Orlicz log-Aleksandrov-Fenchel inequality

Chang-Jian Zhao

Department of Mathematics, China Jiliang University, Hangzhou 310018, P. R.
China

Email<EMAIL_ADDRESS>
Abstract In this paper, we establish an Orlicz log-Aleksandrov-Fenchel
inequality by introducing new concepts of mixed volume measure and Orlicz
multiple mixed volume measure, and using the Orlicz-Aleksandrov-Fenchel
inequality. The Orlicz log-Aleksandrov-Fenchel inequality in special cases
yield the classical Aleksandrov-Fenchel inequality and Orlicz log-Minkowski
type inequality, respectively.

Keywords mixed volumes, $L_{p}$-mixed volume, Minkowski inequality,
logarithmic Minkowski inequality, Aleksandrov-Fenchel inequality, Orlicz-
Aleksandrov-Fenchel inequality.

2010 Mathematics Subject Classification 46E30 52A39

1 Introducation

In 2012, Böröczky, Lutwak, Yang, and Zhang [1] conjecture a logarithmic
Minkowski inequality for origin-symmetric convex bodies, the logarithmic
Minkowski inequality was stated the following.

The conjectured logarithmic Minkowski inequality If $K$ and $L$ are convex
bodies in ${\mathbb{R}}^{n}$ which are symmetric with respect to the origin,
then

$\int_{S^{n-1}}\ln\left(\frac{h_{K}}{h_{L}}\right)d\bar{V}_{L}\geq\frac{1}{n}\ln\left(\frac{V(K)}{V(L)}\right),$
$None$

where $dv_{L}=\frac{1}{n}h_{L}dS_{L}$ is the cone-volume measure of $L$, and
$d\bar{V}_{L}=\frac{1}{V(L)}dv_{L}$ is its normalization, and
$S_{L}=S(L,\cdot)$ is the mixed surface area measure of $L$.

The functions are the support functions. If $K$ is a nonempty closed (not
necessarily bounded) convex set in ${\mathbb{R}}^{n}$, then

$h_{K}=\max\\{x\cdot y:y\in K\\},$

for $x\in{\mathbb{R}}^{n},$ defines the support function $h_{K}$ of $K$. A
nonempty closed convex set is uniquely determined by its support function.

In 2016, Stancu [2] proved a modified logarithmic Minkowski inequality for
non-symmetric convex bodies not symmetric with respect to the origin. The
logarithmic Minkowski inequality was given in the following theorem.

The logarithmic Minkowski inequality If $K$ and $L$ are convex bodies in
${\mathbb{R}}^{n}$ that containing the origin in their interior, then

$\int_{S^{n-1}}\ln\left(\frac{h_{K}}{h_{L}}\right)d\bar{V}_{1}\geq\frac{1}{n}\ln\left(\frac{V(K)}{V(L)}\right).$
$None$

with equality if and only if $K$ and $L$ are homothetic, where $dv_{1}$ is the
mixed volume measure $dv_{1}=\frac{1}{n}h_{K}dS_{L},$ and
$d\bar{v}_{1}=\frac{1}{V_{1}(L,K)}dv_{1}$ is its normalization, and
$V_{1}(L,K)$ denotes the usual mixed volume of $L$ and $K$, defined by (see
[])

$V_{1}(L,K)=\frac{1}{n}\int_{S^{n-1}}h_{K}dS_{L}.$

Recently, this logarithmic Minkowski inequality and the conjectured
logarithmic Minkowski inequality have attracted extensive attention and
research. The recent research on the logarithmic Minkowski and its dual type
inequalities can be found in the references [3-16].

Associated with the convex bodies $K_{1},\cdots,K_{n-1}$ in ${\mathbb{R}}^{n}$
is a unique positive Borel measure on $S^{n-1}$,
$S(K_{1},\ldots,K_{n-1};\cdot)$, call it the mixed area measure of
$K_{1},\ldots,K_{n-1}$, with the property that any convex body $K_{n}$ one has
the integral representation (see e.g. [17], p. 354).

$V(K_{1},\ldots,K_{n})=\frac{1}{n}\int_{S^{n-1}}h_{K_{n}}dS(K_{1},\ldots,K_{n-1};u).$
$None$

The integration is with respect to the mixed area measure
$S(K_{1},\ldots,K_{n-1};\cdot)$ on $S^{n-1}$. The mixed area measure
$S(K_{1},\ldots,K_{n-1};\cdot)$ is symmetric in its (first $n-1$) arguments.
When $K_{1}=\cdots=K_{n-i-1}=K$ and $K_{n-i}=\cdots=K_{n-1}=B$, the mixed area
measure $S(K,\ldots,K,B,\ldots,B;\cdot)$ with $i$ copies of $B$ and $(n-i-1)$
copies of $K$, will be written as $S_{i}(K,\cdot)$. For
$K_{1}=\cdots=K_{n-1}=K$, $S(K_{1},\ldots,K_{n-1};\cdot)$ reduces to the
surface area measure $S(K;\cdot)$.

It is well known that in Brunn-Minkowski theory, Minkowski inequality and
Aleksandrov-Fenchel inequality appear at the same time, and the latter is a
generalization of the former. So a natural question is raised: is there a
logarithmic Aleksandrov-Fenchel inequality relative to the logarithmic
Minkowski inequality? The main purpose of this article is to answer the above
questions perfectly and obtain a logarithmic Aleksandrov-Fenchel inequality by
introducing two new concepts of mixed volume measure and $L_{p}$-multiple
mixed volume measure, and using the $L_{p}$-Aleksandrov-Fenchel inequality for
the $L_{p}$-multiple mixed volume. The logarithmic Aleksandrov-Fenchel
inequality in special cases yield the classical Aleksandrov-Fenchel
inequality, and four recent logarithmic Minkowski type inequalities, which are
logarithmic Minkowski inequality for mixed volumes, $L_{p}$-mixed volumes,
quermassintegrals and $p$-mixed quermassintergrals, respectively. Our main
result is given in the following theorem.

Orlicz log-Aleksandrov-Fenchel inequality Let $L_{1},\ldots,L_{n},K_{n}$ be
convex bodies in ${\mathbb{R}}^{n}$ that containing the origin in their
interior. If $\varphi:[0,\infty)\rightarrow(0,\infty)$ is a convex and
increasing function such that $\varphi(0)=0$ and $\varphi(1)=1$, and $1\leq
r\leq n$, then

$\int_{S^{n-1}}\ln\left(\varphi\left(\frac{h_{K_{n}}}{{h_{L_{n}}}}\right)\right)d\bar{V}_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})\geq\ln\left(\varphi\left(\frac{\prod_{i=1}^{r}V(L_{i},\ldots,L_{i},L_{r+1},\ldots,L_{n-1},K_{n})^{1/r}}{V(L_{1},\ldots,L_{n})}\right)\right).$
$None$

Here $d\bar{V}_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})$ denotes a new
probability measure call it Orlicz multiple mixed volume probability measure
of convex bodies $L_{1},\ldots,L_{n},K_{n}$, defined by

$d\bar{V}_{\varphi}(L_{1},\cdots,L_{n-1},K_{n},L_{n})=\frac{dv_{\varphi}(L_{1},\cdots,L_{n-1},K_{n},L_{n})}{V_{\varphi}(L_{1},\cdots,L_{n-1},K_{n},L_{n})},$
$None$

and $dv_{\varphi}(L_{1},\cdots,L_{n-1},K_{n},L_{n})$ denotes the Orlicz
multiple mixed volume measure of $L_{1},\ldots,L_{n},K_{n}$, defined by

$dv_{\varphi}(L_{1},\cdots,L_{n-1},K_{n},L_{n})=\frac{1}{n}\cdot\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)h_{L_{n}}dS(L_{1},\ldots,L_{n-1};\cdot).$
$None$

Moreover, $V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})$ is the Orlicz
multiple mixed volume of convex bodies $L_{1},\ldots,L_{n},K_{n}$ defined by
([18])

$V_{\varphi}(L_{1},\cdots,L_{n-1},K_{n},L_{n})=\frac{1}{n}\int_{S^{n-1}}\varphi\left(\frac{h_{L_{n}}}{h_{K_{n}}}\right)h_{K_{n}}dS(L_{1},\ldots,L_{n-1};u).$
$None$

Remark When $K_{n}=L_{n}$, inequality (1.4) becomes the classical Aleksandrov-
Fenchel inequality of convex bodies $L_{1},\ldots,L_{n}$ as follows: If
$L_{1},\cdots,L_{n}$ are convex bodies that containing the origin and $1\leq
r\leq n$, then

$V(L_{1},\cdots,L_{n})\geq\prod_{i=1}^{r}V(L_{i}\ldots,L_{i},L_{r+1},\ldots,L_{n})^{1/r},$
$None$

(see e.g. [19, p. 401]).

On the other hand, when $\varphi(x)=x^{p}$, $p=1$, $r=n-1$,
$L_{1}=\ldots=L_{n-1}=L$, $L_{n}=L$ and $K_{n}=K$, then (1.4) becomes Stancu’s
logarithmic Minkowski inequality (1.2) established in [2].

2 Notations and preliminaries

The setting for this paper is $n$-dimensional Euclidean space
${\mathbb{R}}^{n}$. Let ${\cal K}^{n}$ be the class of nonempty compact convex
subsets of ${\mathbb{R}}^{n}$, let ${\cal K}^{n}_{o}$ be the class of members
of ${\cal K}^{n}$ containing the origin. A set $K\in{\cal K}^{n}$ is called a
convex body if its interior is nonempty. We reserve the letter $u\in S^{n-1}$
for unit vectors, and the letter $B$ for the unit ball centered at the origin.
The surface of $B$ is $S^{n-1}$. For a compact set $K$, we write $V(K)$ for
the ($n$-dimensional) Lebesgue measure of $K$ and call this the volume of $K$.
Let $d$ denote the Hausdorff metric on ${\cal K}^{n}$, i.e., for $K,L\in{\cal
K}^{n},$

$d(K,L)=|h_{K}-h_{L}|_{\infty},$

where $|\cdot|_{\infty}$ denotes the sup-norm on the space of continuous
functions $C(S^{n-1})$.

2.1 Mixed quermassintegrals

If $K_{i}\in{\cal K}^{n}$ $(i=1,2,\ldots,r)$ and $\lambda_{i}$
$(i=1,2,\ldots,r)$ are nonnegative real numbers, then of fundamental
importance is the fact that the volume of $\sum_{i=1}^{r}\lambda_{i}K_{i}$ is
a homogeneous polynomial in $\lambda_{i}$ given by (see e.g. [15])

$V(\lambda_{1}K_{1}+\cdots+\lambda_{n}K_{n})=\sum_{i_{1},\ldots,i_{n}}\lambda_{i_{1}}\ldots\lambda_{i_{n}}V_{i_{1}\ldots
i_{n}},$ $None$

where the sum is taken over all $n$-tuples $(i_{1},\ldots,i_{n})$ of positive
integers not exceeding $r$. The coefficient $V_{i_{1}\ldots i_{n}}$ depends
only on the bodies $K_{i_{1}},\ldots,K_{i_{n}}$ and is uniquely determined by
(2.1), it is called the mixed volume of $K_{i},\ldots,K_{i_{n}}$, and is
written as $V(K_{i_{1}},\ldots,K_{i_{n}}).$ Let $K_{1}=\ldots=K_{n-i}=K$ and
$K_{n-i+1}=\ldots=K_{n}=L$, then the mixed volume $V(K_{1},\ldots,K_{n})$ is
written as $V_{i}(K,L)$. If $K_{1}=\cdots=K_{n-i}=K,$
$K_{n-i+1}=\cdots=K_{n}=B$ The mixed volumes $V_{i}(K,B)$ is written as
$W_{i}(K)$ and call as quermassintegrals (or $i$th mixed quermassintegrals) of
$K$. We write $W_{i}(K,L)$ for the mixed volume
$V(\underbrace{K,\ldots,K}_{n-i-1},\underbrace{B,\ldots,B}_{i},L)$ and call as
mixed quermassintegrals of convex bodies $K$ and $L$. Aleksandrov [20] and
Fenchel and Jessen [21] (also see Busemann [22] and Schneider [23]) have shown
that for $K\in{\cal K}^{n}_{o}$, and $i=0,1,\ldots,n-1,$ there exists a
regular Borel measure $S_{i}(K,\cdot)$ on $S^{n-1}$, such that the mixed
quermassintegrals $W_{i}(K,L)$ has the following representation:

$W_{i}(K,L)=\frac{1}{n-i}\lim_{\varepsilon\rightarrow
0^{+}}\frac{W_{i}(K+\varepsilon
L)-W_{i}(K)}{\varepsilon}=\frac{1}{n}\int_{S^{n-1}}h_{L}dS_{i}(K,u).$ $None$

Associated with $K_{1},\ldots,K_{n}\in{\cal K}^{n}$ is a Borel measure
$S(K_{1},\ldots,K_{n-1},\cdot)$ on $S^{n-1}$, called the mixed surface area
measure of $K_{1},\ldots,K_{n-1}$, which has the property that for each
$K\in{\cal K}^{n}$ (see e.g. [24], p.353),

$V(K_{1},\ldots,K_{n-1},K)=\frac{1}{n}\int_{S^{n-1}}h_{K}dS(K_{1},\ldots,K_{n-1},u).$
$None$

In fact, the measure $S(K_{1},\ldots,K_{n-1},\cdot)$ can be defined by the
propter that (2.3) holds for all $K\in{\cal K}^{n}.$ Let
$K_{1}=\ldots=K_{n-i-1}=K$ and $K_{n-i}=\ldots=K_{n-1}=L$, then the mixed
surface area measure $S(K_{1},\ldots,K_{n-1},\cdot)$ is written as
$S_{i}(K,L;\cdot)$. When $L=B$, $S_{i}(K,L;\cdot)$ is written as
$S_{i}(K,\cdot)$ and called as $i$th mixed surface area measure. A fundamental
inequality for mixed quermassintegrals stats that: For $K,L\in{\cal K}^{n}$
and $0\leq i<n-1$,

$W_{i}(K,L)^{n-i}\geq W_{i}(K)^{n-i-1}W_{i}(L),$ $None$

with equality if and only if $K$ and $L$ are homothetic and $L=\\{o\\}$.

2.2 Mixed $p$-quermassintegrals

Mixed quermassintegrals are, of course, the first variation of the ordinary
quermassintegrals, with respect to Minkowski addition. The mixed
quermassintegrals $W_{p,0}(K,L),W_{p,1}(K,L),\ldots,W_{p,n-1}(K,L)$, as the
first variation of the ordinary quermassintegrals, with respect to Firey
addition: For $K,L\in{\cal K}_{o}^{n}$, and real $p\geq 1$, defined by (see
e.g. [25])

$W_{p,i}(K,L)=\frac{p}{n-i}\lim_{\varepsilon\rightarrow_{0^{+}}}\frac{W_{i}(K+_{p}\varepsilon\cdot
L)-W_{i}(K)}{\varepsilon}.$ $None$

The mixed $p$-quermassintegrals $W_{p,i}(K,L)$, for all $K,L\in{\cal
K}^{n}_{o}$, has the following integral representation:

$W_{p,i}(K,L)=\frac{1}{n}\int_{S^{n-1}}h_{L}^{p}dS_{p,i}(K,u),$ $None$

where $S_{p,i}(K,\cdot)$ denotes the Boel measure on $S^{n-1}$. The measure
$S_{p,i}(K,\cdot)$ is absolutely continuous with respect to $S_{i}(K,\cdot)$,
and has Radon-Nikodym derivative

$\frac{dS_{p,i}(K,\cdot)}{dS_{i}(K,\cdot)}=h_{K}^{1-p},$ $None$

where $S_{i}(K,\cdot)$ is a regular Boel measure on $S^{n-1}$. The measure
$S^{n-1}(K,\cdot)$ is independent of the body $K$, and is just ordinary
Lebesgue measure, $S$, on $S^{n-1}$. $S_{i}(B,\cdot)$ denotes the $i$-th
surface area measure of the unit ball in ${\mathbb{R}}^{n}$. In fact,
$S_{i}(B,\cdot)=S$ for all $i$. The surface area measure $S_{0}(K,\cdot)$ just
is $S(K,\cdot)$. When $i=0$, $S_{p,i}(K,\cdot)$ is written as $S_{p}(K,\cdot)$
(see [26] and [27]). A fundamental inequality for mixed $p$-quermassintegrals
stats that: For $K,L\in{\cal K}_{o}^{n},p>1$ and $0\leq i<n-1$,

$W_{p,i}(K,L)^{n-i}\geq W_{i}(K)^{n-i-p}W_{i}(L)^{p},$ $None$

with equality if and only if $K$ and $L$ are homothetic. $L_{p}$-Brunn-
Minkowski inequality for quermassintegrals established by Lutwak [25]. If
$K,L\in{\cal K}_{o}^{n}$ and $p\geq 1$ and $0\leq i\leq n$, then

$W_{i}(K+_{p}L)^{p/(n-i)}\geq W_{i}(K)^{p/(n-i)}+W_{i}(L)^{p/(n-i)},$ $None$

with equality if and only if $K$ and $L$ are dilates or $L=\\{o\\}$.
Obviously, putting $i=0$ in (2.6), the mixed $p$-quermassintegrals
$W_{p,i}(K,L)$ become the well-known $L_{p}$-mixed volume $V_{p}(K,L)$,
defined by (see e.g. [26])

$V_{p}(K,L)=\frac{1}{n}\int_{S^{n-1}}h_{L}^{p}dS_{p}(K,u).$ $None$

3 Orlicz multiple mixed volumes

In [18], the Orlicz multiple mixed volume was introduced as follows:

Definition 3.1 For $\varphi\in\Phi$, the Orlicz multiple mixed volume of
convex bodies $K_{1},\cdots,K_{n},L_{n}$, denoted by
$V_{\varphi}(K_{1},\cdots,K_{n},L_{n})$, defined by

$V_{\varphi}(K_{1},\cdots,K_{n},L_{n})=\frac{1}{n}\int_{S^{n-1}}\varphi\left(\frac{h_{L_{n}}}{h_{K_{n}}}\right)h_{K_{n}}dS(K_{1},\ldots,K_{n-1};u).$
$None$

When $K_{1}=\cdots=K_{n-i-1}=K$, $K_{n-i}=\cdots=K_{n-1}=B$, $K_{n}=L$ and
$L_{n}=K$, $V_{\varphi}(K_{1},\cdots,K_{n},L_{n})$ becomes the $i$-th Orlicz
mixed volume $W_{\varphi,i}(K,L)$. When $L_{n}=K_{n}$
$V_{\varphi}(K_{1},\cdots,K_{n},L_{n})$ becomes the classical mixed volume
$V(K_{1},\cdots,K_{n}).$ When $K_{1}=\cdots=K_{n-1}=K$, $K_{n}=L$ and
$L_{n}=K$, $V_{\varphi}(K_{1},\cdots,K_{n},L_{n})$ becomes the well-known
Orlicz mixed volume $V_{\varphi}(K,L).$

The Orlicz-Aleksandrov-Fenchel inequality If $K_{1},\cdots,K_{n},L_{n}\in{\cal
K}_{o}^{n}$, $1\leq r\leq n$ and ${\varphi}\in\Phi$, then (see [18])

$V_{\varphi}(K_{1},\cdots,K_{n},L_{n})\geq
V(K_{1},\cdots,K_{n-1},L_{n})\cdot\varphi\left(\frac{\prod_{i=1}^{r}V(K_{i}\ldots,K_{i},K_{r+1},\ldots,K_{n})^{\frac{1}{r}}}{V(K_{1},\cdots,K_{n-1},L_{n})}\right).$
$None$

When $r=n$, $K_{1}=\cdots=K_{n-1}=K$, $K_{n}=L$ and $L_{n}=K$, (3.2) becomes
the following inequality which was established by Gardner, Hug and Weil [28].
If $K$ is a convex body containing the origin in its interior, $L$ is a convex
body containing the origin, and ${\varphi}\in\Phi$, then

$V_{\varphi}(K,L)\geq
V(K)\cdot\varphi\left(\left(\frac{V(L)}{V(K)}\right)^{1/n}\right).$ $None$

If $\varphi$ is strictly convex, equality holds if and only if $K$ and $L$ are
homothetic. Obviously, (3.2) in special case yields the following result. If
$K$ is a convex body containing the origin in its interior, $L$ is a convex
body containing the origin, $0\leq i<n$ and ${\varphi}\in\Phi$, then

$W_{\varphi,i}(K,L)\geq
W_{i}(K)\cdot\varphi\left(\left(\frac{W_{i}(L)}{W_{i}(K)}\right)^{1/(n-i)}\right).$
$None$

If $\varphi$ is strictly convex, equality holds if and only if $K$ and $L$ are
homothetic, where $W_{i}(K)$ is the usual quermassintegral of convex body $K$.
(3.4) in special cases yield the following two well-known $L_{p}$-Minkowski
type inequalities. $L_{p}$-Minkowski inequality for mixed
$p$-quermassintegrals established by Lutwak [26].

$W_{p,i}(K,L)^{n-i}\geq W_{i}(K)^{n-i-p}W_{i}(L)^{p},$ $None$

for $p>1$ and $0\leq i\leq n$, with equality if and only if $K$ and $L$ are
dilates or $L=\\{o\\}$. $L_{p}$-Minkowski inequality established by Firey [29]
. For $p>1$,

$V_{p}(K,L)\geq V(K)^{(n-p)/n}V(L)^{p/n},$ $None$

with equality if and only if $K$ and $L$ are dilates or $L=\\{o\\}$.

4 Orlicz log-Aleksandrov-Fenchel inequality

In the section, in order to prove the Orlicz-Aleksandrov-Fenchel inequality,
we need to define some new mixed volume measures.

Associated with the convex bodies $L_{1},\cdots,L_{n-1}$ in ${\mathbb{R}}^{n}$
is a unique positive Borel measure on $S^{n-1}$,
$S(L_{1},\ldots,L_{n-1};\cdot)$, call it the mixed area measure of
$L_{1},\ldots,L_{n-1}$, with the property that any convex body $L_{n}$ one has
the integral representation.

$V(L_{1},\ldots,L_{n})=\frac{1}{n}\int_{S^{n-1}}h_{L_{n}}dS(L_{1},\ldots,L_{n-1};u).$
$None$

From (4.1), we introduce the mixed volume measure of convex bodies
$L_{1},\ldots,L_{n}$.

Definition 4.1 (mixed volume measure) For $L_{1},\cdots,L_{n}\in{\cal
K}^{n}_{o}$, the mixed volume measure of $L_{1},\ldots,L_{n}$, denoted by
$dv(L_{1},\ldots,L_{n})$, defined by

$dv(L_{1},\ldots,L_{n})=\frac{1}{n}h_{L_{n}}dS(L_{1},\cdots,L_{n-1};\cdot).$
$None$

From Definition 4.1, it is not difficult to find the following mixed volume
probability measure.

$d\bar{V}(L_{1},\ldots,L_{n})=\frac{1}{V(L_{1},\ldots,L_{n})}dv(L_{1},\ldots,L_{n}).$
$None$

For $p\geq 1$, $L_{p}$-multiple mixed volume of $L_{1},\cdots,L_{n},K_{n}$,
denoted by $V_{p}(L_{1},\cdots,L_{n-1},K_{n},L_{n})$, defined by

$V_{p}(L_{1},\cdots,L_{n-1},K_{n},L_{n})=\frac{1}{n}\int_{S^{n-1}}\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)^{p}h_{L_{n}}dS(L_{1},\ldots,L_{n-1};u).$
$None$

From (4.4), we introduce $L_{p}$-multiple mixed volume measure of convex
bodies $L_{1},\cdots,L_{n},K_{n}$ as follows.

Definition 4.2 (Orlicz multiple mixed volume measure) For
$L_{1},\cdots,L_{n},K_{n}\in{\cal K}^{n}_{o}$, the Orlicz mixed volume measure
of $L_{1},\ldots,L_{n},K_{n}$, denoted by
$dv_{\varphi}(L_{1},\cdots,L_{n-1},K_{n},L_{n})$, defined by

$dv_{\varphi}(L_{1},\cdots,L_{n-1},K_{n},L_{n})=\frac{1}{n}\cdot\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)\cdot
h_{L_{n}}dS(L_{1},\ldots,L_{n-1};\cdot).$ $None$

From Definition 4.2, $L_{p}$-multiple mixed volume probability measure is
defined by

$d\bar{V}_{\varphi}(L_{1},\cdots,L_{n-1},K_{n},L_{n})=\frac{1}{V_{\varphi}(L_{1},\cdots,L_{n-1},K_{n},L_{n})}dv_{\varphi}(L_{1},\cdots,L_{n-1},K_{n},L_{n}).$
$None$

Obviously, the mixed volume measure $dv(L_{1},\ldots,L_{n})$ is special case
of the Orlicz multiple mixed volume measure. When $K_{n}=L_{n}$, we have

$dv_{\varphi}(L_{1},\cdots,L_{n},L_{n})=dv(L_{1},\cdots,L_{n}).$ $None$

Theorem 4.3 (Orlicz log-Aleksandrov-Fenchel inequality) If
$L_{1},\ldots,L_{n},K_{n}\in{\cal K}^{n}_{o}$, $1\leq r\leq n$ and
$\varphi\in\Phi$, then

$\int_{S^{n-1}}\ln\left(\varphi\left(\frac{h_{K_{n}}}{{h_{L_{n}}}}\right)\right)d\bar{V}_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})\geq\ln\left(\varphi\left(\frac{\prod_{i=1}^{r}V(L_{i},\ldots,L_{i},L_{r+1},\ldots,L_{n-1},K_{n})^{1/r}}{V(L_{1},\ldots,L_{n})}\right)\right).$
$None$

Proof From (4.2), (4.5) and (4.6), we have

$\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)\ln\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)dv(L_{1},\ldots,L_{n})=\int_{S^{n-1}}\ln\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)dv_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n}).$
$None$

Noting that

$V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})=\frac{1}{n}\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)h_{L_{n}}dS(L_{1},\ldots,L_{n-1};u),$

and from Lebesgue’s dominated convergence theorem, we obtain

$\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)^{\frac{q}{q+n}}dv(L_{1},\ldots,L_{n})\rightarrow
V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})$

as $q\rightarrow\infty,$ and

$\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)^{\frac{q}{q+n}}\ln\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)dv(L_{1},\ldots,L_{n})\rightarrow\int_{S^{n-1}}\ln\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)dv_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})$

as $q\rightarrow\infty.$

Considering the function
$f_{L_{1},\ldots,L_{n-1},K_{n},L_{n}}:[1,\infty]\rightarrow{\mathbb{R}}$,
defined by

$f_{L_{1},\ldots,L_{n-1},K_{n},L_{n}}(q)=\frac{1}{V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})}\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)^{\frac{q}{q+n}}dv(L_{1},\ldots,L_{n}).$
$None$

By calculating the derivative and limit of this function, we have

$\frac{df_{L_{1},\ldots,L_{n-1},K_{n},L_{n}}(q)}{dq}=\frac{n}{(q+n)^{2}}\cdot\frac{1}{V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\times\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)^{\frac{q}{q+n}}\ln\left(\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)\right)dv(L_{1},\ldots,L_{n}).$
$None$

and

$\lim_{q\rightarrow\infty}f_{L_{1},\ldots,L_{n-1},K_{n},L_{n}}(q)=1.$ $None$

From (4.11), (4.12) and (4.13), and by using L’Hôpital’s rule, we have

$\displaystyle\lim_{q\rightarrow\infty}\ln\left(f_{L_{1},\ldots,L_{n-1},K_{n},L_{n}}(q)\right)^{q+n}$
$\displaystyle=$
$\displaystyle-(q+n)^{2}\lim_{q\rightarrow\infty}\frac{\frac{\displaystyle
df_{L_{1},\ldots,L_{n-1},K_{n},L_{n}}(q)}{\displaystyle
dq}}{f_{L_{1},\ldots,L_{n-1},K_{n},L_{n}}(q)}$ $\displaystyle=$
$\displaystyle-\frac{n}{V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})}$
$\displaystyle\times$
$\displaystyle\lim_{q\rightarrow\infty}\frac{\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)^{\frac{q}{q+n}}\ln\left(\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)\right)dv(L_{1},\ldots,L_{n})}{f_{L_{1},\ldots,L_{n-1},K_{n},L_{n}}(q)}$
$\displaystyle=$
$\displaystyle-\frac{n}{V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})}$
$\displaystyle\times$
$\displaystyle\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)\ln\left(\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)\right)dv(L_{1},\ldots,L_{n}).$

Hence

$\exp\left(-\frac{n}{V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})}\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)\ln\left(\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)\right)dv(L_{1},\ldots,L_{n})\right)$
$=\lim_{q\rightarrow\infty}(f_{L_{1},\ldots,L_{n-1},K_{n},L_{n}})^{q+n}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}=\lim_{q\rightarrow\infty}\left(\frac{1}{V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})}\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)^{\frac{q}{q+n}}dv(L_{1},\ldots,L_{n})\right)^{q+n}.$
$None$

On the other hand, from Hölder’s inequality

$\left(\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)^{\frac{q}{q+n}}dv(L_{1},\ldots,L_{n})\right)^{(q+n)/q}\left(\int_{S^{n-1}}dv(L_{1},\ldots,L_{n})\right)^{-n/q}$
$\leq\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)dv(L_{1},\ldots,L_{n})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$=V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n}).~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$None$

From the equality condition of Hölder’s inequality, it follows the equality in
(4.15) holds if and only if $h_{K_{n}}$ and $h_{L_{n}}$ are proportional. This
yiels equality in (4.15) holds if and only if $K_{n}$ are $L_{n}$ homothetic.

Namely

$\left(\frac{1}{V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})}\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)^{\frac{q}{q+n}}dv(L_{1},\ldots,L_{n})\right)^{q+n}\leq\left(\frac{V(L_{1},\ldots,L_{n})}{V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})}\right)^{n},$

with equality if and only if $K_{n}$ are $L_{n}$ homothetic.

Hence

$\exp\left(-\frac{n}{V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})}\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)\ln\left(\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)\right)dv(L_{1},\ldots,L_{n})\right)$
$\leq\left(\frac{V(L_{1},\ldots,L_{n})}{V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})}\right)^{n},$

with equality if and only if $K_{n}$ are $L_{n}$ homothetic.

That is

$\frac{1}{V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})}\int_{S^{n-1}}\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)\ln\left(\varphi\left(\frac{h_{K_{n}}}{h_{L_{n}}}\right)\right)dv(L_{1},\ldots,L_{n})$
$\geq\ln\left(\frac{V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})}{V(L_{1},\ldots,L_{n})}\right),$

with equality if and only if $K_{n}$ are $L_{n}$ homothetic.

Therefore

$\int_{S^{n-1}}\ln\left(\varphi\left(\frac{h_{K_{n}}}{{h_{L_{n}}}}\right)\right)d\bar{V}_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})\geq\ln\left(\frac{V_{\varphi}(L_{1},\ldots,L_{n-1},K_{n},L_{n})}{V(L_{1},\ldots,L_{n})}\right),$
$None$

with equality if and only if $K_{n}$ are $L_{n}$ homothetic.

Further, on the right side of (4.16), by using the Orlicz-Aleksandrov-Fenchel
inequality (3.2), (4.16) becomes (4.9).

This completes the proof. $\Box$

Unfortunately, precise equality for logarithmic Aleksandrov-Fenchel inequality
(the second inequality in (4.9)) are also unknown in general, because the
precise equality for the classical Aleksandrov-Fenchel inequality is unknown
in general, see [19, p.376] for a full discussion.

Corollary 4.4 If $K,L\in{\cal K}^{n}_{0}$ and $\varphi\in\Phi$, then

$\int_{S^{n-1}}\ln\left(\frac{h_{K}}{h_{L}}\right)d\overline{W}_{\varphi,i}(L,K)\geq\frac{1}{n-i}\ln\left(\frac{W_{i}(K)}{W_{i}(L)}\right).$
$None$

each equality holds if and only if $K$ and $L$ are homothetic. Here

$dw_{\varphi,i}(L,K)=\frac{1}{n}\varphi\left(\frac{h_{K}}{h_{L}}\right)h_{L}dS_{i}(L,u),$
$None$

and

$d\overline{W}_{\varphi,i}(L,K)=\frac{1}{W_{\varphi,i}(L,K)}dw_{\varphi,i}(L,K),$
$None$

denotes its normalization.

Proof This follows immediately from Theorem 4.3. $\Box$

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# Linear Quadratic Dual Control

Anders Rantzer The author is affiliated with Automatic Control LTH, Lund
University, Box 118, SE-221 00 Lund, Sweden. He is a member of the Excellence
Center ELLIIT and Wallenberg AI, Autonomous Systems and Software Program
(WASP). Support was received from the European Research Council (Advanced
Grant 834142)

###### Abstract

This is a draft paper posted on Arxiv as a documentation of a plenary lecture
at CDC2023. Some of the core material has been submitted for publication at
L4DC 2024.

An adaptive controller subject to (unknown) linear dynamics and a (known)
quadratic objective is derived based on a “data-driven Riccati equation”. The
main result quantifies closed loop performance in terms of input excitation
level and the degree of the plant stabilizability.

##  I Introduction

Adaptive control has a long history, dating back to aircraft autopilot
development in the 1950s. Following the landmark paper [1], a surge of
research activity during the 1970s derived conditions for convergence,
stability, robustness and performance under various assumptions. For example,
[12] analysed adaptive algorithms using averaging, [7] derived an algorithm
that gives mean square stability with probability one, while [9] gave
conditions for the optimal asymptotic rate of convergence. On the other hand,
conditions that may cause instability were studied in [6], [10] and [16].
Altogether, the subject has a rich history documented in numerous textbooks,
such as [2], [8], and [17].

In this paper, the focus is on worst-case models for disturbances and
uncertain parameters, as discussed in [5, 18, 19, 13] and more recently in
[14, 4, 11]. However, the disturbances in this paper are assumend to be
bounded in terms of past states and inputs. This causality constraint is
different from above mentioned references.

##  II Notation

The set of $n\times m$ matrices with real coefficients is denoted
$\mathbb{R}^{n\times m}$. The transpose of a matrix $A$ is denoted $A^{\top}$.
For a symmetric matrix $A\in\mathbb{R}^{n\times n}$, we write $A\succ 0$ to
say that $A$ is positive definite, while $A\succeq 0$ means positive semi-
definite. Given $x\in\mathbb{R}^{n}$ and $A\in\mathbb{R}^{n\times n}$, the
notation $|x|^{2}_{A}$ means $x^{\top}Ax$. The expression
$\min_{K}{\tiny\begin{bmatrix}I\\\
K\end{bmatrix}^{\top}}Q{\tiny\begin{bmatrix}I\\\ K\end{bmatrix}}$ is
equivalent to $Q^{xx}-Q^{xu}(Q^{uu})^{-1}Q^{ux}$ where
$Q={\tiny\begin{bmatrix}Q^{xx}&Q^{ux}\\\ Q^{xu}&Q^{uu}\end{bmatrix}}$.

##  III A Data-driven Riccati Equation

Consider a linear quadratic optimal control problem:

Minimize
$\displaystyle\sum_{t=0}^{\infty}\left(|x_{t}|^{2}+|u_{t}|^{2}\right)$ subject
to $\displaystyle x_{t+1}=Ax_{t}+Bx_{t}.$

Assuming that the system is stabilizable, the optimal value has the form
$|x_{0}|^{2}_{P}$ where $P$ can be obtained by solving the Riccati equation

$\displaystyle|x|^{2}_{P}$
$\displaystyle=\min_{u}\left[|x|^{2}+|u|^{2}+|Ax+Bu|^{2}_{P}\right].$ (1)

Define $Q$ by ${\tiny\begin{bmatrix}x\\\
u\end{bmatrix}}^{\top}Q{\tiny\begin{bmatrix}x\\\
u\end{bmatrix}}=|x|^{2}+|u|^{2}+|Ax+Bu|^{2}_{P}$. Then (1) can alternatively
be written as

$\displaystyle{\begin{bmatrix}x\\\
u\end{bmatrix}^{\top}(Q-I)\begin{bmatrix}x\\\ u\end{bmatrix}}$
$\displaystyle=x_{+}^{\top}\min_{K}{\begin{bmatrix}I\\\
K\end{bmatrix}^{\top}}Q{\begin{bmatrix}I\\\ K\end{bmatrix}x_{+}}$ (2)

where $x_{+}=Ax+Bu$. Without knowing the model parameters $(A,B)$, it is
possible to collect data points $(x,u,x_{+})$ and use (2) to get information
about $Q$. In fact, the total matrix $Q$ can be computed from a trajectory
$x_{0},u_{0},\ldots,x_{n},u_{N}$ spanning all directions of $(x_{t},u_{t})$,
using the equation

$\displaystyle{\small\begin{bmatrix}x_{0}&\ldots&x_{t}\\\
u_{0}&\ldots&u_{t}\end{bmatrix}^{\top}}(Q-I){\small\begin{bmatrix}x_{0}&\ldots&x_{t}\\\
u_{0}&\ldots&u_{t}\end{bmatrix}}$
$\displaystyle={\small\begin{bmatrix}x_{1}&\ldots&x_{t+1}\end{bmatrix}^{\top}\min_{K}\begin{bmatrix}I\\\
K\end{bmatrix}^{\top}}Q{\small\begin{bmatrix}I\\\
K\end{bmatrix}{\small\begin{bmatrix}x_{1}&\ldots&x_{t+1}\end{bmatrix}}}$

This is essentially equation (3) in [3] and (14) in [15]. However, rather than
iterating over $Q$ as in most reinforcement learning algorithms, we multiply
from the left by

$\displaystyle\begin{bmatrix}\lambda^{t}x_{0}&\lambda^{t-1}x_{1}&\ldots&x_{t-1}\\\
\lambda^{t}u_{0}&\lambda^{t-1}u_{1}&\ldots&u_{t-1}\end{bmatrix},$

its transpose from the right. This gives a _data driven Riccati equation_

$\displaystyle{\Sigma}_{t}\left(Q-I\right){\Sigma}_{t}$
$\displaystyle=\hat{{\Sigma}}_{t}^{\top}\min_{K}\left(\begin{bmatrix}I\\\
K\end{bmatrix}^{\top}Q\begin{bmatrix}I\\\
K\end{bmatrix}\right)\hat{{\Sigma}}_{t}$ (3)

where $\lambda$ is a forgetting factor and

$\displaystyle{\Sigma}_{t}$
$\displaystyle=\sum_{k=0}^{t-1}\lambda^{t-1-k}{\small\begin{bmatrix}x_{k}\\\
u_{k}\end{bmatrix}\begin{bmatrix}x_{k}\\\ u_{k}\end{bmatrix}^{\top}}$
$\displaystyle\hat{{\Sigma}}_{t}$
$\displaystyle=\sum_{k=0}^{t-1}\lambda^{t-1-k}{\small
x_{k+1}\begin{bmatrix}x_{k}\\\ u_{k}\end{bmatrix}^{\top}}$

are defined by past data. When ${\Sigma}_{t}$ has full rank, this corresponds
to weighted least squares estimation of $A$ and $B$ according to
$\begin{bmatrix}\hat{A}_{t}&\hat{B}_{t}\end{bmatrix}:=\hat{{\Sigma}}_{t}{\Sigma}_{t}^{-1}$,
followed by solution of a standard Riccati equation. However, the analysis of
the next sections becomes more straightforward by avoiding the detour over $A$
and $B$, especially when ${\Sigma}_{t}$ is singular.

##  IV Problem formulation

Given $\beta\geq 1$, define $\mathcal{M}_{\beta}$ as the set of all pairs
$(A,B)$ such that the Riccati equation

$\displaystyle
Q-I=\min_{K}\left(\begin{bmatrix}A&B\end{bmatrix}^{\top}\begin{bmatrix}I\\\
K\end{bmatrix}^{\top}Q\begin{bmatrix}I\\\
K\end{bmatrix}\begin{bmatrix}A&B\end{bmatrix}\right)$ (4)

has a solution $Q$ with $I\preceq Q\preceq\beta^{2}I$. Our objective is to
find a controller $\mu:(x_{0},\ldots,x_{t})\mapsto u_{t}$, that stabilizes the
system

$\displaystyle x_{t+1}$ $\displaystyle=Ax_{t}+Bu_{t}+w_{t}$ $\displaystyle t$
$\displaystyle\geq 0$ (5)

for all $(A,B)\in\mathcal{M}_{\beta}$ subject to a bound on the disturbances.
Moreover, if $\lim_{t\to\infty}w_{t}=0$, we expect the controller to approach
the optimal state feedback behavior.

Causal
---
Controller
$x_{+}=Ax+Bu+w$$(x,u)$$w$$u$$x$ Unmodelled
---
dynamics

##  V The Linear Quadratic Dual Controller

We will study a controller of the form

$\displaystyle{\Sigma}_{t+1}$
$\displaystyle=\lambda\underbrace{\begin{bmatrix}{\Sigma}^{xx}_{t}&{\Sigma}^{xu}_{t}\\\
{\Sigma}^{ux}_{t}&{\Sigma}^{uu}_{t}\end{bmatrix}}_{{\Sigma}_{t}}+\begin{bmatrix}x_{t}\\\
u_{t}\end{bmatrix}\begin{bmatrix}x_{t}\\\
u_{t}\end{bmatrix}^{\top}\qquad{\Sigma}_{0}=0$ (6)
$\displaystyle\hat{{\Sigma}}_{t+1}$
$\displaystyle=\lambda\hat{{\Sigma}}_{t}+x_{t+1}\begin{bmatrix}x_{t}^{\top}&u_{t}^{\top}\end{bmatrix}\qquad\quad\quad\;\;\;\,\hat{{\Sigma}}_{0}=0$
(7) $\displaystyle K_{t}$
$\displaystyle=\mathbf{K}({\Sigma}_{t},\hat{{\Sigma}}_{t})$ (8) $\displaystyle
u_{t}$
$\displaystyle=K_{t}x_{t}+\epsilon|x_{t}|\mathbf{E}\left({\Sigma}_{t},K_{t}x_{t}-{\Sigma}_{t}^{ux}({\Sigma}_{t}^{xx})^{\dagger}x_{t}\right)$
(9)

The controller states ${\Sigma}_{t},\hat{{\Sigma}}_{t}$, collect correlation
data with forgetting factor $\lambda\in[0,1]$. The _controller map_
$\mathbf{K}$ determines the feedback matrix $K_{t}$:

###### Definition 1

Given ${\Sigma}_{t},\hat{{\Sigma}}_{t}$, solve (3) for $Q_{t}\succeq I$. Let
$K_{t}$ be a corresponding minimizing $K$. Define
$\mathbf{K}({\Sigma}_{t},\hat{{\Sigma}}_{t}):=K_{t}$.

###### Definition 2

Given a positive definite matrix

$\displaystyle{\Sigma}=\begin{bmatrix}{\Sigma}^{xx}&{\Sigma}^{xu}\\\
{\Sigma}^{ux}&{\Sigma}^{uu}\end{bmatrix}\in\mathbb{R}^{(n+m)\times(n+m)},$

let the matrix ${\Sigma}^{uu}-{\Sigma}^{ux}({\Sigma}^{xx})^{-1}{\Sigma}^{xu}$
have the eigenvalues $\sigma_{1}\geq\sigma_{2}\geq\cdots\geq\sigma_{m}\geq 0$
with corresponding eigenvectors $e_{1},\ldots,e_{m}$. For $v\in\mathbb{R}^{m}$
with $v=v_{1}e_{1}+\cdots+v_{m}e_{m}$, define the _excitation map_
$\mathbf{E}({\Sigma},v):=\operatorname{sign}(v_{m})e_{m}$.

##  VI Main Result

Defining

$\displaystyle{\Sigma}_{t+1}^{wx}$
$\displaystyle:=\lambda{\Sigma}_{t}^{wx}+w_{t}x_{t}^{\top},$
$\displaystyle{\Sigma}_{0}^{wx}=0$ $\displaystyle{\Sigma}_{t+1}^{xw}$
$\displaystyle:=\lambda{\Sigma}_{t}^{xw}+x_{t}w_{t}^{\top},$
$\displaystyle{\Sigma}_{0}^{xw}=0$ $\displaystyle{\Sigma}_{t+1}^{ww}$
$\displaystyle:=\lambda{\Sigma}_{t}^{xw}+w_{t}w_{t}^{\top},$
$\displaystyle{\Sigma}_{0}^{ww}=0,$

we will assume that

$\displaystyle{\Sigma}^{ww}_{t}$
$\displaystyle\preceq\gamma^{-2}{\Sigma}^{xx}_{t}$ $\displaystyle 0$
$\displaystyle\preceq\begin{bmatrix}({\Sigma}^{xx}_{t})^{2}&{\Sigma}^{xw}_{t}\\\
{\Sigma}^{wx}_{t}&\gamma^{-2}I\end{bmatrix}.$ (10)

The system (5)-(9) is said to have _excitation level_ $\delta$ at time $t$ if

$\displaystyle{\Sigma}_{t}$
$\displaystyle\succeq\delta\begin{bmatrix}{\Sigma}^{xx}_{t}&0\\\
0&\|{\Sigma}_{t}\|I\end{bmatrix}.$ (11)

The following main result can now be stated.

###### Theorem 3

Suppose that $(A,B)\in\mathcal{M}_{\beta}$ with $Q$ and $K$ being the
corresponding solutions to the Riccati equation (4). Let
$\alpha:=1-2\sqrt{2}\beta(\beta^{2}+1)^{2}\gamma^{-1}\delta^{-1}$ be positive.
Suppose that the system (5)-(9) satisfies the disturbance bound (10) and has
excitation level $\delta$ for $t\geq t_{0}\geq n+m$. Then it has a well-
defined solution satisfying the dissipation inequality

$\displaystyle\begin{bmatrix}x_{t+1}-w_{t+1}\\\
K_{t+1}x_{t+1}\end{bmatrix}^{\top}Q\begin{bmatrix}x_{t+1}-w_{t+1}\\\
K_{t+1}x_{t+1}\end{bmatrix}$
$\displaystyle\leq\alpha\left(|x_{t}|^{2}+|K_{t}x_{t}|^{2}|\right)+\begin{bmatrix}x_{t}\\\
K_{t}x_{t}\end{bmatrix}^{\top}Q\begin{bmatrix}x_{t}\\\
K_{t}x_{t}\end{bmatrix}.$ (12)

The proof will rely on a few preliminary results:

###### Proposition 4

Given ${\Sigma}_{t},\hat{{\Sigma}}_{t}$, consider $Q_{t}$ and $K_{t}$ as in
Definition 1. Let
$\tilde{\Sigma}_{t}:=\begin{bmatrix}{\Sigma}_{t}^{wx}&{\Sigma}_{t}^{wu}\end{bmatrix}:=\sum_{\tau=0}^{t-1}\begin{bmatrix}w_{\tau}x_{\tau}^{\top}&w_{\tau}u_{\tau}^{\top}\end{bmatrix}$
and suppose that (11) holds. Then

$\displaystyle\tilde{\Sigma}_{t}^{\top}\tilde{\Sigma}_{t}$
$\displaystyle\preceq 2\gamma^{-2}\delta^{-2}{\Sigma}_{t}^{2}.$ (13)

Proof.  First note that (10) gives

$\displaystyle\left\|{\Sigma}_{t}^{wu}\right\|^{2}$
$\displaystyle\leq\left\|{\Sigma}_{t}^{ww}\right\|\cdot\left\|{\Sigma}_{t}^{uu}\right\|$
$\displaystyle\leq\gamma^{-2}\left\|{\Sigma}_{t}^{xx}\right\|\cdot\left\|{\Sigma}_{t}^{uu}\right\|$
$\displaystyle\leq\gamma^{-2}\left\|{\Sigma}_{t}\right\|^{2}$

Together with (10) and (11), this gives

$\displaystyle 0$
$\displaystyle\preceq\begin{bmatrix}\gamma^{-2}I&{\Sigma}_{t}^{wx}&0\\\
{\Sigma}_{t}^{xw}&({\Sigma}_{t}^{xx})^{2}&0\\\
0&0&0\end{bmatrix}+\begin{bmatrix}\gamma^{-2}I&0&{\Sigma}_{t}^{wu}\\\ 0&0&0\\\
{\Sigma}_{t}^{uw}&0&\|\Sigma_{t}\|^{2}\end{bmatrix}$
$\displaystyle=\begin{bmatrix}2\gamma^{-2}I&{\Sigma}_{t}^{wx}&{\Sigma}_{t}^{wu}\\\
{\Sigma}_{t}^{xw}&({\Sigma}_{t}^{xx})^{2}&0\\\
{\Sigma}_{t}^{uw}&0&\|\Sigma_{t}\|^{2}\end{bmatrix}$
$\displaystyle\preceq\begin{bmatrix}2\gamma^{-2}I&\tilde{\Sigma}_{t}\\\
\tilde{\Sigma}_{t}^{\top}&\delta^{-2}{\Sigma}_{t}^{2}\end{bmatrix}$

and the desired inequality follows by Schur complement. $\Box$

###### Proposition 5

Consider $E,F\in\mathbb{R}^{n\times n}$ and $\theta>0$. Then

$\displaystyle(E+F)^{\top}M(E+F)$
$\displaystyle\preceq(1+\theta)E^{\top}ME+(1+\theta^{-1})F^{\top}MF.$

Proof.

$\displaystyle(1+\theta)E^{\top}ME+(1+\theta^{-1})F^{\top}MF$
$\displaystyle=\theta^{-1}(\theta E-F)^{\top}M(\theta E-F)+(E+F)^{\top}M(E+F)$
$\displaystyle\succeq(E+F)^{\top}M(E+F)$

$\Box$

###### Proposition 6

Suppose that $0\prec T\in\mathbb{R}^{n\times n}$ and

$\displaystyle 0\preceq\Sigma$
$\displaystyle=\begin{bmatrix}\Sigma^{xx}&\Sigma^{xu}\\\
\Sigma^{ux}&\Sigma^{uu}\end{bmatrix}\in\mathbb{R}^{(n+m)\times(n+m)}$

with $\Sigma^{uu}\succ 0$. Let $\hat{\Sigma}\in\mathbb{R}^{n\times(n+m)}$ with

$\displaystyle 0$
$\displaystyle\preceq\begin{bmatrix}\tau\Sigma^{xx}&\hat{\Sigma}\\\
\hat{\Sigma}^{\top}&\Sigma\end{bmatrix}\text{ for some }\tau\geq 0.$ (14)

Then $\Sigma Q\Sigma$ has the same value for all solutions to

$\displaystyle\Sigma(Q-T)\Sigma$
$\displaystyle=\min_{K}\hat{\Sigma}^{\top}\begin{bmatrix}I\\\
K\end{bmatrix}^{\top}Q\begin{bmatrix}I\\\ K\end{bmatrix}\hat{\Sigma}.$ (15)

Moreover, if $Q_{1}$ and $Q_{2}$ solves (15) for $T_{1}$ and $T_{2}$
respectively and $0\prec T_{1}\preceq T_{2}$, then $\Sigma
Q_{1}\Sigma\preceq\Sigma Q_{2}\Sigma$.

Proof.  Let $\Sigma^{xx}-\Sigma^{xu}(\Sigma^{uu})^{-1}\Sigma^{ux}$ have the
eigenvalue decomposition $U\Lambda U^{\top}$ where $U=[U_{1}\;U_{2}]$ is
unitary and $\Lambda$ is diagonal with all zero eigenvalues in the upper left
corner and the columns of $U_{1}$ as the corresponding eigenvectors. Define

$\displaystyle\bar{\Sigma}$ $\displaystyle:=\begin{bmatrix}U_{2}&0\\\
0&I\end{bmatrix}^{\top}{\Sigma}\begin{bmatrix}U_{2}&0\\\ 0&I\end{bmatrix}$
$\displaystyle\bar{Q}$ $\displaystyle:=\begin{bmatrix}U_{2}&0\\\
0&I\end{bmatrix}^{\top}Q\begin{bmatrix}U_{2}&0\\\ 0&I\end{bmatrix}$
$\displaystyle\bar{T}$ $\displaystyle:=\begin{bmatrix}U_{2}&0\\\
0&I\end{bmatrix}^{\top}T\begin{bmatrix}U_{2}&0\\\ 0&I\end{bmatrix}.$

Then

$\displaystyle\left[\begin{array}[]{c|cc}U_{1}&U_{2}&0\\\
0&0&I\end{array}\right]^{\top}\Sigma\left[\begin{array}[]{c|cc}U_{1}&U_{2}&0\\\
0&0&I\end{array}\right]=\left[\begin{array}[]{c|c}0&0\\\
\hline\cr\\\\[-8.53581pt] 0&\bar{\Sigma}\end{array}\right]$

and (14) gives

$\displaystyle\hat{\Sigma}\begin{bmatrix}U&0\\\ 0&I\end{bmatrix}$
$\displaystyle=\begin{bmatrix}0\\\ \check{\Sigma}\end{bmatrix}$

for some choice of $\check{\Sigma}$. It follows that

$\displaystyle\bar{Q}-\bar{T}$
$\displaystyle=\min_{\bar{K}}\left(\bar{\Sigma}^{-1}\check{\Sigma}^{\top}\begin{bmatrix}I\\\
\bar{K}\end{bmatrix}^{\top}\bar{Q}\begin{bmatrix}I\\\
\bar{K}\end{bmatrix}\check{\Sigma}\bar{\Sigma}^{-1}\right)$

where $\bar{K}$ has the same number of columns as $U_{2}$. Standard dynamic
programming arguments give that $\bar{Q}$ is unique, computable by fixed point
iteration and monotonically dependent on $\bar{T}$. Hence the proof is
completed by noting that

$\displaystyle\Sigma Q\Sigma$ $\displaystyle=\begin{bmatrix}U_{2}&0\\\
0&I\end{bmatrix}\bar{\Sigma}\bar{Q}\bar{\Sigma}\begin{bmatrix}U_{2}&0\\\
0&I\end{bmatrix}^{\top}.$

$\Box$

Proof of Theorem 3. Define $\theta:=\sqrt{2}\beta\gamma^{-1}\delta^{-1}$ and
$\tilde{\Sigma}_{t}$ as in Proposition 4. Define the pair $(Q_{t},K_{t})$ as
in Definition 1. The proof will have the following steps:

  1. 1.

Verify that $Q_{t}\preceq(1-\beta^{2}\theta-\theta^{2})^{-1}Q$.

  2. 2.

Verify that $Q\preceq(1-\beta^{2}\theta-\theta^{2})^{-1}Q_{t}$.

  3. 3.

Verify the Lyapunov inequality

$\displaystyle 0$ $\displaystyle\preceq\Sigma_{t}\left(Q-\alpha
I-\begin{bmatrix}A&B\end{bmatrix}^{\top}\begin{bmatrix}I\\\
K_{t}\end{bmatrix}^{\top}Q\begin{bmatrix}I\\\
K_{t}\end{bmatrix}\begin{bmatrix}A&B\end{bmatrix}\right)\Sigma_{t}.$

Step 1:  Recall that $Q$ and $K$ are obtained from
$(A,B)\in\mathcal{M}_{\beta}$ by solving (4). Hence, the assignments

$\displaystyle M$ $\displaystyle:=\begin{bmatrix}I\\\
K\end{bmatrix}^{\top}Q\begin{bmatrix}I\\\ K\end{bmatrix},$ $\displaystyle E$
$\displaystyle:=\begin{bmatrix}A&B\end{bmatrix}\Sigma_{t},$ $\displaystyle F$
$\displaystyle:=\tilde{\Sigma}_{t},$

together with (4) gives

$\displaystyle E^{\top}ME$ $\displaystyle=\Sigma_{t}(Q-I)\Sigma_{t}$
$\displaystyle F^{\top}MF$
$\displaystyle\preceq\beta^{2}F^{\top}F=\beta^{2}\tilde{\Sigma}_{t}^{\top}\tilde{\Sigma}_{t}\preceq
2\beta^{2}\gamma^{-2}\delta^{-2}{\Sigma}_{t}^{2}=\theta^{2}\Sigma_{t}^{2}.$

so Proposition 5 gives

$\displaystyle\hat{\Sigma}_{t}^{\top}\begin{bmatrix}I\\\
K\end{bmatrix}^{\top}Q\begin{bmatrix}I\\\ K\end{bmatrix}\hat{\Sigma}_{t}$
$\displaystyle=(E+F)^{\top}M(E+F)$
$\displaystyle\preceq(1+\theta)E^{\top}ME+(1+\theta^{-1})F^{\top}MF$
$\displaystyle\preceq(1+\theta)\Sigma_{t}(Q-I)\Sigma_{t}+(\theta^{2}+\theta)\Sigma_{t}^{2}$
$\displaystyle=(1+\theta)\Sigma_{t}Q\Sigma_{t}+(\theta^{2}-1)\Sigma_{t}^{2}$
$\displaystyle\preceq\Sigma_{t}\left[Q+(\beta^{2}\theta+\theta^{2}-1)I\right]\Sigma_{t}.$

Hence the inequality $Q_{t}\preceq(1-\beta^{2}\theta-\theta^{2})^{-1}Q$
follows from Proposition 6. This finishes Step 1 of the proof.

Step 2: The inequality is obtained as in Step 1, by reversing the roles of $Q$
and $Q_{t}$: With

$\displaystyle M$ $\displaystyle:=\begin{bmatrix}I\\\
K_{t}\end{bmatrix}^{\top}Q_{t}\begin{bmatrix}I\\\ K_{t}\end{bmatrix},$
$\displaystyle E$ $\displaystyle:=\hat{\Sigma}_{t},$ $\displaystyle F$
$\displaystyle:=-\tilde{\Sigma}_{t},$

Proposition 5 gives

$\displaystyle\Sigma_{t}\left(\begin{bmatrix}A&B\end{bmatrix}^{\top}\begin{bmatrix}I\\\
K_{t}\end{bmatrix}^{\top}Q_{t}\begin{bmatrix}I\\\
K_{t}\end{bmatrix}\begin{bmatrix}A&B\end{bmatrix}\right)\Sigma_{t}$
$\displaystyle=(E+F)^{\top}M(E+F)$
$\displaystyle\preceq(1+\theta)E^{\top}ME+(1+\theta^{-1})F^{\top}MF$
$\displaystyle\preceq\Sigma_{t}\left[(1+\theta)Q_{t}+(\theta^{2}-1)I\right]\Sigma_{t}$
$\displaystyle\preceq\Sigma_{t}\left[Q_{t}+(\beta^{2}\theta+\theta^{2}-1)I\right]\Sigma_{t}.$

Together with (4) this allows for application of Proposition 6 to get
$Q\preceq(1-\beta^{2}\theta-\theta^{2})^{-1}Q_{t}$ and Step 2 is complete.

Step 3: Combining the outcomes of Step 1 and Step 2 gives

$\displaystyle\Sigma_{t}\left(\begin{bmatrix}A&B\end{bmatrix}^{\top}\begin{bmatrix}I\\\
K_{t}\end{bmatrix}^{\top}Q\begin{bmatrix}I\\\
K_{t}\end{bmatrix}\begin{bmatrix}A&B\end{bmatrix}\right)\Sigma_{t}$
$\displaystyle\preceq\Sigma_{t}\left(\begin{bmatrix}A&B\end{bmatrix}^{\top}\begin{bmatrix}I\\\
K_{t}\end{bmatrix}^{\top}\frac{Q_{t}}{1-\beta^{2}\theta-\theta^{2}}\begin{bmatrix}I\\\
K_{t}\end{bmatrix}\begin{bmatrix}A&B\end{bmatrix}\right)\Sigma_{t}$
$\displaystyle\preceq\frac{1+\theta}{1-\beta^{2}\theta-\theta^{2}}\Sigma_{t}\left[Q_{t}+\theta
I-I\right]\Sigma_{t}$
$\displaystyle\preceq\frac{1+\theta}{(1-\beta^{2}\theta-\theta^{2})^{2}}\Sigma_{t}Q\Sigma_{t}+\frac{\theta^{2}-1}{1-\beta^{2}\theta-\theta^{2}}\Sigma_{t}^{2}$
$\displaystyle\preceq\Sigma_{t}\left(Q-\alpha_{0}I\right)\Sigma_{t},$ (16)

where $\alpha_{0}$ is defined by

$\displaystyle\alpha_{0}$
$\displaystyle:=\frac{1-\theta^{2}}{1-\beta^{2}\theta-\theta^{2}}-\beta^{2}\left[\frac{(1+\theta)}{(1-\beta^{2}\theta-\theta^{2})^{2}}-1\right]$

and bounded by

$\displaystyle\alpha_{0}$
$\displaystyle\geq\alpha_{0}(1-\beta^{2}\theta-\theta^{2})^{2}$
$\displaystyle=(1-\theta^{2})(1-\beta^{2}\theta-\theta^{2})+\beta^{2}(1-\beta^{2}\theta-\theta^{2})^{2}-\beta^{2}(1+\theta)$
$\displaystyle\geq
1-\beta^{2}\theta-2\theta^{2}+\beta^{2}(1-2\beta^{2}\theta-2\theta^{2})-\beta^{2}(1+\theta)$
$\displaystyle=1-\theta\left[\beta^{2}+2\theta+2\beta^{4}+2\beta^{2}\theta+\beta^{2}\right]$
$\displaystyle\geq 1-2\theta(\beta^{2}+1)^{2}$
$\displaystyle=1-2\sqrt{2}\beta(\beta^{2}+1)^{2}\gamma^{-1}\delta^{-1}.$

For $t\geq n+m$, $(x_{t},u_{t})$ must be in the image of $\Sigma_{t}$, so (16)
implies that

$\displaystyle 0$ $\displaystyle\geq\begin{bmatrix}x_{t}\\\
u_{t}\end{bmatrix}^{\top}\left(Q-\alpha
I-\begin{bmatrix}A&B\end{bmatrix}^{\top}\begin{bmatrix}I\\\
K_{t}\end{bmatrix}^{\top}Q\begin{bmatrix}I\\\
K_{t}\end{bmatrix}\begin{bmatrix}A&B\end{bmatrix}\right)\begin{bmatrix}x_{t}\\\
u_{t}\end{bmatrix}$ $\displaystyle=\begin{bmatrix}x_{t}\\\
u_{t}\end{bmatrix}^{\top}\left(Q-\alpha I\right)\begin{bmatrix}x_{t}\\\
u_{t}\end{bmatrix}-\begin{bmatrix}x_{t+1}-w_{t+1}\\\
u_{t+1}\end{bmatrix}^{\top}Q\begin{bmatrix}x_{t+1}-w_{t+1}\\\
u_{t+1}\end{bmatrix}$

and the proof is complete. $\Box$

##  VII Conclusions

A lot remains to be done. In particular, we have still not documented the
relationship between $\epsilon$ in the excitation term of the controller and
$\delta$ in the excitation condition of the main theorem. This is also where
the forgetting factor enters the analysis. Hopefully this will be done soon,
if not before CDC lecture…

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