# 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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### Numerical Model Predictive Control: Optimize states or Optimize control input?

I have read a number of different sources on numerical model predictive control. Something that comes up is if you optimize over states or optimize over controls. I have seen videos where people talk ...
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### Determine the first variation of (second-kind/type) of the solution of $\dot x = ax + ub$

In an optimal control course H/W I came across the following problem which I can't seem to get my mind across. Consider the LTI: $$\dot x = a x + u b$$ Make a complete analysis in order to determine ...
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### Why does covariance $P$ matrix become non positive definite in Unscented Kalman Filter?

I'm doing Unscented Kalman Filter in MATLAB code and I have followed this tutorial how to create one. First I initilize the $\hat x$ vector and covariance $P$ matrix first. In MATLAB code, I just set ...
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### A sufficient and necessary condition for the controllability of a system

Let $A\in\mathbb{R}^{n\times n}$ be symmetric positive definite and $B\in\mathbb{R}^{n\times m}$. Construct the complex matrix $$Z=BB^{\prime} +jA \text{ where } j^2=-1.$$ Consider now the following ...
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### Zero overshoot criterion from the initial point $x_0$ to the final $x_*$, $x_*$ unknown in advance

Let's say there is an ODE: $\dot{x}=f(t,x)+u$ Condition: variable $x$ passes from initial state $x_0$ to final state $x_*$, that do not know in advance. Is it possible to make a transient in such a ...
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### A sufficient condition of infinite horizon HJB equation

I found a lecture note and book describing Hamilton-Jacobi-Bellman (HJB) equation. In the references, the sufficient condition of HJB for optimality seems C1 condition of the value function (optimal ...
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### What is a recurrent vector field?

In pg. 10 of  they define the notion of a recurrent vector field as: A vector field $f$ on a manifold, say $\mathbb{R}^n$, is recurrent if for every point $x_0\in\mathbb{R}^n$ and every ...
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### What is the difference between optimization and optimal control?

What is the difference between optimization and optimal control? I am trying to understand the difference between those two, but it is still confusing to me. Could anyone please explain it briefly?
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### Proof that co-state is a linear function of state LQR

I am studying optimal control and I have a problem on the classical continuous LQR problem with finite time. Let me first recall the situation: \begin{equation} \text{min}_{u\in C^{inf}(\mathbb{R}^d)}[...
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### optimization or nonlinear optimal control solver?

I am looking for some solvers on "optimization or nonlinear optimal control(along with constraints)". Could you recommend me some solver candidates? Best Regards Jie
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