Optimal Control of the Fokker–Planck Equation with Space-Dependent Controls

Fleig, Arthur; Guglielmi, Roberto

doi:10.1007/s10957-017-1120-5

Cited by 36 publications

(45 citation statements)

References 31 publications

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“…The adjoint equation is in this setting an HJB equation. We refer the reader to [1], [2], [4], and [15] for this approach. In this article, optimal control problems of the following form:…”

Section: Introductionmentioning

confidence: 99%

Optimality conditions in variational form for non-linear constrained stochastic control problems

Pfeiffer¹

2020

Mathematical Control &Amp; Related Fields

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Optimality conditions in the form of a variational inequality are proved for a class of constrained optimal control problems of stochastic differential equations. The cost function and the inequality constraints are functions of the probability distribution of the state variable at the final time. The analysis uses in an essential manner a convexity property of the set of reachable probability distributions. An augmented Lagrangian method based on the obtained optimality conditions is proposed and analyzed for solving iteratively the problem. At each iteration of the method, a standard stochastic optimal control problem is solved by dynamic programming. Two academical examples are investigated.

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“…The adjoint equation is in this setting an HJB equation. We refer the reader to [1], [2], [4], and [15] for this approach. In this article, optimal control problems of the following form:…”

Section: Introductionmentioning

confidence: 99%

Optimality conditions in variational form for non-linear constrained stochastic control problems

Pfeiffer¹

2020

Mathematical Control &Amp; Related Fields

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“…Hence, we only point out the key results needed. Particularly, for a C 1,1 domain, the W 2,p regularity estimate (20) also holds for the equation…”

Section: A Controllability Of Pde Systemmentioning

confidence: 99%

“…A similar argument can be used when Ω is convex. However, it is not clear if the W 2,p regularity estimate (20) holds for general convex domains. On the other hand, it can be established that the L p regularity estimate of the PDE (19) holds for such domains.…”

Section: A Controllability Of Pde Systemmentioning

confidence: 99%

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

Elamvazhuthi

Kuiper

Kawski

et al. 2019

IEEE Trans. Automat. Contr.

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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 hybridswitching 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. Lastly, we provide constructive solutions to the problem of asymptotically stabilizing an HSDP to a target non-negative stationary distribution using timeindependent state feedback laws, which correspond to spatiallydependent coefficients of the associated system of PDEs.

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“…With this, the conservation of mass property in (4) holds. Another possibility is to use homogeneous Dirichlet boundary conditions, which, while appropriate in some scenarios [2,3,12], in general, do not guarantee conservation of mass in space. See also [19,Chapter 5] for a comparison between the Gihman-Skorohod [14] and the Feller classification of boundary conditions.…”

Section: Problem Formulation and Assumptionsmentioning

confidence: 99%

“…The optimal control problem to be solved in each step of the MPC scheme belongs to the class of tracking type optimal control problems governed by partial differential equations (PDEs) and the usual norm for measuring the distance to a reference in PDE-based optimal tracking control is the L 2 -norm [31]. The L 2 -norm is advantageous because existence and well posedness of the solution of the resulting optimal control problem for the Fokker-Planck equation were recently established [12]. Moreover, the fact that L 2 is a Hilbert space significantly simplifies, e.g., the computation of gradients, which is crucial for the implementation of numerical optimization algorithms.…”

Section: Introductionmentioning

confidence: 99%

L2-Tracking of Gaussian Distributions via Model Predictive Control for the Fokker–Planck Equation

Fleig

Grüne

2018

Vietnam J. Math.

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This paper presents first results for the stability analysis of Model Predictive Control schemes applied to the Fokker-Planck equation for tracking probability density functions. The analysis is carried out for linear dynamics and Gaussian distributions, where the distance to the desired reference is measured in the L 2 -norm. We present results for general such systems with and without control penalization. Refined results are given for the special case of the Ornstein-Uhlenbeck process. Some of the results establish stability for the shortest possible (discrete time) optimization horizon N = 2. X t (t = 0) = X 0 a.s.,

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Optimal Control of the Fokker–Planck Equation with Space-Dependent Controls

Cited by 36 publications

References 31 publications

Optimality conditions in variational form for non-linear constrained stochastic control problems

Optimality conditions in variational form for non-linear constrained stochastic control problems

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

L2-Tracking of Gaussian Distributions via Model Predictive Control for the Fokker–Planck Equation

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