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## week 1

### Introduction

Lange, Ch. 1.

Basic notions: feasible points; criterion of optimality; maxima and minima.

Existence and nonexistence of optimal points; uniqueness and nonuniqueness.

Univariate problems. Critical points. Classical inequalities.

Multivariate problems (unconstrained).

Constrained problems; Lagrange multipliers.

Sample problems:

• On

## manifolds

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

Manifolds are topological spaces that locally look like Euclidean spaces.

Formalism: charts and atlases. Beware of pathologies (long line; double point…)! Smooth manifolds vs. topological ones.

Inverse function theorem (for a …

## final

$$\def\Real{\mathbb{R}} \def\Comp{\mathbb{C}} \def\Rat{\mathbb{Q}} \def\Field{\mathbb{F}} \def\Fun{\mathbf{Fun}} \def\e{\mathbf{e}} \def\f{\mathbf{f}} \def\bv{\mathbf{v}}$$
Due by noon 12.18.

Numbers in brackets indicate the points for each (sub)problem.

Questions? Misprints suspected? Post as a comment here!

1. Consider the vector field $$\bv$$ given by
$\dot{x}=-xy^2-x^5;\\ \dot{y}=-x^2y-y^5. ## december 5 1. Solve the Dirichlet problem in the disk $$D=\{x^2+y^2\leq 4\}$$ (i.e. find a harmonic function $$u$$ such that on the boundary $$u$$ is equal to 1. \[ x^3;$
2. $2x^2+y^2;$
3. $y^3-y.$
2. Let $$u$$ be a harmonic function in

## 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