6. Consider a control system given by1,2+3 u,1 3133.with r(0) = (1,0,0). Find a control input u : [0, +o0)-+ R that minimizesI (1)||? + u?(t)dt.If you are using any computational tools, you may express any relevant values or matrices rounded tothree decimals.

Answered: 6. Consider a control system given by 1…

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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Consider the following optimisation problemmin f(x, y) = x + y − x2subject to x + y ≤ 1x ≥ 0, y ≥ 0.a) Find a critical point of the Lagrangian.b) Find a better solution to the problem above than the critical point ofthe Lagrangian calculated in a).c) What sufficient condition for the optimality of the Lagrangian solutionis violated by the problem.
Give a clear and detailed solution of the intergral of e^-ct/m 2)
Given the following R2→R function:b) f(x,y)= x2+5y2-2x-20y+24Find and analyze the nature of the critical points using the algebraic method and Hessian matrix and show that both methods lead to the same results.
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This problem is exercise 5 from Robert L. Devaney's An Introduction to Chaotic Dynamical Systems
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10.The average sizes of the prey and predator populations are defined as ¯x=1T∫A+TAx(t)dt,¯y=1T∫A+TAy(t)dt,x¯=1T∫AA+Txtdt,y¯=1T∫AA+Tytdt, respectively, where T is the period of a full cycle, and A is any nonnegative constant. a.Using the approximation (24), which is valid near the critical point, show that ¯x=c/γx¯=c/γ and ¯y=a/αy¯=a/α. b.For the solution of the nonlinear system (2) shown in Figure 9.5.3, estimate ¯xx¯ and ¯yy¯ as well as you can. Try to determine whether ¯xx¯ and ¯yy¯ are given by c/γ and a/α, respectively, in this case. Hint: Consider how you might estimate the value of an integral even though you do not have a formula for the integrand. c.Calculate other solutions of the system (2)—that is, solutions satisfying other initial conditions—and determine ¯xx¯ and ¯yy¯ for these solutions. Are the values of ¯xx¯ and ¯yy¯ the same for all solutions? In Problems 11 and 12, we consider the effect of modifying the equation for the prey x by including a term −σx2 so that…
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the consideration of the Rosenzweig–MacArthur predator–prey model introduced in Problem 13 of Section 9.5. Consider the system x′=x(a−0.2x−2yx+6),y′=y(−0.25+xx+6),x′=xa−0.2x−2yx+6,y′=y−0.25+xx+6, where a is a positive parameter. Observe that this system includes the one in Problem 18 above and the one in Problem 13 in Section 9.5. a.Find all of the critical points. b.Consider the critical point in the interior of the first quadrant. Find the eigenvalues of the approximate linear system. Determine the value a0 where this critical point changes from asymptotically stable to unstable. c.Draw a phase portrait for a value of a slightly greater than a0. Observe that a limit cycle has appeared. How does the limit cycle change as a increases further?
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6. Consider a control system given by 1 2 2 2 1 3 3 () u, ra, with r(0) (1,0,0). Find a control input u : [0, +o0) R that minimizes If you are using any computational tools, you may express any relevant values or matrices rounded to three decimals.