Consider a resource allocation problem for a Martian base. A fleet of N reconfigurable, general purpose robots is sent to Mars at t= 0. The robots can (i) replicate or (ii) make human habitats. We model this setting as a dynamical system. Let z be the number of robots and b be the number of buildings. Assume that decision variable u is the proportion of robots building new robots (so, u(t) C [0,1]). Then, z(0) N, 6(0) = 0, and z(t)=au(t)r(1), b(1)=8(1 u(t))x(1) where a > 0, and 3> 0 are given constants. Determine how to optimize the tradeoff between (i) and (ii) to result in maximal number of buildings at time T. Find the optimal policy for general constants a>0, 8>0, and T≥ 0.

Answers

Answer 1

Overall, this policy balances the tradeoff between (i) and (ii) by allocating robots between replicating and building human habitats in a way that maximizes the number of buildings at time T using Bernoulli differential equation.

To optimize the tradeoff between (i) and (ii) and achieve maximal number of buildings at time T, we need to find the optimal value of u(t) over the time interval [0, T]. We can do this using the calculus of variations.

First, we need to define the objective function that we want to optimize. In this case, we want to maximize the number of buildings at time T, which is given by b(T). Therefore, our objective function is:

J(u) = b(T)

Next, we need to formulate the problem as a constrained optimization problem. The constraints in this case are that the number of robots cannot be negative and the total proportion of robots allocated to building new robots and making buildings must be equal to 1. Mathematically, we can express this as:

z(t) ≥ 0

u(t) + x(t) = 1

where x(t) is the proportion of robots allocated to making buildings.

Now, we can apply the Euler-Lagrange equation to find the optimal value of u(t). The Euler-Lagrange equation is:

d/dt (∂L/∂u') - ∂L/∂u = 0

where L is the Lagrangian, which is given by:

L = J(u) + λ(z(t) - z(0)) + μ(u(t) + x(t) - 1)

where λ and μ are Lagrange multipliers.

We can compute the partial derivatives of L with respect to u and u', and then use the Euler-Lagrange equation to find the optimal value of u(t).

After some algebraic manipulations, we obtain the following differential equation for u(t):

d/dt (u^2(t) (1-u(t))^2) = 4a^2u(t)^2 (1-u(t))^2

This is a Bernoulli differential equation, which can be solved by making the substitution v(t) = u(t) / (1-u(t)). After some further algebraic manipulations, we obtain:

v(t) = C / (1 + C exp(-2at))

where C is a constant of integration.

Finally, we can solve for u(t) in terms of v(t) using the equation u(t) = v(t) / (1 + v(t)).

Therefore, the optimal policy for maximizing the number of buildings at time T is given by:

u*(t) = v*(t) / (1 + v*(t))

where v*(t) is given by v*(t) = C / (1 + C exp(-2at)) with the constant C determined by the initial condition z(0) = N.

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Related Questions

survey of a company's employees explored the relationship between employee status and support of a new workplace safety policy The results are displayed in the following two-way frequency table. Workplace Safety Policy For Against Total Part-Time
Employee 20 60 80 Full-Time Employee 100 20 120 Total 120 BO 200 used on the table above, which of the following statements are true? There are twice as many part-time employees against the workplace safety policy than full-time employees against the workplace safety policy IL There are more employees for the workplace safety policy than against the workplace safety policy There is an equal number of part-time employees and full-time employees.​

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what is the HCF prime factorization of 30 and 36​

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Answer:

We found the factors and prime factorization of 30 and 36. The biggest common factor number is the GCF number. So the greatest common factor 30 and 36 is 6.

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X+3y=-7 5x+6y=-8​

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Step-by-step explanation:

I have used substitution method here to find the value of x and y.

Let
cos
(
2

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0.9994
What is the the value of
x
, in degrees, for
sin
(
x
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0.9994
?

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Answer:

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Proving trapezoid theorems Geometry

Answers

Answer:

ΔABD ≅ ΔACD by SAS, therefore;

[tex]\overline {BD} \cong \overline {CA}[/tex] by CPCTC

Step-by-step explanation:

The two column proof is presented as follows;

Statement [tex]{}[/tex]                   Reason

ABCD is a trapezoid [tex]{}[/tex]  Given

[tex]\overline {BA} \cong \overline {CD}[/tex]  [tex]{}[/tex]                   Given

[tex]\overline {BC} \parallel \overline {AD}[/tex]   [tex]{}[/tex]                   Definition of a trapezoid

ABCD is an isosceles trapezoid  [tex]{}[/tex]    Left and right leg are equal

∠BAD ≅ ∠CDA  [tex]{}[/tex]          Base angle of an isosceles trapezoid are congruent

[tex]\overline {AD} \cong \overline {AD}[/tex]   [tex]{}[/tex]                  Reflexive property

ΔABD ≅ ΔACD   [tex]{}[/tex]         By SAS rule of congruency

[tex]\overline {BD} \cong \overline {CA}[/tex]   [tex]{}[/tex]                  CPCTC

CPCTC; Congruent Parts of Congruent Triangles are Congruent

SAS; Side Angle Side rule of congruency

Answer:

Step-by-step explanation:

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mr. russell questions:

1: false.

2: false.

3: true

4: false.

5: true.

the last question: D.

PLEASE HELP WILL MARK BRAINLY

Use a ratio box to solve this problem: Five hundred fifty students attended the assembly. If the ratio of boys to girls in the assembly was 6 to 5, how many girls attended the assembly?

Answers

Answer:

250 girls

Step-by-step explanation:

Please let me know if you want me to write an explanation for my answer :)

Determine which fraction is smallest given 21/64, 5/16, 11/32, 1,2

Answers

Answer :

5/16

Explanation :

5 divided by 16 is less than the other one equations

Is this a, b, c, or d?

Answers

Answer:

Alright so let's start by elimination.

Step-by-step explanation:

It cannot be 1/4 (unless it's y or x)

Cannot be 1/2 unless it's the bottom part.

So that leaves you with 1/3 and 1/6, 1/6 is to small so it has to be 1/3.

Hope this helps!

Find the measure of each angle in the angle pair described.
The measure of one angle is twice the measure of its supplement. What are the measures of the two angles?

Answers

Answer:

The best way to measure an angle is to use a protractor. To do this, you'll start by lining up one ray along the 0-degree line on the protractor. Then, line up the vertex with the midpoint of the protractor. Follow the second ray to determine the angle's measurement to the nearest degree

You did not attach a photo of the angles!

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which of the following is a irratinol number
A. 1 squared
B. 41 squared
C.- 9 squared
D.-64 squared

Answers

B is the correct answerLet me know if I am wrong.

I dont understand the work my teacher gave me and I Have 45 pages to do ​

Answers

Answer:

just send them here , I will answer the ones I know .

Martha spent $12.35 at the local farmer's market. She bought 3 bundles of collards and paid $4.30 for some fruit. How much did each bundle of collards cost?

Answers

Answer:

12.35-4.30=8.05

8.05÷3= 2.68333333333333....

            ≈2.68

What’s 12 times 12divided by 58 ?

Answers

12 x 12 = 144
144 / 58 = 2.66

Answer:

2.48........is your answer

Step-by-step explanation:

12×12 =144

144÷58

5+5-66564+6367347646-4848

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Answer:

6367276244 6367276244

6367276244 6367276244

Answer:

6367276244

Step-by-step explanation:

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Answer:

djchdjcycuchrivivurtiytvivytivi8fr cgdyhcd8w

C, B, D, D, A, C, B, D, C, B

3.5 m = ______cm

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Answers

Answer:

Formula *10

Step-by-step explanation:

35

Two siblings' ages have a sum of 64 years. The older sibling was born 6 years before their young sibling.

Write a system of 2 equations. Solve the system using substitution. Report your answer in the form of (older, younger).

Answers

Hi!

So we have to make a system of equations here.

Let sibling 1 be O and sibling 2 be Y (for older and younger)

So:

O + Y = 64

O = 64 - Y

(Either one works)

The older sibling is 6 years older.

O = Y + 6

Make everything in terms of Y.

64 - Y = Y + 6

Same to both:

58 = 2y

2y = 58

Divide by 2 on both sides:

y = 29

Since the older sibling is 6 years older:

29 + 6 = 35

Y = 29

O = 35

Let me know if you have any questions.

Combine the like terms to create an equivalent expression:2r+1+(−4r)+7

Answers

Answer:

2r+1+(-4r)+7

= -2r+8

= -2(r+4)

Step-by-step explanation:

PLZ HELP!!! WILL GIVE HIGHEST POINTS IF THE ANSWER IS CORRECT!!!

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Answer: -5 Hope this helps :)

Step-by-step explanation:

Brainlist Plz?

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DIVIDING EXPONETS

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Here is the answer
I can’t seem to do the symbols here so the answer is in the picture.

Answer:

[tex]-0.5n^{4}[/tex] or [tex]-\frac{1}{2} n^{4}[/tex]

Step-by-step explanation:

To simplify a dividing exponent, use the "Quotient of Powers" law: [tex]\frac{a^{m} }{a^{n} } = a^{m-n}[/tex]. In this situation, the equation would be: [tex]\frac{-9}{18} n^{6} -n^{2}[/tex] or [tex]-0.5n^{4}[/tex] (-0.5 is also [tex]-\frac{1}{2}[/tex]).

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The points L, M, N and O all lie on the same line segment, in that order, such that the ratio of LM:MN:NOLM:MN:NO is equal to 3:1:2.3:1:2. If LO=30,LO=30, find LN.LN.

Answers

Answer:

[tex]|LN| = 24[/tex]

Step-by-step explanation:

Given

[tex]LM:MN:NO = 3:1:2[/tex]

[tex]LO = 30[/tex]

Required

Find LN

[tex]LN = LM + MN[/tex]

From the ratio, LM = 3 and MN = 1

So:

[tex]LN = 3 + 1[/tex]

[tex]LN = 4[/tex]

LN and NO can be represented as:

[tex]LN : NO = 4 : 1[/tex]

The length of LN is then calculated as:

[tex]|LN| = \frac{LN}{LN+NO}*|LO|[/tex]

[tex]|LN| = \frac{4}{4+1}*30[/tex]

[tex]|LN| = \frac{4}{5}*30[/tex]

[tex]|LN| = \frac{4*30}{5}[/tex]

[tex]|LN| = \frac{120}{5}[/tex]

[tex]|LN| = 24[/tex]

In circle O, AC and BE are diameters. The measure of arc DC is 50°. What is the measure of EBC? ​

Answers

Answer:

The angle measure would be 40°

Step-by-step explanation:

Right angles have a measure of 90. Since DC is 50, 90-50 is 40.

The angle measure of  ∠EBC would be 40°.

What is arc?

The arc is a portion of the circumference of a circle

Right angles have a measure of 90°.

∵∠BOD=90°

Since ∠DC is 50°,

∠EBC = ∠BOD-∠DC

∠EBC = 90°-50° is 40°.

The angle measure of  ∠EBC would be 40°.

learn more about arc of circle

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A rental car company offers a rental package for a mid-size car. ​The cost is comprised of a fixed \$ 40$40 administrative fee for the cleaning and maintenance of the car plus a rental cost of \$ 45$45 per day. ​ ​Using xx for the number of days and yy for the total cost in dollars, construct a linear function to model the relationship between the number of days and the total cost of renting a mid-size car. ​

Answers

Answer:

y = 40 + 45x

Step-by-step explanation:

Fixed administrative fee for the cleaning and maintenance of the car = $40

rental cost = $45 per day

x = number of days

y = total cost in dollars

y = 40 + 45x

This shows the relationship between the number of days and the total cost of renting a mid-size car.

Work out
51.43% of 377.59cm
Give your answer founded to 2 DP

Answers

Answer:

194.194537 centimeters

Step-by-step explanation:

Answer:

194.19cm

Step-by-step explanation:

yeah yeah yeah yeah

The speed limit on a road is 45 mph what is the rate in miles per minute

Answers

The answer is 320 minutes

The rate in miles per minute is 320

The school that Stefan goes to is selling tickets to a choral performance. On the first day of ticket sales the school sold 3 senior citizen tickets and 1 child ticket for a total of $38. The school took in $52 on the second day by selling 3 senior citizen tickets and 2 child tickets. Find the price of a senior citizen ticket and the price of a child ticket

Answers

Answer:

senior citizen tickets are $10 and child tickets are $8

Answer:

senior is 10 and child is 8

Step-by-step explanation:

Julie has $80,000 to invest. She invests part at 5%, one fourth this amount at 6%, and the balance 7%. Her total annual income from interest is $4,700. Find the amount invested at each rate.

Answers

Answer:

Julie invested $ 25,000 at 7%, $ 20,000 at 6%, and $ 35,000 at 5%.

Step-by-step explanation:

Given that Julie has $ 80,000 to invest, and she invests part at 5%, one fourth this amount at 6%, and the balance 7%, knowing that her total annual income from interest is $ 4,700, to find the amount invested at each rate se you must perform the following calculation:

80,000 / 4 = 20,000

20,000 x 0.06 = 1,200

4,700 - 1,200 = 3,500

60,000 x 0.07 + 0 x 0.05 = 4,200

50,000 x 0.07 + 10,000 x 0.05 = 4,000

40,000 x 0.07 + 20,000 x 0.05 = 3,800

30,000 x 0.07 + 20,000 x 0.05 = 3,600

25,000 x 0.07 + 35,000 x 0.05 = 3,500

Thus, Julie invested $ 25,000 at 7%, $ 20,000 at 6%, and $ 35,000 at 5%.

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