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Control Systems Assignments

Assignment 6: Calculus of variations

6.1 Find the extremal for the following functionals:
(i) $J(x)=\int_0^1[x^2(t)+\dot{x}^2(t)]dt;$with boundary condition (a) $ x(0)=0, x(1)=1$, (b) $x(0)=1, x(1)=free$.

(ii) $J(x)=\int_0^2[x^2(t)+2\dot{x}(t)x(t)+\dot{x}^2(t)]dt; x(0)=1, x(2)=-3$

(iii) $J(X)=\int_0^{\pi/2}[\dot{x}_1^2(t)+ \dot{x}_2^2(t) +2x_1(t)x_2(t)]dt$
(a)$X(0)=[0 \;0]^T, X(\pi/2)=[1 \;1]^T$
(b) $X(0)=[0 \;0]^T, x_1(\pi/2)=free, x_2(\pi/2)=1$

(iv) $J(x)=\int_0^1[0.5\dot{x}^2(t)+3x(t)\dot{x}(t)+2x^2(t)+4x(t)]dt; x(0)=1, x(1)=4$

(v) $J(x)=\int_0^1[0.5\dot{x}^2(t)+x(t)\dot{x}(t)+\dot{x}(t)+x(t)]dt; x(0)=1/2, x(1)=free$

(vi) $J(x)=\int_{-2}^0[12tx(t)+\dot{x}^2(t)]dt; x(-2)=3, x(0)=0$

(vii) $J(x)=\int_{1}^2[\dot{x}^2(t)/2t^3]dt; x(1)=1, x(2)=10$

6.2 Using Euler equation find the extremal for the following functionals:
(a)$J(x)=\int_0^1[x(t)\dot{x}(t)+\ddot{x}^2(t)]dt; x(0)=0, x(1)=2, \dot{x}(0)=1, \dot{x}(1)=4$

(b)$J(x)=\int_0^{\infty}[\dot{x}^2(t)+x^2(t)+(\ddot{x}(t)+\dot{x}(t))^2]dt;$ $x(0)=1, x(\infty)=0, \dot{x}(0)=2, \dot{x}(\infty)=0 $

6.3 Determine an extremal of $J(x)=\int_0^{t_f}\sqrt{1+\dot{x}^2(t)}dt;$
(a) $x(0)=0$, and terminate on the curve $\theta(t)=-4t+5$.
(b) $x(0)=0$, and terminate on the circle $(t-9)^2+x^2(t)=25$.
(c) $x(0)=0$, and terminal points lie on the circle $(t-5)^2+x^2(t)-4=0$.


6.4 Determine an extremal for $J(x)=\int_0^{t_f}[\sqrt{1+\dot{x}^2(t)}/x(t)]dt; x(0)=0$, and $x(t_f) $ lies on $\theta(t)=t-5$.

6.5 Find a point on the curve $y_2=y_1^2-4.5$, that minimizes the function $f(y_1,y_2)=y_1^2+y_2^2.$

6.6 Find the point in three dimensional Euclidean space that is nearest to the origin and satisfies the constraints $y_1+y_2+y_3=5; y_1^2+y_2^2+y_3=9$.

6.7 Determine the necessary conditions (excluding boundary condition) for which the following cost functions (with constraints) are extremal.
(a) $J(X)=\int_{t_0}^{t_f}[x_1^2(t)+x_1(t)x_2(t)+x_2^2(t)+x_3^2(t)]dt$ with the constrains $\dot{x}_1(t)=x_2(t)$; $\dot{x}_2(t)=-x_1(t)+[1-x_1^2(t)]x_2(t)+x_3(t)$.
(b) $J(X)=\int_{t_0}^{t_f}[\lambda +x_3^2(t)]dt; \lambda>0$ with the constrains $\dot{x}_1(t)=x_2(t)$; $\dot{x}_2(t)=x_3(t)$.

6.8 Brachistochrone Problem: Find the shape of the curve down which a bead sliding from rest and accelerated by gravity will slip (without friction) from one point to another in the least time.
Hints: First determine the cost function. It would be $J(x)=\int_{t_0}^{t_f}[\sqrt{1+\dot{x}^2(t)}/\sqrt{2gx(t)}]dt;$ Then solve with Euler equation.

6.9 Find the curve of length $L$ from $(x, t)=(0, 0)$, to $(x, t)=(0, b)$ point such that the area between the $x$ and the $t$ is maximal. Assume that $b$ and $L$ are fixed, with $L>b$.


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