Consider the problem minimize 1[r(-)] = 2 / r,(t)2 dt subject to the conditions r(0) - r(T)0 and ...
(5) Consider the problem: minimize I[r(.)] - /r2 dt 0 subject to the conditions x(0)-x()-0 and the constraint 0 R is a C2 function that solves the above Suppose that x : [0, π] Let y : [0, π] → R be any other C2 function such that y(0) = Define problem y(n) 0. 0 an a(s) a. Explain why α(0)-1 and i'(0) b. Show that 0. i'(0)r'(t) y'(t) dt -X /x(t) y(t) dt 0 0 for some constant λ,...
Consider the optimization problem 5-6 5-6 F=(X-I)2 + (X Minimize: Subject to: 2-1) X +X-0.5s 0 a. Write the expression for the augmented Lagrangian using r'p = 1. b. Beginning with λ 1 0 and λ2-0 , perform three iterations of the ALM method. c. Repeat part (b), beginning with λ 1-1 and λ2-1 d. Repeat part (b), beginning with λι--I and λ2--1
Exercise 7.3. Consider the nonlinearly constrained problem minimize xER2 to (7.1) a x2 1 = 0. subject 1)T is a feasible path for the nonlinear constraint (a) Show that x(a) x x - 1 = 0 of problem (7.1). Compute the tangent to the feasible path at E = (0, 0)7 (sin a, cos a - + X (b) Find another feasible path for the constraint x? + (x2 + 1)2 - 1 = 0. Compute the tangent to the...
(6). The quantities x(t) and y(t) satisfy the simultaneous equations dt dt dx dt where x(0)-y(0)-ay (0)-0, and ax (0)-λ. Here n, μ, and λ are all positive real numbers. This problem involves Laplace transforms, has three parts, and is continued on the next page. You must use Laplace transforms where instructed to receive credit for your solution (a). Define the Laplace Transforms X(s) -|e"x(t)dt and Y(s) -e-"y(t)dt Laplace Transform the differential equations for x(t) and y(t) above, and incorporate...
Problem 2. Solve the given initial-value problem: dx = -xt, r(0) = 1/VT 1. dt dy 2. dt y(0) = 4 y – t?y'
(a) Let θ : R-+ R be a smooth function. Find the (signed) curvature of the curve a:R- R2 given by cos(θ(t)) dt,I α(s) sin(θ(t)) dt Use your result to give another geometric interpretation to the (signed) curva- ture and its sign? to) rindy,R-- parmetrised with unit speed suchhat y -0and kt) - s for all seR. (a) Let θ : R-+ R be a smooth function. Find the (signed) curvature of the curve a:R- R2 given by cos(θ(t)) dt,I...
(1 point) A. Let g(t) be the solution of the initial value problem dy dt with g(1)1 Find g(t) B. Let f(t) be the solution of the initial value problem dy dt with f(0) 0 Find f(t). C. Find a constant c so that solves the differential equation in part B and k(1) 13. cE (1 point) A. Let g(t) be the solution of the initial value problem dy dt with g(1)1 Find g(t) B. Let f(t) be the solution...
(3) Solve the IVP + 6y(t) + 9 Sy()dt = 1, y(0) = 0. (4) Find a(t) that satisfies e(t) = e-t +S* sinh(t – 7)2(7) dt.
Assume u E C2 (B (0,r)) solves the boundary-value problem where g E C(OB(0,r)). Show that gry.ndS(y) (хев"(0. т.)) which is called Poisson's formula with Poisson's kernel Assume u E C2 (B (0,r)) solves the boundary-value problem where g E C(OB(0,r)). Show that gry.ndS(y) (хев"(0. т.)) which is called Poisson's formula with Poisson's kernel