Abstract
The characterization and numerical solution of two non-smooth optimal control problems governed by a Fokker–Planck (FP) equation are investigated in the framework of the Pontryagin maximum principle (PMP). The two FP control problems are related to the problem of determining open- and closed-loop controls for a stochastic process whose probability density function is modelled by the FP equation. In both cases, existence and PMP characterisation of optimal controls are proved, and PMP-based numerical optimization schemes are implemented that solve the PMP optimality conditions to determine the controls sought. Results of experiments are presented that successfully validate the proposed computational framework and allow to compare the two control strategies.
Highlights
In the framework of stochastic optimal control theory [9, 23, 24], given a stochastic process X(t) subject to a control function u, a control problem is defined by introducing a general objective functional to be minimized that has the following structure1 3 Vol.:(0123456789)T
A fundamental tool for analysing stochastic processes is the fact that the evolution of the probability density function (PDF) associated to X(t) is governed by the so-called Fokker–Planck (FP) equation, which is a time-dependent partial differential equation (PDE) with an initial PDF configuration; see, e.g., [6] and references therein
Our choice of a specific openloop control structure is motivated by the discussion in [15, 16] in the framework of ensemble controls, where it is pointed out that a composite linear–bilinear openloop control mechanism in the stochastic differential equation (SDE) may provide a reasonable approximation of a closed-loop control; we investigate this fact with numerical experiments
Summary
In the framework of stochastic optimal control theory [9, 23, 24], given a stochastic process X(t) subject to a control function u, a control problem is defined by introducing a general objective functional to be minimized that has the following structure. If a uniformly parabolicity condition for the FP equation holds, the solution of the FP problem, with f0(x) ≥ 0 (strictly in some open set, and in the case f0(x) = (x − x0) ) and with the given boundary- and initial conditions, remains positive in the sense that f (x, t) > 0 for t > t0 and almost everywhere in Ω Based on this fact, we see that (13) represents the first-order optimality condition for the minimization problem for the control function in (7). We see that (13) represents the first-order optimality condition for the minimization problem for the control function in (7) This remark is the starting point to establish a formal connection between the HJB equation and the adjoint equation (12) at optimality [6, 7].
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