Decomposition of differential equations

Abstract

Consider the linear system written in the matrical form dz/dt = (A (t) + B(t)) z, z(to) = z,where A (t) and B (t) are n x n matrices and z an n X 1 matrix. It is well known that the solution takes the form z = u (t) v (t) z0,We may demand that both U (to) and V (to) are the identity matrix. The purpose of this paper is to establish the above type of decomposition formula for nonlinear cases and to study its application.Differential equations in this paper will be written in the form Du= A (t)uwhere Du is the tangent vector to the solution path u(t) on a manifold ~ and A (t)u denotes the vector of the time-dependent vector field A (t) at the point u(t). The solution of (0.1) with the initial condition u(to)= Po will be written as u = T(A;t, to)Po. If a system of local coordinates x is chosen, then (0.1) takes the usual form du/dt = F (u, t), where both u and F are vectors. After some preparatory work in § 1, we define in § 2 the operation Adj a on vector fields, a being a homeomorphism from one open set to another of ~. (A + B;t, to)Po= T (A;t, to) T(B*;t, to)po,where B* (t) = Adj T (A ; to, t) B (t). We also assert more or less to the effect that, for any vector field X on ~, if Y (t) = Adj T (A; t, to)X, then r (t)/dt = [ Y (t), A (t)],where the right hand side is the usual bracket product of two infinitesimal transformations (vector fields).

On lie algebras and finite dimensional filtering

“…(1.6) looks much simpler than (1.2); indeed, (1.6) is an (infinite dimensional) bilinear differential equation [11] in p, with z ·considered as the input. This is the first indication (given work on the roles of Lie algebras in solving finite dimensional bilinear equations [32], [33]) that the Lie algebraic and differential geometric techniques developed for finite dimensional systems of this type may be brought to bear here. Modulo some conjectured infinite dimensional extensions of some known results in the finite dimensional case (to be discussed below) this can be made more precise as follows: suppose that, for some given initial density, some statistic of the conditional distribution of x1 given z 1 can be calculated with a finite dimensional recursive estimator of the form (l.4Hl.5), where a, b;, and y are C"° or analytic.…”

Towards robust Lie-algebraic stability conditions for switched linear systems

“…In view of (13), we can apply Proposition 1 with L instead of N to each induced norm D 0 A j1 · · · A jK D K appearing in (15), concluding that this norm does not exceed ce −λ|W | . Pulling out this common bound and then returning the sum of products of |α ij |'s into the original factored form similar to (14), we arrive at (16) and the result now follows from (12).…”

A brief remark on the topological entropy for linear switched systems

“…Note that if the constituent systems are linear and stable, it should be noted that the switched system remains stable under an arbitrary switching if the Lie-algebra is nilpotent (e.g., see [17]) or a compact semi-simple subalgebra (e.g., see [15]). Moreover, it should be noted that each of these classes of Lie-algebras strictly contains the others (e.g., see [18], [16]).…”