A hybrid stochastic river environmental restoration modeling with discrete and costly observations

Abstract

Summary A recent river environmental restoration problem is approached from a standpoint of stochastic control of hybrid regime‐switching diffusion processes with discrete and costly observations. This setting harmonizes with real problems because continuously obtaining environmental and ecological information is difficult, and is often costly. The main problem is to decide when and how much of the sediment should be supplied into a river environment to effectively suppress bloom of benthic algae. The interventions are allowed only at observation times. Finding the optimal river restoration policy ultimately reduces to solving an optimality equation in an unconventional form due to the observation cost and discrete observations. We show its unique solvability and present a connection with degenerate parabolic partial differential equations, with which we can construct an effective algorithm for its approximation. An uncertainty‐averse optimization problem is also considered as an advanced problem. Coefficients and parameter values are identified from experimental and observation results to numerically compute the optimal restoration policy of an existing river environment with and without uncertainty.

Stochastic control of single-species population dynamics model subject to jump ambiguity

“…Based on related mathematical frameworks of stochastic control under ambiguity, the authors have considered impulsive observation/control of stochastic population dynamics (e.g. the stochastic control models under partial observations [73,74]). They can be effectively combined with the presented model to formulate a more realistic model concurrently considering impulsive and continuous-time control variables under ambiguity.…”

Stochastic impulse control of nonsmooth dynamics with partial observation and execution delay: Application to an environmental restoration problem

“…These are satisfied for example if S(i, x) = S i {x > 0} and S i 0 > 0 with some i 0 ∈ M . 23,46 The solution is relevant for considering the replenishment problem because of its nonincreasing nature.…”

“…Continuously observing environmental and biological dynamics is a difficult task, and scheduled discrete observations of environmental and biological dynamics are more reasonable and feasible alternatives. [21][22][23] Under such situations, the decision-maker must make decisions based only on partial information, and the HJBE becomes more complicated because of using strictly smaller filtrations. 24,25 In some models, even the assumption of the scheduled observations is relaxed and only random observation processes are assumed.…”

“…The objective of this article is to formulate a stochastic impulse control problem based on discrete and random observation subject to execution delay, with a focus on a river environmental restoration problem by wisely using earth and soils. 23,38 These previous studies considered coupled algae-sediment dynamics, while we solely focus on the sediment dynamics to build a simpler model such that we can handle it analytically under certain assumptions. The mathematical problem analyzed in this article has not been explored well to the best of the authors' knowledge.…”

Control for hybrid systems: Applications and methods for adaptation and optimality

“…New stability conditions and control designs based on numerical methods, including linear matrix inequalities and gradient based optimization methods, are introduced to address time delays, parametric uncertainties, and the nonlinearities involved in the switching dynamics with stability and performance guarantees. The second group of articles, 5‐14 constituting the major category of this issue, applies adaptive control, optimization methods, and hybrid system theories to various practical fields, ranging from UAVs, AGVs, robotics, and water systems to batch processes. The last group 15,16 considers distributed cyber physical systems, which deals with parametric uncertainties, communication constraints, and signal estimations by resorting to adaptive control and fast Kalman filtering.…”