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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 $P(a)=0$.

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    $\begingroup$ Is it the case that $F(b,P(b),P'(b))=G(b,P(b),P'(b))$? If so and their derivatives also match at $b$ then you can define a piecwise $C^1$ function and use the Euler Lagrange equations. $\endgroup$ – Basel J. Jul 10 at 19:45
  • $\begingroup$ Comment to the post (v1): Does $P$ satisfies any conditions at $b$ and/or $c$? $\endgroup$ – Qmechanic Jul 11 at 11:02
  • $\begingroup$ Yes: I need $F(b,P(b),P'(b))=0=G(c,P(c),P'(c))$ $\endgroup$ – asd Jul 12 at 19:54
  • $\begingroup$ Those seem not necessary. Other conditions? $\endgroup$ – Qmechanic Jul 13 at 8:20
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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.

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