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THE CALCULUS OF VARIATIONS
BOUNDARY CONDITIONS IN VARIATIONAL PROBLEMS
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arbitrary constants assumed Ax(t bang bang-bang bang-bang control boundary conditions Calculus of Variations capital Chapter Clearly coefficients concave Consider the problem consumption utility control variables control vector cost curve defined determined difference equations discount discussed dynamic system Economic Applications eigen values Euler equation gives examined example extremum functional J(x fundamental matrix given Hamiltonian hence implies inequality constraints initial conditions investment isoperimetric labour Liapunov function linear Linear Regulator marginal utility maximize the present Maximum Principle minimize minimum problem n-vector necessary condition neo-classical non-renewable resources Note objective functional obtained Optimal Growth Model output parameter phase diagram Pontryagin positive definite positive definite matrix present value production function profit functional Riccati equation saddle point satisfied scalar sensitivity singular control solution stable Substitution sufficient conditions switching function theorem transversality conditions unspecified utility function vanishes variational problem Weierstrass x(tQ