Bilinear Controllability of a Class of Advection–Diffusion–Reaction Systems

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.

[1]  R. Nagel,et al.  One-parameter semigroups for linear evolution equations , 1999 .

[2]  R. Nagel,et al.  One-parameter Semigroups of Positive Operators , 1986 .

[3]  Spring Berman,et al.  PDE-Based Optimization for Stochastic Mapping and Coverage Strategies using Robotic Ensembles , 2017, Autom..

[4]  E. Ouhabaz.  Analysis of Heat Equations on Domains. (LMS-31) , 2009 .

[5]  Spring Berman,et al.  Optimal control of stochastic coverage strategies for robotic swarms , 2015, 2015 IEEE International Conference on Robotics and Automation (ICRA).

[6]  Tryphon T. Georgiou,et al.  Optimal Transport Over a Linear Dynamical System , 2015, IEEE Transactions on Automatic Control.

[7]  Spring Berman,et al.  Coverage and field estimation on bounded domains by diffusive swarms , 2016, 2016 IEEE 55th Conference on Decision and Control (CDC).

[8]  Suzanne Lenhart,et al.  Optimal control of advective direction in reaction-diffusion population models , 2012 .

[9]  G. Leoni.  A First Course in Sobolev Spaces , 2009 .

[10]  Andrey Pilipenko,et al.  An introduction to stochastic differential equations with reflection , 2014 .

[11]  M. Choulli,et al.  Gaussian lower bound for the Neumann Green function of a general parabolic operator , 2013, 1301.2906.

[12]  K. Schmüdgen.  Unbounded Self-adjoint Operators on Hilbert Space , 2012 .

[13]  Alexander Y. Khapalov,et al.  Controllability of Partial Differential Equations Governed by Multiplicative Controls , 2010 .

[14]  Paolo Dai Pra,et al.  A stochastic control approach to reciprocal diffusion processes , 1991 .

[15]  Arthur Fleig,et al.  Optimal Control of the Fokker–Planck Equation with Space-Dependent Controls , 2017, J. Optim. Theory Appl..

[16]  S. Yau,et al.  On the parabolic kernel of the Schrödinger operator , 1986 .

[17]  Steven R. Finch,et al.  Mathematical constants , 2005, Encyclopedia of mathematics and its applications.

[18]  L. Ambrosio,et al.  Existence and stability for Fokker–Planck equations with log-concave reference measure , 2007, Probability Theory and Related Fields.

[19]  J. Ball.  Strongly continuous semigroups, weak solutions, and the variation of constants formula , 1977 .

[20]  Simon M. J. Lyons.  Introduction to stochastic differential equations , 2011 .

[21]  Pedro U. Lima,et al.  Modeling and Optimal Centralized Control of a Large-Size Robotic Population , 2006, IEEE Transactions on Robotics.

[22]  Spring Berman,et al.  Controllability to equilibria of the 1-D fokker-planck equation with zero-flux boundary condition , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[23]  Alcherio Martinoli,et al.  Modeling Swarm Robotic Systems: a Case Study in Collaborative Distributed Manipulation , 2004, Int. J. Robotics Res..

[24]  Tad Hogg,et al.  Modeling and mathematical analysis of swarms of microscopic robots , 2005, Proceedings 2005 IEEE Swarm Intelligence Symposium, 2005. SIS 2005..

[25]  Spring Berman,et al.  Performance Bounds on Spatial Coverage Tasks by Stochastic Robotic Swarms , 2016, IEEE Transactions on Automatic Control.

[26]  F. Otto.  THE GEOMETRY OF DISSIPATIVE EVOLUTION EQUATIONS: THE POROUS MEDIUM EQUATION , 2001 .

[27]  M. Agranovich,et al.  Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains , 2015 .

[28]  G. Yin,et al.  Hybrid Switching Diffusions: Properties and Applications , 2009 .

[29]  W. Ziemer.  Weakly Differentiable Functions: Sobolev Spaces and Functions of Bounded Variation , 1989 .

[30]  Richard F. Bass,et al.  Some Potential Theory for Reflecting Brownian Motion in Holder and Lipschitz Domains , 1991 .

[31]  Spring Berman,et al.  Mean-field controllability and decentralized stabilization of Markov chains , 2017, 2017 IEEE 56th Annual Conference on Decision and Control (CDC).

[32]  Roger W. Brockett,et al.  Notes on the Control of the Liouville Equation , 2012 .

[33]  C. Villani,et al.  ON THE TREND TO EQUILIBRIUM FOR THE FOKKER-PLANCK EQUATION : AN INTERPLAY BETWEEN PHYSICS AND FUNCTIONAL ANALYSIS , 2004 .

[34]  M. Ledoux,et al.  Analysis and Geometry of Markov Diffusion Operators , 2013 .

[35]  Alessio Porretta,et al.  On the Planning Problem for the Mean Field Games System , 2014, Dyn. Games Appl..

[36]  Jun Geng.  Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains , 2018 .

[37]  On the boundedness of first derivatives for solutions to the Neumann–Laplace problem in a convex domain , 2009 .

[38]  A. Blaquière,et al.  Controllability of a Fokker-Planck equation, the Schrödinger system, and a related stochastic optimal control (revised version) , 1992 .

[39]  P. Grisvard.  Elliptic Problems in Nonsmooth Domains , 1985 .

[40]  A. Lunardi.  Analytic Semigroups and Optimal Regularity in Parabolic Problems , 2003 .

[41]  João Pedro Hespanha,et al.  Jump Control of Probability Densities With Applications to Autonomous Vehicle Motion , 2012, IEEE Transactions on Automatic Control.

[42]  D. Stroock.  Logarithmic Sobolev inequalities for gibbs states , 1993 .

[43]  The functional calculus for the laplacian on Lipschitz domains , 1989 .