Take a discrete Lyapunov equation
$$
P = R + \beta A' P A
$$
for $\beta \in (0,1)$, $R$ and $A$ symmetric.
Quetsion: Can I put an upper bound on $tr(P)$ given properties of the primitives? In particular, I have the $tr(R)$ and bounds on the maximum eigenvalue of $A$.
I can probably deal with something that doesn't generate the tightest bound exploiting the $A$ as long as the $tr(R)$ is involved.
Alternatively: I am happy to use similar results for the solution to the Discrete matrix ricatti equation that comes out of optimal LQ control if that is available.
Best Answer
By using a cyclic permutation it can be shown that the matrix part of the trace of the second term of the right hand side can also be written as
$$ \text{tr}(A'\,P\,A) = \text{tr}(A\,A'\,P). \tag{1} $$
When assuming that $\sqrt{\beta} A$ is Hurwitz (this would be the case for all $\beta$ if $A$ is Hurwitz) and $R$ positive definite, then the solution for $P$ should also be positive definite. In that case one can use the Cauchy-Schwarz inequality, such that $(1)$ satisfies
$$ \text{tr}(A'\,P\,A) \leq \text{tr}(A\,A')\,\text{tr}(P). \tag{2} $$
The trace of a matrix is equal to the sum of its eigenvalues, thus an upper bound for $\text{tr}(A\,A')$ would be $n$ times the maximum eigenvalue of $A\,A'$. If $A$ is symmetric then the maximum eigenvalue of $A\,A'$ would simply be the square of the the maximum eigenvalue of $A$, so
$$ \text{tr}(A\,A') \leq n\,\lambda_\text{max}\!(A)^2, \tag{3} $$
with $A\in\mathbb{R}^{n\times n}$.
Taking the trace of the Riccati equation and substituting in $(2)$ and $(3)$ yields
$$ \text{tr}(P) \leq \text{tr}(R) + \beta\,n\,\lambda_\text{max}\!(A)^2\,\text{tr}(P). \tag{4} $$
Solving $(4)$ for $\text{tr}(P)$ thus yields
$$ \text{tr}(P) \leq \frac{\text{tr}(R)}{1 - \beta\,n\,\lambda_\text{max}\!(A)^2}, \tag{5} $$
assuming that $\beta\,n\,\lambda_\text{max}\!(A)^2 < 1$.
Or a less tight bound could be obtained by only substituting in $(2)$ which yields
$$ \text{tr}(P) \leq \frac{\text{tr}(R)}{1 - \beta\,\text{tr}(A\,A')}. \tag{6} $$