Amita,

I see that this question is attached to a project, whose stated goal is to increase ground water levels.  Personally, I would not get involved in mathematical analyses and proofs before outlining the basic problem.  Is this water project for a specific place, or do you want to attack the global water availablity problem?

I have had to explain complex issues to people with little or no mathematical or statistical experience.  Most people are familiar with spreadsheets. Yes, there are more complex tools, but they are not easy to explain.  Put everything you know about the project into a spreadsheet, the size of the reservoirs, the depths and costs of pumping and water treatment, piping, investments costs, labor costs. Essentially work out a detailed budget for the project.   A detailed model of everything you know or can guess.

I use a standard coloring scheme.  I mark all my assumptions, the numbers I make up as rough beginnings, in green.  These can be changed.  The formulas are blue.  The indicators that I want to watch are orange.  Using the Excel Solver you can maximize or minimize the target goals (like total cost of the project, or total cost per person), and try to maximize benefits - allowing certain variables (green usually) to vary within specified bounds.  The greatest benefit is from writing it down so you can talk to others.

A few hours building a back-of-the-envelope estimate like this is much more useful than months of mathematical analysis.  Plus you can immediately sit down with individual experts and go through their part of the overall project looking at the sheet together.  If they tell you the reservoir numbers are too optimistic, you can change your sheet and add a comment to the cell indicating who told you and when, with links to resources. The sheet becomes a common map and communication device.

The World Bank has some really outstanding people working on constrained optimization of projects like this.  But keeping the assumptions and details in a form where they can be easily audited and communicated - is golden.   I worked at Georgetown University Center for Population Research on population impacts on the futures of countries, for USAID.  There is an extensive body of resources for quantitative analysis of economic, social, engineering and technical projects on the scale of states, regions, countries or the whole world.  USAID asked me to help them with their central international database, so I worked with all the international organizations on these sorts of things.  Out of all that I learn particularly KISS - keep it simple.

Back to your question.  Once you have complete project spreadsheets (or their equivalents systems of equations and constraints in some other modelling language), you can apply sensitivity analysis to quantitatively exercise the model to learn where it is stable and where your points of leverage might be.  The stochastic functions you choose (the statistical properties of distributions and processes you use to model real world processes) can be quite subtle and beautiful.  Unfortunately, most people are just trying to find the money for their project and trying to get by.  They have little time for elegant mathematics, or beautiful patterns evolving in complex models.

Good luck with your project.  If you decide to clarify your purposes and to tackle a real water project, please contact me.  I am retired and extremely busy with many projects, but I will help you sketch out a path, it it will benefit many people.

Finally, it is fairly simple to export the data and formulas from Excel into text files, and from there into things like Mathematica or the World Bank's General Algebraic Modelling Language (GAMS).  There are lots of languages and tools for this sort of thing.

Richard

Dear Amita,

It is possible to use stochastic Lyapunov's second method (stochastic Lyapunov functions). See, for example, links below.

http://personal.strath.ac.uk/x.mao/talks/Berlin.pdf

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.44.1147&rep=rep1&type=pdf

http://www6.cityu.edu.hk/ma/ws2010/doc/mao_notes.pdf

Thank you Sergiy,