Analyze the stability of the critical point of the following system, proposing as a Lyapunov function V(x,y)=ax2m+by2n
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Analyze the stability of the critical point of the following system, proposing as a Lyapunov function V(x,y)=ax2m+by2n
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- 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.3. Find all critical points of the functionsf (x, y) = e2y (x² + y²) and determine whether the critical point(s) is a local maximum, local minimum or saddle point.(a) Determine whether the set of functions f₁(x) = x, f₂(x) = is linearly independent on the interval (-∞, ∞). = x= 1, f3(x) = x+3
- Let g(x, y) = x³ – 6x² – y* + 5. Find and classify the critical point(s) of g.A linear continuous time system is given by a state space model with A-[ ]), a-[1]. B = В C = [1 - 1], D=0 4 Analyze the following properties of the above system for different a: (i) asymptotic stability (ii) controllability, and (iii) observability.Analyze the stability of the critical point of the following system, proposing as a Lyapunov function V(x,y)=ax2m+by2n. a) x′=−2xy, y′=x2−y3 b) x′=xy2−(x3/2), y′=−(y3/2)+(yx2/5) c) Show that x′=−2x+xy3, y′=−x2y2−y3 is asymptotically stable. Pleas be as clear as possible and legible. Show and explain all the steps in detail. Thank you very much.
- Consider the Keynesian consumption function Yt = B₁ + B₂x2t + &t where yt is per capita consumption, and x2+ is per capita income. The coefficient ₂ is interpreted causally as the marginal propensity to consume, and we expect 0We intend to design a system for optimal control (u₁(t), u2(t)) which represent the Tuberculosis (TB) treatment rates for latent class and infectious class, respectively. Our target is to minimize the TB infected individuals (including latent and infectious individuals) as well as the costs required to con- trol TB by treating latent and infectious individuals, over a certain time horizon [0,tf]. The cost of each intervention is assumed to be proportional to the square of its intensity. Thus we formulate the optimization problem below. Minimize the objective function ' A₁E(t) + A₂I(t) + ¹¹³u²(t) + Bu²(t) J(u(.)), u(·) = "h 1 subject to dS = A(1p) BSI+ kV - ds, dt dV dt dE ུ|ཙ⪜ཊྛི|༴ཁྱི|ཙ dt dt = Ap-kV - dV, = BSI - (u(t) ++ d)E + OR, dR = E-(d--u2(t))I, = u₁(t)Eu(t) (d+ 0)R S(0) = So≥0, V(0) = √≥0, E(0) = Eo ≥0,1(0) = 10 ≥0, R(0) = Ro≥ 0. The control variables are assumed to be bounded: Given any t > 0, 0Describe the long run behavior of f(p)=p8−p7+2p3+5As p→−∞, f(p)= As p→∞, f(p)=Recommended textbooks for youAdvanced Engineering MathematicsAdvanced MathISBN:9780470458365Author:Erwin KreyszigPublisher:Wiley, John & Sons, Incorporated
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