Refer to page 34 for deriving and applying Pontryagin's Maximum Principle. Instructions: ⚫ Define the Hamiltonian for the given control problem. • • Derive the necessary conditions for optimality step-by-step, including state and co-state equations. Solve the resulting system of equations explicitly, showing all intermediate steps. Link: [https://drive.google.com/file/d/1wKSrun-GlxirS31Z9qoHazb9tC440AZF/view?usp=sharing]

After the availability of covid 19 vaccinces in the Phiippines, the data below shows the number of senior citizens who received the vacines in City of Manila. Week No. 1 5 9 Senior Citizens who are Vaccinated 100 120 210 300 250 Based on the above data, determine the model of the vaccinated senior citizens. Determine also the vaccinated senior citizens in the 15th week.

Consider one application in which either a first order or second order IVP is formed to find a solution in a model problem.

Find the minimum value of x +y +z² givenx+y+z=3a.,

Traffic Control  The accompanying figure shows the flow of downtown traffic during the rush hours on a typical weekday. Each avenue can handle up to 1500 vehicles/hour without causing congestion, whereas the maximum capacity of each street is 1000 vehicles/hour. The flow of traffic is controlled by traffic lights at each of the six intersections. (A) Suggest two possible traffic-flow patterns that ensure no traffic congestion. (B) Suppose the traffic flow along 9th Street between 5th and 6th Avenues, x6, is restricted due to sewer construction. What is the minimum permissible traffic flow along this road that will not result in traffic congestion?

An ecologist models the interaction between the tree frog (P) and insect (N) populations of a small region of a rainforest using the Lotka-Volterra predator prey model. The insects are food for the tree frogs. The model has nullclines at N=0, N=500, P=0, and P=75. Suppose the small region of the rainforest currently has 800 insects and 50 tree frogs. In the short term, the model predicts the insect population will • and the tree frog population will At another point time, a researcher finds the region has 300 insects and 70 tree frogs. In the short term, the model predicts the insect population will * and the tree frog population will

How might the competitive-hunter model be validated? Include adiscussion of how the various constants a, b, m, and n might beestimated. How could state conservation authorities use the modelto ensure the survival of both species?

1.2.4. Consider the spreading of a highly communicable disease on an isolated island with population size N. A portion of the population travels abroad and returns to the island infected with the disease. Formulate a dynamical system to approximate the change in the number of people in the population who have the disease.

please help me to do the first question,thank you!