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After learning about the calculus of variations and optimal control for a bit this semester, I've decided to tackle a "simple" (in the words of my professor) problem meant to illustrate a simplified example of highway construction.

Suppose the cost of constructing a road of length L over uneven terrain is proportional to the difference between the unknown height of the road and the known height of the existing terrain, i.e. COST = k*abs(y(x) - h(x)) for some constant k>0 and non-negative functions y(x) (the height of the road to be built at point x, 0<=x<=L) and h(x) (the known height of the terrain at point x, 0<=x<=L).

I also want to consider the constraints

1. abs(y'(x)) <= M1 > 0,

implying that the grade (slope) of the road is never too excessive, and

2. abs(y''(x)) <= M2 > 0,

implying that the grade itself doesn't change too rapidly.

Finally, the terminal values y(0) and y(L) are free.

Hence the problem is

minimize integral k*abs(y(x) - h(x)) dt on the interval [0,L]

subject to abs(y'(x)) <= M1, abs(y''(x)) <= M2, y(0) = y_0 free, y(L) = y_L free.

I'm pretty comfortable with the methods of solving these problems, but this one has me stumped because of the constraints. My textbook doesn't feature anything about incorporating bounded derivative constraints for a CoV problem. Could someone run me through the basics, or maybe point me in a direction?

Would it be easier to formulate this problem via optimal control? If so, I could use a little nudge there too.

Again, I won't need help actually determining the solution, outside of figuring out how to incorporate these constraints. I hope I'm not missing something glaringly obvious. Thanks in advance!

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# Variational calculus with bounded derivative constraints

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