Closed trajectories and global controllability in the plane

In the plane, results are given about the structure of closedtrajectories which may occur as simple closed curves or generalclosed curves with self-intersections. Necessary conditionsfor the global controllability of nonlinear systems that arein the so-called linear-analytic form = f(x)+ug(x), where x R² and u 1, are given. It is provedthat, if there exists a closed trajectory of the system, then either contains a point where f and g are linearly dependent,or encloses some zeros of f + ug for all u [-1,1]. Then thisresult is used to prove that, if the linear-analytic systemis controllable and the vector field g is never zero in W R²,then W contains some zeros of f + ug for some u [-1,1]. A topologicalapproach is taken. Remarks are made about the size of the regionwhere a closed trajectory can lie, and about the shape of theclosed trajectories. Further implications are discussed.     

Linearized control systems and small-time reachable sets

Using linear approximations of nonlinear systems has long beena practice to design control laws. In this paper, an analysisis given involving linear approximation of the nonlinear controlsystem and small-time reachable sets in R². A useful concept,the swing–out, which is a measure of nonlinearity, isdefined. This is used to examine the relationship between thesmall-time reachable sets of the nonlinear control system andits linear approximation. Behaviour of the nonlinear systemunder a control law is examined within this context. More factsare given about the swing-out for some special cases.     

Cubic Splines on Curved Spaces

We consider a second-order problem in the calculus of variations,with an application to robotics in mind. The analysis is carriedout on a general Riemannian manifold M and then specializedto the case where M is the Lie group SO(3) of rotations in R³.For SO(3), the Euler-Lagrange equations reduce to interestingnonlinear systems of ordinary differential equations in R³.     

DYNAMIC LINEAR DICHROISM IN CHROMOPROTEINS

The time-dependent polarized fluorescence and optical density transients arising from electronic excitation transport are derived for solutions of randomly oriented chromoproteins in which the chromophore transition moments have well-defined orientations with respect to the protein. The treatment is extended to oriented monolayers of chromoproteins, in which the molecules are aligned with a particular protein-fixed axis perpendicular to the surface plane and with random azimuthal angles about the surface normal. The initial ratio of parallel to perpendicular transients is invariably 3:1, irrespective of system dimensionality and chromophore organization. The residual anisotropy observed at long times is directly related to the relative orientations of the donor and acceptor chromophores. These calculations form a basis for detailed interpretation of ultrafast fluoresence and pump-probe linear dichroism studies in photosynthetic systems, for which 3-dimensional structures are becoming increasingly available.