Solve the following optimal control problem:minimize (y(t)² + ku(t) ²) dtsubject to x(t) = [1]x(t) + [q]u(t), y(t) = x₁(t), x₁(0) = 1,x₂(0) = 1where k is a positive parameter.a) Determine the optimal control and the corresponding optimal cost.b) Determine the location of closed-loop poles (eigenvalues) depending on k.(You can consider this part of the question as a root-locus plot problem!)

Solution: Determine the optimal control and corresponding optimal cost Given: minimize…

Answered: Solve the following optimal control…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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This extreme value problem has a solution with both a maximum value and a minimum value. Use Lagrange multipliers to find the extreme values of the function subject to the given constraint. f(x, y, z) = xy2z; x2 + y2 + z2 = 16
Find the maximum and minimum values of the function f(x, y, z) = 3x – y – 3z subject to the constraints x? + 2z = 361 and a + y – z = 10. Maximum value is occuring at ( ). Minimum value is occuring at ( ).
3. Use the method of Lagrange multipliers to find the minimum value of f (x, y) = x² + 4y² + 2x + 5 subject to y² + 2x = 10. (You may assume that the function f(x,y) subject to the given constraint attains a minimum.)
Given an annual capital investment of RM x million, along with the employment of y million unskilled labour hours and z million skilled labour hours per year, the annual sugar production output (in tonnes) of a sugarcane mill can be modelled by the function 16 Q(x, y, z) = - -enz 100 in which the values of x, y, and z should adhere to the budget constraint of 2x² + y² + 2z² = 21 Apply both the D-test method AND the method of Lagrange multipliers to determine the optimal allocation of capital investment and the most effective utilization of unskilled and skilled labour force to maximize sugar production. Then, compare and comment on the two methods used.
4. Find the stationary points of the following functions, and determine whether each is a maximum, minimum, or a saddle point. (a) (b) 1 ƒ (x) = ₁ ¹ x, x # 1 f(x,y)=x²-xy+²³ 1
Suppose that Marie produces milk q using her own labor l and cattle k using the production functionq = f(k, l) = k2/3ℓ1/3Although Marie does not need to pay anyone to use either input, the opportunity costs of labor and cattle are w = 1 and v = 16, respectively, and P is the price of milk. a) Set up Marie’s short run profit maximization problem and find her short run supply curve q(P). b) Now consider Marie’s problem in the long run, where her stock of cattle may vary. Set up her long run cost minimization problem and find her labor ℓc(q) and capital kc(q) demands and cost function C(q). c) Set up Marie’s long run profit maximization problem and find her long run supply curve q(P).
Find the minimum or maximum value of f (as indicated) subject to the given constraint. Minimum of f(x, y) = x – 14x + y“ – 16y, subject to 2x + 3y = 12 %3D O A. Minimum = - 24 at (2,0) O B. Minimum = - 68 at (1,5) O C. Minimum = -61 at (3,2) O D. Minimum = - 15 at (0,1)
A small company produces three types of goods. The profit function is given by P(x, y, z) = 5 ln(x + 1) + 8 ln(y + 1) + 12 ln(z + 1), where x, y and z are the monthly quantities sold of goods of type 1, 2 and 3, respectively. The production cost per unit of type 1 goods is $ 10; the cost per unit of type 2 goods is $ 15; the cost per unit of type 3 goods is $ 30. The monthly budget of the company (which is spent entirely on the 3 types of goods) is $2945. How should the budget be allocated to maximize the monthly profit? You do not need to investigate boundary points and it suffices to find a relative maximum. You might want to work with the variables x = x + 1, y = y + 1 and z = z + 1 to make the calculations easier.
The function f(x,y) = e 1xy has an absolute maximum value and absolute minimum value subject to the constraint x + xy + y = 36. Use Lagrange multipliers to find these values. The absolute maximum is (Type an exact answer in terms of e.)
(a) Let f(x, y) = x² + 2y². Find the extreme values of f(x,y) subject to the constraint equation x² + y? = 1.
c) A student wishes to allocate her available study time of 60 hours per week between two subjects in such a way as to maximize her grade average. She formulates two functions governing the grades of each module which can be taken as: 9₁ (t₁) = 20 + 20√₁, with t₁ > 0 and 9₂ (t₂) = -80 + 3t2, with t₂ > 80/3 maximize the grade average subject to the time constraint t₁ + t₂ = 60. You do not need to show that you have obtained a maximum.

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