Linear Quadratic Stochastic Differential Games: Open-Loop and Closed-Loop Saddle Points

In this paper, we consider a linear quadratic stochastic two-person zero-sum differential game. The controls for both players are allowed to appear in both drift and diffusion of the state equation. The weighting matrices in the performance functional are not assumed to be definite/non-singular. A necessary and sufficient condition for the existence of a closed-loop saddle point is established in terms of the solvability of a Riccati differential equation with certain regularity. It is possible that the closed-loop saddle point fails to exist, and at the same time, the corresponding Riccati equation admits a solution (which does not have needed regularity). Also, we will indicate that the solution of the Riccati equation may be non-unique.Comment: 28 page

CONNECTIVITY RESTORATION FOR FISHES IN POST-INDUSTRIAL RIVERS OF NORTH EAST ENGLAND

Many rivers in developed regions experienced a strong decline in ecological function during the Industrial Revolution, due to poor water quality, degraded habitat and diminished hydrological connectivity. Post-industrially, water quality has dramatically improved in many rivers, and clean-water indicator species have returned, yet such rivers often remain very fragmented by river engineering, with locally degraded habitat and resultant effects on ecological communities, especially of fishes. River restoration activities are widespread, but their effectiveness in restoring biodiversity and ecological function remain poorly known. This study explores the causes of decline of fish populations in rivers of industrial North East England, their partial recovery, and the role of river restoration, especially through removal and mitigation of anthropogenic river barriers. In a historical review of the decline and partial recovery of the rivers Tyne, Wear and Tees, and their fish stocks, it was found that before the 19th Century Atlantic salmon (Salmo salar) and sea trout (Salmo trutta) were abundant in all three rivers. These catchments were subject to heavy industry and urbanization, instream barrier construction, and industrial pollution from the 19th Century to the mid-20th Century, during which time their fish stocks dramatically declined. Following decreased heavy industry, closure of mines and improvements in wastewater treatment, salmon and sea trout started to recover in the Tyne and Wear from the 1960s onwards and stabilized in recent years; these rivers are now the first and second best salmon rivers in England, in terms of angler catches. By comparison, anadromous salmonid numbers in the Tees increased much more slowly, potentially and partly due to impacts from the Tees Barrage. In general, the potential for recovery of anadromous salmonid stocks in post-industrial Pennine rivers appears driven by both accessibility and survival in the river, through effects of barriers, pollution and predators. Since river reconnection programmes require barrier inventories for restoration planning,the adequacy of the current national barrier inventory was assessed by field surveying two medium-sized catchments, the Wear and the Tees. The national river barrier inventory was found to be highly incomplete. From surveyed reaches across both catchments, 77.3% of barriers were found to be missing from the national database, including 68.6% of artificial barriers and 82.6% of natural barriers. Only 21.5% of artificial barriers had been removed or mitigated in both catchments, suggesting that river restoration in Northeast England, and perhaps in England more generally, still has a long way to go. The effectiveness of barrier removal on habitat change and responses of fish and invertebrate communities was studied in a small stream joining the Tees estuary. Removal of a small tidal barrier increased habitat diversity immediately upstream, while changes in the invertebrate community up- and downstream were minor and transitory. A dramatic and sustained increase in fish density occurred in the previously impounded zone. The upstream recolonization of European eel (Anguilla anguilla) was greatly increased within two years. The eel density in the previously impounded zone increased from 0.5 per 100 m2 before barrier removal to 32.5 per 100 m2 five months after removal. In contrast, the population of brown/sea trout (S. trutta) has not yet benefitted from barrier removal, suggesting wider catchment management such as habitat and water quality improvements are required to complement connectivity restoration. In rivers or tributaries with multiple barriers, catchment-scale connectivity restoration may be needed to help restore the density and distribution of diadromous and river-resident fish species. Extensive within-tributary fish sampling was used to determine local and sub-catchment responses to partial connectivity restoration. It was found that benefits of connectivity restoration in streams with many barriers may take several years to develop and that stochastic events on fish populations can obscure restoration responses. Compared with fish pass installation, barrier removal was found to be more effective in restoring lotic habitat and fish species, and facilitating movement of poorly dispersing species such as bullhead (Cottus perifretum). Findings of this thesis underline the importance of managing in-stream barriers sensitively, and have contributed to our understanding of the effects of connectivity restoration on post-industrial rivers

Open-Loop and Closed-Loop Solvabilities for Stochastic Linear Quadratic Optimal Control Problems

This paper is concerned with a stochastic linear quadratic (LQ, for short) optimal control problem. The notions of open-loop and closed-loop solvabilities are introduced. A simple example shows that these two solvabilities are different. Closed-loop solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. Conditions ensuring the convexity of the cost functional are discussed, including the issue that how negative the control weighting matrix-valued function R(s) can be. Finiteness of the LQ problem is characterized by the convergence of the solutions to a family of Riccati equations. Then, a minimizing sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. Finally, an illustrative example is presented.Comment: 40 page

Mean-Field Stochastic Linear Quadratic Optimal Control Problems: Closed-Loop Solvability

An optimal control problem is studied for a linear mean-field stochastic differential equation with a quadratic cost functional. The coefficients and the weighting matrices in the cost functional are all assumed to be deterministic. Closed-loop strategies are introduced, which require to be independent of initial states; and such a nature makes it very useful and convenient in applications. In this paper, the existence of an optimal closed-loop strategy for the system (also called the closed-loop solvability of the problem) is characterized by the existence of a regular solution to the coupled two (generalized) Riccati equations, together with some constraints on the adapted solution to a linear backward stochastic differential equation and a linear terminal value problem of an ordinary differential equation.Comment: 23 page