Liapunov's direct method for Birkhoffian systems: Applications to electrical networks
aa r X i v : . [ m a t h . D S ] D ec Liapunov’s direct method for Birkhoffian systems:Applications to electrical networks
Delia Ionescu-Kruse,
Institute of Mathematics of the Romanian AcademyP.O. Box 1-764, RO-014700, Bucharest, Romania,[email protected]
Abstract
In this paper, the concepts and the direct theorems of stability inthe sense of Liapunov, within the framework of Birkhoffian dynamicalsystems on manifolds, are considered. The Liapunov-type functions areconstructed for linear and nonlinear LC and RLC electrical networks, toprove stability under certain conditions.
Keywords : Liapunov stability, geometric methods in differential equa-tions, Birkhoffian differential systems, electrical networks.MSC: 34A26, 34D20, 58A20, 94C
During the last years, a far reaching generalization of the Hamiltonian frame-work has been developed in a series of papers. This generalization, which isbased on the geometric notion of generalized Dirac structure (see Courant [5]and Dorfman [6]), gives rise to implicit Hamiltonian systems (see, for example,the papers by Maschke and van der Schaft [12], [14]). Applications to non-holonomic systems and electrical circuits (see Bloch and Crouch [2], Maschkeand van der Schaft [12]) illustrate this theory. Recently, the notion of implicitLagrangian system has been developed by Yoshimura and Marsden [16]. Non-holonomic mechanical systems and degenerate Lagrangian systems such as LCcircuits can be systematically formulated in the implicit Lagrangian context inwhich Dirac structures are also used.An alternative approach to the study of dynamical systems is the Birkhoffianformalism. This is a global formalism of implicit systems of second order ordi-nary differential equations on a manifold. It applies to a wide class of systems,among them, nonholonomic systems, degenerate systems as well as dissipativesystems. Kobayashi and Oliva developed in [9] the framework of Birkhoffiandynamical systems on manifolds, following Birkhoff’s ideas presented locally in[1]. The space of configurations is a smooth m -dimensional differentiable con-nected manifold and the covariant character of the Birkhoff generalized forces isobtained by defining the notion of elementary work, called Birkhoffian, a special1faffian form defined on the 2-jets manifold. The dynamical system associatedto this Pfaffian form is a subset of the 2-jets manifold which defines an implicitsecond order ordinary differential system. The notion of Birkhoffian allows theintroduction of the intrinsic concepts of reciprocity, regularity, affine structurein the accelerations, conservativeness [9], dissipativeness [7].The Birkhoffian formalism in the context of electrical circuits was discussedby Ionescu and Scheurle [8] for the case of LC circuits, and Ionescu [7] forthe case of RLC circuits. An LC/RLC circuit, with no assumptions placed onits topology, will be described by a family of Birkhoffian systems, parameter-ized by a finite number of real constants which correspond to initial values ofcertain state variables of the circuit. It is shown that the Birkhoffian systemassociated to an LC circuit is conservative. Under certain assumptions on thevoltage-current characteristic for resistors, it is shown that a Birkhoffian systemassociated to an RLC circuit is dissipative. For LC/RLC networks which con-tain a number of loops formed only from capacitors, the Birkhoffian associatedis never regular. A procedure to reduce the original configuration space to alower dimensional one, thereby regularizing the Birkhoffian, is presented as well.For RLC electrical networks, Brayton and Moser [3] proved under a specialhypothesis, that there exists a mixed potential function which can be used toput the system of differential equations describing the dynamics of such a net-work, into a special form (see § § Liapunov’s direct method for Birkhoffian sys-tems
In order to present the ideas in a coordinate free fashion, we consider the for-malism of 2-jets. Let M be a m -dimensional differentiable connected manifold.We consider the tangent bundles ( T M , π M , M ) and ( T T M , π T M , T M ).The ( J ( M ), π J , T M ) is defined by J ( M ) := { z ∈ T T M / T π M ( z ) = π T M ( z ) } (2.1)where ( T π M ) v : T v T M → T π M ( v ) M is the tangent map and π J := π T M | J ( M ) = T π M | J ( M ) (2.2)A local system of coordinates ( q ) = ( q j ) j =1 ,...,m on M induces natural local co-ordinates on J ( M ), denoted by ( q, ˙ q, ¨ q ) = ( q j , ˙ q j , ¨ q j ) j =1 ,...,m (see for example[9], [13]).A Birkhoffian corresponding to the configuration manifold M is a smooth1-form ω on J ( M ) such that, for any x ∈ M , we have ι ∗ x ω = 0 (2.3)where ι x : β − ( x ) → J ( M ) is the embedding of the submanifold β − ( x ) into J ( M ), β = π M ◦ π J . From this definition it follows that, in the natural localcoordinate system ( q, ˙ q, ¨ q ) of J ( M ), a Birkhoffian ω is given by ω = m X j =1 Q j ( q, ˙ q, ¨ q ) dq j (2.4)with certain functions Q j : J ( M ) → R . The pair ( M, ω ) is said to be a
Birkhoffian system (see [9]).The differential system associated to a Birkhoffian ω is the set (maybe empty) D ( ω ), given by D ( ω ) := (cid:8) z ∈ J ( M ) | ω ( z ) = 0 (cid:9) (2.5)The manifold M is the space of configurations of D ( ω ), and D ( ω ) is said to have m ’degrees of freedom’. The Q i are the ’generalized external forces’ associatedto the local coordinate system. In the natural local coordinate system, D ( ω ) ischaracterized by the following implicit system of second order ODE’s Q j ( q, ˙ q, ¨ q ) = 0 for all j = 1 , m (2.6)The Birkhoffian formalism is a global formalism for the dynamics of implicitsystems of second order differential equations on a manifold.A cross section X of the affine bundle ( J ( M ) , π J , T M ), that is, a smoothfunction X : T M → J ( M ) such that π J ◦ X =id, can be identified with a specialvector field on T M , namely, the second order vector field Y on T M , that is, a3mooth function Y : T M → T T M such that π T M ◦ Y =id and T π M ◦ Y =id.Using the canonical embedding i : J ( M ) → T T M , we write Y = i ◦ X .In natural local coordinates, a second order vector field can be represented as Y = m X j =1 (cid:20) ˙ q j ∂∂q j + ¨ q j ( q, ˙ q ) ∂∂ ˙ q j (cid:21) (2.7)A Birkhoffian vector field associated to a Birkhoffian ω of M (see [9]) is a smoothsecond order vector field on T M , Y = i ◦ X , with X : T M → J ( M ), such that ImX ⊂ D ( ω ), that is, X ∗ ω = 0 (2.8)In the natural local coordinate system, a Birkhoffian vector field is given by theexpression (2.7), such that Q j ( q, ˙ q, ¨ q ( q, ˙ q )) = 0.A Birkhoffian ω is regular if and only ifdet (cid:20) ∂ Q j ∂ ¨ q i ( q, ˙ q, ¨ q ) (cid:21) j,i =1 ,...,m = 0 (2.9)for all ( q, ˙ q, ¨ q ), and for each ( q, ˙ q ), there exists ( q, ˙ q, ¨ q ) ∈ J ( M ) such that Q j ( q, ˙ q, ¨ q ) = 0 , j = 1 , ..., m. If ω is a regular Birkhoffian corresponding to the configuration manifold M ,then, the principle of determinism is satisfied, that is, there exists an unique Birkhoffian vector field Y = i ◦ X associated to ω such that Im X = D ( ω ) (see[9]).A Birkhoffian ω of M is called conservative (see [9]) if and only if there existsa smooth function E ω : T M → R such that( X ∗ ω ) Y = dE ω ( Y ) (2.10)for all second order vector fields Y = i ◦ X , which is equivalent, in the naturallocal coordinate system, to the identity m X j =1 Q j ( q ˙ q, ¨ q ) ˙ q j = m X j =1 (cid:20) ∂E ω ∂q j ˙ q j + ∂E ω ∂ ˙ q j ¨ q j (cid:21) (2.11)If ω is conservative and Y is a Birkhoffian vector field, then (2.10) becomes dE ω ( Y ) = 0 (2.12)This means that E ω is constant along the trajectories of Y .A Birkhoffian ω of the configuration space M is called dissipative (see [7]) ifand only if there exists a smooth function E ω : T M → R such that( X ∗ ω ) Y = dE ω ( Y ) + D ( Y ) (2.13)for all second order vector fields Y = i ◦ X on T M , D being a dissipative 1-formon T M , that is, D = P mj =1 D j ( q, ˙ q ) dq j and m X j =1 D j ( q, ˙ q ) ˙ q j > m X j =1 Q j ( q ˙ q, ¨ q ) ˙ q j = m X j =1 (cid:20) ∂E ω ∂q j ˙ q j + ∂E ω ∂ ˙ q j ¨ q j + D j ( q, ˙ q ) ˙ q j (cid:21) (2.15)In view of (2.14), we obtain from (2.13),( X ∗ ω ) Y > dE ω ( Y ) (2.16)for all second order vector fields Y = i ◦ X . That is equivalent, in local coordi-nates, to the dissipation inequality m X j =1 Q j ( q ˙ q, ¨ q ) ˙ q j > m X j =1 (cid:20) ∂E ω ∂q j ˙ q j + ∂E ω ∂ ˙ q j ¨ q j (cid:21) (2.17)If ω is a dissipative Birkhoffian and Y is the Birkhoffian vector field, then (2.16)becomes dE ω ( Y ) < E ω is nonincreasing along the trajectories of Y .Let us introduce now the concepts of stability for a Birkhoffian system.The equilibrium points of the system, that is, the points in which the systemcan remain permanently at rest, are to be found as the solutions of the system Q j ( q, ,
0) = 0 , j = 1 , ..., m (2.19)Let us denote an equilibrium point by ( q e , ∈ Ω ⊂ T M , and an initial state ofthe system by ( q , ˙ q ), with q (0) = q , ˙ q (0) = ˙ q .For regular Birkhoffians, we can define the equilibrium points using the notionof Birkhoffian vector field, that is, a point ( q e ,
0) is an equilibrium point of theBirkhoffian vector field Y if and only if Y ( q e ,
0) = 0 (2.20)An equilibrium point ( q e ,
0) is said to be stable (or Liapunov stable) if forevery open neighborhood Ω of ( q e , ⊂ Ω such thata motion ( q ( t ) , ˙ q ( t )) starting at ( q , ˙ q ) ∈ Ω , remains in Ω. If in addition, Ω can be chosen such that, for any ( q , ˙ q ) ∈ Ω , ( q ( t ) , ˙ q ( t )) converges to ( q e ,
0) as t → ∞ , then ( q e ,
0) is said to be asymptotically stable .In the memoir [11], Liapunov presents geometric theorems, generally referredto as the direct method of Liapunov (see, for example, [10]), for deciding thestability or instability of an equilibrium point of a differential equation.In what follows we consider
Liapunov’s direct method for Birkhoffiansystems . This is based on finding a function V ∈ C ( T M, R ) such that( i ) V ( q e ,
0) = 0( ii ) V ( q, ˙ q ) > q, ˙ q ) = ( q e ,
0) in Ω( iii ) dV ( Y ) ≤ Y defined on Ω (2.21)5ith Ω an open neighborhood of ( q e , V is called Liapunovfunction .One can prove the following theorems (completely analogous to the theoremsproved in [10] for a Liapunov function defined on U ⊂ M ): Stability theorem.
If there exists in a neighborhood Ω of ( q e ,
0) a Lia-punov function V ( q, ˙ q ), then ( q e ,
0) is stable.
Asymptotic stability theorem.
If there exists in a neighborhood Ω of( q e ,
0) a Liapunov function V ( q, ˙ q ), such that dV ( Y ) < Y defined on Ω, then ( q e ,
0) is asymptotically stable.From the condition ( ii ) in (2.21) we get that there exists c > { ( q, ˙ q ) ∈ Ω , V ( q, ˙ q ) = c } (2.22)is a closed curve for every constant 0 ≤ c ≤ c . Sketching in the m -plane ( q, ˙ q )these level curves of the function V , we obtain surfaces like ”ellipsoids” centeredat the equilibrium point.If dV = 0, then the equilibrium point ( q e ,
0) is a center and the motion of thesystem is periodic.If dV <
0, then each trajectory keeps moving to lower c and hence penetratessmaller and smaller ”ellipsoids” as t → ∞ . Thus, the equilibrium point is asymp-totically stable. This exclude the existence of periodic motions of the system.
A simple electrical circuit provides us with an oriented connected graph. Thegraph will be assumed to be planar . Let b be the total number of branches inthe graph, n be one less than the number of nodes and m be the cardinality ofa selection of loops that cover the whole graph. By Euler’s polyhedron formula, b = m + n . We choose a reference node and a current direction in each l -branchof the graph, l = 1 , ..., b . We also consider a covering of the graph with m loops, and a current direction in each j -loop, j = 1 , ..., m . We assume thatthe associated graph has at least one loop, meaning that m >
0. An orientedconnected graph can be described by matrices which contain only 0, ±
1, theseare: the incidence matrix B ∈ M bn ( R ), rank( B ) = n , and the loop matrix A ∈ M bm ( R ), rank( A ) = m . For the fundamentals of electrical circuit theory,see, for example, [4].Let us now consider an RLC electrical circuit consisting of r resistors, k inductors and p capacitors, such that to each branch of the associated graphthere corresponds just one electrical device, that is, b = r + k + p . For LCelectrical circuits r = 0. Using the matrices A and B , Kirchhoff’s current lawand Kirchhoff’s voltage law can be expressed by the equations B T i = 0 ( KCL ) , A T v = 0 ( KV L ) (3.1)6here i = ( i [Γ] , i ( a ) , i α ) ∈ R r × R k × R p ≃ R b is the current vector and v =( v [Γ] , v ( a ) , v α ) ∈ R r × R k × R p ≃ R b is the voltage drop vector. Tellegen’stheorem establishes a relation between the matrices A T and B T : the kernel ofthe matrix B T is orthogonal to the kernel of the matrix A T (see, for example,page 5 of [3]).We consider the voltage-current laws for nonlinear devices given by v [Γ] = R Γ ( i [Γ] ) , v ( a ) = L a ( i ( a ) ) d i ( a ) dt , v α = C α ( q α ) , (3.2) R Γ , L a , C α : R −→ R \{ } being smooth functions, q α denote the charges ofthe capacitors, with i α = d q α dt . If the capacitors and the inductors are linearthen the relations above become, respectively, v [Γ] = RΓ i [Γ] , v ( a ) = L a d i ( a ) dt v α = q α C α , (3.3)where RΓ = 0, C α = 0 and L a = 0 are distinct constants.Summing up, the equations governing the network are B T i [Γ] i ( a ) d q α dt = 0 , A T R Γ ( i [Γ] ) L a ( i ( a ) ) d i ( a ) dt C α ( q α ) = 0 . (3.4)Using the first set of equations (3.4), one defines (see [7], [8]) a family of m -dimensional affine-linear configuration spaces M c ⊂ R b , parameterized by aconstant vector c in R n which corresponds to initial values of certain statevariables of the circuit. Since the matrix B is constant, integrating the first setof equations (3.4), one gets B T x = c , with i = ˙ x , c a constant vector in R n .Thus, one defines M c := { x ∈ R b | B T x = c } (3.5)Its dimension is m = b − n , because rank( B ) = n . Local coordinates on M c aredenoted by q = ( q , .., q m ). Solving the system in (3.5), one expresses any ofthe x -variables in terms of q -s, namely, x = N q + K (3.6)where N = N Γ j N aj N αj Γ=1 ,r,a = r +1 ,r + k,α = r + k +1 ,b,j =1 ,m is a matrix of constants and K = K Γ K a K α a constant vector in R b .By Tellegen’s theorem and a fundamental theorem of linear algebra, one obtainsthat Ker ( A T ) = Ker ( N T ) (see [7] , [8]).A Birkhoffian ω c on the configuration space M c arises from a linear combination7f the second set of equations (3.4), by replacing the matrix A T with the matrixof constants N T .I) For a linear LC network ( r = 0) we have the following expression of theBirkhoffian (see [8]) Q j ( q, ˙ q, ¨ q ) = k X a =1 m X i =1 L a N aj N ai ¨ q i + b X α = k +1 m X i =1 N αj N αi C α − k q i + (const) j (3.7)with const ∈ R m a constant vector.A linear LC network is conservative (see [8]). The function E ω : T M c → R satisfying (2.11) has the following expression E ω ( q, ˙ q ) = 12 k X a =1 m X j,i =1 L a N aj N ai ˙ q j ˙ q i + 12 b X α = k +1 m X j,i =1 N αj N αi C α − k q j q i + m X j =1 (const) j q j (3.8)In what follows we assume thatdet " k X a =1 L a N aj N ai j,i =1 ,...,m = 0 , det " b X α = k +1 N αj N αi C α − k j,i =1 ,...,m = 0 (3.9)that is, the network does not contain loops formed only by capacitors and re-spectively, loops formed only by inductors (see [8]). If the network containscapacitor loops and inductor loops, we will reduce first the configuration spaceto a lower dimensional configuration space. On the reduced configuration spacethe corresponding Birkhoffian is still conservative (see [8]) and the correspond-ing determinants (3.9) will be different from zero. The inductor loops can beconsidered as some conserved quantities of the network. Theorem 1.
Let ( q e , be an equilibrium point of a linear LC network withthe Birkhoffian components given by (3.7). Then q e satisfies the system b X α = k +1 m X i =1 N αj N αi C α − k q i + (const) j = 0 , j = 1 , ..., m (3.10) To each const which is related to the initial data for the considered network, weget a unique equilibrium point . If L a > , ∀ a = 1 , ..., k, C α > , ∀ α = 1 , ..., p (3.11) the equilibrium point is a stable center , and the motion of the system is peri-odic. Indeed, the equilibrium points of a linear LC network are obtained as solu-tions of the system Q j ( q, ,
0) = 0, j = 1 , ..., m , where Q j ( q, ˙ q, ¨ q ) is given by83.7). Thus, we get that q e has to fulfill the system (3.10). Under the secondcondition in (3.9), this system has for each const ∈ R m a unique solution.The stability of this equilibrium point is obtained by the Stability Theorempresented in section 2. We define a Liapunov function V ∈ C ( T M c , R ) by V ( q, ˙ q ) = E ω ( q, ˙ q ) − E ω ( q e , k X a =1 m X j,i =1 L a N aj N ai ˙ q j ˙ q i + 12 b X α = k +1 m X j,i =1 N αj N αi C α − k ( q j − q je )( q i − q ie )(3.12)where q e satisfies the system (3.10). Indeed, this function satisfies the condi-tions (2.21). Taking into account (3.11), the matrices (cid:16)P ka =1 L a N aj N ai (cid:17) j,i and (cid:16)P bα = k +1 N αj N αi C α − k (cid:17) j,i are positive definite. Thus, the condition ( ii ) in (2.21) isfulfilled. The first determinant in (3.9) being different from zero implies thatthe corresponding Birkhoffian is regular. Therefore, along the trajectories ofthe unique (principle of determinism) Birkhoffian vector field, the function E ω defined in (3.8), satisfies (2.12). Thus, the function (3.12) satisfies the condition( iii ) in (2.21). In this case, sketching in the m -plane ( q, ˙ q ) the level curves ofthe function (3.12), we obtain ellipsoids centered at the equilibrium point. Theequilibrium point is a center and the motion of the system is periodic. (cid:3) II) For a nonlinear LC network we have the following expression of theBirkhoffian (see [8]) Q j ( q, ˙ q, ¨ q ) = k X a =1 N aj L a m X l =1 N al ˙ q l ! m X i =1 N ai ¨ q i + b X α = k +1 N αj C α − k m X l =1 N αl q l + K α ! = m X i =1 k X a =1 N aj N ai e L a ( ˙ q ) ! ¨ q i + b X α = k +1 N αj e C α − k ( q ) (3.13)A nonlinear LC network is conservative (see [8]). In this case, the function E ω : T M c → R is given by E ω ( q, ˙ q ) = E ( ˙ q ) + E ( q ) (3.14)with E ( ˙ q ) = k X a =1 m X l =1 m X i <...
Let ( q e , be an equilibrium point of a nonlinear LC networkwith the Birkhoffian components given by (3.13). Then q e satisfies the system b X α = k +1 N αj C α − k m X l =1 N αl q l + K α ! = 0 , j = 1 , ..., m (3.17) A nonlinear LC networks can have several equilibrium points . If L a (0) > , ∀ a = 1 , ..., k, C ′ α ( q e ) > , ∀ α = 1 , ..., p (3.18) then the equilibrium points are locally stable centers . Indeed, the equilibrium points of a nonlinear LC network are obtained assolutions of the system Q j ( q, ,
0) = 0, j = 1 , ..., m , where Q j ( q, ˙ q, ¨ q ) is givenby (3.13). Thus, we see that q e has to fulfill the system (3.17).The local stability of the equilibrium points is obtained by the Stability Theorempresented in section 2. We define a Liapunov function V ∈ C ( T M c , R ) by V ( q, ˙ q ) = E ω ( q, ˙ q ) − E ω ( q e ,
0) (3.19)with E ω given by (3.14) and q e satisfying the system (3.17).Let us now evaluate the Hessian matrix of the function V in (3.19) at an equi-librium point ( q e , H V ( q e ,
0) = ∂ E ( ˙ q ) ∂ ˙ q i ∂ ˙ q j | ( q e , ∂ E ( q ) ∂q i ∂q j | ( q e , ! (3.20)For the Birkhoffian (3.13), the function E ω in (3.14) satisfies the identity (2.11)(see [8]), that is, ∂ E ( ˙ q ) ∂ ˙ q i = k X a =1 m X l =1 e L a ( ˙ q ) N ai N al ˙ q l (3.21) ∂ E ( q ) ∂q i = b X α = k +1 e C α − k ( q ) N αi (3.22)10herefore, we get ∂ E ( ˙ q ) ∂ ˙ q i ∂ ˙ q j = k X a =1 " m X l =1 e L ′ a ( ˙ q ) N aj N ai N al ˙ q l + e L a ( ˙ q ) N ai N aj (3.23) ∂ E ( q ) ∂q i ∂q j = b X α = k +1 e C ′ α − k ( q ) N αi N αj (3.24)From (3.23), (3.24), the matrix (3.20) writes as H V ( q e ,
0) = P ka =1 L a (0) N ai N aj P bα = k +1 e C ′ α − k ( q e ) N αi N αj ! (3.25)In view of the conditions (3.18), the matrices (cid:16)P ka =1 L a (0) N ai N aj (cid:17) i,j and (cid:16)P bα = k +1 e C ′ α − k ( q e ) N αi N αj (cid:17) i,j are positive definite. Therefore, the Hessian ma-trix (3.25) is positive definite. The centers of the level curves of the function(3.19) have the coordinates ( q e , q e satisfies the system (3.17). Thus,in a neighborhood of an equilibrium point, the condition ( ii ) in (2.21) is fulfilledby the function V in (3.19). The determinant (3.16) being different from zeroimplies that the corresponding Birkhoffian is regular. Therefore, along the tra-jectories of the unique (principle of determinism) Birkhoffian vector field, thefunction E ω satisfies (2.12). Thus, the function (3.19) satisfies the condition( iii ) in (2.21). By the Stability Theorem, the equilibrium points are locallystable centers. (cid:3) III) For a linear RLC network we have the following expression of theBirkhoffian (see [7]) Q j ( q, ˙ q, ¨ q ) = r + k X a = r +1 m X i =1 L a − r N aj N ai ¨ q i + r X Γ=1 m X i =1 RΓ N Γ j N Γ i ˙ q i + b X α = r + k +1 m X i =1 N αj N αi C α − r − k q i + (const) j (3.26)with const ∈ R m a constant vector.A linear RLC network with RΓ > , Γ = 1 , ..., r, (3.27)is dissipative (see [7]). The function E ω : T M c → R and the dissipative 1-formsatisfying (2.15), are given by E ω ( q, ˙ q ) = 12 r + k X a = r +1 m X j,i =1 L a − r N aj N ai ˙ q j ˙ q i + 12 b X α = r + k +1 m X j,i =1 N αj N αi C α − r − k q j q i + m X j =1 (const) j q j (3.28)11 = m X j,i =1 r X Γ=1 RΓ N Γ j N Γ i ˙ q i dq j (3.29)In what follows we assume thatdet " r + k X a = r +1 L a − r N aj N ai j,i =1 ,...,m = 0 , det " b X α = r + k +1 N αj N αi C α − r − k j,i =1 ,...,m = 0 (3.30)that is, the network does not contain capacitor loops and inductor loops, re-spectively. If the network contains capacitor loops and inductor loops, we willreduce first the configuration space to a lower dimensional configuration space.On the reduced configuration space the corresponding Birkhoffian is still dissi-pative (see [7]) and the corresponding determinants above will be different fromzero. Theorem 3.
Let ( q e , ) be an equilibrium point of a linear RLC networkwith the Birkhoffian given by (3.26). Then q e satisfies the system b X α = r + k +1 m X i =1 N αj N αi C α − r − k q i + (const) j = 0 , j = 1 , ..., m (3.31) To each const which is related to the initial data for the considered network, weget a unique equilibrium point . If L a > , ∀ a = 1 , ..., k, C α > , ∀ α = 1 , ..., p (3.32) the equilibrium point is asymptotically stable .Indeed, the equilibrium points of a linear RLC network are obtained as solutionsof the system Q j ( q, ,
0) = 0 , j = 1 , ..., m , where Q j ( q, ˙ q, ¨ q ) is given by (3.26).Thus, we see that q e has to fulfill the system (3.31). Under the second conditionin (3.30), this system has for each const ∈ R m a unique solution.The asymptotic stability of this equilibrium point is obtained by the AsymptoticStability Theorem presented in section 2. We define a Liapunov function V ∈ C ( T M c , R ) by V ( q, ˙ q ) = E ω ( q, ˙ q ) − E ω ( q e ,
0) = 12 r + k X a = r +1 m X j,i =1 L a − r N aj N ai ˙ q j ˙ q i + 12 b X α = r + k +1 m X j,i =1 N αj N αi C α − r − k ( q j − q je )( q i − q ie ) (3.33)where q e satisfies the system (3.31). Indeed, this function satisfies the conditions(2.21). Taking into account (3.32), the matrices (cid:16)P r + ka = r +1 L a − r N aj N ai (cid:17) j,i and (cid:16)P bα = r + k +1 N αj N αi C α − r − k (cid:17) j,i are positive definite. Thus, the condition ( ii ) in (2.21)is fulfilled. The first determinant in (3.30) being different from zero implies12hat the corresponding Birkhoffian is regular. Therefore, along the trajectoriesof the unique (principle of determinism) Birkhoffian vector field, the function E ω satisfies (2.18). Thus, the function (3.33) also satisfies (2.18). In this case,sketching the level curves of the function (3.33) in the m -plane ( q, ˙ q ), we ob-tain ellipsoids centered at the equilibrium point. From the Asymptotic StabilityTheorem we conclude that the equilibrium point is asymptotically stable. Thisexcludes the existence of periodic motions of the system. (cid:3) IV) For a nonlinear RLC network we have the following expression of theBirkhoffian (see [7]) Q j ( q, ˙ q, ¨ q ) = r + k X a = r +1 N aj L a − r m X l =1 N al ˙ q l ! m X i =1 N ai ¨ q i ! + r X Γ=1 N Γ j R Γ m X l =1 N Γ l ˙ q l ! + b X α = r + k +1 N αj C α − r − k m X l =1 N αl q l + K α ! = m X i =1 r + k X a = r +1 N aj N ai e L a − r ( ˙ q ) ¨ q i + r X Γ=1 N Γ j e R Γ ( ˙ q ) + b X α = r + k +1 N αj e C α − r − k ( q )(3.34)In order to obtain a dissipative Birkhoffian (see [7]), we assume that, for all x = 0, xR Γ ( x ) > , ∀ Γ = 1 , ..., r (3.35)that is, for each nonlinear resistor, the graph of the function R Γ lies in the firstand third quadrants. The function E ω : T M c → R and the dissipative 1-formsatisfying (2.15), are given by E ω ( q, ˙ q ) = E ( ˙ q ) + E ( q ) (3.36)with E ( ˙ q ) = r + k X a = r +1 m X l =1 m X i <...
Let ( q e , ) be an equilibrium point of a nonlinear RLC networkwith the Birkhoffian given by (3.34). Then q e satisfies the system r X Γ=1 N Γ j R Γ (0) + b X α = r + k +1 N αj C α − r − k m X l =1 N αl q l + K α ! = 0 , j = 1 , ..., m (3.40) A nonlinear RLC network can have several equilibrium points .1) If R Γ (0) = 0 , ∀ Γ = 1 , ..., r (3.41) L a (0) > , ∀ a = 1 , ..., k, C ′ α ( q e ) > , ∀ α = 1 , ..., p (3.42) the equilibrium points are locally asymptotically stable . If there exists
Γ = 1 , ..., r such that R Γ (0) = 0 , but, for all x = 0 , x ( R Γ ( x ) − R Γ (0)) > , ∀ Γ = 1 , ..., r (3.43) and the conditions (3.42) are fulfilled, then the equilibrium points are locallyasymptotically stable . Indeed, the equilibrium points of a nonlinear RLC network are obtained assolutions of the system Q j ( q, ,
0) = 0, j = 1 , ..., m , where Q j ( q, ˙ q, ¨ q ) is givenby (3.34). Thus, we see that q e has to fulfill the system (3.40). The localasymptotic stability of the equilibrium points now follows from the AsymptoticStability Theorem presented in section 2.First we assume condition (3.41) to be satisfied. Then the system (3.40) writesas b X α = r + k +1 N αj C α − r − k m X l =1 N αl q l + K α ! = 0 , j = 1 , ..., m (3.44)In order to show 1), we define a Liapunov function V ∈ C ( T M c , R ) by V ( q, ˙ q ) = E ω ( q, ˙ q ) − E ω ( q e ,
0) (3.45)with E ω given by (3.36) and q e satisfying the system (3.44). In the neigh-borhood of any equilibrium point, this function satisfies the conditions (2.21).14aking into account the conditions (3.42), the Hessian matrix of the function V in (3.45), at the equilibrium point ( q e , H V ( q e ,
0) = P r + ka = r +1 L a − r (0) N ai N aj P bα = r + k +1 e C ′ α − r − k ( q e ) N αi N αj ! (3.46)is positive definite. The centers of the level curves of the function (3.45) have thecoordinates ( q e , q e satisfies the system (3.44). Thus, in a neighborhoodof an equilibrium point the condition ( ii ) in (2.21) is fulfilled by the function V in (3.45). The determinant (3.39) being different from zero implies that thecorresponding Birkhoffian is regular. Therefore, along the trajectories of theunique (principle of determinism) Birkhoffian vector field, the function E ω sat-isfies (2.18). Thus, the function (3.45) also satisfies (2.18). By the AsymptoticStability Theorem, the equilibrium points are locally asymptotically stable.We assume now that there exists Γ = 1 , ..., r such that R Γ (0) = 0. In or-der to show 2), we consider instead of the function E ω the following function E ω : T M c → R E ω ( q, ˙ q ) = E ( ˙ q ) + E ( q ) + r X Γ=1 N Γ j R Γ (0) q j (3.47) E ( ˙ q ), E ( q ) being given by (3.37), and instead of D the following dissipative1-form D = m X j =1 r X Γ=1 N Γ j h e R Γ ( ˙ q ) − R Γ (0) i dq j (3.48)In view of assumption (3.43), the vertical 1-form (3.48) is indeed dissipative,that is, m X j =1 r X Γ=1 (cid:0) N Γ j ˙ q j (cid:1) " R Γ m X l =1 N Γ l ˙ q l ! − R Γ (0) > E ω ( q, ˙ q ) given by (3.47) and the dis-sipative 1-form (3.48), the Birkhoffian (3.34) is dissipative, that is, the identity m X j =1 Q j ( q ˙ q, ¨ q ) ˙ q j = m X j =1 " ∂ E ω ∂q j ˙ q j + ∂ E ω ∂ ˙ q j ¨ q j + D j ( q, ˙ q ) ˙ q j (3.50)is fulfilled.We define now a Liapunov function V ∈ C ( T M c , R ) by V ( q, ˙ q ) = E ω ( q, ˙ q ) − E ω ( q e ,
0) (3.51)If ω is a dissipative Birkhoffian and Y is the Birkhoffian vector field, then (2.18)becomes d E ω ( Y ) < V in (3.51) satisfies (3.52) as well.The centers of the level curves of the function (3.51) have the coordinates ( q e , q e satisfies the system (3.40). The Hessian matrix of the function V in (3.51) has at the equilibrium point the same expression (3.46). By theAsymptotic Stability Theorem, the equilibrium points are locally asymptoti-cally stable. (cid:3) We consider an electrical circuit with the associated oriented connected graphas in Figure 1.
II C CRC VVVV
12 321 1 12 4
L L=0V V Figure 1
We have r = 1, k = 2, p = 3, n = 4, m = 2 , b = 6. We choose the referencenode to be V and the current directions as indicated in Figure 1. We cover theassociated graph with the loops I , I . The branches in Figure 1 are labelledas follows: the first branch is the resistive branch r , the second and the thirdbranch are the inductive branches l , l and the last three branches are thecapacitor branches c , c , c . The incidence and loop matrices, B ∈ M ( R )and A ∈ M ( R ), write as B = − − − − − , A = −
11 00 1 (4.1)One has rank( B ) = 4, rank( A ) = 2. Kirchhoff’s current law and Kirchhoff’svoltage law can be expressed by the equations B T i = 0 ( KCL ) , A T v = 0 ( KV L ) (4.2)where i = ( i [Γ] , i ( a ) , i α ) ∈ R × R × R and v = ( v [Γ] , v ( a ) , v α ) ∈ R × R × R is the voltage drop vector.We define the configuration space by M c := { x ∈ R | B T x = c } (4.3)16ith c a constant vector in R . M c is an affine-linear subspace in R , itsdimension is 2. The system in (4.3) writes as x − x = c − x + x + x = c − x + x = c (4.4) x − x − x = c We denote local coordinates on M c by q = ( q , q ). If we take, for example, q := x , q := x (4.5)we get x = q + c x = q + c + c + c + c x = q − c (4.6) x = q − q + c + c + c Thus, the matrix of constants N in (3.6) is exactly the matrix A and the constant K = c c + c + c + c − c c + c + c .First we consider the case that all the electrical devices in the circuit arelinear, they are described by the relations (3.3). In this case, in terms of the q -coordinates (4.5), the Birkhoffian ω c on M c writes as Q ( q, ˙ q, ¨ q ) = L1 ¨ q + R1 ˙ q + (cid:18) C1 + 1 C2 (cid:19) q − C1 q + c + c + c Q ( q, ˙ q, ¨ q ) = L2 ¨ q − C1 q + (cid:18) C1 + 1 C3 (cid:19) q − c + c + c (4.7)Let us see now how the constants are related to the initial conditions thatmay be specified for the considered network.The differential system associated to the Birkhoffian (4.7) is written Q ( q, ˙ q, ¨ q ) = 0 , Q ( q, ˙ q, ¨ q ) = 0 (4.8)For each capacitor we are able to specify the initial charge, that is, q (0) , q (0) , q (0) , and for each inductor the initial current, that is, i (1) (0), i (2) (0). Takinginto account (4.5), the relation i = ˙ x and the relations two and three in (4.7),we have the following initial conditions for the differential system (4.8) q (0) = q (0)17 (0) = q (0)˙ q (0) = i (1) (0)˙ q (0) = i (2) (0) (4.9)Besides, taking into account the notations (4.5) and the last relation in (4.7),we find c + c + c = q (0) − q (0) + q (0) (4.10)Thus, the Birkhoffian (4.7) becomes Q ( q, ˙ q, ¨ q ) = L1 ¨ q + R1 ˙ q + (cid:18) C1 + 1 C2 (cid:19) q − C1 q + q (0) − q (0) + q (0) C1 Q ( q, ˙ q, ¨ q ) = L2 ¨ q − C1 q + (cid:18) C1 + 1 C3 (cid:19) q − q (0) − q (0) + q (0) C1 (4.11)If the constant R1 >
0, the Birkhoffian (4.11) is dissipative . The function E ω : T M c → R and the dissipative 1-form satisfying (2.15) have the expressions E ω ( q, ˙ q ) = 12 L1 ( ˙ q ) + 12 L2 ( ˙ q ) + 12 C1 ( q − q ) + 12 C2 ( q ) + 12 C3 ( q ) + q (0) − q (0) + q (0) C1 q − q (0) − q (0) + q (0) C1 q (4.12) D = R1 dq (4.13)The equilibrium point of the considered linear network is the solution of thesystem (cid:18) C1 + 1 C2 (cid:19) q − C1 q + q (0) − q (0) + q (0) C1 = 0 − C1 q + (cid:18) C1 + 1 C3 (cid:19) q − q (0) − q (0) + q (0) C1 = 0 (4.14)If the constants L1 , L2 , C1 , C2 , C3 satisfy the conditions (3.32), this equilibriumpoint is asymptotically stable . We define a Liapunov function V by V ( q, ˙ q ) = E ω ( q, ˙ q ) − E ω ( q e ,
0) = 12 L1 ( ˙ q ) + 12 L2 ( ˙ q ) + 12 C1 (cid:2) ( q − q ) − ( q e − q e ) (cid:3) + 12 C2 ( q − q e ) + 12 C3 ( q − q e ) (4.15)where q e satisfies the system (4.14). The level curves of the function (4.14)represent a set of ellipsoids surrounding the equilibrium point. Because theBirkhoffian (4.11) is dissipative, it follows that dE ω <
0, and therefore dV < M c given by(4.5), the Birkhoffian becomes Q ( q, ˙ q, ¨ q ) = L ( ˙ q )¨ q + R ( ˙ q ) + C ( q − q + K ) + C ( q ) Q ( q, ˙ q, ¨ q ) = L ( ˙ q )¨ q − C ( q − q + K ) + C ( q ) (4.16)18ith K = c + c + c = q (0) − q (0) + q (0).If R satisfies the condition (3.35), the Birkhoffian (4.16) is dissipative .The function E ω : T M c → R and the dissipative 1-form satisfying (2.15) aregiven by E ω ( q, ˙ q ) = Z L ( ˙ q ) ˙ q d ˙ q + Z L ( ˙ q ) ˙ q d ˙ q + Z C ( q − q + K )( dq − dq ) + Z C ( q ) dq + Z C ( q ) dq − Z Z C ′ ( q − q + K ) dq dq (4.17) D = R ( ˙ q ) dq (4.18)The equilibrium points of the considered nonlinear network are the solutions ofthe system R (0) + C ( q − q + K ) + C ( q ) = 0 − C ( q − q + K ) + C ( q ) = 0 (4.19)1) If R (0) = 0 , and L , L , C , C , C satisfies (3.42), then the equilibriumpoints are locally asymptotically stable . We define a Liapunov function V by V ( q, ˙ q ) = E ω ( q, ˙ q ) − E ω ( q e ,
0) (4.20)with E ω given by (4.17) and q e satisfying (4.19). The Hessian matrix of V ata equilibrium point ( q e ,
0) has the expression H V ( q e ,
0) = L (0) 0 0 00 L (0) 0 00 0 ˜ C ′ ( q e , q e ) + C ′ ( q e ) − ˜ C ′ ( q e , q e )0 0 − ˜ C ′ ( q e , q e ) ˜ C ′ ( q e , q e ) + C ′ ( q e ) (4.21)Under the assumptions we made, this matrix is positive definite. The centersof the level curves of the function (4.20) have the coordinates ( q e , q e satisfies the system (4.19) with R (0) = 0. Because the Birkhoffian (4.16) isdissipative, it follows that dE ω <
0, and therefore dV < R (0) = 0 , but for all x = 0 x ( R ( x ) − R (0)) > L , L , C , C , C satisfies (3.42), then the equilibrium points are locallyasymptotically stable . We define now a Liapunov function V by V ( q, ˙ q ) = E ω ( q, ˙ q ) − E ω ( q e ,
0) (4.23)with E ω : T M c → R given by E ω ( q, ˙ q ) = E ω ( q, ˙ q ) + R (0) q (4.24)19 ω ( q, ˙ q ) having the expression (4.17). The centers of the level curves of thefunction (4.23) have the coordinates ( q e , q e satisfies the system (4.19).The Hessian matrix of the function V in (4.23) has at the equilibrium point thesame expression (4.21). It remains to prove that d V <
0. This yields from thedissipativeness of the Birkhoffian (4.16). We consider the following dissipative1-form D = (cid:2) R (cid:0) ˙ q (cid:1) − R (0) (cid:3) dq (4.25)In view of assumption (4.22), this vertical 1-form is indeed dissipative. One caneasily check that for the function E ω ( q, ˙ q ) in (4.24) and the dissipative 1-formin (4.25), the following identity is fulfilled X j =1 Q j ( q ˙ q, ¨ q ) ˙ q j = X j =1 " ∂ E ω ∂q j ˙ q j + ∂ E ω ∂ ˙ q j ¨ q j + D j ( q, ˙ q ) ˙ q j (4.26)The Birkhoffian (4.16) being dissipative we have d E ω <
0, therefore d V < Acknowledgement.
This work was supported by IMAR through the con-tract of excellency CEx 06-11-12/ 25.07.06.
References [1] G. D. Birkhoff,
Dynamical Systems in: American Mathematical SocietyColloquium Publications, vol. IX, New York, 1927.[2] A.M. Bloch, P.E. Crouch, Representations of Dirac structures on vectorspaces and nonlinear LC circuits, in:
Differential Geometry and Control,Proceedings of Symposia in Pure Mathematics , in: Amer. Math. Soc., vol. (1999), 103-117.[3] R.K. Brayton, J.K. Moser, A theory of nonlinear networks I, II. Quarterlyof Applied Mathematics (1964), 1-33, 81-104.[4] L. O. Chua, C. A. Desoer, D. A. Kuh, Linear and Nonlinear Circuits ,McGraw-Hill Inc., 1987.[5] T. Courant, Dirac manifolds,
Trans. Amer. Math. Soc. (1990), 631-661.[6] I. Dorfman, Dirac structures of integrable evolution equations,
Physics Let-ters A. (1987), 240-246.[7] D. Ionescu, A geometric Birkhoffian formalism for nonlinear RLC networks,
J. Geom. Phys. (2006), 2545-2572.[8] D. Ionescu, J. Scheurle, Birkhoffian formulation of the dynamics of LCcircuits, Z. Angew. Math. Phys. (2007), 175-208.[9] M.H. Kobayashi, W.M. Oliva, On the Birkhoff approach to classical me-chanics, Resenhas IME-USP (2003), 1-71.2010] J.P. LaSalle, S. Lefschetz, Stability by Liapunov’s direct Method with Ap-plications , Academic Press, New York, 1961.[11] A. M. Liapunov, Probl´eme G´eneral de la Stabilit´e du Mouvement,
Ann.Math. Studies , vol. , Princeton University Press, 1949.[12] B.M. Maschke, A.J. van der Schaft, The Hamiltonian formulation of energyconserving physical systems with external ports, Archiv f¨ur Elektronik undUbertragungstechnik (1995), 362-371.[13] D.J. Saunders, The Geometry of Jet Bundles ,in: London MathematicalSociety Lecture Note Series, vol. 142, Cambridge University Press, 1989.[14] A.J. van der Schaft. Implicit Hamiltonian systems with symmetry.
Rep.Math. Phys. (1998), 203-221.[15] S. Smale. On the mechanical foundations of electrical circuit theory. J.Differential Geometry (1972), 193-210.[16] H. Yoshimura, J. E. Marsden. Dirac structures in Lagrangian mechanics, J. Geom. Phys.
I, II,57