Euler Lagrange equation in variational calculus for a sum of integrals Let $F, G: \mathbb{R}^3\rightarrow{}\mathbb{R}$ be two continuously differentiable functions and let $a\leq b \leq c$. I want to know if there exists some known method to find a function that maximizes the functional $J(P):=\int_a^b F(q,P(q),P'(q))dq+ \int_b^c G(q,P(q),P'(q))dq$ between all the continuously differentiable functions $P: \mathbb{R}\rightarrow{}\mathbb{R}$ satisfying $P(a)=0$.
This is known as a multiphase optimal control problem, and there is software available online for this kind of problems. Examples are DIDO (matlab, proprietary), GPOPS-II (matlab, proprietary), PROPT(matlab, proprietary - i think), ICLOCS2 (matlab, free) and PSOPT (C++, free).
They all use pseudospectral methods, which are great to feed a computer, but not very practical to solve by hand. I don't know any method to solve analytically. However, I am certain that if $F$ and $G$ are not identical, then the Hamiltonian will (most likely) have a jump discontinuity at q = b.
This is known as a multiphase optimal control problem, and there is software available online for this kind of problems. Examples are DIDO (matlab, proprietary), GPOPS-II (matlab, proprietary), PROPT(matlab, proprietary - i think), ICLOCS2 (matlab, free) and PSOPT (C++, free).
They all use pseudospectral methods, which are great to feed a computer, but not very practical to solve by hand. I don't know any method to solve analytically. However, I am certain that if $F$ and $G$ are not identical, then the Hamiltonian will (most likely) have a jump discontinuity at q = b.