One method to optimize the integral $$\int_{\mathcal T} L(t,x,\dot{x}) \; dt $$ of a functional over a curve is the calculus of variations, which leads to ordinary differential equations: the Euler-Lagrange equation $$-\frac{d}{dt} L_{\dot{x}}(t,x(t),\dot{x}(t)) + L_x(t,x(t),\dot{x}(t)) =0$$ or its generalization, Pontryagin's maximum principle. An alternative is Bellman's optimality principle, which leads to Hamilton-Jacobi-Bellman partial differential equations. Each of the methods has advantages and disadvantages depending on the application, and there are numerous technical differences between them, but in the cases when both are applicable the answers are broadly similar.

The calculus of variations can also be used to optimize a functional $$\int_{\mathcal X} L(x,u,p) \; dx $$ integrated over a multidimensional space. The resulting Euler-Lagrange equations $$-\frac{\partial}{\partial x} L_{p}(x,u(x),p(x)) + L_u(x,u(x),p(x))$$ are partial differential equations with the space coordinates as independent variables. Is an alternative approach using value functions, leading to optimality conditions along the lines of Bellman's optimality principle, known?