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assume autonomous boundary conditions calculus of variations capital stock coefficient comparison path compute concave function conditions for solution constants of integration constraint consumption continuous function continuously differentiable function control variable convex convex function curve defined denote dt subject dynamic endpoint equivalent Euler equation evaluated Example Exercise feasible modifications Figure Find necessary conditions fixed function h FURTHER READING given gives Hamiltonian Hence holds implies increasing inequality initial condition integrand interval inventory investment jump Lagrange multiplier Legendre condition Lemma line integral linear differential equation linearly locus marginal valuation maximize maximum minimizing MRAP multiplier nonnegative nonpositive obeyed optimal control optimal path optimal solution output parameter partial derivatives particular solution profit rate of change resource roots saddlepoint satisfies the Euler second order Section solve stationary steady subject to x(0 subjectto Substituting Suppose terminal value theorem tion transversality condition utility yields zero