Questions tagged [optimal-control]
Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. (Def: http://en.m.wikipedia.org/wiki/Optimal_control)
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Running Supremum of standard Brownian motion and probability distribution [closed]
I am reading Optimal Stopping and Free-Boundary Problems by Peskir and Shiryaev and noticed a result on page 151 as follows:
$\mathbb{P} (\sup_{t \geq 0} (B_t - \alpha t) \geq \beta) = \exp(-2\alpha \...
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2D rigid body dynamics thrust vector-controlled rocket and optimal control
I have to do some calculations about the optimal control of a powered landing of a thrust vector-controlled rocket. As a start I found [1] which is almost what I need, which is the system (1) but have ...
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Uniqueness of optimal control in infinite horizon LQR
Consider the following discrete finite horizon LQR problem with dynamics
$$
x_{t+1} = Ax_t + B u_t$$
and cost matrices $Q$ and $R$. The goal is to find a linear control $K$ which optimizes
\begin{...
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An Optimization Problem With Permutation Function [closed]
When I tried to solve an one-to-one assignment problem, I constructed it as the following optimization problem, which is a min-max optimization problem with the optimization objective being functions.
...
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Proving the two optimization are equivalent or not
Here are two optimization problems:
$$
(P) : \inf_{x} [ f(x)|g(x) \leq 0 ] \text{ where } g(x) = \inf_{y} [ h(x,y)|y \in Y ]
$$
$$
(Q) : \inf_{x,y} [f(x)|h(x, y) \leq 0, y \in Y ]
$$
Are they ...
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2
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Show that if $(A, B)$ is controllable then the D.T described by $x(k+1) = Ax(k-1) + Bu(k)$ with initial states $x(0), x(1)$ is also controllable
I show this statement on some lecture notes :
It is obvious that if $(A, B)$ is controllable then the D.T with initial conditions $x(0), x(1)$ described by
$$ x(k+1)= Ax(k-1) + Bu(k)$$ is also ...
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What does "Mean field" really mean (across domains)?
I have a terminology question about the use of the phrase "mean-field", coming from someone in control. Other examples of "mean-field" objects include
mean-field physics
mean-...
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Discrete time linear quadratic regulator - uniqueness of feedback gain given Riccati solution
Suppose we have a controllable discrete time linear system
\begin{align*}
x_{t+1} = Ax_t + Bu_{t}
\end{align*}
In order to design a stabilizing LQR controller with respect to supply rate $l(x,u)=x^{\...
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Reading material for Bellman's optimality principle in dynamic programming - for someone who is not familiar with control theory
I am trying to understand the underlying theory behind dynamic programming, given the fact that I am not familiar with control theory, I fail to find relevant reading material that addresses the this ...
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Wrong sign in co-state of optimal control problem
Consider the following deterministic optimisation problem
\begin{align}
J(t) = \min_{c(t)} \ & \frac{1}{2} \int_0^\infty e^{-\delta u} \left( x(u)^2 + \lambda y(u)^2 \right) du \\
s.t. \ &c(t) ...
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Numerically solving Pontryagin's Maximum Principle
I am trying to understand why when solving this boundary problem in matlab is significantly impacted by the $A_0$ matrix.
The optimal control problem I am trying to solve reads:
$$
\begin{aligned}
...
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Proof that excessive function are also regular (super-harmonic)
In page 117 of Shiryaev's book optimal stopping rules, he claims that the excessive functions are also regular under some condition and state the proof is analogous to another proof, but I fail to ...
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Definition of the value function in control theory
The following definition stems from the notes "An Introduction to Mathematical Optimal Control Theory" by Lawrence C. Evans (they are available online for free).
In defining the value ...
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answer
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Stable Kalman Filter estimator with given covariance matrices
I asked this question a while back. Essentially considering the follow basic Kalman Filter, following the Wikipedia convention.
\begin{equation}
\begin{split}
x_k &= F_kx_{k-1} + B_k u_k +w_k\\
...
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Function $\frac{x}{1+x}g(a-x) + \frac{1}{1+x}g(-x)$ has a unique maximizer over $x\ge0$, for any concave increasing function $g$ and $a>0$
Let $g:(-L,\infty) \to \mathbb{R}$ be a strictly concave, increasing, and differentiable function with $L > 0$ such that $g(x)\to -\infty$ as $x \to -L$. Also, let the map $f$ be defined on $[0,L)$ ...
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