Finite-time stochastic control for complex dynamical systems: The estimate for control time and energy consumption

TL;DR

This work introduces a finite-time, closed-loop stochastic controller for high-dimensional, noisy dynamical systems by injecting the control into the diffusion term $\bm{u}(\bm{x})\,dB_t$ with a piecewise, state-dependent form $\bm{u}(\bm{x}) = k\bm{x}\mathbf{1}_{\|\bm{x}\|\ge 1} + k\|\bm{x}\|^{\alpha-1}\bm{x}\mathbf{1}_{\|\bm{x}\|<1}$ and parameter constraints $k>\sqrt{2L}$, $\alpha\in(0,1)$. The authors establish finite-time stability and synchronization and derive explicit upper bounds on the expected convergence time $\mathbb{E}\tau$ and the energy $\mathbb{E}\mathcal{E}_q$ using a generalized Itô formula and a piecewise Lyapunov function $V_p$. Theoretical results are complemented by numerical demonstrations on random ecosystems, neural networks, Hindmarsh-Rose models, and the Lorenz system, highlighting a favorable time-energy trade-off relative to deterministic finite-time schemes, albeit with stochastic fluctuations and discretization sensitivity. The work provides practical bounds and insights for designing diffusion-based finite-time controllers in chaotic and networked settings, with avenues for tighter analyses and robust implementations.

Abstract

Controlling complex dynamical systems has been a topic of considerable interest in academic circles in recent decades. While existing works have primarily focused on closed-loop control schemes with infinite-time durations, this paper introduces a novel finite-time, closed-loop stochastic controller that pays special attention to control time and energy and their dependence on system parameters. This technique of stochastic control not only enables finite-time control in chaotic dynamical systems but also facilitates finite-time synchronization in unidirectionally coupled systems. Notably, our new scheme offers several advantages over existing deterministic finite-time controllers from a physical standpoint of time and energy consumption. Using numerical experiments based on random ecosystems, neural networks, and Lorenz systems, we demonstrate the effectiveness of our analytical results. It is anticipated that this proposed stochastic scheme will have widespread applicability in controlling complex dynamical systems and achieving network synchronization.

Figures (9)

• Figure 1: Physical underpinning of $\bm u(\bm x)$ for the three- and one-dimension case, respectively. Here, $\bm u^{a}(\bm{x})=k\bm x \Vert\bm x\Vert^{\alpha-1}$, $\bm u^{b}(\bm{x})=k\bm x$. Since controller \ref{['2']} is designed as a stochastic controller, it is possible for the trajectory to enter and exit the unit ball multiple times before eventually converging to zero within a finite time (refer to Remark \ref{['remark2']}). This behavior is significantly different from the deterministic controller discussed in b15.
• Figure 2: The solid (blue) cureves indicate the dependence of the $T_{f}^{\rm Sup}$ on $k$ and $\alpha$. And the dashed (red) curves indicate $\alpha=3/4+L/(2k^{2})$. The parameters are set as $L=8$, $\Vert \bm x_{0} \Vert=3$.
• Figure 3: The convergence time and energy consumption for system \ref{['newexample']} for different values of $\alpha$ and $k$. The solid (blue) curves represent the numerical evaluation of convergence time and energy consumption, which is counted as the average of 1000 random realization of system \ref{['newexample']}. The dashed (red) curves indicate the upper bound obtained from Theorem \ref{['theorem1']} and \ref{['theorem2']}. (a) When fix $\alpha$, convergence time and energy consumption shows a decreasing tendency when $k$ increases. The parameters are $\alpha=0.5$, $q=1/2$. (b)When fix $k$, convergence time shows an increasing tendency when $\alpha$ increases while energy consumption does not have an obvious monotonic trend when $\alpha$ varies. The parameters are $k=5, q=1/2$. For both (a) and (b), the initial value is $x_0=10$. Here, Euler-Maruyama scheme (see b29b30) with step size $\triangle t=10^{-6}$ is used for integrating the system of stochastic differentian equation \ref{['newexample']}.
• Figure 4: Dependence of the mean value of the convergence time $\tau_k$ on the coupling gain $k$ and $\alpha$. The solid (blue) curves indicate the numerical evaluation of $\tau_k$, which is counted as the average of 100 random realizations of system \ref{['example1']}. The dashed (red) curves indicate upper bound obtained from Theorem \ref{['theorem1']}. The initial states are $\bm{x}_0=[10,10]^{\rm T}$.
• Figure 5: The convergence time and the required energy consumption for \ref{['example2']}, vary respectively, with the increase of $N$. The parameters are $p=\dfrac{1}{3}$, $\sigma=1$, $r=1$, $q=0.1$, $k=1.1\sqrt{2\eta_{\max}}$, and the initial states are $\bm{x}_0=[1,1\cdots,1]^{\rm T}$. The solid (blue) curves indicate the numerical evaluation of $\tau$ and $\mathcal{E}_q$, which is counted as the average of 100 random realizations of system \ref{['example2']}. The dashed (red) curves indicate upper bound obtained from Theorem \ref{['theorem1']} and \ref{['theorem2']}.

Theorems & Definitions (17)

• Definition 1
• Definition 2
• Remark 1
• Remark 2
• Remark 3
• Theorem 1
• Theorem 2
• Example 1
• Example 2
• Example 3