Polynomial inverse integrating factors for quadratic differential systems

We characterize all the quadratic polynomial differential systems having a polynomial inverse integrating factor and provide explicit normal forms for such systems and for their associated first integrals. We also prove that these families of quadratic systems have no limit cycles.     

Appell systems on Lie groups

Using the probabilistic interpretation of Appell polynomials as systems of moments, we show how to define them in the noncommutative case. The method is based on certain infinite-dimensional representations of local Lie groups. For processes, limit theorems play an essential role in the construction. Polynomial matrix representations of convolution semigroups are a principal feature.     

Modular Lie representations of groups of prime order

Let K be a field of prime characteristic p and let G be a group of order p. For any finite-dimensional KG-module V and any positive integer n let L ⁿ (V) denote the nth homogeneous component of the free Lie K-algebra generated by (a basis of) V. Then L ⁿ (V) can be considered as a KG-module, called the nth Lie power of V. The main result of the paper is a formula which describes the module structure of L ⁿ (V) up to isomorphism. Mathematics Subject Classification (2000): 17B01, 20C20.     

One-parameter groups of regular quasimultipliers

Groups of unbounded operators are approached in the setting of the Esterle quasimultiplier theory. We introduce groups of regular quasimultipliers of growth ω, or ω-groups for short, where ω is a continuous weight on the real line. We study the relationship of ω-groups with families of operators and homomorphisms such as regularized, distribution and integrated groups, holomorphic semigroups, and functional calculi. Some convolution Banach algebras of functions with derivatives to fractional order are needed, which we construct using the Weyl fractional calculus.     

Non-overlapping control systems on Lie groups

LetG be a Lie group with Lie algebraL(G) and let Ω be a non-empty subset ofL(G). If Ω is interpreted as the set of controls, then the set of elements attainable from the identity for the system Ω is a subsemigroup ofG. A system Ω is called anon-overlapping control system if any element attainable for Ω is only attainable at one time. In this paper, we show that a compact convex generating nonoverlapping control systems on a connected Lie group must be contained inX+E for someX∈L(G)\E, where E is a subspace of codimension one containing the commutator, and the homomorphism from the attainable semigroup intoR ⁺ extends continuously to the whole group in the case of solvable Lie groups. This work is done under the support of TGRC-KOSEF.     

Special symplectic Lie groups and hypersymplectic Lie groups

A special symplectic Lie group is a triple ${(G,\omega,\nabla)}$ such that G is a finite-dimensional real Lie group and ω is a left invariant symplectic form on G which is parallel with respect to a left invariant affine structure ${\nabla}$ . In this paper starting from a special symplectic Lie group we show how to “deform” the standard Lie group structure on the (co)tangent bundle through the left invariant affine structure ${\nabla}$ such that the resulting Lie group admits families of left invariant hypersymplectic structures and thus becomes a hypersymplectic Lie group. We consider the affine cotangent extension problem and then introduce notions of post-affine structure and post-left-symmetric algebra which is the underlying algebraic structure of a special symplectic Lie algebra. Furthermore, we give a kind of double extensions of special symplectic Lie groups in terms of post-left-symmetric algebras.     

Conditions for the coefficients of Runge-Kutta methods for systems of n-th order differential equations

To derive order conditions for Runge-Kutta methods of Nyström or Fehlberg type, applicable to arbitrary order differential equations, a theory similar to that about Runge-Kutta methods for first order systems, due to Butcher [1], is developed. By a new definition of elementary differentials, which is independent of the order of the given system, each condition to be satisfied by the coefficients of the method directly follows from the representation of the corresponding elementary differential.     

Linear feedback systems and the groups of Galois and Lie

Our purpose in this paper is to delineate precisely the extent to which one can make explicit calculations involving the most basic linear feedback systems. Our results center around the Galois theory of the “root-locus” equation p(s) + kq(s) = 0 and the Lie symmetries associated with the related differential equation p(D)x + k(t)q(D)x = 0, D = d/dt. We show that the Galois theory leads to a more refined classification, but that these theories are related in a substantial way. Considerable insight into this is obtained through the study of the monodromy group associated with algebraic curve defined by p(s) + kq(s) = 0.     

Optimal control on Lie groups and integrable Hamiltonian systems

Control theory, initially conceived in the 1950’s as an engineering subject motivated by the needs of automatic control, has undergone an important mathematical transformation since then, in which its basic question, understood in a larger geometric context, led to a theory that provides distinctive and innovative insights, not only to the original problems of engineering, but also to the problems of differential geometry and mechanics. This paper elaborates the contributions of control theory to geometry and mechanics by focusing on the class of problems which have played an important part in the evolution of integrable systems. In particular the paper identifies a large class of Hamiltonians obtained by the Maximum principle that admit isospectral representation on the Lie algebras \(\mathfrak{g} = \mathfrak{p} \oplus \mathfrak{k}\) of the form

$\frac{{dL_\lambda }}{{dt}} = \left[ {\Omega _\lambda ,L_\lambda } \right]L_\lambda = L_\mathfrak{p} - \lambda L_\mathfrak{k} - \left( {\lambda ^2 - s} \right)A, L_\mathfrak{p} \in \mathfrak{p}, L_\mathfrak{k} \in \mathfrak{k}.$

The spectral invariants associated with L _λ recover the integrability results of C.G.J. Jacobi concerning the geodesics on an ellipsoid as well as the results of C. Newmann for mechanical problem on the sphere with a quadratic potential. More significantly, this study reveals a large class of integrable systems in which these classical examples appear only as very special cases.

     

Periodic solutions of some autonomous second order Hamiltonian systems

In this paper, we study the existence of periodic solutions of some autonomous second order Hamiltonian systems $$\left\{\begin{array}{l}\ddot{u}(t)=\nabla{H(u(t)),}\\[3pt]u(0)-u(T)=\dot{u}{(0)}-\dot{u}{(T)}=0.\end{array}\right.$$ We obtain some new existence theorems by the least action principle.     

Control of mechanical systems on Lie groups and ideal hydrodynamics

In contrast to the Euler–Poincaré reduction of geodesic flows of left- or right-invariant metrics on Lie groups to the corresponding Lie algebra (or its dual), one can consider the reduction of the geodesic flows to the group itself. The reduced vector field has a remarkable hydrodynamic interpretation: it is the velocity field for a stationary flow of an ideal fluid. Right- or left-invariant symmetry fields of the reduced field define vortex manifolds for such flows. Now we consider a mechanical system, whose configuration space is a Lie group and whose Lagrangian is invariant with respect to left translations on this group, and assume that the mass geometry f the system may change under the action of internal control forces. Such a system can also be reduced to a Lie group. Without controls, this mechanical system describes a geodesic flow of the left-invariant metric, given by the Lagrangian, and, therefore, its reduced flow is a stationary ideal fluid flow on the Lie group. The standard control problem for such system is to find the conditions under which the system can be brought from any initial position in the configuration space to another preassigned position by changing its mass geometry. We show that under these conditions, by changing the mass geometry, one can also bring one vortex manifold to any other preassigned vortex manifold. Translated from Sovremennaya Matematika i Ee Prilozheniya (Contemporary Mathematics and Its Applications), Vol. 61, Optimal Control, 2008.