## Nominal Dynamics in Control: A Theory for Hamiltonian Systems and Nanoscale Applications

**March 24, 2006**

Figure 1. Invariant sets of the standard map and the control mechanism.

**Igor Mezic**

Some of the most interesting applications of nanotechnology require manipulation of matter at the nanoscale. An example is control of molecular dynamics, where dissipation mechanisms can be weak and many systems have Hamiltonian structure. Hamiltonian structure typically provides rich internal dynamics for a molecule, making it hard to control. The central control theoretic problem for nonlinear systems, in turn, is controllability. Given the initial and final states of the system, what is the set of inputs that the controller should apply to get the system to move from the initial to the final state? It is usually difficult to persuade a system with strong nominal dynamics (the part of the dynamics that is not affected by the control, and exists in the absence of the control) to start moving in the direction we want it to, especially if the persuasive powers (or actuation) are limited.

In the work briefly described here, this problem was addressed through a coupling of ergodic theory and methods from control theory. The results are part of a program that submits that ergodic properties of the drift are central to the development of controllability results. The key concept is the *ergodic partition* of the drift: Partition of the phase space into subsets on which drift is ergodic (roughly speaking, *ergodic* means that the drift will thoroughly sample the subset), called ergodic subsets. The resulting control design, intuitively appealing and particularly appropriate when control authority is not large, unfolds as follows:

The controller waits for the nominal part of the system (drift) to bring the system to the state in which it is maximally responsive to control input. The control is turned on at that moment; it is turned off again when "control power" is lost and the system is in another ergodic subset. Drift then takes over, and the controller waits until the system is again positioned by the drift in the state of maximal responsiveness, and so on.

It turns out that rigorous results along these lines can be proved in several interesting contexts. The resulting control can be thought of as "go-with-the-flow control" and is related to the concept of Ott–Grebogi–Yorke (OGY) control of dissipative systems.

For general systems with Hamiltonian structure, ergodic partition of the phase space can be very complicated. Figure 1 is a visualization of invariant sets (ergodic subsets are invariant, but invariant subsets need not be ergodic). In the figure, the complexity of invariant sets, such as the large chaotic region in the background (on which the dynamics is likely ergodic, although this is an open problem) and the periodic islands (the lighter areas), reveals complications that can arise in determining effective methods for the control of systems with drifts in the mixed (neither integrable nor ergodic) regime. The arrow from the chaotic to the regular zone illustrates the control mechanism described above. The system jumps around chaotically in the chaotic zone until it reaches the black circle, within which responsiveness to control is highest. On application of the control, the system jumps from the chaotic to the regular zone.

These concepts can be applied within a family of models for molecular motions; localized, targeted control perturbations have been found to be an efficient means for transferring a macromolecule from one conformation to another.

*Igor Mezic of the Department of Mechanical and Environmental Engineering and the Department of Mathematics at the University of California, Santa Barbara, presented the work briefly described here in an invited talk at the Sixth SIAM Conference on Control and its Applications.*