Dynamical Control: Comparison of Map and Continuous-Flow Approaches

Continuous and pulsed forms of control of a multistable system are compared directly, both theoretically and numerically, taking as an example the switching of a periodically driven class-B laser between its stable and unstable pulsing regimes. It is shown that continuous control is the more energy efficient. This result is illuminated by making use of the close correspondence that exists between the problems of energy-optimal control and the stability of a steady state.

Figure 1

(a) Basins of attraction for the flow system (1), indicating some stable cycles $C$ and saddle cycles $S$. Different colors correspond to the basins of different cycles. The cycles $C_1$, $C_2$, $C_3$ and $S_2$, $S_3$ correspond to one pulse periodic motion, i.e., a fixed point in the Poincaré section; the cycles $C_P$ and $S_P$ correspond to three pulses periodic motion, i.e., a period-3 fixed point. Time realizations of the coordinate $x(t)$ (b) and control force $f(t)$ (c) are shown during migration from $C_3$ to $S_3$. The stable cycle $C_3$ and saddle cycle $S_3$ are marked by × and ●, respectively.

Figure 2

(a) Basins of attraction for the map (2), indicating some stable cycles $C$ and saddle cycles $S$. Different colors correspond to the basins of different cycles. (b) Migration trajectory from $C_3$ to $S_3$ in the map (2), and (c) realizations of the corresponding control force. Dashed lines and markers + correspond to solutions of the boundary problem, and the solid line with ○ indicates the solution based on fluctuational prehistory analysis.