# Stability estimates for semigroups
in the Banach case.

B. Helffer

###### Abstract

The purpose of this paper is to revisit previous works of the author with J.
Sjöstrand (2010–2021) proved in the Hilbert case by considering the Banach
case at the light of a paper by Y. Latushkin and V. Yurov (2013).

##  1 Introduction

Let ${\cal B}$ be a complex Banach space and let $[0,+\infty[\ni t\mapsto
S(t)\in{\cal L}({\cal B},{\cal B})$ be a strongly continuous semigroup with
$S(0)=I$.

The purpose of this note is to revisit some results of [6] and [7] which have
been established in the case of a Hilbert space and to consider their
extension to the Banach case. The idea in the Hilbert space is essentially to
use the property of the inhomogeneous equation $(\partial_{t}-A)u=w$ in
exponentially weighted spaces which are related via Fourier-Laplace transform
and then use Plancherel’s formula. This approach cannot be used in the Banach
case (see [11] for a discussion) but one can see in [9] an alternative
approach proposed by Y. Latushkin and V. Yurov which combined with the
approach of [6] permits this extension. We will describe how to extend also
all the results of [7].

The main result in [6] was established in the Hilbertian case:

###### Theorem 1.1

Let assume that $\mathcal{B}$ is a Hilbert space. Let $r(\omega)$ defined by

$\frac{1}{r(\omega)}:=\sup_{\Re
z\geq\omega}||(A-z)^{-1}||_{\mathcal{L}(\mathcal{B})}\,.$

Let $m(t)\geq\|S(t)\|_{\mathcal{L}(\mathcal{B})}$ be a continuous positive
function. Then for all $t,a,b>0$, such that $t\geq a+b$, we have

$\|S(t)\|\leq\frac{e^{\omega
t}}{r(\omega)\|\frac{1}{m}\|_{e^{-\omega\cdot}L^{2}(0,a)}\|\frac{1}{m}\|_{e^{-\omega\cdot}L^{2}(0,b)}}.$
(1.1)

Here the norms are always the natural ones obtained from ${\cal B}$, $L^{2}$,
thus for instance $\|S(t)\|=\|S(t)\|_{{\cal L}({\cal B},{\cal B})}$, if $u$ is
a function on $\mathbb{R}$ with values in $\mathbb{C}$ or in ${\cal B}$,
$\|u\|$ denotes the natural $L^{2}$ norm. In (1.1) we also have the natural
norm in the exponentially weighted space $e^{-\omega\cdot}L^{2}(0,a)$ and
similarly with $b$ instead of $a$;
$\|f\|_{e^{-\omega\cdot}L^{2}(0,a)}=\|e^{\omega\cdot}f(\cdot)\|_{L^{2}(0,a)}$.
The extension proposed by [9] (with less generality111The authors consider the
case when $m(t):=Le^{\lambda t}$ and play instead with the consequence of this
theorem as in [6].) can be obtained by introducing $K_{\omega,p}$ which is
defined by

$\frac{1}{\hat{r}_{p}(\omega)}:=K_{\omega,p}:=||\mathcal{K}^{+}||_{\mathcal{L}(L^{p}_{\omega}(\mathbb{R}_{+};X))}<+\infty$
(1.2)

where

$(\mathcal{K}^{+}u)(t)=\int_{0}^{t}\,S(t-s)u(s)ds\,,\,\mbox{ for }t\geq 0\,.$
(1.3)

Our first new theorem reads:

###### Theorem 1.2

Suppose that $\omega\in\mathbb{R}$, $p>1$, and that $K_{\omega,p}$ is finite.
Let $m(t):[0,+\infty[\to]0,+\infty[$ be a continuous positive function such
that

$\|S(t)\|\leq m(t)\mbox{ for all }t\geq 0\,.$ (1.4)

Then for all $t,a,b>0$ such that $t\geq a+b$,

$\|S(t)\|\leq K_{\omega,p}\frac{e^{\omega
t}}{\|\frac{1}{m}\|_{e^{-\omega\cdot}L^{q}(]0,a[)}\|\frac{1}{m}\|_{e^{-\omega\cdot}L^{p}(]0,b[)}}\,,$
(1.5)

where $q$ is such that $\frac{1}{p}+\frac{1}{q}=1$ .

The idea behind this extension by [9] is to replace the use of the Laplace
transform and the assumptions on the resolvent of $A$ by more directly the
estimate for $\mathcal{K}^{+}$. Hence $1/r(\omega)$ is replaced by
$K_{\omega,p}$.
The proof will be given in Section 2.

Another aim in this note is to see if we can also extend to the Banach case
the results of [7]. Some of the steps are independent of the Banach
assumption, so we just explain the modifications to do in the case of a
reflexive Banach space.
Let $\Phi$ satisfy

$0\leq\Phi\in C^{1}([0,+\infty[)\mbox{ with }\Phi(0)=0\mbox{ and
}\Phi(t)>0\mbox{ for }t>0\,,$ (1.6)

and assume that $\Psi$ has the same properties222By a density argument we can
replace $C^{1}([0,+\infty[)$ in (1.6) by the space of locally Lipschitz
functions on $[0,+\infty[$.. For $t>0$, let $\iota_{t}$ be the reflection with
respect to $t/2$: $\iota_{t}u(s)=u(t-s)$. With this notation, we have the
following theorem:

###### Theorem 1.3

We assume that $\mathcal{B}$ is a complex reflexive Banach space. Under the
assumptions of Theorem 1.2, for any $\Phi$ and $\Psi$ satisfying (1.6) and for
any $\epsilon_{1},\epsilon_{2}\in\\{-,+\\}$, we have

$||S(t)||_{\mathcal{L}(\cal B)}\leq e^{\omega
t}\frac{\|(\hat{r}_{p}(\omega)^{p}\Phi^{p}-|\Phi^{\prime}|^{p})^{\frac{1}{p}}_{-}\,m\|_{e^{\omega\cdot}L^{p}(]0,t[)}\|(\hat{r}_{p}(\omega)^{q}\Psi^{q}-|\Psi^{\prime}|^{q})^{\frac{1}{q}}_{-}\,m\|_{e^{\omega\cdot}L^{q}(]0,t[)}}{\int_{0}^{t}(\hat{r}_{p}(\omega)^{p}\Phi^{p}-|\Phi^{\prime}|^{p})^{\frac{1}{p}}_{\epsilon_{1}}(\hat{r}_{p}(\omega)^{q}(\iota_{t}\Psi)^{q}-|\iota_{t}\Psi^{\prime}|^{q})^{\frac{1}{q}}_{\epsilon_{2}}ds}\,.$
(1.7)

Here for $a\in\mathbb{R}$, $a_{+}=\max(a,0)$ and $a_{-}=\max(-a,0)$. This
theorem was established in [7] (Theorem 1.6) in the Hilbert case with $p=2$
and $\hat{r}_{2}(\omega)$ replaced by $r(\omega)$.
With this generalization, the other sections of [7] hold in the case $p=2$, in
particular all the consequences considered as $\epsilon_{1}\epsilon_{2}=-$ in
[7] (Theorem 1.7, Proposition 1.8, Theorem 1.9). Notice that in the Hilbert
case the replacement of $r(\omega)$ by $\hat{r}_{2}(\omega)=1/K_{\omega,2}$
which could appear as stronger but is probably equivalent (see the discussion
in [9]).The case $p\neq 2$ will be further discussed in the last sections,
with in particular an extension of Proposition 1.8 ([7] and an extension of
Wei’s theorem.
These applications will be stated in Theorem 4.6. and in Theorem 6.2.

###### Remark 1.4

There is a huge litterature on the subject and we refer for example to the
recent survey by [12] for a description of the different approachs. In
particular J. Rozendaal and M. Veraar [13] obtain more general results as here
(see for example their Theorem 3.2) albeit with different constants when
considering our particular case. The question of the finiteness of
$K_{\omega,p}$ is also an interesting open question. Notice333We thank J.
Rozendahl for this remark that the inequality $K_{\omega,p}\leq 1/r(\omega)$
also holds, by application of Proposition 3.7 in [13] when $S$ is a positive
semi-group on an $L^{p}$-space.

Acknowledgements
The author was motivated by a question of L. Boulton at the Banff conference
(2022) and encouraged later by discussions with Y. Latushkin during the Aspect
Conference in Oldenburg (2022) organized by K. Pankrashkin.
We thank J. Rozendaal for comments on a previous version of the manuscript and
Y. Latushkin for his careful reading and suggestions of simplification.

##  2 Proof of Theorem 1.2 and applications.

###  2.1 Proof of Theorem 1.2

Following [6] and a simplification proposed by Y. Latushkin, we consider $v\in
D(A)$ and $u(t)=S(t)v$, for solving the Cauchy problem

$\begin{array}[]{l}(\partial_{t}-A)u=0\,,\,t\geq 0\,\,,\\\
u(0)=v\,.\end{array}$

Assume $t>a+b$, and let $\chi$, $\tilde{\chi}$ be decreasing Lipschitz
functions on $\mathbb{R}$, equal to $1$ on $]-\infty,0]$ and such that ${\rm
supp\,}\chi\in(-\infty,a)$ and ${\rm supp\,}\tilde{\chi}\in(-\infty,b)$. We
notice that

$u(t)=S(t-s)S(s)v=S(t-s)u(s)\,,\,\mbox{ for }t\geq s\geq 0\,.$ (2.8)

Then, using (2.8) between the first line and the second line, we obtain, for
$t\geq 0$,

$\begin{array}[]{ll}(1-\chi(t))u(t)&=-(\int_{0}^{t}\chi^{\prime}(s)ds)\,u(t)\\\
&=-\int_{0}^{t}S(t-s)\chi^{\prime}(s)u(s)ds\\\
&=-\mathcal{K}^{+}(\chi^{\prime}(\cdot)u(\cdot))(t)\,.\end{array}$

On the other hand, introducing $\tilde{\chi}$, we first write

$u(t)=\tilde{\chi}(0)u(t)=-\big{(}\int_{t-b}^{t}\tilde{\chi}^{\prime}(t-s)ds\big{)}u(t)\,.$

Proceeding similarly as above we then write (using that $t-b>a$ at the third
line)

$\begin{array}[]{ll}u(t)&=-\int_{t-b}^{t}\tilde{\chi}^{\prime}(t-s)S(t-s)u(s)ds\\\
&=-\int_{t-b}^{t}\tilde{\chi}^{\prime}(t-s)S(t-s)u(s)ds\\\
&=-\int_{t-b}^{t}\tilde{\chi}^{\prime}(t-s)S(t-s)(1-\chi(s))u(s)ds\\\
&=-\int_{t-b}^{t}\tilde{\chi}^{\prime}(t-s)S(t-s)\mathcal{K}^{+}(\chi^{\prime}(\cdot)u(\cdot))(s)ds\,.\end{array}$

Thus, with in mind the assumption on $||S(t)||$, we get

$||e^{-\omega
t}S(t)v||\leq\Big{(}\int_{t-b}^{t}\big{(}e^{-\omega(t-s)}|\tilde{\chi}^{\prime}(t-s)|m(t-s)\big{)}\big{(}||\mathcal{K}^{+}(\chi^{\prime}(\cdot)u(\cdot))(s)||\big{)}ds\Big{)}\,,$

and Hölder’s inequality yields

$e^{-\omega t}||S(t)||_{\mathcal{L}(\mathcal{B})}\leq
K_{\omega,p}||m\chi^{\prime}||_{e^{\omega}L^{p}}||e^{-\omega
s}m\tilde{\chi}^{\prime}||_{e^{\omega}L^{q}}\,,$ (2.9)

as

$||\mathcal{K}^{+}(\chi^{\prime}(\cdot)u(\cdot))||_{e^{\omega\cdot}L^{p}}\leq
K_{\omega,p}||\chi^{\prime}(\cdot)S(\cdot)v||_{e^{\omega\cdot}L^{p}}\ \leq
K_{\omega,p}||\chi^{\prime}m||_{e^{\omega\cdot}L^{p}}||v||\,.$

This is the inequality (3.8) in [9] but with the general weight $m(t)$
replacing the particular weight $Le^{\lambda t}$. When $p=2$, (2.9) is just
the Banach analog of (4.12) in [6] where $1/r(\omega)$ is replaced by
$K_{\omega,2}$.
Following the strategy of [6] it remains to optimize the right hand side by
choosing $\chi$ and $\tilde{\chi}$ optimally. We look for $\chi$ such that
$\|m\chi^{\prime}\|_{e^{\omega}L^{p}(0,a)}$ is as small as possible. By the
Hölder inequality,

$1=\int_{0}^{a}|\chi^{\prime}(s)|ds\leq\|\chi^{\prime}m\|_{e^{\omega\cdot}L^{p}}\|\frac{1}{m}\|_{e^{-\omega\cdot}L^{q}(]0,a[)}\,,$
(2.10)

so

$\|\chi^{\prime}m\|_{e^{\omega\cdot}L^{p}}\geq\frac{1}{\|\frac{1}{m}\|_{e^{-\omega\cdot}L^{q}(]0,a[)}}\,.$
(2.11)

As classical, we get equality in (2.10) if for some constant $C$,

$(|\chi^{\prime}(s)|m(s)e^{-\omega s})^{p}=C\big{(}\frac{1}{m(s)}e^{\omega
s}\big{)}^{q}\hbox{ on }[0,a],$

i.e.

$\chi^{\prime}(s)m(s)e^{-\omega s}=-C^{1/p}\Big{(}\frac{1}{m(s)}e^{\omega
s}\Big{)}^{q/p}\hbox{ on }[0,a],$

where $C$ is determined by the condition

$1=\int_{0}^{a}|\chi^{\prime}(s)|ds\,.$

Doing the same job with $\tilde{\chi}$, we obtain the theorem.

###  2.2 The result of Latushkin-Yurov and extensions

In [9], the authors prove directly the following statement:

###### Theorem 2.1

Let $\omega,\lambda$, $p>1$ and $L>0$. Let $\\{S(t)\\}_{t\geq 0}$ be a
strongly continuous semigroup on a Banach space $\mathcal{B}$. If
$\omega<\lambda$, $||S(t)||\leq Le^{\lambda t}\mbox{ for all }t\geq 0$ and
$K_{\omega,p}<+\infty$ , then

$||S(t)||\leq Me^{\omega t}\mbox{ for all }t\geq 0\,,$

with

$\frac{1}{p}+\frac{1}{q}=1\,,$

and

$M=L(1+4p^{-1/p}q^{-1/q}LK_{\omega,p}(\lambda-\omega))\,.$

The theorem can also be obtained as in [6] for the case $p=2$ as a corollary
of Theorem 1.2.

When $p=2$ and $\mathcal{B}$ is an Hilbert space it is possible to prove (see
[9]) that

$K_{\omega,2}\leq\sup_{s\in\mathbb{R}}||R(A,\omega+is)||_{\mathcal{L}(\mathcal{B})}\,,$

where $R(A,\lambda)=(\lambda-A)^{-1}$ and we recover the statement of [7],
which was obtained as a consequence of the $L^{2}$ version of Theorem 1.2 with
$p=2$.

##  3 Proof of Theorem 1.3 in the reflexive Banach case.

###  3.1 Flux

We assume that $\mathcal{B}$ is a reflexive Banach space and we denote by
$\mathcal{B}^{*}$ its dual444The more common notation is
$\mathcal{B}^{\prime}$.. As before, let $A$ the generator of a strongly
continuous semi-group and $u(t)\in C^{1}([0,+\infty[;{\cal B})\cap
C^{0}([0,+\infty[;{\cal D}(A))$ solve $(A-\partial_{t})u=0$ on $[0,+\infty[$.
As $\mathcal{B}$ is reflexive, we can define $A^{*}$ as the infinitesimal
generator of the dual semi-group which is a strongly continuous semi-group on
$\mathcal{B}^{*}$. Let $u^{*}(t)\in C^{1}(]-\infty,T];{\cal B^{*}})\cap
C^{0}(]-\infty,T];{\cal D}(A^{*}))$ solve $(A^{*}+\partial_{t})u^{*}=0$ on
$]-\infty,T]$. We refer to [1] (and references therein) for the properties of
the dual semi-group. Then the flux (or Wronskian)
$<u(t),u^{*}(t)>_{\mathcal{B},\mathcal{B}^{*}}$ (where the bracket indicates
the duality bracket between $\mathcal{B}$ and $\mathcal{B}^{*}$) is constant
on $[0,T]$ as can be seen by computing the derivative with respect to $t$.

###  3.2 $L^{p}$ estimate

Write $L^{p}_{\phi}(I)=L^{p}(I;e^{-p\phi}dt)=e^{\phi}L^{p}(I)$,
$\|u\|_{p,\phi}=\|u\|_{p,\phi,I}=\|u\|_{L^{p}_{\phi}(I)}$, where $I$ is an
interval and our functions take values in ${\cal B}$. We assume (see (1.2))
that $K_{p,\omega}=1/\hat{r}_{p}(\omega)<+\infty$.

Consider $(A-\partial_{t})u=0$ on $[0,+\infty[$ with $u\in
L^{p}_{\omega\cdot}([0,+\infty[)$.
Let $\Phi$ satisfy (1.6) and add temporarily the assumption that $\Phi(s)$ is
constant for $s\gg 0$. Then $\Phi u$, $\Phi^{\prime}u$ can be viewed as
elements of $L^{p}_{\omega\cdot}(\mathbb{R})$ and from

$(A-\partial_{t})\Phi u=-\Phi^{\prime}u\,,$

we get, by the definition of $\hat{r}_{p}(\omega)$,

$\|\Phi
u\|_{p,\omega\cdot}\leq\frac{1}{\hat{r}_{p}(\omega)}\|\Phi^{\prime}u\|_{p,\omega\cdot}\,.$

Taking the power $p$, we get

$\int_{-\infty}^{+\infty}(\hat{r}_{p}(\omega)^{2}|\Phi|^{p}-|\Phi^{\prime}|^{p})||u(t)||^{p}_{\mathcal{B}}e^{-p\omega
t}dt\leq 0\,,$

which can be rewritten as

$\int_{-\infty}^{+\infty}(\hat{r}_{p}(\omega)^{2}|\Phi|^{p}-|\Phi^{\prime}|^{p})_{+}||u(t)||^{p}_{\mathcal{B}}e^{-p\omega
t}dt\leq\int_{-\infty}^{+\infty}(\hat{r}_{p}(\omega)^{p}|\Phi|^{p}-|\Phi^{\prime}|^{p})_{-}||u(t)||^{p}_{\mathcal{B}}e^{-p\omega
t}dt\,,$

or finally in. the form

$\|(\hat{r}_{p}(\omega)^{p}|\Phi|^{p}-|\Phi^{\prime}|^{p})_{+}^{1/p}u\|_{p,\omega\cdot}\leq\|(\hat{r}_{p}(\omega)^{p}|\Phi|^{p}-|\Phi^{\prime}|^{p})_{-}^{1/p}u\|_{p,\omega\cdot}.$
(3.1)

Writing $\Phi=e^{\phi}$, $\phi\in C^{1}(]0,+\infty[)$, $\phi(t)\to-\infty$
when $t\to 0$, we have

$\hat{r}_{p}(\omega)^{p}|\Phi|^{p}-|\Phi^{\prime}|^{p}=(\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p})e^{p\phi}\,,$

and (3.1) becomes

$\|\big{(}\hat{r}_{p}(\omega))^{p}-|\phi^{\prime}|^{p}\big{)}_{+}^{1/p}u\|_{p,\omega\cdot-\phi}\leq\|\big{(}\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p}\big{)}_{-}^{1/p}u\|_{\omega\cdot-\phi}\,.$
(3.2)

Let $S(t)=e^{tA}$, $t\geq 0$ and let $m(t)>0$ be a continuous function such
that

$\|S(t)\|\leq m(t),\ t\geq 0\,.$ (3.3)

Then we get

$\|\big{(}\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p}\big{)}_{+}^{1/p}u\|_{p,\omega\cdot-\phi}\leq\|\big{(}\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p}\big{)}_{-}^{1/p}m\|_{p,\omega\cdot-\phi}|u(0)|_{\cal
B}\,.$ (3.4)

Note that we have also trivially

$\|(\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p})_{-}^{1/p}u\|_{p,\omega\cdot-\phi}\leq\|\big{(}\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p}\big{)}_{-}^{1/p}m\|_{p,\omega\cdot-\phi}|u(0)|_{\cal
B}\,.$ (3.5)

We get the same bound for the forward solution of $A^{*}-\partial_{t}$ and,
after changing the orientation of time, for the backward solution of
$A^{*}+\partial_{t}=(A-\partial_{t})^{*}$. Then for $u^{*}(s)$, solving

$(A^{*}+\partial_{s})u^{*}(s)=0,\ s\leq t,$

with $u^{*}(t)$ prescribed in $\mathcal{B}^{*}$, we get

$\|\big{(}\hat{r}_{q}^{*}(\omega)^{q}-|\iota_{t}\phi^{\prime}|^{q}\big{)}_{+}^{1/q}u^{*}\|_{q,\omega(t-\cdot)-\iota_{t}\phi}\leq\|\big{(}\hat{r}_{q}^{*}(\omega)^{q}-|\iota_{t}\phi^{\prime}|^{q}\big{)}_{-}^{1/q}\iota_{t}m\|_{q,\omega(t-\cdot)-\iota_{t}\phi}\,|u^{*}(t)|_{\mathcal{B}^{*}}\,,$

where $\iota_{t}\phi$ and $\iota_{t}m$ denote the compositions of $\phi$ and
$m$ respectively with the reflection $\iota_{t}$ in $t/2$ so that

$\iota_{t}m(s)=m(t-s),\ \ \iota_{t}\phi(s)=\phi(t-s)\,.$

Here $\hat{r}_{q}^{*}(\omega)$ is associated with the dual semi-group like in
(1.2)-(1.3). By duality, one can show in the reflexive case that

$\hat{r}_{q}^{*}(\omega)=\hat{r}_{p}(\omega)\,.$ (3.6)

More generally, we can replace $\phi$ by $\psi$ with the same properties (see
(1.6)) and consider $\Psi=\exp\psi\,.$

Note that we have

$\|\big{(}\hat{r}_{q}^{*}(\omega)^{q}-|\iota_{t}\psi^{\prime}|^{q}\big{)}_{+}^{1/q}u^{*}\|_{q,\omega(t-\cdot)-\iota_{t}\psi}\leq\|\big{(}\hat{r}_{q}^{*}(\omega)^{q}-|\psi^{\prime}|^{q}\big{)}_{-}^{1/q}m\|_{q,\omega\cdot-\psi}|u^{*}(t)|_{\cal
B^{*}}.$ (3.7)

and also trivially

$\|\big{(}\hat{r}_{q}^{*}(\omega)^{q}-|\iota_{t}\psi^{\prime}|^{q}\big{)}_{-}^{1/q}u^{*}\|_{q,\omega(t-\cdot)-\iota_{t}\psi}\leq\|\big{(}\hat{r}_{q}^{*}(\omega)^{q}-|\psi^{\prime}|^{q}\big{)}_{-}^{1/q}m\|_{q,\omega\cdot-\psi}|u^{*}(t)|_{\cal
B^{*}}.$ (3.8)

###  3.3 From $L^{p}$ to $L^{\infty}$ bounds

In order to estimate $|u(t)|_{{\cal B}}$ for a given $u(0)$ it suffices to
estimate $|<u(t),u^{*}(t)>_{\cal B,\cal B^{*}}|$ for arbitrary
$u^{*}(t)\in{\cal B}^{*}$. Extend $u^{*}(t)$ to a backward solution $u^{*}(s)$
of $(A^{*}+\partial_{s})u^{*}(s)=0$, so that

$<u(s),u^{*}(s)>_{\cal B,\cal B^{*}}=<u(t),u^{*}(t)>_{\cal B,\cal B^{*}},\
\forall s\in[0,t].$ (3.9)

Let $M=M_{t}:[0,t]\to[0,+\infty[$ have mass 1:

$\int_{0}^{t}M(s)ds=1\,.$ (3.10)

Then, using (3.9),

$|<u(t),u^{*}(t)>_{\cal B,\cal
B^{*}}|=\left|\int_{0}^{t}M(s)|<u(s),u^{*}(s)>_{\cal B,\cal
B^{*}}|ds\right|\\\ \leq\int_{0}^{t}M(s)|u(s)|_{\cal B}|u^{*}(s)|_{\cal
B^{*}}ds.$ (3.11)

Let $\epsilon_{1},\epsilon_{2}\in\\{-,+\\}$. Assume that

$\mathrm{supp\,}M\subset\\{s;\epsilon_{1}(\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}(s)|^{p})>0,\
\epsilon_{2}(\hat{r}_{q}^{*}(\omega)^{2}-\iota_{t}|\psi^{\prime}(s)|^{q})>0\\}.$
(3.12)

Then multiplying and dividing with suitable factors in the last member of
(3.11), we get (in the reflexive case)

$\begin{split}|<u(t),u^{*}(t)>_{\cal B,\cal B^{*}}|&\leq e^{\omega
t}\int_{0}^{t}\frac{M(s)e^{-\phi(s)-\iota_{t}\psi(s)}}{(\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}(s)|^{p})^{\frac{1}{p}}_{\epsilon_{1}}(\hat{r}_{q}^{*}(\omega)^{q}-|\iota_{t}\psi^{\prime}(s)|^{q})^{\frac{1}{q}}_{\epsilon_{2}}}\times\\\
&\qquad\times e^{\phi(s)-\omega
s}(\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}(s)|^{p})^{\frac{1}{p}}_{\epsilon_{1}}|u(s)|_{\cal
B}\times\\\ &\qquad\times
e^{\iota_{t}\psi(s)-\omega(t-s)}(\hat{r}_{q}^{*}(\omega)^{q}-|\iota_{t}\psi^{\prime}(s)|^{q})^{\frac{1}{q}}_{\epsilon_{2}}|u^{*}(s)|_{\cal
B^{*}}ds\\\ &\leq e^{\omega
t}\sup_{[0,t]}\frac{Me^{-\phi-\iota_{t}\psi}}{(\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p})^{\frac{1}{p}}_{\epsilon_{1}}(\hat{r}_{q}^{*}(\omega)^{q}-|\iota_{t}\psi^{\prime}|^{q})^{\frac{1}{q}}_{\epsilon_{2}}}\times\\\
&\qquad\times\|(\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p})^{\frac{1}{p}}_{\epsilon_{1}}u\|_{p,\omega\cdot-\phi}\|(\hat{r}_{q}^{*}(\omega)^{q}-|\iota_{t}\psi^{\prime}|^{q})_{\epsilon_{2}}^{\frac{1}{q}}u^{*}\|_{\omega(q,t-\cdot)-\iota_{t}\psi}.\end{split}$

Using (3.4), (3.7) when $\epsilon_{j}=+$ or (3.5), (3.8) when
$\epsilon_{j}=-$, we get

$|<u(t),u^{*}(t)>_{\cal B,\cal B^{*}}|\leq e^{\omega
t}\sup_{[0,t]}\frac{Me^{-\phi-\iota_{t}\psi}}{(\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p})^{\frac{1}{p}}_{\epsilon_{1}}(\hat{r}_{q}^{*}(\omega)^{q}-|\iota_{t}\psi^{\prime}|^{q})^{\frac{1}{q}}_{\epsilon_{2}}}\times\\\
\times\|(\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p})^{\frac{1}{p}}_{-}m\|_{p,\omega\cdot-\phi}\|(\hat{r}_{q}^{*}(\omega)^{q}-|\psi^{\prime}|^{q})_{-}^{\frac{1}{q}}\,m\|_{q,\omega\cdot-\psi}|u(0)|_{\cal
B}\,|u^{*}(t)|_{\cal B^{*}}.$ (3.13)

This estimate holding for any $u^{*}(t)$, we get

$\begin{split}|u(t)|_{\cal B}\leq e^{\omega
t}\sup_{[0,t]}\frac{Me^{-\phi-\iota_{t}\psi}}{(\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p})^{\frac{1}{p}}_{\epsilon_{1}}(\hat{r}_{q}^{*}(\omega)^{q}-|\iota_{t}\psi^{\prime}|^{q})^{\frac{1}{q}}_{\epsilon_{2}}}\times\\\
\times\|(\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p})^{\frac{1}{p}}_{-}m\|_{p,\omega\cdot-\phi}\|(\hat{r}_{q}^{*}(\omega)^{q}-|\psi^{\prime}|^{q})_{-}^{\frac{1}{q}}\,m\|_{q,\omega\cdot-\psi}|u(0)|_{\cal
B}\,.\end{split}$ (3.14)

In order to optimize the choice of $M$, we let $0\not\equiv F\in
C([0,t];[0,+\infty[)$ and study

$\inf_{0\leq M\in C([0,t]),\atop\,\int Mds=1}\sup_{s}\frac{M(s)}{F(s)}.$
(3.15)

We first notice that

$1=\int Mds=\int\frac{M}{F}Fds\leq\left(\sup_{s}\frac{M}{F}\right)\int Fds\,,$

and hence the quantity (3.15) is $\geq 1/\int Fds$.
Choosing $M=\theta F$ with $\theta=1/\int F(s)\,ds$, we get equality.

###### Lemma 3.1

For any continuous function $F\geq 0$, non identically $0$,

$\inf_{0\leq M\in C([0,t]),\atop\,\int M(s)\,ds=1}\
\left(\sup_{s}\frac{M}{F}\right)=1/\int Fds\,.$

Applying the lemma to the supremum in (3.14) with

$F=e^{\phi+\iota_{t}\psi}\,(\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p})^{\frac{1}{p}}_{\epsilon_{1}}(\hat{r}_{q}(\omega)^{q}-|\iota_{t}\psi^{\prime}|^{q})^{\frac{1}{q}}_{\epsilon_{2}},$

we get

$|u(t)|_{\cal B}\leq e^{\omega
t}\frac{\|(\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p})^{\frac{1}{p}}_{-}\,m\|_{p,\omega\cdot-\phi}\|(\hat{r}_{q}^{*}(\omega)^{q}-|\psi^{\prime}|^{q})^{\frac{1}{q}}_{-}\,m\|_{q,\omega\cdot-\psi}}{\int_{0}^{t}e^{\phi+\iota_{t}\psi}(\hat{r}_{p}(\omega)^{p}-|\phi^{\prime}|^{p})^{\frac{1}{p}}_{\epsilon_{1}}(\hat{r}_{q}^{*}(\omega)^{q}-|\iota_{t}\psi^{\prime}|^{q})^{\frac{1}{q}}_{\epsilon_{2}}ds}|u(0)|_{\cal
B}.$ (3.16)

Since $u(0)$ is arbitrary, and noting (3.6) we get Theorem 1.3.

##  4 Consequences of Theorem 1.3

###  4.1 Main proposition

An important step is to prove (we assume $\omega=0$, $\hat{r}_{p}(0)=1$) as a
consequence of Theorem 1.3 with $\epsilon_{1}=+$ and $\epsilon_{2}=-$, the
following key proposition:

###### Proposition 4.1

Assume that $\omega=0$, $\hat{r}_{p}(\omega)=1$. Let $a,b$ positive. Then for
$t\geq a+b$,

$||S(t)||\leq\exp-(t-a-b)\,\frac{\left(\inf_{u}\int_{0}^{a}m(s)^{p}(|u^{\prime}(s)|^{p}-u^{p}(s))_{+}ds\right)^{1/p}}{\left(\sup_{\theta}\int_{0}^{b}\frac{1}{m^{p}}(\theta(s)^{p}-|\theta^{\prime}(s)|^{p})\,ds\right)^{1/p}}\,,$
(4.17)

where

  * •

$u\in W^{1,p}(]0,a[)$ satisfies $u(0)=0$, $u(a)=1$ ;

  * •

$\theta\in W^{1,p}((]0,b[)$ satisfies $\theta(b)=1$ and
$|\theta^{\prime}|\leq\theta$ .

The analysis of the minimizers of

$I_{inf,p}:=\inf_{u}\int_{0}^{a}m(s)^{p}(|u^{\prime}(s)|^{p}-u^{p}(s))_{+}ds$
(4.18)

and the maximizers of

$J_{sup,p}:=\sup_{\theta}\int_{0}^{b}\frac{1}{m^{p}}(\theta(s)^{p}-|\theta^{\prime}(s)|^{p})\,ds$
(4.19)

is close to what was done in Section 3 of [7] and we will sketch what has been
modified in Subsection 4.3.

###  4.2 From Proposition 4.1 to Theorem 1.2

This proposition implies rather directly Theorem 1.2 in the following way. We
first observe the trivial lower bound (take $\theta(s)=1$)

$\sup_{\theta}\int_{0}^{b}\frac{1}{m^{p}}(\theta(s)^{p}-|\theta^{\prime}(s)|^{p})\,ds\geq\int_{0}^{b}\frac{1}{m^{p}}ds\,.$
(4.20)

A more tricky argument based on the equality case in Hölder’s inequality555In
[7] we were using instead Cauchy-Schwarz gives

$\inf_{u}\int_{0}^{a}m(s)^{p}(u^{\prime}(s)^{p}-u(s)^{p})_{+}ds\leq\inf_{u}\int_{0}^{a}m(s)^{p}u^{\prime}(s)^{p}ds\leq\Big{(}1/\int_{0}^{a}\frac{1}{m^{q}}ds\Big{)}^{p/q}\,,$
(4.21)

More precisely we start from the upper bound

$\int_{0}^{a}(|u^{\prime}|^{p}-u^{p})_{+}m(s)^{p}ds\leq\int_{0}^{a}|u^{\prime}(s)|^{p}m(s)^{p}\,ds$

and minimize the right hand side.
Observing that

$\begin{split}1&=u(a)=\int_{0}^{a}u^{\prime}(s)\,ds=\int_{0}^{a}u^{\prime}(s)m(s)\,m(s)^{-1}\,ds\\\
&\leq\left(\int_{0}^{a}(u^{\prime}(s)m(s)^{p}ds\right)^{\frac{1}{p}}\,\,\left(\int_{0}^{a}\frac{1}{m(s)^{q}}ds\right)^{\frac{1}{q}}\,,\end{split}$

we look for a $u$ for which we have equality.
By a standard criterion for the optimality of the Hölder inequality, this is
the case if, for some constant $C>0$,

$(|u^{\prime}(s)|m(s))^{p}=\frac{C}{m(s)^{q}}\,,$

or

$|u^{\prime}|=C^{1/p}\frac{1}{m^{q}}\,.$

Hence, we choose

$u(s)=\hat{C}\int_{0}^{s}\frac{1}{m(\tau)^{q}}\,d\tau\,,$

where the choice of $\hat{C}$ is determined by imposing $u(a)=1$. We obtain

###### Lemma 4.2

For any $a>0$,

$\inf_{\\{u\in
W^{1,p}(]0,a[),u(0)=0,u(a)=1\\}}\int_{0}^{a}(|u^{\prime}|^{p}-u^{p})_{+}m^{p}\,ds\leq\left(\int_{0}^{a}\frac{1}{m(s)^{q}}ds\right)^{-p/q}\,.$
(4.22)

Note here that we have no condition on $a>0\,$.

###### Remark 4.3

Nevertheless, one can observe that this implication holds only under the
additional condition that $\mathcal{B}$ is reflexive. This was not needed in
the direct proof of Theorem 1.2 given in Section 2.

###  4.3 Proof of Proposition 4.1

We now assume $\omega=0$ and $\hat{r}_{p}(0)=1$. In this case, (1.7) takes the
form

$||S(t)||_{\mathcal{L}(\cal B)}\leq e^{\omega
t}\frac{\|(\Phi^{p}-|\Phi^{\prime}|^{p})^{\frac{1}{p}}_{-}\,m\|_{L^{p}([0,t[)}\|(\Psi^{q}-|\Psi^{\prime}|^{q})^{\frac{1}{q}}_{-}\,m\|_{L^{q}([0,t[)}}{\int_{0}^{t}(\Phi^{p}-|\Phi^{\prime}|^{p})^{\frac{1}{p}}_{+}(\iota_{t}\Psi^{q}-|\iota_{t}\Psi^{\prime}|^{q})^{\frac{1}{q}}_{-}ds}\,.$
(4.23)

Replacing $(\Phi,\Psi)$ by $(\lambda\Phi,\mu\Psi)$ give for any
$(\lambda,\mu)\in(\mathbb{R}\setminus\\{0\\})^{2}$ does not change the right
hand side. Hence we may choose a suitable normalization without loss of
generality. We also choose $\Phi$ and $\Psi$ to be piecewise $C^{1}([0,t])$.
If $0\leq\sigma<\tau<+\infty$, $p>1$, and $S,T\in\mathbb{R}$ we put

$W_{S,T}^{1,p}(]\sigma,\tau[)=\\{u\in W^{1,p}(]\sigma,\tau[);\,u(\sigma)=S,\
u(\tau)=T\\}\,,$ (4.24)

where $W^{1,p}$ denotes the classical Sobolev space associated with $L^{p}$.
Here and in the following all functions are assumed to be real-valued unless
stated otherwise. As in [7] we can here replace $W_{0,1}^{1,p}$ by a subspace
that allows to avoid the use of positive parts. Put

${\cal H}^{a,p}={\cal W}_{0,a}^{0,1,p}\,,$ (4.25)

where for $\sigma,\tau,S,T$ as above,

${\cal W}_{\sigma,\tau}^{S,T,p}=\\{u\in W_{S,T}^{1,p}(]\sigma,\tau[);0\leq
u\leq u^{\prime}\\}.$ (4.26) ${\cal G}^{b,p}=\\{\theta\in
W^{1,p}(]0,b[);\,|\theta^{\prime}|\leq\theta,\ \theta(b)=1\\}\,.$ (4.27)

Given some $t>a+b$, we now give the conditions satisfied by $\Phi$:

######  Property 4.4 ($P_{a,b}^{p}$)

  1. 1.

$\Phi=e^{a}u$ on $]0,a]$ and $u\in\mathcal{H}^{a,p}$.

  2. 2.

On $[a,t-b]$, we take $\Phi(s)=e^{s}$, so
$|\Phi^{\prime}|^{p}(s)-\Phi(s)^{p}=0\,.$

  3. 3.

On $[t-b,t]$ we take $\Phi(s)=e^{t-b}\theta(t-s)$ with
$\theta\in\mathcal{G}^{b,p}$ .

Hence, we have

${\rm supp\,}(\Phi^{p}-|\Phi^{\prime}|^{p})_{+}\subset[t-b,t]\,.$

Similarly we assume that $\Psi$ satisfies property $(P_{b,a}^{q})$ but with
$\theta=1$, hence

  1. 1.

$\Psi(s)=e^{b}v(s)\mbox{ on }]0,b[\mbox{ with }v\in\mathcal{H}^{b,q}\,,$

  2. 2.

On $[b,t-a]$, we take $\Psi(s)=e^{s}$.

  3. 3.

On $[t-a,t]$, $\Psi(s)=e^{t-a}\,.$

Recalling the definition of $\iota_{t}$, we get for $\iota_{t}\Psi$:

  1. 1.

On $[0,a]$, $\iota_{t}\Psi=e^{(t-a)}$, satisfying

$(\iota_{t}\Psi)^{\prime q}-(\iota_{t}\Psi)^{q}=-e^{q(t-a)}\,.$

  2. 2.

On $[a,t-b]$, we have $\iota_{t}\Psi(s)=e^{t-s}$, hence

$|(\iota_{t}\Psi)^{\prime}(s)|^{q}-\iota_{t}\Psi(s)^{q}=0\,.$

  3. 3.

On $]t-b,t[$, we have

$|(\iota_{t}\Psi)^{\prime}(s)|^{q}-(\iota_{t}\Psi)^{q}(s)\geq 0\,.$

Assuming that $t>a+b$, we have under these assumptions on $\Phi$ and $\Psi$

$\\{s;\Phi(s)^{p}-|\Phi^{\prime}(s)|^{p}>0,\iota_{t}\Psi(s)^{q}-|\iota_{t}\Psi^{\prime}(s)|^{q}<0\\}\subset[t-b,b]\,.$

We now compute or estimate the various quantities appearing in (1.7).

We have

$\|(\Phi^{p}-|\Phi^{\prime}|^{p})^{\frac{1}{p}}_{-}\,m\|=e^{a}\left(\int_{0}^{a}(|u^{\prime}(s)|^{p}-u^{p}(s))m(s)^{p}ds\right)^{1/p}\,,$
(4.28)
$\|(\Psi^{q}-|\Psi^{\prime}|^{q})^{\frac{1}{q}}_{-}m\|=e^{b}\left(\int_{0}^{b}(|v^{\prime}(s)|^{q}-v(s)^{q})m(s)^{q}ds\right)^{1/q}\,,$
(4.29)

and

$\begin{split}&\hskip-28.45274pt\int_{0}^{t}(\Phi^{p}-\Phi^{\prime
p})^{\frac{1}{p}}_{+}((\iota_{t}\Psi)^{q}-(\iota_{t}\Psi^{\prime})^{q})^{\frac{1}{q}}_{-}ds\\\
&=\int_{t-b}^{t}(\Phi^{p}-|\Phi^{\prime}|^{p})^{\frac{1}{p}}_{+}((\iota_{t}\Psi)^{q}-|\iota_{t}\Psi^{\prime}|^{q})^{\frac{1}{q}}_{-}ds\\\
&=e^{t-b}\int_{t-b}^{t}(\theta(t-s)^{p}-|\theta^{\prime}(t-s)|^{p})_{+}^{\frac{1}{2}}\,((\iota_{t}\Psi)^{q}-|\iota_{t}\Psi^{\prime}|^{q})^{\frac{1}{q}}_{-}ds\\\
&=e^{t}\int_{0}^{b}(\theta(s)^{p}-|\theta^{\prime}(s)|^{p})^{\frac{1}{p}}(|v^{\prime}(s)|^{q}-v(s)^{q})^{\frac{1}{q}}ds\,.\end{split}$
(4.30)

So we get from (1.7)

$||e^{t}S(t)||_{\mathcal{L}(\cal B)}\leq
e^{a+b}\left(\int_{0}^{a}(|u^{\prime}(s)|^{p}-u^{p}(s))m(s)^{p}ds\right)^{\frac{1}{p}}\,K^{p}(b,\theta,v)\,,$
(4.31)

where

$K^{p}(b,\theta,v):=\frac{\left(\int_{0}^{b}(|v^{\prime}(s)|^{q}-v(s)^{q})m(s)^{q}ds\right)^{1/q}}{\int_{0}^{b}(\theta(s)^{p}-|\theta^{\prime}(s)|^{p})^{\frac{1}{p}}(|v^{\prime}(s)|^{q}-v(s)^{q})^{\frac{1}{q}}ds}\,.$
(4.32)

We start by considering for a given $\theta\in\mathcal{G}^{b,p}$

$K_{\mathrm{inf}}^{p}(b,\theta):=\inf_{v\in\mathcal{H}^{b,q}}K^{p}(b,\theta,v)\,,$

and get the following:

###### Lemma 4.5

If $\theta\in\mathcal{G}^{b,p}$ and $\theta-\theta^{\prime}$ is not
identically $0$ on $]0,b$[, we have

$K^{p}_{\mathrm{inf}}(b,\theta)=\frac{1}{\big{(}\int_{0}^{b}(\theta(s)^{p}-|\theta^{\prime}(s)|^{p})\frac{1}{m^{p}}ds\big{)}^{1/p}}\,.$
(4.33)

Proof. We consider with

$h(s)=\big{(}\theta(s)^{p}-|\theta^{\prime}(s)|^{p}\big{)}^{\frac{1}{p}}\geq
0$

the denominator in (4.32),

$\int_{0}^{b}h(s)(|v^{\prime}(s)|^{q}-v(s)^{q})^{\frac{1}{q}}ds=\int_{0}^{b}\big{(}h(s)/m(s)\big{)}\,\big{(}m^{q}(|v^{\prime}(s)|^{q}-v(s)^{q})^{\frac{1}{q}}\big{)}\,ds\,.$

By the Hölder inequality, we have

$\int_{0}^{b}h(s)(|v^{\prime}(s)|^{q}-v(s)^{q})^{\frac{1}{q}}ds\leq\\\
\left(\int_{0}^{b}(h(s)/m(s))^{p}ds\right)^{\frac{1}{p}}\,\left(\int_{0}^{b}m(s)^{q}(|v^{\prime}(s)|^{q}-v(s)^{q})ds\right)^{\frac{1}{q}}\,,$

which implies that $K^{p}_{\mathrm{inf}}(b,\theta)$ is bounded from below by
the right hand side of (4.33).

We have equality for some $v$ in $\mathcal{H}^{b,q}$ if and only if, for some
constant $c>0$,

$m(s)^{q}(|v^{\prime}(s)|^{q}-v(s)^{q})=c\,h(s)^{p}/m(s)^{p}\,.$

In order to get such a $v$, we first consider $w\in W^{1,p}$ defined by

$w^{\prime}=(w^{q}+h^{p}m^{-pq})^{1/q}\,,\,w(0)=0\,,$

noticing that the right hand side of the differential equation is Lipschitz
continuous in $w$, so that the Cauchy-Lipschitz theorem applies. According to
our assumption on $\theta$, we verify that $w(b)>0$ and we choose

$v=\frac{1}{w(b)}w\,,\,c=\frac{1}{w(b)}\,.$

For this pair $(c,v)$ we get

$\begin{split}&\left(\int_{0}^{b}(|v^{\prime}(s)|^{q}-v(s)^{q})m(s)^{q}ds\right)^{\frac{1}{2}}\Big{/}\int_{0}^{b}\left(\theta(s)^{p}-|\theta^{\prime}(s)|^{p}\right)^{\frac{1}{2}}\left(|v^{\prime}(s)|^{q}-v(s)^{q}\right)^{\frac{1}{q}}ds\\\
&=1\big{/}\left(\int_{0}^{b}h^{p}(s)m^{-p}(s)\,ds\right)^{\frac{1}{p}}\,.\end{split}$
(4.34)

Returning to the definition of $h$ shows that $K^{p}_{\mathrm{inf}}(b,\theta)$
is bounded from above by the right hand side of (4.33) and we get the
announced result. $\Box$

To conclude the proof of Proposition 4.1, we just combine Lemma 4.5 and
(4.31).

###  4.4 Application of Proposition 4.1: Wei’s theorem

When $p=2$, we can directly apply [7] where $r(\omega)$ is replaced by
$\hat{r}_{2}(\omega)$. The case $p\neq 2$ is less clear. Already, in the case
$m=1$ this involves new questions related to the $p$-Laplacian instead of the
Laplacian. Nevertheless, one can get (probably non optimal) upper bounds by
using the optimizers obtained for $p=2$. If we consider $m=1$, (4.17),
$a\leq\frac{\pi}{4}$, $b\leq\frac{\pi}{4}$ and take therein $u(s)=\sin s/\sin
a$ and $\theta(s)=\cos s/\cos b$, we obtain

$||S(t)||\leq\cos b\sin a^{-1}\exp-(t-a-b)\,\frac{\left(\int_{0}^{a}((\cos
s)^{p}-(\sin s)^{p})ds\right)^{1/p}}{\left(\int_{0}^{b}((\cos s)^{p}-(\sin
s)^{p})\,ds\right)^{1/p}}\,,$ (4.35)

When $a=b$, we obtain

$||S(t)||\leq\cot a\exp-(t-2a)\,.$ (4.36)

For $a=b=\frac{\pi}{4}$ we get an extension of Wei’s theorem to the reflexive
Banach case.

###### Theorem 4.6

Let $p>1$ and $S(t)$ a $C_{0}$-semigroup of generator $A$ in a reflexive
Banach space $\mathcal{B}$ such that

$\|S(t)\|\leq 1\mbox{ for all }t\geq 0\,.$ (4.37)

holds and $\hat{r}_{p}(0)>0$. Then we have,

$||S(t)||\leq e^{-\hat{r}_{p}(0)t+\frac{\pi}{2}}\,,\,\forall t\geq 0\,.$
(4.38)

##  5 Modified Riccati equation and application to the optimization problems.

When analyzing the optimality of the statements in [7], an important tool was
a fine analysis of a natural Riccati equation (see more precisely Subsection
3.4.2 in [7]). Let us see what is going on for $p\in(1,+\infty)$ and we start
with the assumption that

$\hat{r}_{p}(0)=1\,.$

Let $f$ be defined on $]\sigma,\tau[$ such that

$0<f\leq f^{\prime}.$ (5.39)

Put

$\mu=m^{\prime}/m\,.$

We now assume that $f$ satisfies

$(\partial_{s}\circ m^{p}\circ\partial_{s}+m^{p})f^{p-1}=0\,,$

we get

$(\partial_{s}^{2}+p\mu\partial_{s}+1)f^{p-1}=0\,.$ (5.40)

In this case, we say that $f$ is $(m,p)$-harmonic.
Writing

$\phi=\log f\mbox{ and }\psi=\phi^{\prime}=f^{\prime}/f\,,$

we get, noting that

$f^{\prime\prime}/f=\psi^{\prime}+\psi^{2}\,,\,\psi\geq 1$

and

$\psi^{\prime}=-((p-1)\psi^{2}+p\mu\psi+\frac{1}{p-1})\,,$ (5.41)

or equivalently

$\frac{\psi^{\prime}}{\psi}=-((p-1)\psi+p\mu+\frac{1}{(p-1)\psi})\,.$ (5.42)

As in [7], we note that $\tilde{\psi}:=-1/\psi$ satisfies

$\frac{\tilde{\psi}^{\prime}}{\tilde{\psi}}=-((p-1)\tilde{\psi}-p\mu+\frac{1}{(p-1)\tilde{\psi}})\,.$
(5.43)

We consider the condition

$\lim_{x\mapsto 0^{+}}\psi_{p}(x)=+\infty\,.$ (5.44)

With the conditions (5.41) and (5.44), $\psi=\psi_{m,p}$ is uniquely defined
and and we introduce

$a^{*}=a^{*}(m,p)=\sup\\{a>0\mbox{ and }\psi_{m,p}>1\mbox{ on }(0,a)\\}\,,$
(5.45)

so that $a^{*}(m,p)\in]0,+\infty]$.
Following [7], one can prove that in (4.18) the infimum is realized by the
$(m,p)$-harmonic function $u_{p}$ such that $u^{\prime}_{p}/u_{p}=\psi_{p}$.
Hence we get with in mind that in (4.18) the infimum is for
$u\in\mathcal{H}^{a,p}$

$I_{inf,p}=\int_{0}^{a}(m(s)^{p}u_{p}^{\prime}(s)^{p}-u_{p}^{p}(s))ds$

After an integration by parts, we get (since $u_{p}(0)=0$ and $u_{p}(a)=1$)

$I_{inf,p}:=m(a)^{p}(u_{p}^{\prime}(a))^{p-1}(a)u_{p}^{p}(a)=m(a)^{p}\psi_{p}^{p-1}(a)\,.$

Similarly one can prove that in (4.19) the supremum is realized by the
$(1/m,p)$-harmonic function $\theta_{p}\in\mathcal{G}^{b,p}$ such that
$\theta^{\prime}_{p}/\theta_{p}=-1/\psi_{p}$.
Hence we get, with in mind that in (4.19) the infimum is for
$\theta\in\mathcal{G}^{b,p}$ satisfying $\theta^{\prime}(0)=0$,

$J_{sup,p}=\int_{0}^{b}\frac{1}{m(s)^{p}}(\theta_{p}(s)^{p}-|\theta_{p}^{\prime}(s)|^{p})\,ds$
(5.46)

After an integration by parts, we get (since $\theta^{\prime}_{p}(0)=0$ and
$\theta_{p}(b)=1$)

$J_{sup,p}=-m(b)^{-p}|\theta_{p}^{\prime}(b)|^{p-1}(a)\theta_{p}^{p}(b)=m(b)^{-p}\psi_{p}^{-(p-1)}(b)\,.$

##  6 Final theorem

Like in [7] and coming back to Proposition 4.1, we immediately get from the
previous section:

###### Proposition 6.1

Let $p>1$, $\omega=0$, $\hat{r}_{p}(0)=1$ and
$a^{*}:=a^{*}(m,p)\in]0,+\infty]$. When $a,b\in]0,+\infty[\cap]0,a^{*}]$ and
$t>a+b$, we have

$||e^{t}S(t)||\leq\exp(a+b)m(a)m(b)\psi_{p}(a)^{\frac{p-1}{p}}\psi_{p}(b)^{\frac{p-1}{p}}\,.$
(6.47)

In particular, when $a^{*}<+\infty$, we have

$||e^{t}S(t)||\leq\exp(2a^{*})\,m(a^{*})^{2}\,,\ \ t>2a^{*}\,.$ (6.48)

This proposition is the analog of Wei’s theorem for general weights $m$ and
the $L^{p}$-Banach version of Theorem 1.9 in [7].
By the same rescaling procedure, we have actually a more general statement. We
consider $\hat{A}$ with the same properties as $A$ where the hat’s are
introduced to make easier the transition between the particular case above to
the general case below. As before, we introduce $\hat{\omega}$ and
$\hat{r}=\hat{r}_{p}(\hat{\omega})$.

###### Theorem 6.2

Let $p>1$, $\hat{r}_{p}(\hat{\omega})<+\infty$. Let
$\hat{S}(\hat{t})=e^{\hat{t}\,\hat{A}}$ satisfying

$||\hat{S}(\hat{t})||\leq\hat{m}(\hat{t})\,,\,\forall\hat{t}>0\,.$

Then there exist uniquely defined
$\hat{a}^{*}:=\hat{a}^{*}(\hat{m},\hat{\omega},\hat{r},p)>0$ and
$\hat{\psi}_{p}:=\hat{\psi}_{p}(\cdot;\hat{m},\hat{\omega},\hat{r})$ on
$]0,\hat{a}^{*}[$ with the same general properties as above such that, if
$\hat{a},\hat{b}\in]0,+\infty[\cap]0,\hat{a}^{*}]$ and
$\hat{t}>\hat{a}+\hat{b}$, we have

$||S(\hat{t})||\leq\exp\left((\hat{\omega}-\hat{r}_{p}(\hat{\omega}))(\hat{t}-(\hat{a}+\hat{b}))\right)\hat{m}(\hat{a})\hat{m}(\hat{b})\hat{\psi}_{p}(\hat{a})^{\frac{p-1}{p}}\,\hat{\psi}_{p}(\hat{b})^{\frac{p-1}{p}}\,.$
(6.49)

Moreover, when $\hat{a}^{*}<+\infty$, the estimate is optimal for
$\hat{a}=\hat{b}=\hat{a}^{*}$ and reads

$||\hat{S}(\hat{t})||\leq\exp((\hat{\omega}-\hat{r}_{p}(\hat{\omega})(\hat{t}-2\hat{a}^{*}))\,\hat{m}(\hat{a}^{*})^{2}\,,\
\ t>2\hat{a}^{*}\,.$ (6.50)

Note that in the statement

$\hat{a}^{*}(\hat{m},\hat{\omega})=\hat{r}\,a^{*}(e^{-\hat{\omega}\cdot}\hat{m})\,,\,\hat{\psi}_{p}(\hat{s};\hat{m},\hat{\omega},\hat{r})=\psi_{p}(\hat{r}\hat{s};e^{-\hat{\omega}\cdot}\hat{m})\,.$

This theorem is the $L^{p}$-Banach version of Theorem 1.10 in [7].

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# Smooth-Trajectron++: Augmenting the Trajectron++ behaviour prediction model
with smooth attention

Frederik S.B. Westerhout1 Julian F. Schumann1 Arkady Zgonnikov1

###### Abstract

Understanding traffic participants’ behaviour is crucial for predicting their
future trajectories, aiding in developing safe and reliable planning systems
for autonomous vehicles. Integrating cognitive processes and machine learning
models has shown promise in other domains but is lacking in the trajectory
forecasting of multiple traffic agents in large-scale autonomous driving
datasets. This work investigates the state-of-the-art trajectory forecasting
model Trajectron++ which we enhance by incorporating a smoothing term in its
attention module. This attention mechanism mimics human attention inspired by
cognitive science research indicating limits to attention switching. We
evaluate the performance of the resulting Smooth-Trajectron++ model and
compare it to the original model on various benchmarks, revealing the
potential of incorporating insights from human cognition into trajectory
prediction models.

11footnotetext: Department of Cognitive Robotics, Delft University of
Technology, Delft, The Netherlands

##  I Introduction

In a world where the demand for intelligent vehicles is increasing rapidly
[1], the concern for the safety of passengers and other road users should grow
with it [2]. According to the World Health Organization, approximately 1.3
million people die each year due to road traffic accidents, and this number is
expected to increase if proper measures are not taken [3]. Therefore, it
should be paramount for future autonomous vehicles to improve traffic safety.

One of the most critical factors for ensuring a safe environment around
intelligent vehicles is accurately predicting the future movements of
surrounding traffic participants. These predictions allow for a better
assessment of the environment and anticipation of potentially dangerous
situations at an early stage, lowering the risk of accidents. The accurate
predictions of interactive behaviours are especially important, as those
comprise the most challenging situations.

Numerous methods have been used to tackle the human behaviour prediction
problem [4, 5, 6], with examples ranging from reasoning-based methods to data-
driven techniques. Over the last few years, data-driven approaches have shown
great potential [7, 8, 9, 10, 11, 12, 13], using machine learning algorithms
to learn from large amounts of data to predict the trajectories of traffic
participants. One of these data-driven models is Trajectron++ [10], which
stands out due to its public code availability, the general applicability and
the results it achieved on multiple datasets (including _nuScenes_ [14], and
_highD_ [15]).

Other methods to predict future behaviour of traffic participants are based on
theories from cognitive science. Instead of learning merely from data, the
model is constructed to mimic human cognition. One class of such models based
on the concept of evidence accumulation [16] have proved useful specifically
in predicting binary decisions in traffic interactions [17, 18]. However,
these models are not yet applicable to trajectory forecasting in a more
general setting. Another example of using insights from cognitive science for
behaviour prediction in traffic is the use of a quantum-like Bayesian model, a
mathematical framework that combines elements of quantum theory and Bayesian
probability theory to describe decision-making and information processing in
complex and uncertain environments [19]. It is used in [20] to more accurately
predict human street crossing behaviour, compared to the more data-driven
model _Social-LSTM_ [7]. Yet another insight from cognitive science suggests
that the brain has a limited capacity for shifting attention rapidly between
different tasks [21]. This is used in [22], where the application of the
smoothing term to the attention module of a machine-learning prediction model
– referred to as _Smooth-Attention_ – which mimics human cognition, allows for
better predictive performance.

Recent work demonstrated that integrating insights from cognitive science is a
promising way of improving the performance of trajectory prediction models,
but such cognitively inspired models need to be explored in a much more
comprehensive way. Specifically, Cao et al. [22] emphasize the need to combine
smooth attention with more advanced interaction modelling network
architectures. Here we aim to address this challenge by applying smooth
attention to a state-of-the-art behaviour prediction model. Namely, we aim to
improve upon the performance of Trajectron++ (T++) [10] by leveraging the
method of _smooth attention_ proposed in [22]. Applying a smoothness
constraint on the attention module significantly reduces changes in attention,
thereby ensuring that the module’s output mimics the natural human cognitive
processing. We name our approach of this combined model Smooth-Trajectron++.
We test this new model on the _nuScenes_ [14] and _highD_ [15] datasets.

##  II Methods

In this section, we provide an overview of the various elements of the
_Trajectron++_ model (_T++_) as well as the functioning of the _Smooth-
Attention_ module.

###  II-A Trajectron++

We use Trajectron++ [10] as the baseline model, for several reasons. Firstly,
the model showed state-of-the-art performance on various public datasets [10]
while including an attention module. Secondly, the authors have made the
source code publicly available, including proper documentation regarding its
application. This offers the opportunity to potentially reproduce the
originally reported results while minimizing deviations from the original
setup used by the authors. Lastly, the model has been tested on a large-scale
public autonomous driving dataset, _nuScenes_ [14]. This provides evidence of
the applicability of the model to real-world scenarios concerning interactions
between multiple traffic participants.

Figure 1: Encoder part of _Trajectron++_ [10] that encodes various past input
information into the representation vector $e_{x}$.

_Trajectron++_ is a graph-based conditional variational autoencoder model
comprising an encoder and a decoder. The encoder uses various modules
representing different influences on the trajectory forecast Figure 1. First,
the past location and speed of the chosen traffic agent $i$, for which
predictions must be made, is fed into the Node History Encoder, whose main
component is a long-short-term memory cell (LSTM). Second, the T++ model also
uses road map data to make its predictions more feasible and realistic. The
Map Encoder module takes relevant environmental information, fed to a
convolutional neural network (CNN) and then added to the feature vector.
Third, the Edge Influence Encoder models the influence of neighbouring agents
on a given agent in the considered past time from ${t-T}$ to the current time
${t}$. To encode graph edges, the method follows a two-step process. Firstly,
edge information is collected from neighbouring agents belonging to the same
semantic class, such as pedestrian-pedestrian and car-car semantic classes.
Summation is used for feature aggregation inside these classes to handle
varying numbers of neighbouring nodes while preserving count information,
following [23]. The encodings of all the connections between the modelled
agent and its neighbours are combined to create an ”influence” representation
vector, which represents the overall impact of the neighbouring nodes, which
is done using an additive attention module [24]. Finally, the output is
concatenated with the node’s history and road map data to produce a unified
node representation vector $e_{x}$ fed to a decoder that constructs a
predicted trajectory.

###  II-B Smooth Attention

The _smooth attention_ approach [22] presents a novel perspective on attention
modules. Unlike traditional methods, it applies attention at each time step,
following [25]. Emulating human attention during deliberate tasks incorporates
a smoothness constraint based on the hypothesis that attention does not
frequently change over time. Previous research [21] shows that deliberate
attention movements are slower due to internal limitations. This implies that
attention does not frequently fluctuate during driving, as it falls under
intentional movement. By incorporating the smoothness constraint, the _smooth
attention_ approach enhances the attention mechanism, improving the selection
of important information while disregarding less relevant input variables and
aligning better with the characteristics of human attention.

##  III Smooth-Trajectron++

Figure 2: Edge Influence Encoder including Smooth-Attention (in green)

In this section, we propose a way to apply the smooth attention module [22]
specifically to the Trajectron++ model. To do this, we alter the Edge
Influence Encoder module of Trajectron++ Figure 1, as this is the part where
the social interactions are modelled, and the attention is applied.

Our approach222The source code is available at GitHub/Smooth-Trajectron++,
which we call _Smooth-Trajectron++_ , is illustrated in Figure 2. At a high
level, the original Edge Influence Encoder is expanded by applying the
attention module at each time step in a similar fashion as in the _smooth
attention_ model (the green highlighted box in Figure 2), where the outputs
$\alpha^{\tau}_{i,j_{ab}}$ are the attention weights that are used to rank the
importance the human agent $i$ assigns to the semantic class $j_{ab}$ for
neighbouring agents of types $a$ and $b$ ($a$ and $b$ can stand for agent
types such as cars or pedestrians) at the time $\tau$. All these attention
weights from every time step are then used as an input for the added smoothing
term in the loss function to incorporate the regularising of the attention by
imposing a vectorial total variation penalty:

$\mathcal{L}_{\mathrm{smooth}}(\alpha)=\sum_{\tau=t-T+1}^{t}\sum_{i=1}^{N}\sqrt{\sum_{j}\left(\alpha_{ij}^{t}-\alpha_{ij}^{t-1}\right)^{2}}.$
(1)

To ensure that the attention weights are utilised during the model training
process, we incorporate $\mathcal{L}_{\mathrm{smooth}}$ into the original loss
function $\mathcal{L}_{0}$ [10]:

$\mathcal{L}_{new}=\mathcal{L}_{0}+\beta\mathcal{L}_{smooth}.$ (2)

The scaling factor $\beta$ is introduced to fine-tune the influence of
$\mathcal{L}_{smooth}$. By adjusting $\beta$, the model can be trained to
effectively balance the contribution of the attention weights with the
original loss function.

The extra loss term $\mathcal{L}_{smooth}$ and associated additional calls to
the attention module increase the number of computations and therefore have an
effect on the training time, which is approximately 1.5 times slower compared
to the original version of _Trajectron++_.

##  IV Results

Our method is evaluated on two publicly available datasets: _nuScenes_ [14]
and _highD_ [15]. In both scenarios, we trained and assessed both the original
_Trajectron++_ model (which is a special case of Smooth-Trajectron++ for
$\beta=0$) and the expanded model, with multiple versions of the latter,
differentiated by five $\beta$-values ranging from $0.01$ to $10$. We also
compared the obtained results with those originally reported for T++ [10], as
these turned out to be substantially different from the results we obtained
after directly reproducing T++ using the available source code and the same
hyperparameters mentioned in the original article for the model’s training.

The experiments were performed on DelftBlue, the high-performance cluster of
Delft University of Technology.

###  IV-A nuScenes dataset

The _nuScenes_ dataset [14] consists of 1000 driving scenes in Boston and
Singapore, characterised by their high traffic volumes and challenging driving
situations. The driving scenes span 20 seconds each and are annotated at
$2\text{\,}\mathrm{Hz}$.

For this dataset, we evaluated the models according to three metrics: Final
Displacement Error (FDE), Average Displacement Error (ADE) and Kernel Density
Estimation of Negative Log Likelihood (KDE-NLL). These metrics are chosen as
they were used in the original paper [10]. First, FDE indicates how far off
the model’s predicted location is from the actual location at the end of a
predicted trajectory. Second, ADE is particularly useful for evaluating the
overall accuracy of a model’s trajectory predictions, as it considers the
entire predicted trajectory rather than just the final location. Finally, KDE-
NLL is a valuable metric for evaluating the uncertainty of a model’s
predictions, as it measures how well the model can capture the true
distribution of the data. Following [10], we calculate the three above metrics
at prediction horizons of 1, 2, 3, and 4s. The FDE and the ADE outputs
comprise the most likely single trajectory prediction, using the ”Most Likely”
output configuration as in [10].

There are two main agent classes in _nuScenes_ , pedestrians and vehicles. As
their behaviour is significantly different, we evaluate the models on these
classes separately.

####  IV-A1 Pedestrian-only predictions

Leftmost numbers in each column of Table I-Table III show the results for the
predicted pedestrian trajectories. The numbers in bold represent the lowest
metric values per prediction horizon, compared to the reproduced T++ results
(the ”T++ (rep)” row), which serve as a reference for comparative analysis.

First, we found a significant gap between the reproduced T++ performance and
the results reported and T++ paper. The FDE and ADE exhibit notable
differences, especially at shorter prediction horizons. The reproduced KDE-NLL
values also diverge significantly from the reported values. Several factors
may contribute to this deviation; for example, a different version of
_nuScenes_ , a discrepancy in used and reported hyper-parameters and model
settings, or possible deviations introduced during the reproduction process,
such as data downloading or package installation. Future research should more
closely examine reproducibility of the original results and clarify potential
causes of mismatches with the original findings.

Second, the smooth attention extensions of the reproduced T++ ($\beta=0.01$ to
$\beta=10$) consistently outperform the baseline reproduced version of T++.
Tuning the scaling factor $\beta$ influences the error. Regarding the FDE, the
parameter $\beta=0.1$ has the lowest error in all cases, except for the shared
lowest error at the first prediction horizon (@1s) of $\beta=1.0$. The higher
the $\beta$-value, the more it resembles the ”T++ (rep)” reference values.
However, in Table III, the opposite seems to be happening; the ”T++ (rep)” row
shows the lowest values for almost all cases. An exception is the smooth
version with $\beta=0.01$ @4s, where a marginal performance increase is seen.
However, in general, in this pedestrian-only case, smooth attention does not
improve this metric, although the decline for the smooth versions is minimal.
The smoothing term might decrease the variety of predicted trajectory
distributions, affecting the average and making it less similar to the ground
truth. Further research is needed to explore this hypothesis in other
pedestrian-only scenarios.

TABLE I: Results _nuScenes_ T++ pedestrian-only/vehicle-only: FDE (m)

Model | @1s | @2s | @3s | @4s
---|---|---|---|---
T++ [10] | 0.014/0.07 | 0.17/0.45 | 0.37/1.14 | 0.62/2.20
T++ (rep.) | 0.168/0.430 | 0.369/1.168 | 0.608/2.323 | 0.886/3.868
$\beta=0.01$ | 0.157/0.413 | 0.353/1.102 | 0.586/2.141 | 0.855/3.546
$\beta=0.1$ | 0.155/0.419 | 0.350/1.081 | 0.580/2.122 | 0.842/3.496
$\beta=0.5$ | 0.159/0.421 | 0.354/1.123 | 0.588/2.181 | 0.857/3.560
$\beta=1.0$ | 0.155/0.448 | 0.351/1.128 | 0.582/2.165 | 0.845/3.507
$\beta=10$ | 0.160/0.425 | 0.366/1.149 | 0.607/2.190 | 0.876/3.539

TABLE II: Results _nuScenes_ T++ pedestrian-only/vehicle-only: ADE (m)

Model | @1s | @2s | @3s | @4s
---|---|---|---|---
T++ [10] | 0.021/- | 0.073/- | 0.15/- | 0.25/-
T++ (rep) | 0.126/0.307 | 0.221/0.632 | 0.329/1.092 | 0.450/1.689
$\beta=0.01$ | 0.116/0.296 | 0.208/0.602 | 0.314/1.021 | 0.432/1.559
$\beta=0.1$ | 0.114/0.302 | 0.206/0.597 | 0.311/1.012 | 0.427/0.543
$\beta=0.5$ | 0.118/0.301 | 0.210/0.613 | 0.316/1.041 | 0.434/1.580
$\beta=1.0$ | 0.114/0.319 | 0.207/0.630 | 0.312/1.048 | 0.428/1.575
$\beta=10$ | 0.118/0.303 | 0.215/0.628 | 0.325/1.055 | 0.445/1.586

TABLE III: Results _nuScenes_ T++ pedestrian-only/vehicle-only: KDE NLL

Model | @1s | @2s | @3s | @4s
---|---|---|---|---
T++ [10] | -5.58/-4.17 | -3.96/-2.74 | -2.77/-1.62 | -1.89/-0.71
T++ (rep) | -2.575/-1.760 | -1.530/-0.604 | -0.797/0.235 | -0.230/0.927
$\beta=0.01$ | -2.560/-1.861 | -1.519/-0.726 | -0.795/0.108 | -0.240/0.801
$\beta=0.1$ | -2.541/-1.856 | -1.502/-0.679 | -0.776/0.176 | -0.216/0.875
$\beta=0.5$ | -2.542/-1.885 | -1.505/-0.690 | -0.779/0.150 | -0.211/0.818
$\beta=1.0$ | -2.549/-1.880 | -1.507/-0.679 | -0.785/0.173 | -0.226/0.868
$\beta=10$ | -2.480/-1.861 | -1.471/-0.661 | -0.759/0.173 | -0.205/0.845

####  IV-A2 Vehicle-only predictions

Rightmost numbers in each column of Table I-Table III show the results for the
predicted vehicle trajectories. Similarly to the previous pedestrian-only
case, a general FDE and ADE decline is seen along the $\beta$-versions of the
Smooth-Trajectron++. The ADE values of the T++ paper are missing in Table II,
as they are not reported by the authors in the original article. The version
with $\beta=0.01$ holds the lowest value for the prediction horizon of 1
second, while $\beta=0.1$ has a minor error for the remaining prediction
horizons. Contrary to the pedestrian-only predictions in Table III, Smooth-
Trajectron++ on the vehicle-only forecasts has better KDE-NLL numbers than the
reproduced model, which indicates that the model is better able to match the
original distribution of predicted trajectories with the inclusion of the
smooth-attention term in the loss function. Furthermore, this can be said for
all $\beta$-factors, while in this case, the $\beta=0.01$ has the lowest
values.

###  IV-B highD dataset

To evaluate the models on the _highD_ dataset, we used the previously proposed
benchmarking framework [26].This framework was designed to benchmark
prediction models in gap acceptance scenarios, i.e. situations where drivers
decide whether to enter a gap in traffic or wait for the next opportunity,
such as when a car approaches an intersection and decides whether to turn left
immediately or wait for a break in oncoming traffic. In case of highD dataset,
we investigated the predictions of gap acceptance in lane-change decisions
using a restricted version of highD (see [26] for details).

The framework [26] allowed us to use two methods of splitting the highD data
into training and testing sets: the random split and the critical split. The
first method randomly splits the data for testing and training. In contrast,
the second method deliberately selects the most unusual behaviour for testing,
such as accepting a very small gap or rejecting a large gap. This latter
testing scenario, therefore, tests the model’s ability to extrapolate to
situations that lie outside its training distribution, which is generally
considered to be a more difficult task [27]. Also, small accepted gaps can be
regarded as safety-critical scenarios, which is especially important when
developing safe and reliable prediction models.

Furthermore, the framework allowed us to test the models with varying number
of input time steps($n_{I}$) to study the input-dependability of the tested
models; we used $n_{I}=2$ and $n_{I}=10$.

In addition to the metrics used for the _nuScenes_ dataset, the gap acceptance
benchmark includes an additional metric, the Area under the Receiver-Operator
Curve (AUC), used to evaluate the performance of binary classification models
(here between accepted and rejected gaps).

Initial gapsFixed-sized gaps$0.0$$1.0$_AUC_ $0.0$$1.0$_AUC_ $3.0$$12.0$_ADE_
$T_{\beta_{1}}$$T_{\beta_{2}}$$T_{\beta_{3}}$$T_{\beta_{4}}$$T_{\beta_{5}}$$T+$$9.0$$35.0$_FDE_
$n_{I}=2$, Random split$n_{I}=2$, Critical split$n_{I}=10$, Random
split$n_{I}=10$, Critical split Figure 3: The performance of Smooth-
Trajectron++ and _T++_ on the _highD_ dataset: $\beta_{1-5}$ refer to the
$\beta$-values of 0.01, 0.1, 0.5, 1.0 and 10 respectively

First, we analyse the performance of the models at predicting lane-change
decisions at initial gaps, i.e. at the start of the interaction. Here, only in
the case of $\beta=0.5$ there is a notable increase in AUC for the random
split compared to $T+$ in both $n_{I}$-instances. For the critical split,
almost all AUC-values are lower than $T+$, except for $\beta=0.1$ and
$\beta=0.5$ at $n_{I}=10$ where it is slightly higher. Generally, the
$\beta$-term does not seem to increase the performance of the base model.

Second, we investigated models’ predictions of lane changes in _highD_ at the
fixed-sized gaps, as defined in [26]. Looking at the random splits, again for
$\beta=0.5$ there is an increase in AUC for both $n_{I}$-situations. The
changes for the other $\beta$-versions are not consistently different when
compared to $T+$, having minor fluctuations to perform slightly better or
slightly worse. Concerning the critical split, all $\beta$-versions but
$\beta=10$ perform better than the base model, where $\beta=10$ performs very
similarly to $T+$. Also, the difference between $n_{I}=2$ and $n_{I}=10$ is
logical, as the latter consistently has a higher AUC than the former.

For both the FDE and ADE, all the $\beta$-versions are outperforming the
$T+$-model at $n_{I}=10$ on the random split. However, this seems due to one
extremely high value of one of the random splits of $T+$ (each random split
consists of three sub-splits, which are averaged to minimize the effect of
randomness). This could be an outlier, caused by an error in the training
process. At $n_{I}=2$, the $\beta$-values under-perform compared to $T+$ for
the random split, indicating no significant improvement. At the critical
split, only at $\beta=0.1$ and $\beta=10$ both ADE and FDE values are lower at
$n_{I}=10$. In general, there is no clear improvement regarding these metrics
across the various $\beta$-values.

Overall, in _highD_ lane-change prediction experiments, there are instances of
both better and worse performance of Smooth-Trajectron++ compared to T++,
indicating no consistent benefits of adding smooth attention to T++. This is
in contrast to the nuScenes results, which may stem from fundamental
differences in the datasets. While the _nuScenes_ dataset encompasses a wide
range of data with cars and pedestrians, _highD_ mainly focuses on cars. The
application of the smoothing term $\beta$ in the Smooth-Trajectron++ model
relies on the attention module that compares different semantic classes of
traffic participants. In datasets where one class dominates, the smoothing
term may not yield tangible improvements.

##  V Conclusion

This paper proposed Smooth-Trajectron++, a trajectory prediction model based
on an existing state-of-the-art model Trajectron++ [10] in which we
incorporated a cognitively-inspired smooth attention module [22]. We
demonstrated that our smooth-attention version of T++ can achieve increased
performance on the large-scale dataset nuScenes, but does not result in
tangible improvements on the highD dataset. This suggests that the smooth
attention approach seems to be more suitable for large-scale multi-agent
datasets with multiple agent types rather than on datasets with few traffic
agents of mostly the same type. Hence, the concept of _smooth attention_ might
be better applied to models where the attention module is implemented over
individual agents and not semantic classes. Nevertheless, our results further
strengthen previous work [18, 20, 22], indicating that including cognitive
insights can allow better predictions of human behavior in traffic.

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