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This work offers the mathematical foundation and theory required for studying in depth the stability and optimal control of systems whose history is taken into account. In this second edition the economic application is enlarged, and explored in some depth. The application holds out the hope that full employment and high income growth will be compatible with low prices and low inflation, provided that the control matrix has full rank - in other words, that the existing controls are fully effectively used. The text concludes with a new appendix containing complete programmes, data, graphs and quantitative results for the US economy.