( x )) > L ( x ) = ∂ψ∂H ▽ H · ˙ x + ∂ψ∂C ▽ C · ˙ x = ( ∂ψ∂H ▽ H + ∂ψ∂C ▽ C )(Π ▽ H + G ▽ C )= ∂ψ∂H ▽ H · Π ▽ H + ∂ψ∂H ▽ H · G ▽ C + ∂ψ∂C ▽ C · Π ▽ H + ∂ψ∂C ▽ C · G ▽ C = ∂ψ∂H ▽ H · Π ▽ H + ∂ψ∂H G ▽ H · ▽ C − ∂ψ∂C Π ▽ C · ▽ H + ∂ψ∂C ▽ C · G ▽ C Using Lemma 2.1 (i), antisymmetry of Π and the fact that C is a Casimir function for Π we obtainthat ˙ L ( x ) = ∂ψ∂C ▽ C · G ▽ C From the hypothesis ∂ψ∂C > U and by Lemma 2.1 (ii) we have˙ L ( x ) ≤ . Using again Lemma 2.1 (ii) we obtain that E from the Theorem 3.2 equals E ξ ∪ C ∗ which give us theresult. Remark 3.4.
Observe that L is also a Lyapunov function for the unperturbed system ξ Π . Consequently,by adding the dissipation we render the stable points of ξ Π into asymptotically stable points for ξ . Alsothe Casimir function C can be considered as a Lyapunov function. Corollary 3.5.
Let x e ∈ E ξ and a function ψ ( H, C ) ∈ C ∞ ( R , R ) such that ∂ψ∂C ( H ( x e ) , C ( x e )) > .Suppose that the function L ( x ) = ψ ( H ( x ) , C ( x )) − ψ ( H ( x e ) , C ( x e )) has the properties:i) δL ( x e ) = 0 ;ii) δ L ( x e ) is positive definite. Then there exists a small compact neighborhood U of x e and another neighborhood V of x e with V ⊂ U such that every solution of ξ starting in V approaches the largest invariant set M in U T ( E ξ S C ∗ ) as t → ∞ . Moreover H remains constant along these solutions and C decreases along these solutions.Proof. It is easy to observe that the point x e is a strict relative minimum of L . All the condition of theTheorem 3.3 are satisfied and we obtain the desired result. Remark 3.6.
If we consider, in the Corolarry 3.5, the function ψ ( H, C ) = H + C we obtain the stabilityresult of [3]. 4 orollary 3.7. In the hypotheses of the Theorem 3.3 we consider a point x ∈ V . Suppose that the set H − ( { h } ) T U T ( E ξ S C ∗ ) has a unique point x h ( h = H ( x ) ). The solution x ( t, x ) of the system ξ which verifies the initial condition x (0 , x ) = x satisfies the property lim t →∞ x ( t, x ) = x h . We can interpret the above result as follows, using the Lyapunov function of Theorem 3.3 we obtainthat every equilibrium point in the neighborhood V is asymptotically stable for the dynamical systemon the corresponding level set. The motion of a rigid body can be reduced to the translation of center of mass and rotation about it.Rotation is conveniently described, in a coordinate system with the origin at the center of mass and theaxes along principal central axes of inertia, by Euler’s equations. This equations can be written in thefollowing form ˙ x = ( I − I ) x x + u ˙ x = ( I − I ) x x + u ˙ x = ( I − I ) x x + u where x = I ω , x = I ω , x = I ω are the components of x , I , I , I are the principal momentsof inertia, ω , ω , ω are the components of the angular velocity and u , u , u are the components ofapplied torques u . In this paper we suppose that I > I > I .The system of free rotations, denoted by ξ Π , has the property that u = . It has the followingHamilton-Poisson realization (( so (3)) ∗ ≈ R , {· , ·} − , H )where {· , ·} − is minus-Lie-Poisson structure on ( so (3)) ∗ ≈ R generated by the matrixΠ − = − x x x − x − x x and the Hamiltonian H is given by H ( x , x , x ) = 12 ( x I + x I + x I ) . It is easy to see that the function C ∈ C ∞ ( R , R ) given by C ( x , x , x ) = 12 ( x + x + x )is a Casimir of our configuration (( so (3)) ∗ ≈ R , {· , ·} − ), called the standard Casimir , i.e. { C , f } = 0for each f ∈ C ∞ ( R , R ) where { f, g } = ( ▽ f ) T Π − ▽ g. The set of the Casimir functions is given by { C = ϕ ( C ) | ϕ ∈ C ∞ ( R , R ) } . It is known that the set of equilibria is given by E ξ Π = { ( M , , | M ∈ R } [ { (0 , M , | M ∈ R } [ { (0 , , M ) | M ∈ R } M , ,
0) or (0 , , M ) are stable and the equilibrium points ofthe form (0 , M ,
0) with M = 0 are not stable.In this paper we consider a torque of the form u = G ▽ C, with C a Casimir function and G the matrix G = ▽ H ⊗ ▽ H − || ▽ H || I = − x I − x I x x I I x x I I x x I I − x I − x I x x I I x x I I x x I I − x I − x I . For the case of the free rigid body the above matrix is the one also used in [15].
Case I.
We first consider the standard Casimir C in order to construct the perturbation. We obtaina torque u with the components: u = x [( I − I ) x I + ( I − I ) x I ] u = x [( I − I ) x I + ( I − I ) x I ] u = x [( I − I ) x I + ( I − I ) x I ]Let ξ be the rotation system with the torque u . It is easy to see that the set of the equilibriumpoints of the dissipated system is E ξ = E ξ Π . Also we have C ∗ = { (0 , , } . We consider M ∈ R ∗ . The point x e = (0 , , M ) is an equilibrium point of the rotation system ξ .We define the function ψ : R → R by ψ ( H, C ) = ( C − M + C − I H. This function has the properties: ∂ψ∂C ( H, C ) = 2( C − M ∂ψ∂C ( H ( x e ) , C ( x e )) = 1 > . Using our notations we introduce the Lyapunov function L ( x , x , x ) = ( C ( x , x , x ) − M + C ( x , x , x ) − I H ( x , x , x )= ( 12 ( x + x + x ) − M + 12 ( x + x + x ) − I x I + x I + x I ) . It is easy to see that δL ( x e ) = 0and δ L ( x e ) = − I I − I I
00 0 2 M . Using our hypotheses we observe that δ L ( x e ) is positive definite. Consequently, the hypotheses of theCorollary 3.5 are satisfied and we have the following stability result.6 heorem 4.1. There exists a small compact neighborhood U of x e and an other V ⊂ U such that everysolution of ξ starting in V approaches U ∩ { (0 , , x ) | x ∈ R } as t → ∞ . Moreover H remains constantalong solutions and C decreases along this solutions. If U is sufficiently small then U ∩ E ξ ⊂ { (0 , , x ) | sgn ( x ) = constant } . Suppose that sgn ( x ) = 1.Using the Corollary 3.7 we have the following asymptotic stability result. Theorem 4.2. If x ∈ V then the solution x ( t, x ) of the system ξ which verifies the initial condition x (0 , x ) = x satisfies the property lim t →∞ x ( t, x ) = (0 , , p I H ( x )) . We present the simulation of the rotation of the rigid body in the case when the principal momentsof inertia are I = 4, I = 1 . I = 1 and the initial conditions are ( − . , . , . -0.05 x2(t)0 0.05 0.1 0.15-0.08-0.040 x1(t)0.04 Figure 1: The rotation of the rigid body
Remark 4.3.
We see that the function C = C + ( C − M ) is also a Casimir function. If weconsider ψ ( H, C ) = C − I H we have ∂ψ ∂C = 1 > ψ ( H, C ) = ψ ( H, C ). It is possible to applyCorollary 3.5 for the system ξ of the form (1.1) with the Casimir function C and the equilibrium point x e = (0 , , M ). Case II.
We study the case of the Casimir C = − C .We obtain a torque u with the components: u = − x [( I − I ) x I + ( I − I ) x I ] u = − x [( I − I ) x I + ( I − I ) x I ] u = − x [( I − I ) x I + ( I − I ) x I ]Let ξ the rotation system with the torque u . It is easy to see that the set of the equilibrium pointsof the dissipated system is E ξ = E ξ Π . Also we have C ∗ = { (0 , , } .
7e consider the equilibrium point x e = ( M , ,
0) with M ∈ R ∗ .If we define ψ ( H, C ) := H + ( C + M + C I , we have that ∂ψ ∂C ( H, C ) = 2( C + M I and ∂ψ ∂C ( H ( x e ) , C ( x e )) = 1 I > . In this situation we can apply Corollary 3.5 and we obtain the stability result.
Theorem 4.4.
There exists a small compact neighborhood U of x e and an other V ⊂ U such that everysolution of ξ starting in V approaches U ∩ { ( x, , | x ∈ R } as t → ∞ . Moreover H remains constantalong solutions and C = − C decreases along this solutions. If U is sufficiently small then U ∩ E ξ ⊂ { ( x, , | sgn ( x ) = constant } . Suppose that sgn ( x ) = 1.Using the Corollary 3.7 we obtain the asymptotic stability result. Theorem 4.5. If x ∈ V then the solution x ( t, x ) of the system ξ which verifies the initial condition x (0 , x ) = x satisfies the property lim t →∞ x ( t, x ) = ( p I H ( x ) , , . Case III.
In [3] were introduced the functions C = ( C − M − C I = ( C + M + C I and L ( x ) = H ( x ) + C ( x ) − H ( x e ) − C ( x e ) . It is easy to see that:i) δL ( x e ) = 0;ii) δ L ( x e ) is positive definite because the Hessian matrix of L in x e is M I − I
00 0 I − I . We denote by ξ the system of the form (1.1) with the Casimir C . In [3] has been proved thefollowing result. Theorem 4.6.
There exists a small compact neighborhood U of x e and an other V ⊂ U such that everysolution of ξ starting in V approaches U ∩ { ( x, , | x ∈ R } as t → ∞ . Our results applies to equilibria that are already stable. By adding the dissipation we can makethem asymptotically stable if certain conditions are satisfied. Going from asymptotic stability on thelevel set to stability in the whole space have been studied in [2]. By adding other type of controls onecan stabilize the unstable equilibria. This subject is studied for example in [1], [4], [6], [17].8 eferences [1]
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