Engineering Optimization: Theory and Practice, Fourth Edition

Problems 663 11.5 What is a random variable? 11.6 Give two examples of random design parameters. 11.7 What is the difference bet ...

664 Stochastic Programming distribution, fV(ν)=    1 V 0 e−(ν/V^0 ) forν≥ 0 0 forν < 0 whereV 0 is the mean velocity, de ...

Problems 665 11.13 The range (R)of a projectile is given by R= V 02 g sin 2φ whereV 0 is the initial velocity of the projectile, ...

666 Stochastic Programming Time per unit (min) for product: Stage capacity A B C (mins/day) Stage Mean Standard deviation Mean S ...

Problems 667 11.19 An article is to be restocked every three months in a year. The quarterly demandUis random and its law of pro ...

12 Optimal Control and Optimality Criteria Methods 12.1 Introduction In this chapter we give a brief introduction to the followi ...

12.2 Calculus of Variations 669 12.2.2 Problem of Calculus of Variations A simple problem in the theory of the calculus of varia ...

670 Optimal Control and Optimality Criteria Methods Figure 12.1 Tentative and exact solutions. Also, we define the variation of ...

12.2 Calculus of Variations 671 Thus δA= ∫x 2 x 1 [ ∂F ∂u − d dx ( ∂F ∂u′ ) + d^2 dx^2 ( ∂F ∂u′′ )] δu dx + [ ∂F ∂u′ − d dx ( ∂F ...

672 Optimal Control and Optimality Criteria Methods Figure 12.2 Curve of minimum time of descent. Since potential energy is conv ...

12.2 Calculus of Variations 673 depend on the shape of the body and the relative velocity in a very complex manner. However, if ...

674 Optimal Control and Optimality Criteria Methods Figure 12.4 Element of surface area acted on by the pressurep. Findy(x)which ...

12.2 Calculus of Variations 675 Hence the shape of the solid body having minimum drag is given by the equation y(x)=R (x L ) 3 / ...

676 Optimal Control and Optimality Criteria Methods To formulate the problem, we first write the heat balance equation for an el ...

12.2 Calculus of Variations 677 By substituting Eq. (E 9 ) in (E 7 ), the variational problem can be restated as Findy(x)which m ...

678 Optimal Control and Optimality Criteria Methods The value of the unknown constantλcan be found by using Eq. (E 7 ) as m= 2 ρ ...

12.3 Optimal Control Theory 679 which minimizes the functional, called theperformance index, J= ∫T 0 f 0 ( x,u,t) dt (12.21) whe ...

680 Optimal Control and Optimality Criteria Methods is a function of the two variablesxandu, we can write the Euler–Lagrange equ ...

12.3 Optimal Control Theory 681 Differentiation of Eq. (E 5 ) leads to 2 u ̇+ ̇λ= 0 (E 6 ) Equations(E 4 ) and (E 6 ) yield u ̇= ...

682 Optimal Control and Optimality Criteria Methods Now we introduce aLagrange multiplier pi, also known as theadjoint variable, ...