Mahesh Nerurkar

The topological dynamics of semigroup actions

In these notes we explore the fine structure of recurrence for semigroup actions, using the algebraic structure of compactifications of the acting semigroup.

An example of almost universally observable vector fields on P 1 R

Abstract Using methods of dynamical systems, we construct examples of smooth, almost universally observable vector fields on the projective 1-space. These vector fields are non-linear, non-autonomous and their time dependence is almost periodic.

Controllability, stabilization, and the regulator problem for random differential systems

Introduction Basic dynamical notions Random linear control processes Some facts about random linear systems Sufficiency conditions for uniform controllability Dependence of controllability on the dynamics of the flow Global null controllability The feedback stabilization problem for random linear systems The rotation number The solution of the linear regulator and the stabilization problem Linearization of the regulator and the stabilization problem.

Construction of minimal cocycles arising from specific differential equations

Blending methods of Topological Dynamics and Control Theory, we develop a new technique to construct compact-Lie-group-valued minimal cocycles arising as fundamental matrix solutions of linear differential equations with recurrent coefficients subject to a given constraint. The precise requirement on the coefficients is that they belong to a specified closed convex subsetS of the Lie algebraL of the Lie group. Our result is proved for a very thin class of cocycles, since the dimension ofS is allowed to be much smaller than that ofL, and the only assumption onS is thatL0(S) =L, whereL0(S) is the ideal ofL(S) generated by the difference setS − S, andL(S) is the Lie subalgebra ofL generated byS. This covers a number of differential equations arising in Mathematical Physics, and applies in particular to the widely studied example of the Rabi oscillator.

On null controllability of linear systems with recurrent coefficients and constrained controls

Given a family of time-dependent linear control processes, we study conditions under which local null controllability implies global null controllability. This is done by employing methods of dynamical systems and the Sacker-Sell spectral theory. We show that the above implication holds “almost surely” for recurrent families provided the spectrum of the associated linear system is contained in (−∞,0].

Observability and topological dynamics

Using methods of topological dynamics, we study sufficiency conditions for observability of a given dynamical system (X, T) by a continuous output function. In particular, we show that ifX has either finite topological or covering dimension andT does not have infinitely many periodic points with same period, then (X, T) admits an observable. The paper also gives a partial survey of various related results and improves some of them.

Spectral and stability questions concerning evolution of non-autonomous linear systems

The study of long time behaviour of quantum systems with time dependent Hamiltonians has received increasing attention. A number of numerical and theoretical studies are carried out for special systems (primarily rotors and oscillators) with time dependent external field. In this article we survey analytical results about such systems driven by stationary ergodic external fields.

Controllability and the nature of quasi frequencies

Abstract Using methods of dynamical systems, we construct an example of a time dependent linear control process with quasi periodic coefficients for which the controllability properties of the system are very sensitive to the Diophantine properties of the quasi frequencies.