I was looking into Parker Solar Probes trajectory and was wondering what would be the most optimal way to utilize a retrograde thrust at perihelion to reduce the number of needed gravity assists. I think it would be best to apply the speed reducing thrust at perihelion since, by the Oberth effect, the faster you are moving the more you can benefit from a thrust.
If PSP were to apply a thrust at its first perihelion it would require an impossible amount of fuel for it to get it to its target distance of 0.05 AU from the Sun. If it applied this thrust at its second perihelion since it is closer it would require slightly less fuel per mass to get it to its target distance and also since it is moving faster, for the same thrust we should need less fuel (still an impossible amount for the given mass though).
So say we are able to add enough fuel, keeping the mass the same, such that if we were to burn it all at PSP's 3rd perihelion then we wouldn't need to go back to Venus for another assist (PSP would achieve its target perihelion distance). I was wondering if it would require less fuel to instead of burning all the fuel we have at the 3rd assist to instead burn 1/3 of the fuel on the 1st, 1/3 on the 2nd, and 1/3 on the 3rd such that the velocities are increasing but less fuel is being burned.
This seems like a simple problem to me but I am unsure how to solve it. For example, are there any formulas where I could estimate the effect of a certain retrograde thrust on the resulting aphelion velocity and then the perihelion velocity of the following orbit and then perihelion distance?