Mathematics Stack Exchange
Mathematics Stack Exchange

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)

Optimal control theory, an extension of the calculus of variations, is a mathematical optimization method for deriving control policies. Reference: Wikipedia.

The method is largely due to the work of Lev Pontryagin and Richard Bellman.

1022 questions
6
votes
0 answers

Hamilton-Jacobi-Bellman under Lévy driven Ornstein Uhlenbeck process

I am trying to solve a HJB equation with terminal condition under mean reverting process (Ornstein-Uhlenbeck process). I am pretty confused on how to account for the terminal condition and how to guess solution form for the value function. Here are…
Ikemou
  • 61
5
votes
1 answer

Whether Pontryagin's maximum principle valid or not?

Let consider this system of the equation like $\dot{x}=p(1-e^{2x})$ where $x$ is position and $p$ is momentum. We want to minimize this payoff function $\int_{0}^{t}2a_1^2$ where $a_1\in[-\alpha,\alpha]$. Now can we compute…
CR7
  • 71
4
votes
1 answer

Obtaining all possible transitions based on the estimation of a subset of transitions through the linearity property in an optimal control problem

I found an article in which the authors formulated an optimal control problem based on a linear time-invariant system $$ \dot{x}(t)=Ax(t)+Bu(t) $$ with initial state $ x(0)=x_0 $ and final state $ x(T)=x_T $. The performance measure to be…
holistic
  • 1,039
3
votes
1 answer

Linear Quadratic Regulator Matrices Positivity

Shortly, the LQR problem says that: for $\begin{cases} x'=Ax+Bu \\ x(t_0)=x_0\end{cases}$ find: $$\min_{u\in L^2(t_0,T; \mathbb{R}^m)} J(u)=\frac{1}{2}\left\{\int_{t_0}^T x^TQx+u^TRu+2x^TNu\ dt + x(T)^TPx(T)\right\}$$ where $Q,P$ are positive…
Bogdan
  • 1,867
3
votes
1 answer

Help with an optimal control problem

Given a linear time-invariant system: $$ \dot{x}(t)=Ax(t)+Bu(t) $$ with initial state $ x(0)=x_0 $ and final state $ x(T)=x_T $. The performance measure to be minimized is: $$ \int_{0}^{T} ((x_T-x(t))^T(x_T-x(t))+\rho u(t)^Tu(t) dt $$ with $…
holistic
  • 1,039
3
votes
1 answer

How to numerically solve an optimal control problem with an indirect method (in MATLAB)?

This question was also posted at the Space Exploration StackExchange, but I thought it might have a better chance of receiving an answer here. For an in-plane non-impulsive orbital maneuver, I'd like to find the thrust-direction history $\beta(t)$…
woeterb
  • 133
3
votes
0 answers

Optimal Control - Nyquist Plot

The following picture shows Nyquist Plot according to optimal control LQR. Nyquist Plot for LQR It shows regions that when avoided, it guarantees gain margin $GM = \infty$, phase margin $PM = 60°$ (at least). I need a little bit of clarification of…
Martin G
  • 319
2
votes
0 answers

Control constraints in weaker norm

I've just a short question (at least I think so): I want to study optimal control problems of the form: $\min_{\omega\in L^2(\Omega)} J(\omega)+\frac{\epsilon}{2} \|\omega\|^2_{L^2{\Omega}}$ s.t. $\omega\geq 0 ~a.e.$, $\|\omega\|_{L^1(\Omega)}\leq…
Waldi
  • 21
2
votes
1 answer

H2 control v.s LQG

H2 control is try to find K to minimize the transfer function from w -> z. LQG is try to find kalman gain K to minimize the steady-state covariance of the error. What is the same things and different things between them? When I read some papers, it…
sleeve chen
  • 8,281
2
votes
1 answer

Optimal control with integral constraint

I am faced with the following optimization problem: $$\max_{x(v_1,v_2)}\int_0^1\int_0^1f_1(v_1,v_2)x(v_1,v_2)\,dv_1\,dv_2,$$ subject to: $$\int_0^1\int_0^1f_2(v_1,v_2)x(v_1,v_2) \, dv_1 \, dv_2\geq 0,\; x(v_1,v_2)\in[0,1].$$ $f_1$ and $f_2$ are some…
Brqano
  • 90
  • 6
2
votes
0 answers

Optimal Control Problem to Maximize the Horizon

$w > 0$ and $m > 0$ are known and $w \ge m$. $\mu$ and $\lambda$ are known and $\mu > \lambda$. \begin{align} \max_{H_t} & \, T \\ s.t. & \, \frac{dW_t}{dt} = \mu W_t - \mu H_t, W_T = 0, W_0 = w \\ & \, \frac{dM_t}{dt} = \lambda M_t - \lambda H_t,…
ftor
  • 229
2
votes
0 answers

Replace costate with momentum for Pontryagin's principle

According to the Maximum principle, we know that there exists a costate $\hat{w}:[0,t]\to\mathbb{R}$ such that $\dot{x}=H_{\hat{w}}(x,\hat{w},\alpha)$ and $\dot{\hat{w}}=-H_x(x,\hat{w},\alpha)$ and…
CR7
  • 71
2
votes
1 answer

Is Lyapunov theoreum only valid for equilibrium points, if so what if we want to do the analysis about a non-equilibrium point?

For a simple pendulum case, the dynamics is non linear and to study the stability of the system we tend to linearize the system about the equilibrium point ($0$ and $\pi$) and then for stability analysis use lyapunov theorem. What if we want to do…
Sameer Purwar
  • 21
2
votes
0 answers

Target Sets in Optimal Control Theory

I am kind of confused by the following problem, and wondering if someone could give me some hints. Many thanks! In optimal control theory, the target set is a description of restrictions on the endpoints. For example, a fixed-time, free-endpoint…
OnoL
  • 2,685
2
votes
1 answer

Existence Optimal Control - Binary Control Set

Can you point me at an existence theorem for an optimal control problem with binary control set? In particular, $$ \max_u \int_0^T u(t) e^{-f(A(t))}(v(A(t))-c) \, dt $$ $$\text{s.t. } \dot{A}(t)=u $$ where $u \in \{0,1\}$. The existence results…
ctmntlds
  • 51
1
2 3 4