Optimization—Theory and Applications pp 116-158 | Cite as

# Examples and Exercises on Classical Problems

## Abstract

Let us briefly consider, in terms of the integral \( {\text{I}} = {{\int_{{{{t}_{1}}}}^{{{{t}_{2}}}} {\left( {1 + {{{x'}}^{2}}} \right)} }^{{1/2}}}dt, \), the question of the path of minimum length between two fixed points, or between a fixed point and a given curve in the *tx*-plane. Here \( {{f}_{0}} = {{\left( {1 + {{{x'}}^{2}}} \right)}^{{1/2}}} \) depends on x’ only, and satisfies condition (S) of Section 2.8. Any extremal must satisfy Euler’s equation in the reduced form (2.2.10), \( {{f}_{{0x'}}} = c, \) or \( x'{{\left( {1 + {{{x'}}^{2}}} \right)}^{{ - 1/2}}} = c \). Here \( {{f}_{{0x'}}} = x'{{\left( {1 + {{{x'}}^{2}}} \right)}^{{ - 1/2}}} \) is a strictly increasing function of *x*′ with range (−1,1). Thus −1 < *c <* 1, and there is one and only one value *x*′ = *m*, depending on *c*, such that \(x{\left( {1 + {{x}^2}} \right)^{ - {1 \mathord{\left/ {\vphantom {1 2}} \right. \kern-\nulldelimiterspace} 2}}} = c\). From (2.6.iii) we know that any optimal solution is of class C^{2} and therefore an extremal. Thus an optimal solution must be a segment *x*′ = *m*, *x*(*t*)= *mt* + *b*,*m*, *b* constants.

## Keywords

Euler Equation Classical Problem Absolute Minimum Linear Manifold Polygonal Line## Preview

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