Tagged Questions
1
vote
0answers
36 views
How do I solve an optimal control problem when the state and control are multiplied?
Suppose I have the following objective function
$$
R = \sum_t^T x_tu_t + ku_t^2
$$
subject to
$$
\Delta x_t = m u_t + n x_t
$$
where $x_t$ is the state and $u_t$ is the control. $x_0$ is known.
How ...
3
votes
1answer
58 views
minimization problem on differential equations - optimal control
I am trying to minimize an time-integral of a linear function with respect to differential equations. The problem is formally defined as follows:
Given $\lambda< \mu_1, \mu_2$ fixed ...
0
votes
1answer
52 views
Enigmatic optimization problem
My problem, which I proposed to myself months ago is based on the simple optimization problem
in which you find the best path for a lifeguard to rescue a drowning victim. Obviously the
shortest ...
4
votes
0answers
63 views
Optimizing a functional with a differential equation as a constraint
I am working on solving the following optimization problem. I think it is well-poised but, if not, please give me some pointers that could make the question make more sense.
We have a parametric ...
0
votes
1answer
30 views
Numeric Differentiation of Analytic funtion
Can anyone validate if my understanding regarding numeric differentiation is correct??
$z = f(x,y)$ is an analytic function.
$$\frac{\partial z}{\partial x} = \frac{f(x+h,y)-f(x,y)}{h}$$
...
1
vote
0answers
97 views
A dynamic Stackelberg game - general characterization
my question is about general representation of a dynamic Stackelberg game which is played in continuous time. We have maximization problems of two agents who play this game. Agents are 'Leader' and ...
0
votes
1answer
58 views
2nd Order Optimal Control Problem
I'm working on a homework problem in optimal controls and my plant model is described as:
$$\ddot{x}(t) = u(t)$$
The performance index (cost function) is described by:
$$J = 1/2\int_0^5u^{2}(t)dt\,$$
...
1
vote
1answer
60 views
how to obtain Euler equation for smoothing spline minimization problem?
The question might be trivial, but I don't understand why this minimization problem in Sobolev space
$$
\min_{g}\int_{0}^{1}\left\{ f(x)-g(x)\right\}^{2} dx+\lambda\int_{0}^{1}\left\{ ...
0
votes
0answers
88 views
Differential Equations, Probability/Statistics, Optimization Problem - Relations?
While I am working on some physical/mathematical problems, I feel strongly that these three areas are almost the identical thing, except that they have different methods/from different aspects to ...
2
votes
1answer
265 views
Going in the direction of the gradient
First, a motivating example. Suppose $f(x)$ is convex, differentiable, with a single minimum $x^*$. Then the differential equation $$\dot{x}(t) = -\nabla f(x(t))$$ drives $x(t)$ to $x^*$.
Now my ...
3
votes
1answer
61 views
Maximizing $f(x,y)$
Could somebody please shed some light on this problem?
Let $x,y \in \mathbb R$, we wish to maximize $f(x,y)=\frac{x^2-y^2}{(x^2+y^2)^2}$ by finding suitable values of $x,y$.
Setting $\partial f\over ...
2
votes
1answer
156 views
Find equation of line such that area formed by line & positive coordinate axis is minimal
Find equation of line passing through $(20,12)$ such that the area of the triangle formed by the line and the positive axis is smallest possible.
Also: $\frac{x}{a}+\frac{x}{b}=1$
where $a, b$ are ...
5
votes
2answers
955 views
Euler-Lagrange, Gradient Descent, Heat Equation and Image Denoising
For an image denoising problem, the author has a functional $E$ defined
$$E(u) = \iint_\Omega F \;\mathrm d\Omega$$
which he wants to minimize. $F$ is defined as
$$F = \|\nabla u \|^2 = u_x^2 + ...
5
votes
2answers
261 views
How to Interpret Time Scales in a Dynamic System
Here I have a question about time scales in dynamic systems - for reference you can look at a previous question that spurs this one:
Minimizing the cost of a path in a dynamic system
That question ...
2
votes
1answer
126 views
Minimizing the cost of a path in a dynamic system
So suppose I want a path from 0 to $c>0$ on the real line, and I am going to use the function $S(t)$ to get there in (discrete) time $T$. That is, my position at time 0 is 0, my position at time $T$ ...
3
votes
3answers
2k views
Karush-Kuhn-Tucker condition - Lagrange multiplier
I was maths student but now I'm a software engineer and almost all my knowledge about maths formulas is perished.
One of my client wants to calculate optimal price for perishable products. He has ...
