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

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.

Automated summary

In this study, we propose a linear-quadratic control method for optimizing streamflow discharge using an infinite-dimensional jump-driven stochastic differential equation. Our model utilizes a superposition of Ornstein-Uhlenbeck processes to generate the observed sub-exponential autocorrelation function in real data, and the optimal control is determined through an integral operator Riccati equation. By parameterizing the process based on actual data and conducting computational experiments, we analyze the convergence of the numerical scheme and demonstrate the application of the proposed model to realistic problems, evaluating the performance of the controls using the Kolmogorov backward equation.