Modeling and computation of an integral operator Riccati equation for an infinite-dimensional stochastic differential equation governing streamflow discharge

TL;DR

This paper develops a control framework for streamflow discharge modeled by an infinite-dimensional jump-driven SDE using a Riccati equation, with numerical methods validated through real data and applications.

Contribution

It introduces a novel Riccati equation approach for controlling an infinite-dimensional supOU process-based SDE in streamflow modeling.

Findings

Finite-dimensional Riccati equation derived and solved numerically.

Convergence of the numerical scheme confirmed through experiments.

Model successfully applied to real river data for control optimization.

Abstract

We propose a linear-quadratic (LQ) control problem of streamflow discharge by optimizing an infinite-dimensional jump-driven stochastic differential equation (SDE). Our SDE is a superposition of Ornstein-Uhlenbeck processes (supOU process), generating a sub-exponential autocorrelation function observed in actual data. The integral operator Riccati equation is heuristically derived to determine the optimal control of the infinite-dimensional system. In addition, its finite-dimensional version is derived with a discretized distribution of the reversion speed and computed by a finite difference scheme. The optimality of the Riccati equation is analyzed by a verification argument. The supOU process is parameterized based on the actual data of a perennial river. The convergence of the numerical scheme is analyzed through computational experiments. Finally, we demonstrate the application of the proposed model to realistic problems along with the Kolmogorov backward equation for the performance evaluation of controls.