By Daniel Liberzon

**Read or Download Calculus of Variations and Optimal Control Theory: A Concise Introduction (Free preliminary copy) PDF**

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**Example text**

D. and contains quite a few values. 33 when light passes from air to water). A satisfactory explanation of this behavior was first given by Fermat around 1650. Fermat’s principle states that, although the light does not take the path of shortest distance any more, it travels along the path of shortest time. Snell’s law can be derived from Fermat’s principle by differential calculus; in fact, this was one of the examples that Leibniz gave to illustrate the power of calculus in his original 1684 calculus monograph.

To give an example of such a situation, we now consider a simple variable-endpoint problem. 9) as before, the initial point of the curve is still fixed by the boundary condition y(a) = y 0 , but the terminal point y(b) is free. 9. The perturbations η must still satisfy η(a) = 0 but η(b) can be arbitrary. 9: Variable terminal point and this must be 0 if y is to be an extremum. Perturbations such that η(b) = 0 are still allowed; let us consider them first. 16). , it is still a necessary condition for optimality.

For example, we can set η(x) = (x − c) 2 (x − d)2 for x ∈ [c, d] and η(x) = 0 otherwise. This b gives a ξ(x)η(x)dx > 0, and we reach a contradiction. 1 that for y(·) to be an extremum, a necessary condition is Ly (x, y(x), y (x)) = d Lz (x, y(x), y (x)) dx ∀ x ∈ [a, b]. 17) This is the celebrated Euler-Lagrange equation providing the first-order necessary condition for optimality. 17): y and y are treated as independent variables when computing the partial derivatives L y and Ly , then one plugs in for these variables the position y(x) and velocity y (x) of the curve, and finally the differentiation with respect to x is performed using the chain rule.