Author Archive | yuliy

parabola construction

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}}$$

Consider the optimal control problem
$\dot{x}=u; x(0)=x_o; \int_0^T u^2/2dt+M(x(T))\to\min.$
The task is to find the cost function $$S(x_o)$$ (the optimal achievable value).

It can be solved as follows. Assume for …

ece515, homework 5

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}}$$
Due by midnight of december 4.

Use MATLAB or octave if needed.

1. Consider the optimal control problem
${x}”’=u; \int_0^\infty u^2+x^2 dt\to\min.$
• Write this system as a LQR;
• Solve the ARE;

november 21

1. Find sin transform of the function on $$[0,1]$$
given by
1. $$1$$;
2. $$\cos(\pi x)$$;
3. $$x$$.
2. Solve the heat equation
$u_t=\frac{1}{2} u_{xx},$
on $$[0,1]$$ with boundary conditions $$u(0,t)=u(1,t)=0$$ and initial values
1. u(x,0)=$$1$$;
2. u(x,0)=$$\cos(\pi x)$$;
3. u(x,0)=$$x$$.
3. Solve the heat equation
$math285, week of november 17 Read the textbook, chapters 9.5 and 9.7. Homework (due by Monday, 12.1): 1. Find the cosine transform of $$\sin(x)$$ on $$[0,\pi]$$. 2. Using separation of variables, solve the heat equation on the interval $$[0,\pi]$$: \[u_{t}=u_{xx}; \quad u_x(0,t)=u_x(\pi,t)=0; u(x,0)=\sin(x) november 14 1. Find the solutions of the wave equation at time $$t=1/6,1/3,1/2$$ \[ u_{tt}=u_{xx}, u(0,t)=u(1,t)=0,$
on $$[0,1]$$
with initial data $$u_t(x,0)=0$$ and $$u(x,0)$$ is as shown:

2. Same question, but the initial data are
$$u(x,0)=0$$ and $$u_t(x,0)$$ is as shown:

3. Sketch the

math 285, week of november 10

Read the textbook, chapters 9.4 and 9.6.

Videos to watch.

Homework (due by Monday, 11.17):

1. Find sine transform of the function (\T(x)\) on $$[0,1]$$ equal to $$x$$ for $$0\leq x\leq 1/2$$ and to $$1-x$$ for $$1/2\leq x \leq1$$.
2. Using

november 6

• Find Fourier series for $$2\pi$$-periodic function defined on $$-\pi,\pi$$ by
1. $\cos(2x)\sin^2(x);$
2. $\cos^3(x);$
3. $f(x)=\left\{\begin{array}{rcl} 1& \mathrm{if}& -\pi\leq x<0;\\ -1 & \mathrm{if}& 0\leq x \leq \pi.\\ \end{array}\right.$
• Find the Fourier series for the function which is
• notes for lectures 10-14

here

Many thanks to James!…

here

ece515, homework 4

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}}$$
Due by midnight of november 8.

1. Consider the system $$\dot{x}=Ax+bu$$,
$A=\left( \begin{array}{ccc} 1&1&0\\ 1&1&1\\ 0&1&1\\ \end{array}\right); b=\left( \begin{array}{c} b_1\\ b_2\\ b_3\\ \end{array}\right).$
• Derive conditions on $$b$$ making the system controllable.