Controlling the Position of Traveling Waves in Reaction-Diffusion Systems

We present a method to control the position as a function of time of one-dimensional traveling wave solutions to reaction-diffusion systems according to a prespecified protocol of motion. Given this protocol, the control function is found as the solution of a perturbatively derived integral equation. Two cases are considered. First, we derive an analytical expression for the space ($x$) and time ($t$) dependent control function $f(x,t)$ that is valid for arbitrary protocols and many reaction-diffusion systems. These results are close to numerically computed optimal controls. Second, for stationary control of traveling waves in one-component systems, the integral equation reduces to a Fredholm integral equation of the first kind. In both cases, the control can be expressed in terms of the uncontrolled wave profile and its propagation velocity, rendering detailed knowledge of the reaction kinetics unnecessary.

Figure 1

Periodic acceleration and deceleration of a Schlögl front realized by multiplicative control. Left: the numerically obtained front position (red dashed line) is in excellent agreement with the protocol of motion $\varphi (t)=B_0+A\sin (2\pi t/T+B_1)$ (black solid line). $B_{0/1}$ are determined by $\varphi (t_0)={\varphi }_0$, $\dot{\varphi }(t_0)=c$ so that the protocol is smooth at $t=t_0$. Right figure shows velocities. See S1 in the Supplemental Material [8] for a movie.

Figure 2

Snapshot of position control of a FHN pulse with an invertible coupling matrix $\mathcal{G}$ taken from movie S2 in the Supplemental Material [8]. Results by analytical control (black solid) agree very well with results obtained by optimal control (red dashed). Clockwise from top left: activator $u$, inhibitor $v$, controls $f_u$, $f_v$.

Figure 3

Snapshot of position control of a FHN pulse with a non-invertible coupling matrix $\mathcal{G}$. The control ${\tilde{f}}_u$ (bottom left) acts solely on the activator equation. The controlled inhibitor pulse profile (top right) is much more deformed than the activator pulse profile (top left). Shown are results of optimal (red dashed) and analytical control (black solid). Bottom right: Analytical protocol (black solid) and numerically obtained position over time data for the maximum activator value of the controlled RD system (red dashed). See movie S3 [8].

Figure 4

Deceleration of a Schlögl front by an additive stationary control. Red dashed line is the result of numerical simulations, black solid line is the pre-given protocol. Shown are the position $\varphi$ and the velocity $\dot{\varphi }$ (right inset). The front profile (blue solid line) is slightly deformed in the region where the control (purple dotted line) is large, see left inset. Compare also S8 in [8].