TY - JOUR

T1 - Bilinear Controllability of a Class of Advection-Diffusion-Reaction Systems

AU - Elamvazhuthi, Karthik

AU - Kuiper, H. J.

AU - Kawski, Matthias

AU - Berman, Spring

N1 - Funding Information:
Manuscript received November 6, 2017; revised May 13, 2018; accepted July 24, 2018. Date of publication December 5, 2018; date of current version May 27, 2019. This work was supported by the National Science Foundation Award CMMI-1436960. Recommended by Associate Editor E. Cerpa. (Corresponding author: Karthik Elamvazhuthi.) K. Elamvazhuthi and S. Berman are with the School for Engineering of Matter, Transport and Energy, Arizona State University, Tempe, AZ 85287 USA (e-mail:,karthikevaz@asu.edu; spring.berman@asu.edu).
Publisher Copyright:
© 1963-2012 IEEE.

PY - 2019/6

Y1 - 2019/6

N2 - In this paper, we investigate the exact controllability properties of an advection-diffusion equation on a bounded domain, using time-and space-dependent velocity fields as the control parameters. This partial differential equation (PDE) is the Kolmogorov forward equation for a reflected diffusion process that models the spatiotemporal evolution of a swarm of agents. We prove that if a target probability density has bounded first-order weak derivatives and is uniformly bounded from below by a positive constant, then it can be reached in finite time using control inputs that are bounded in space and time. We then extend this controllability result to a class of advection-diffusion-reaction PDEs that corresponds to a hybrid switching diffusion process (HSDP), in which case the reaction parameters are additionally incorporated as the control inputs. For the HSDP, we first constructively prove controllability of the associated continuous-time Markov chain (CTMC) system in which the state space is finite. Then, we show that our controllability results for the advection-diffusion equation and the CTMC can be combined to establish controllability of the forward equation of the HSDP. Finally, we provide constructive solutions to the problem of asymptotically stabilizing an HSDP to a target nonnegative stationary distribution using time-independent state feedback laws, which correspond to spatially dependent coefficients of the associated system of PDEs.

AB - In this paper, we investigate the exact controllability properties of an advection-diffusion equation on a bounded domain, using time-and space-dependent velocity fields as the control parameters. This partial differential equation (PDE) is the Kolmogorov forward equation for a reflected diffusion process that models the spatiotemporal evolution of a swarm of agents. We prove that if a target probability density has bounded first-order weak derivatives and is uniformly bounded from below by a positive constant, then it can be reached in finite time using control inputs that are bounded in space and time. We then extend this controllability result to a class of advection-diffusion-reaction PDEs that corresponds to a hybrid switching diffusion process (HSDP), in which case the reaction parameters are additionally incorporated as the control inputs. For the HSDP, we first constructively prove controllability of the associated continuous-time Markov chain (CTMC) system in which the state space is finite. Then, we show that our controllability results for the advection-diffusion equation and the CTMC can be combined to establish controllability of the forward equation of the HSDP. Finally, we provide constructive solutions to the problem of asymptotically stabilizing an HSDP to a target nonnegative stationary distribution using time-independent state feedback laws, which correspond to spatially dependent coefficients of the associated system of PDEs.

KW - Advection-diffusion-reaction partial differential equation (PDE)

KW - continuous-time Markov chains (CTMCs)

KW - controllability

KW - stochastic processes

KW - swarm robotics

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U2 - 10.1109/TAC.2018.2885231

DO - 10.1109/TAC.2018.2885231

M3 - Article

AN - SCOPUS:85058067323

SN - 0018-9286

VL - 64

SP - 2282

EP - 2297

JO - IRE Transactions on Automatic Control

JF - IRE Transactions on Automatic Control

IS - 6

M1 - 8561138

ER -