Intereting Posts

Verifying Hilberts Nullstellensatz on a particular example
complex polynomial satisfying inequality
How is $\dfrac1{(1-x)^5}=\sum_{n\geq0}{n+4\choose4}x^n$
Norm of Gaussian integers: solutions to $N(a) = k$ for $k \in \mathbb{N}$ and $a$ a Guassian integer
Why is $\ell^1(\mathbb{Z})$ not a $C^{*}$-algebra?
Non-zero Conditional Differential Entropy between a random variable and a function of it
Interpolation inequality
Why is a linear transformation a $(1,1)$ tensor?
Give a demonstration that $\sum\limits_{n=1}^\infty\frac{\sin(n)}{n}$ converges.
How to calculate volume given by inequalities?
Elegant proof of $\int_{-\infty}^{\infty} \frac{dx}{x^4 + a x^2 + b ^2} =\frac {\pi} {b \sqrt{2b+a}}$?
Lack of implication and logical quantifiers
Show inequality of integrals (cauchy-schwarz??)
The importance of modular forms
Show that $ a，b，c, \sqrt{a}+ \sqrt{b}+\sqrt{c} \in\mathbb Q \implies \sqrt{a},\sqrt{b},\sqrt{c} \in\mathbb Q $

I am attempting to derive the Riccati Equation for linear-quadratic control. The original equation is:

$-\partial V/\partial t = \min_{u(t)} \{x^TQx + u^TRu + \partial V^T/\partial x(Ax + Bu) \}$

$x \in \Re^n$, $u \in \Re^m$, $Q \in \Re^{n\times n}$, $R \in \Re^{m\times m}$, $A \in \Re^{n\times n}$, $B \in \Re^{n\times m}$.

- Optimal control
- Matrix Identity $(I-G_1G_2)^{-1}G_1=G_1(I-G_2G_1)^{-1}$?
- Robust Control VS Optimal Control
- Strict Inequality in Homogenous LMI
- If matrices $A$ and $B$ commute, $A$ with distinct eigenvalues, then $B$ is a polynomial in $A$
- Control / Feedback Theory

It can be shown that the minimal $u$ is $u^*=-\frac{1}{2}R^{-1}B^T\partial V/\partial x$; also, $V(x,t)$ can be shown to be quadratic in $x$, so it is of the form $V(x(t),t) = x(t)^{T}P(t)x(t)$, so $\partial V/\partial x = 2P(t)x(t)$. Thus $u^*(t) = -R^{-1}B^TP(t)x(t)$. We’d like to solve for $P$, which is symmetrical.

Plugging into the original equation, I obtain

$-\partial V/\partial t = -x^T\dot{P}x \equiv x^TQx + (-R^{-1}B^TPx)^TR(-R^{-1}B^TPx)+2x^TP(Ax+B[-R^{-1}B^TPx])$

Somehow this gets reduced to

$-x^T\dot{P}x = x^T\{A^TP + PA + Q – PBR^{-1}B^TP\}x$

I cannot figure out the manipulation to get to the final equation. In particular, how is there both an $A^TP$ and $PA$ term in the final expression, when I distribute $2x^TP$ into $Ax+Bu^*$? Any insight would be helpful. Thanks.

- If $f(x)$ is continuous and differentiable function , prove that $f(x) \geq e^{3x}, \forall x \geq 0$
- Useful reformulation of Goldbach's conjecture?
- Solving an ordinary differential equation with initial conditions
- How to make a smart guess for this ODE
- second order DE using reduction of order
- A question regarding Frobenius method in ODE
- A reference for existence/uniqueness theorem for an ODE with Carathéodory condition
- Division of differential equations
- About the solution of a difference equation
- Solving the differential equation $(x^2-y^2)y' - 2xy = 0$.

$P$ is real symmetric and so is diagonalizable: $P = QLQ^T$ with some orthogonal $Q$.

Then,

$$\begin{align*}

A^TP+PA &= A^TQLQ^T + QLQ^TA \\

Q^TA^TP+Q^TPA &= Q^TA^TQLQ^T+LQ^TA \\

Q^TA^TPQ+Q^TPAQ &= Q^TA^TQL+LQ^TAQ \\

Q^T(A^TP+PA)Q &=Q^TA^TQL+(Q^TAQ)^TL \\

&= Q^TA^TQL+Q^TA^TQL \\

&= 2(Q^TA^TQ)L

\end{align*}

$$

Moving back the $Q$’s from the left-hand side,

$$\begin{align*}

A^TP+PA &= 2Q(Q^TA^TQ)LQ^T \\

&= 2A^T(QLQ^T)\\

&= 2A^TP \\

&= 2PA

\end{align*}

$$

Associativity of matrix operations and the symmetry of $P$, namely $P^T=P$, are exploited a few times here.

I know it’s a little late but it might help someone in the future.

If you notice that $2x^TPAx$ is a scalar, then it is equivalent to it’s transpose, so:

\begin{align}

2x^TPAx &= x^TPAx + x^TPAx \\

&= x^TPAx + x^TA^TP^Tx,\ \text{but}\ P = P^T \\

&= x^TPAx + x^TA^TPx \\

&= x^T(PA + A^TP)x

\end{align}

which is what you were looking for.

I hope this helps.

- Convex optimization with non-convex objective function
- Algorithms for “solving” $\sqrt{2}$
- Prove that $\lim\limits_{n\to\infty}1 + \frac{1}{1!} + \frac {1}{2!} + \cdots + \frac{1}{n!}\ge\lim\limits_{n\to\infty}(1+\frac{1}{n})^n$
- Why is the number $e$ so important in mathematics?
- Proving that the closure of a subset is the intersection of the closed subsets containing it
- $13\mid4^{2n+1}+3^{n+2}$
- Cauchy problem for nolinear PDE
- Sum of two independent geometric random variables
- $f>0$ on $$ implies $\int_0^1 f >0$
- Prove DeMorgan's Theorem for indexed family of sets.
- Why does rational trigonometry not work over a field of Characteristic 2?
- Surprise exam paradox?
- Bergman-Shilov Boundary and Peak Points
- show that $\int_{-\infty}^{+\infty} \frac{dx}{(x^2+1)^{n+1}}=\frac {(2n)!\pi}{2^{2n}(n!)^2}$
- How do solve this integral $\int_{-1}^1\frac{1}{\sqrt{1-x^2}}\arctan\frac{11-6\,x}{4\,\sqrt{21}}\mathrm dx$?