Equations, adjoint partial differential

The accelerated gradient method is used because of its advantages especially when the control is constrained. The system and its adjoint equations are coupled hyperbolic partial differential equations. They can be solved numerically using the method of characteristics (Lapidus, 1962b Chang and Bankoff, 1969). This method is used with the fourth order Runge-Kutta method (with variable step size to ensure accuracy of the integration) to solve the state and adjoint equations. 

The optimal control problem represents one of the most difficult optimization problems as it involves determination of optimal variables, which are vectors. There are three methods to solve these problems, namely, calculus of variation, which results in second-order differential equations, maximum principle, which adds adjoint variables and adjoint equations, and dynamic programming, which involves partial differential equations. For details of these methods, please refer to [23]. If we can discretize the whole system or use the model as a black box, then we can use NLP techniques. However, this results in discontinuous profiles. Since we need to manipulate the techno-socio-economic poHcy, we can consider the intermediate and integrated model for this purpose as it includes economics in the sustainabiHty models. As stated earlier, when we study the increase in per capita consumption, the system becomes unsustainable. Here we present the derivation of techno-socio-economic poHcies using optimal control appHed to the two models. 

X here represents various variables and the equation is therefore a partial differential equation. L[ j represents a linear, homogeneous, self-adjoint differential expression of second order, ip is the desired function, p x) the density function and A the eigenvalue parameter of this Sturm-Liouville eigenvalue problem. ... 

Pontryagin s adjoint variables i/j are clearly the partial derivatives of Bellman s F and the continuity of the adjoint variables (as solutions of adjoint differential equations) implies the smoothness of the surface that was lacking in the first paper. 

Assume that the Lagrangian has been established and the adjoint equation and boundary conditions imposed as before. Suppose some perturbation in the control is considered, small but arbitrary within the limits set by the constraints on controls. We would, in general, need to consider not the partial derivative of a functional with respect to the perturbation in control variable, but rather the total differential that included the indirect effect of the control perturbation on the state and control variables ... 

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