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adiabatic adiabatic control adjoint variables allocation Bellman boundary calculate calculus of variations chemical reactors choice choose component compute consider constraint continuous decision variables define denote desired steady dimensionality dimensionless discrete dynamic programming equilibrium curve exothermic reaction expression extent of reaction fact feed final value problem find its optimal finite number fixed full cooling curve given Hamiltonian heat removal increase initial value problem integral intermediate control interval Lagrange multiplier linear look maximum reaction rate method minimize minimum non-linear objective function optimal control optimal policy optimal value output parameters partial derivatives partial differential equation plane Pontryagin's positive possible principle of optimality reaction rate reactor region respect restriction shown in Figure situation solution solve steady state point stirred tank Suppose switching surface tangent temperature theorems thing total holding trajectory transformation equation vanish vector want to maximize