# One-particle engine with a porous piston

Carlos E. Álvarez  Escuela de Ingeniería, Ciencia y Tecnología, Universidad
del Rosario, Bogotá, Colombia<EMAIL_ADDRESS>Manuel
Camargo  FIMEB & CICBA, Universidad Antonio Nariño - Campus Farallones, Cali,
Colombia  these authors contributed equally to this work  Gabriel Téllez
Departamento de Física, Universidad de los Andes, Bogotá, Colombia  these
authors contributed equally to this work

###### Abstract

We propose a variation of the classical Szilard engine that uses a porous
piston. Such an engine requires neither information about the position of the
particle, nor the removal and subsequent insertion of the piston when
resetting the engine to continue doing work by lifting a mass against a
gravitational field. Though the engine operates in contact with a single
thermal reservoir, the reset mechanism acts as a second reservoir, dissipating
energy when a mass that has been lifted by the engine is removed to initiate a
new operation cycle.

## Introduction

In 1929 Leo Szilard proposed a version of Maxwell’s demon that appeared to
violate the Kelvin-Planck statement of the second law by lifting a mass
against a gravitational field while in contact with a single heat reservoir
[1]. The demon consists of an engine working with a single particle gas, and
since then several models for single particle engines and processes have been
studied (see for example [2, 3, 4]). Though some variants of this demon, known
as Szilard’s engine, have been proposed [5, 6, 7, 8, 9, 10], the basic design
is composed of the following parts: A particle inside a container along with a
removable piston, a heat bath at temperature $T$ in contact with the container
walls, a hanging mass which is attached to the piston in such a way that work
is performed on it as the piston moves, and an external agent in charge of
resetting the system once the piston has reached one of the ends of the
container.

During a cycle, the agent inserts the piston in the middle of the container
and measures the particle’s position with respect to the piston (left or
right). Next, it attaches the mass to the piston from the same side as the
particle, such that when the piston is released the particle pushes it to the
opposite end of the container, simultaneously lifting the mass in the process.
Finally, the agent removes the piston from the end of the container, taking
away with it the potential energy gained by the mass in the current cycle and
preparing the system for the next one.

However, the agent generates entropy with its actions. The information
obtained from the measurement is stored in a memory and it is used when the
decision to hang the mass, either from the left or right, has to be made.
Landauer found that in order to reset the memory an amount of entropy equal to
the entropy reduction produced by the engine has to be generated [11]. To this
minimal entropy one can add other terms, like the amount of entropy generated
when resetting the piston, which has also been calculated [12]. This
connection between physics and information theory is now the subject of the
field of thermodynamics of information [13].

Currently, there is a growing interest in automating the process by which a
small out-of-equilibrium system like Szilard’s engine generates work, by
replacing a demon that acquires and processes information by an autonomous
agent which is coupled to the system [14, 15, 16]. The second law of
thermodynamics still holds in such schemes, as the autonomous agent has to be
coupled to a thermal reservoir at a temperature which is lower than the lowest
temperature in the system [17].

In this paper we propose a variation of Szilard’s model engine that does work
without the need for an agent to measure the position of the particle, to
decide based on this information from which side of the container to hang the
mass, or to reallocate the piston in the reset step of the process. This
system is not driven by changing a set of variables following some defined
protocol and, according to our analysis, it still respects the Kelvin-Planck
statement of the second law, as even if the engine is technically in contact
with a single thermal reservoir the process dissipates energy through its
reset mechanism.

##  1 Model

We propose a model composed by a single particle of mass $m$ and a piston of
mass $M$ inside a one dimensional container of length $L$ whose ends are in
contact with a heat bath at temperature $T$, as schematically shown in Fig. 1.
The piston has pores in it such that when the particle is close enough, there
is a probability $p$ for it to cross to the other side of the container
instead of colliding with the piston. We refer to $p$ as the porosity of the
piston. As with Szilard’s model, a load mass $M_{l}$ can be hanged from the
piston so that when it is pushed by the particle, the mass is lifted. If $p$
is small enough, most of the time the particle will be able to push the piston
to the opposite side of the container before crossing through the pore and
lift the load mass in the process. Once the particle crosses, it will push the
piston in the opposite direction.

Figure 1: Schematic of the engine with porous piston. (a) The piston is
initially connected to the right load mass and the particle bounces between
the right thermal wall and the piston. (b) The particle pushes the piston to
the left. (c) Once the piston gets to position $a$, the high energy load mass
hanging from its right is removed and a new low energy one is placed on the
right end of the container. (d) As the piston gets to point $b$ it couples
with the left load mass. Once the particle crosses to the left side through
the pore it will push the piston to the right and the cycle repeats.

An external agent is charged with the task of removing the incoming high
potential energy mass once the piston has reached one of the container’s ends,
while another agent prepares a new mass with low potential energy on the
opposite side of the container, which will be coupled to the piston at a later
instant. One can use feedback by measuring the state of the system to pinpoint
the moment at which a mass has been removed from the system and use this
information to place a new mass as soon as possible. To model the coupling
between the piston and the hanging mass we assume that the mass is connected
through an ideal string to a hook that the agent places at one of the ends of
the container. Such a hook can couple to the piston by spending a small fixed
amount of the kinetic energy of the piston. As the particle pushes the piston,
the piston collides elastically with the mass hook, pushing it to the other
side of the container until it gets removed past a given point $a$ (see Fig.
1). When reaching point $b$, the piston hook couples with the mass hook at
that point, dissipating once again the same arbitrary amount of kinetic
energy.

The inner workings of the engine presented here can be understood in the
context of nonequilibrium thermodynamics of feedback control [18]. Let us
consider the system composed of the particle and the piston. Let $x$ be phase-
space point of the system composed of the particle position and momentum and
the piston position and momentum. The Hamiltonian of the system is composed of
the interaction between the piston and the particle, the fixed external
interaction between the particle and the container walls and the potential
interaction $U_{pl}(x,\lambda)$ between the piston and the load masses. As the
system evolves, this interaction potential switches between three different
forms depending on the load masses, labelled by $\lambda\in\\{L,R,U\\}$,
corresponding respectively to a load mass on the left side ($L$), load mass on
the right side ($R$) or the piston is unhooked from the masses ($U$). If
$\lambda$ is kept fixed, i.e., the unhook and hooking mechanism is switched
off, the system will attain thermal equilibrium with an equilibrium free
energy $F(\lambda)$. Due to the symmetry of reflection between left and right
sides $F(L)=F(R)$, therefore the difference of free energy between two states
where a load mass is hooked is $\Delta F=0$.

When the hooking and unhooking mechanism is operating, a given function
$\lambda(t)$ can be though as a protocol controlling the dynamics of the
system. But, contrarily to many small-scale systems studied in stochastic
thermodynamics [19], the protocol in this case is neither fixed nor
deterministically applied to the system on each realization. Rather, the inner
mechanisms of the engine described previously are responsible for switching
$\lambda$ at random times, which can be interpreted as a feedback mechanism as
follows.

Suppose that at each time step, a measure without error is performed on the
system to obtain $x$, from which the position of the piston and its kinetic
energy are obtained. Then the following feedback control is performed:

  * •

If the piston is neither in any of the positions $a$ or $b$ shown in Fig. 1,
then $\lambda$ is unchanged from its current value. Note that there are
actually two $a$ positions and two $b$ positions, located symmetrically at
each side of the container: $a_{L}$ and $b_{L}$ at the left, and $a_{R}$ and
$b_{R}$ at the right.

  * •

If the piston is in a position $a_{L}$ and $\lambda=R$ or the piston is at
$a_{R}$ and $\lambda=L$, then $\lambda$ is changed to $U$ (unhook).

  * •

If the piston is in a position $b_{L}$ (or $b_{R}$) and has enough kinetic
energy, and $\lambda=U$ then $\lambda$ is changed to $L$ (or $R$)
corresponding to the side the piston is at.

Similarly to Szilard’s engine, there is an arbitrary amount of work that the
external agents must perform to remove the masses that have been lifted and to
bring new masses from the surroundings to the position at which they can be
connected to the piston. We consider such agents as part of the surroundings
and concentrate instead on the intrinsic capacity of the engine to perform
work given that the protocol described previously is carried out by the
agents. For simplicity, the new masses are introduced to the system with zero
kinetic energy, as if they were drawn from a thermal reservoir at zero
temperature.

It is not always possible to define cycles for engines at the nanoscale level
in the same way as it is done macroscopically.[20] However, in our case we
find it useful to define a cycle as the process that occurs between the
coupling of a new load mass to the piston and the subsequent removal of an old
one. Note that the duration of such a cycle ($t_{c}$) is variable.

##  2 Simulations

Mesoscale and smaller systems are commonly subject to diffusive effects and
hydrodynamic interactions, as they are in general not found in vacuum. Such
effects are important at the moment of thinking about a possible experimental
realization. However, in this first study we are neglecting them to
concentrate on the basic operation of a simple model of the engine.

We perform event driven simulations [21] for ensembles of 200 systems using
reduced mass ($M$), distance ($l$) and time ($t$) units defined as

$\displaystyle M^{*}$ $\displaystyle=\frac{M}{m},$ (1) $\displaystyle l^{*}$
$\displaystyle=\frac{l}{L},$ (2) $\displaystyle t^{*}$
$\displaystyle=\sqrt{\frac{k_{B}T}{mL^{2}}}t,$ (3)

which, for the energy ($E$) and acceleration ($g$) gives

$\displaystyle E^{*}$ $\displaystyle=\frac{E}{k_{B}T},$ (4) $\displaystyle
g^{*}$ $\displaystyle=\frac{mL}{k_{B}T}g.$ (5)

For ease of notation the asterisks will be dropped in the following.

To simulate the interaction of the particle with the thermal walls, every time
it collides with one of the container’s end walls, its recoil speed $v$ is
randomly drawn from the distribution [3]

$\displaystyle\rho(v)=\beta mve^{-\frac{1}{2}\beta mv^{2}},$ (6)

where $\beta=(k_{B}T)^{-1}$, $k_{B}$ is the Boltzmann constant and $T$ is the
absolute temperature of the thermal walls.

The collisions between the particle and the piston are elastic and occur with
probability $1-p$ every time they meet each other, otherwise the particle
crosses through the pore and both the piston and the particle continue with
their trajectories without interacting. Once the piston reaches a return
point, indicated by $b$ in Fig. 1, a small fixed amount of energy is
subtracted from its kinetic energy to overcome the energy barrier to couple
the hook. The model assumes that this energy is dissipated. If the piston does
not have enough kinetic energy then an elastic collision occurs. Lastly, as
shown in Fig. 1, once a load mass is coupled to the piston, it is pushed by
means of elastic collisions until it gets pass point $a$, where it is removed
from the system. Between collisions the load mass moves under the influence of
the external gravitational field.

The energy dissipated every time a coupling occurs, as well as the kinetic
energy that gets out of the system along with the masses and their hooks, is
accumulated in a variable $E_{dis}$, that accounts for the energy dissipated
by the process. The change in the gravitational potential energy of the load
masses as they are lifted by the engine is accumulated in the variable $W$.
The work done by the engine (particle + piston) on the load masses is $-W$.
The change in kinetic energy of the particle each time a collision with one of
the thermal walls occurs is accumulated in the variable $Q$, that accounts for
the heat exchanged with the bath. Finally, the kinetic energy change of the
piston during the collisions with the mass hooks is accumulated in the
variable $W^{(ph)}$ which represents the work done by the hooks on the piston.

We compute the efficiency of the engine as the ratio between the power
delivered by the engine on the hanging mass $\dot{W}_{t}$ during a time $t$
and the heat exchanged per unit of time $\dot{Q}_{t}$ during the same lapse

$\displaystyle\eta_{t}=-\frac{\dot{W}_{t}}{\dot{Q}_{t}}.$ (7)

In addition to the total efficiency $\eta_{t}$, we obtain the efficiency
$\eta^{(bp)}_{t}$ of the energy transfer between the heat bath and the piston

$\displaystyle\eta^{(bp)}_{t}=-\frac{\dot{W}^{(ph)}_{t}}{\dot{Q}_{t}},$ (8)

as well as

$\displaystyle\eta^{(pl)}_{t}=\frac{\dot{W}_{t}}{\dot{W}^{(ph)}_{t}},$ (9)

that measures the conversion efficiency of work done by the piston on the
hooks of the load masses into work done against the external field.

The parameters explored by the simulations where: $k_{B}T=1$, $M=50$,
$b=0.075$, $a=0.15$ and the gravitational field $g=2$, along with several
values for the porosity $p\in\\{0.036,0.2,0.4,0.6,0.96\\}$ and the load mass
$M_{l}\in\\{0.1,0.5,1.0,2.0,3.0\\}$.

##  3 Results

Figure 2: Example of the system dynamics. (a) Position along the container of
the particle and the piston as a function of time. (b) Position of the piston,
left and right hooks as a function of time, during a simulation of the engine
where $p=0.2$ and $M_{l}=1.0$. The arrows in (a) point to some events where
the particle crosses through the pore, while the arrows in (b) point to events
where one of the hooks is pushed out of the system and reset.

As the engine is a system out of equilibrium featuring predominant
fluctuations, we study the distribution of the efficiency [22, 23] obtained
from the simulations in order to estimate the macroscopic efficiency[24, 3,
25]

$\displaystyle\bar{\eta}=-\frac{\langle\dot{W}\rangle}{\langle\dot{Q}\rangle},$
(10)

where $\langle\cdot\rangle$ denotes the ensemble average in the thermodynamic
limit ($t\rightarrow\infty$). To achieve this, we make use of the large
deviation function[26, 27, 28] defined as

$\displaystyle J(\eta)=\lim_{t\rightarrow\infty}-\frac{1}{t}\ln
P(\eta_{t}\in[\eta,\eta+d\eta]),$ (11)

where $P(\eta_{t}\in[\eta,\eta+d\eta])$ is the probability for $\eta_{t}$ to
take a value between $\eta$ and $\eta+d\eta$. The large deviation function
describes the asymptotic behavior ($t\rightarrow\infty$) of the efficiency
fluctuations, and its minimum, which is located at the most probable value,
indicates the macroscopic efficiency of the engine $\bar{\eta}$. To estimate
$J(\eta)$, one can resort to the extrapolation procedure proposed by Proesmans
and van den Broeck[25], which assumes a general functional form for
$P_{t}(\eta)$ having three fitting parameters, that can be determined by using
three evaluations of $P_{t}(\eta)$ at finite times.

An example of the dynamics of the different components of the engine during a
simulation is displayed in Fig. 2. It is observed in Fig. 2(a) that the piston
bounces against the stops on each side of the engine while it collides with
the particle. In the crossing events, in which the particle goes through the
pore, one observes that the trajectory of the particle crosses that of the
piston without bouncing back. Similarly, Fig. 2(b) shows how the piston pushes
the hooks against the gravitational field by means of elastic collisions until
the hook reaches the opposite end of the engine and then it is reset.

Figure 3(a) shows the accumulated work done on the load mass and the heat
exchanged between the thermal walls and the particle as a function of time for
a single simulation. Note that in this simple model the amount of work done
per cycle by lifting the load mass is constant and equal to $M_{l}gh$, where
$h=0.7$ corresponds to the height that the mass is lifted restricted by the
chosen geometry of the container, which can be seen in the plot as discrete
jumps in the accumulated work. That the work per cycle is constant means that
for fixed $M_{l}$ the power delivered in a cycle ($M_{l}gh/t_{c}$) is a
function of $t_{c}$ only. The oscillations of the work observed over the base
line correspond to the bounces of the hooks observed in Fig. 2(b). Figure 3(b)
shows the ensemble average of the heat and the work.

Figure 3: Work and heat vs. time. Work done to lift the load mass ($W$) and
heat exchanged between the particle and the thermal walls ($Q$) during a
simulation of the engine where $p=0.2$ and $M_{l}=1.0$. (a) Single simulation.
(b) Ensemble average. Figure 4: Cycle duration $t_{c}$ pdf for systems with
$M_{l}=1$ and several values of $p$. The inset shows the logarithm of average
cycle duration time $\bar{t}_{c}$ as a function of $p$. Figure 5: Cycle
duration $t_{c}$ pdf for systems with $p=0.2$ and several values of $M_{l}$.
The inset shows the average cycle duration time $\bar{t}_{c}$ as a function of
$M_{l}$. Figure 6: Work distribution at fixed load mass. (a) Pdf of the work
performed by the engine on the load mass during a time window
$t=40\bar{t}_{c}$ for systems with $M_{l}=1$ and several values of $p$. (b)
Logarithm of the local minima of the curves shown in panel (a). Inset:
Variance of the distribution computed from a Gaussian fit to the data around
its maximum (squares) and from the numerical computation using all the work
data (circles). Figure 7: Work distribution at fixed porosity. (a) Pdf of the
work performed by the engine on the load mass during a time window
$t=40\bar{t}_{c}$ for systems with $p=0.2$ and several values of $M_{l}$. (b)
Logarithm of the local minima of the curves shown in panel (a). Inset:
Variance of the distribution computed from a Gaussian fit to the data around
its maximum (squares) and from the numerical computation using all the work
data (circles).

The probability density function (pdf) of the time per cycle ($t_{c}$) for a
fixed load mass $M_{l}=1$ and various porosities is displayed in Fig. 4. The
pdf presents several peaks before decaying. In contrast with systems like, for
example, a gas within a piston that follows an isothermal process, where it is
compressed from state $A$ to state $B$ and then expanded from $B$ to $A$ to
complete a cycle taking always the same time, the maxima found in the pdf can
be explained by the fact that many different events might occur within one of
the cycles of the engine with porous piston. For example, the piston can
simply couple a mass soon after it was introduced and pull it all the way up,
or it might wiggle for a time, bounce against it without coupling to it and
pick instead the mass in the opposite side of the container. These different
types of cycle might have different characteristic times that reflect in the
pdf as different peaks. The inset of Fig 4 shows that the average time
($\bar{t}_{c}$) gets to a minimum value near $p=0.2$ and as the work delivered
per cycle is constant in this setup, it can be said that the engine operates
at maximum power for a value close to $p=0.2$.

Similarly, Fig. 5 shows the pdf of $t_{c}$ for a fixed porosity $p=0.2$ and
various load mass values. In contrast with its dependence on $p$, the average
cycle duration as a function of $M_{l}$ increases monotonically.

On the other hand, we obtained the pdf of the work performed by the engine on
the load mass during a time window of length $t$. Figure 6(a) shows the pdf of
the work $W_{t}$ normalized by $M_{l}gh$ when a time window of length
$t=40\bar{t}_{c}$ is used for a fixed load mass and several values of the
porosity. The periodic peaks observed are located at integer multiples of the
work done during a single cycle. That the mode of the distributions in Fig.
6(a) is close to $W_{t}/M_{l}gh\approx-40$ tells us simply that the most
probable scenario when the engine runs during a time that spans 40 times the
average cycle duration, is for it to complete 40 cycles.

In microscopic models of the Carnot engine it has been found that the pdf of
the work presents a long tail towards zero, but it approximates a Gaussian in
the quasistatic limit[3], which sometimes has been used to approximate the
distribution of the work[29, 30]. To estimate the importance of these tails in
the case of the engine with porous piston, we obtained a curve consisting of
all the local minima observed in Fig. 6(a) and computed its logarithm, as
shown in Fig. 6(b). It is observed that the distribution deviates from a
Gaussian as it presents the mentioned tail towards zero.

The inset in Fig. 6(b) presents the variance of the distributions obtained
directly from the data, as well as from a Gaussian fit performed around the
mode of the distribution. The variance values computed from the data are
considerably higher that those obtained from the fit because of the large tail
towards zero of the distribution, but this difference becomes smaller as
$p\rightarrow 0$. Also, it is observed that the variance of the data gets to a
maximum around the point at which the engine operates at maximum power.

Figure 7(a) shows the pdf of $W_{t}$, when a time window of length $t=40t_{c}$
was used, for a fixed porosity and several values of the load mass. It is
observed that, when normalized by $M_{l}gh$, the amount of work done depends
slightly on the load mass. As $M_{l}$ increases, the variance of the
distribution decreases and it can be better fitted by a Gaussian function. As
the pdf of $t_{c}$ is skewed towards large times and the accumulated work
($W_{t}$) increases as more cycles fit within the time window, the pdf of
$W_{t}$ is skewed towards small absolute values. Figure 7(b) shows the
logarithm of the pdf of $W_{t}$, where a deviation from a Gaussian behavior is
also observed.

A closely related quantity to the variance of the work distribution is the
entropy production. It has been shown for various systems that the latter is
bounded from below by the inverse of the so-called precision, i.e., the ratio
of the variance to the squared mean value of a current,[31, 32, 33]

$\displaystyle\frac{\Sigma_{t}}{k_{B}}\geq\frac{2\langle
W_{t}\rangle^{2}}{\text{Var}(W_{t})},$ (12)

where $\Sigma_{t}$ indicates the average entropy production during a time
window of length $t$ and the current measured is the work done by the engine
on the load masses. Table 1 shows the precision as a function of averaging
time $t/\bar{t}_{c}$. It is observed that for larger time windows there is
less uncertainty in the value of the work. Assuming that the uncertainty
relation given by Eq. (12) holds for this system, these precision values
indicate a lower bound for the entropy production.

$t/\bar{t}_{c}$ | $2\left<W_{t}\right>^{2}/\mbox{Var}(W_{t})$
---|---
0.1 | 0.22
0.2 | 0.46
0.5 | 0.79
1.0 | 2.59
2.0 | 5.13
3.0 | 7.41
5.0 | 11.76
Table 1: Inverse of the precision as a function of the averaging time
$t/\bar{t}_{c}$ for a system with $M_{l}=1$ and $p=0.2$.

Figures 8 and 9 show the pdf of the heat exchanged between the particle and
the thermal walls. As the change in the accumulated heat is not done in
discrete steps as those present on the accumulated work, the peaks observed at
multiples of $M_{l}gh$ in the work distribution are absent and, more
importantly, the mode of the distributions for the heat varies noticeably as
the parameters $p$ and $M_{l}$ are changed. It is this variation in heat
absorption which produces the variation in the efficiency, as presented in the
following paragraphs.

Figure 8: Heat distribution at fixed load mass. (a) Pdf of the heat exchanged
between the particle and the thermal walls during a time window
$t=40\bar{t}_{c}$ for systems with $M_{l}=1$ and several values of $p$. (b)
Logarithm of the data of shown in panel (a). Inset: Variance of the
distribution computed from a Gaussian fit to the data around its maximum
(squares) and from the numerical computation using all the heat data
(circles). Figure 9: Heat distribution at fixed porosity. (a) Pdf of the heat
exchanged between the particle and the thermal walls during a time window
$t=40\bar{t}_{c}$ for systems with $p=0.2$ and several values of $M_{l}$. (b)
Logarithm of the data shown in panel (a). Inset: Variance of the distribution
computed from a Gaussian fit to the data around its maximum (squares) and from
the numerical computation using all the heat data (circles).

Figure 10 shows an example of the pdf of $\eta_{t}$, $\eta_{t}^{(bp)}$ and
$\eta_{t}^{(pl)}$ obtained from the simulations. The bar plot refers to the
actual efficiency per cycle of the engine, while the lines show the efficiency
pdf computed for several time windows of size $t$. The distributions observed
are heavily skewed towards small values of $t$, while both the spread and
skewedness decrease as $t\rightarrow\infty$. The peak at $\eta=0$ shows that
for values of $t$ near $\bar{t}_{c}$ there is a high probability of the window
falling within the same cycle doing almost no work, as shown in FIG. 3(a).
Note that the shape of these distributions is different from those of
previously studied systems,[22, 24, 25] in that in most of those cases two
maxima are present. The presence of the two maxima can be explained by the
possibility for the systems to function either as an engine or a heat pump
with certain probability[29]. However, due to the reset of the masses, the
engine with porous piston cannot operate in a reverse fashion. The sudden
change in the probability distribution of microstates that the reset produces
amounts to a symmetry breaking,[34] and it has been observed that such events
might affect the distributions and fluctuation relations that the system
obeys.[35]

Figure 10: Pdf of the efficiency obtained from an ensemble of systems with
$p=0.2$ and $M_{l}=1$. The bar plots refer to the actual efficiency per cycle
of the engine, while the lines represent the efficiency pdf obtained by
sampling several time windows of size $t$.

As explained in Section 2, the macroscopic efficiency $\bar{\eta}$ was
computed by means of the large deviation function (Eq. (11)) using several
finite values of $t$ and then extrapolating the asymptotic behavior. Figure 11
shows Eq. (11) as $t$ goes from $\bar{t}_{c}$ to $320\bar{t}_{c}$ for an
ensemble of systems with parameters $p=0.2$ and $M_{l}=1.0$. The values for
$\bar{\eta}$ are shown in Fig. 11 and were computed as an average of
extrapolations using all possible combinations of ordered triplets of the
curves, as described in Ref. 25.

Figure 11: Estimation of the large deviation function. Behavior of the
function $-\ln(P(\eta_{t}))/t$ in Eq. (11) computed at $p=0.2$ and
$M_{l}=1.0$, using several time window sizes.

Figures 12 and 13 plot the macroscopic efficiencies $\bar{\eta}$,
$\bar{\eta}^{(bp)}$ and $\bar{\eta}^{(pl)}$ as functions of the porosity and
load mass, respectively. It is observed that $\bar{\eta}^{(bp)}$ remains at 1,
as on average all the energy that comes from the thermal walls goes to the
piston. We also have that on average
$\bar{\eta}=\bar{\eta}^{(bp)}\bar{\eta}^{(pl)}$, and that the dissipation is
taking place as the work provided by the piston is transformed into potential
energy of the load masses, because part of the energy transferred is converted
into kinetic energy of the load masses that is lost at the instant of their
removal from the system.

Figure 12: Macroscopic efficiency as a function of $p$ for ensembles of
systems with $M_{l}=1$. The solid line plots the average power as a function
of $p$ using a time window of $t=320\bar{t}_{c}$ (note the corresponding scale
on the right side of the figure). Figure 13: Macroscopic efficiency as a
function of $M_{l}$ for ensembles of systems with $p=0.2$. The solid line
plots the average power as a function of $M_{l}$ using a time window of
$t=320\bar{t}_{c}$ (note the corresponding scale on the right side of the
figure).

The associated average power is also shown in Figs. 12 and 13, this time
without the normalization by $M_{l}gh$. As it was previously observed, for
fixed $M_{l}=1$ the engine operates at maximum power around $p=0.2$. On the
other hand, for fixed $p=0.2$ the simulation data shows that the engine
operates at maximum power for a load mass value around $M_{l}=1.0$.

According to the general theory of feedback control in non-equilibrium
systems,[18] the work $W$ done on the system between two states satisfies the
generalized Jarzynski relation[36, 37]

$\langle e^{-\beta(W-\Delta F)-I_{c}}\rangle=1\,,$ (13)

where $\Delta F$ is the free energy difference between the initial and final
state and $I_{c}$ is the mutual information between phase-space point $x$ and
the outcome of its measure $y$. Here the measure is without error, therefore
the average $\langle I_{c}\rangle$ has its maximum value equal to the Shannon
information $H[X]$ of the trajectory $X=\\{x(t)\\}$

$\langle I_{c}\rangle=H[X]=-\int P[X]\ln P[X]\,dX\,,$ (14)

with $P[X]$ is probability of the trajectory $X$ being realized. From the
convexity of the exponential function one obtains

$\langle W\rangle\geq\Delta F-k_{B}T\langle I_{c}\rangle\,.$ (15)

If in the initial and final state a load mass is hooked then $\Delta F=0$. In
this point of view, the work that the machine is able to provide comes from
the information obtained by measuring the piston position and performing the
feedback control. Note that, contrarily to the Szilard engine, the position of
the particle is not used in the feedback control.

To characterize how efficient the feedback control is, one can introduce the
efficacy parameter[18] $\gamma$, which we obtain from our simulations and the
relation

$\langle e^{-\beta(W-\Delta F)}\rangle=\gamma\,.$ (16)

For systems without feedback control $\gamma=1$, whereas for systems with
feedback control $\gamma>1$. Figure 14 shows the efficacy, estimated as the
average of $\exp(-\beta W_{h})$ over the ensemble of simulations, with $W_{h}$
the work done during the intervals where the control parameter $\lambda$ is
either $R$ or $L$, so that $\Delta F=0$. As expected from a feedback
controlled system, the efficacy starts at 1 and increases from there as the
time increases.

Figure 14: Estimation of the efficacy parameter ($\gamma$) as a function of
time for a system with $p=0.2$ and $M_{l}=1$.

##  4 Discussion

By modifying Szilard’s engine with the replacement of the piston by a porous
one, the engine can convert heat into work without the need to store
information about the position of the particle with respect to the piston. If
the engine is not coupled to an external system on which to perform work, one
can expect it to wiggle constantly as it exchanges heat back and forth with
the thermal walls. However, the coupling of the engine with another system on
which to do work introduces the possibility for energy dissipation.

We chose to couple the engine to a system similar to the one proposed in
Szilard’s original model, where work is done by lifting a small mass against a
gravitational field. Energy is dissipated because once the load mass is
removed from the system one does not take away merely the potential energy
gained during the process, but also any kinetic energy that the load mass
happens to have at that instant. This reset operation amounts to a symmetry
breaking,[34, 35] and as with engines in which an autonomous agent ultimately
has to discard heat to a low temperature reservoir,[17] in this case the
mechanical agent that places the new, low-energy load masses with zero kinetic
energy, acts as a low temperature reservoir at $T_{\rm low}=0$ in the sense
that it takes away all the kinetic energy from them. In addition, we chose to
use a feedback control by unhooking and hooking a new load mass when the
piston passes through positions $a$ and $b$ respectively. As observed
previously in a thermal ratchet system,[18] such a feedback allows for an
efficacy parameter $\gamma>1$. In this case, the feedback reduces the amount
of time that the system of spends with no mass hanging from the piston,
enabling a greater power output.

If it were possible to take the system to a quasi-static limit at which the
masses arrive to the extraction point with a velocity close to zero, one could
expect that the amount of dissipated energy would also be very small. Although
the system is not driven by an external parameter which can be made to change
infinitely slowly, it is observed that as the load mass is increased, keeping
all the other parameters constant, the efficiency of the engine does increase
while the power decreases past a certain point. The effect of the porosity of
the piston on the efficiency seems to be more modest, achieving slightly
better efficiency and less power output at higher values of $p$. Finally, an
approximate maximum in the power output of the engine was observed as a
function of both, the load mass and the porosity of the piston. The power
decreases as one moves away from these maxima in both directions, towards
small and large values of parameters $p$ and $M_{l}$.

## Data availability

The code and datasets used and/or analysed during the current study are
available from the corresponding author on reasonable request.

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## Acknowledgements

G. T. acknowledges support from Fondo de Investigaciones de la Facultad de
Ciencias de la Universidad de los Andes INV-2021-128-2267. M.C. thanks
Vicerrectoría de Ciencia, Tecnología e Innovación (VCTI) – Universidad Antonio
Nariño for financial support through Project 2022023. C.E.A. acknowledges
financial support from Universidad del Rosario. The present article has been
published in Scientific Reports: Sci. Rep. 12, 13896 (2022).

## Author contributions statement

C.E.A. and M.C. performed the simulations. All authors analyzed the results
and reviewed the manuscript.


$\mathbb{P}(\mathcal{C}_{o}(p,f(p))=\mathbb{Z}^{d})=0$, so $p\geq p_{c}(f,d)$
as claimed . ∎

## Acknowledgements

Salisbury’s work was supported in part by a grant from NSERC. The work of
Holmes was supported in part by Future Fellowship FT160100166 from the
Australian Research Council. MH thanks the Pacific Institute for the
Mathematical Sciences for hosting part of this research. The authors sincerely
thank Geoffrey Grimmett for several helpful comments, including identifying
some inaccuracies in the introduction.

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# Well-posedness of Partially and Oblique Projected Dynamical Systems

W.P.M.H. Heemels and A. Tanwani Maurice Heemels is with the Control Systems
Technology Group, Department of Mechanical Engineering, Eindhoven University
of Technology, P. O. Box 513, 5600 MB Eindhoven, The Netherlands. Email:
<EMAIL_ADDRESS>Tanwani is with CNRS – LAAS, University of Toulouse,
CNRS, Toulouse.Maurice Heemels is supported by the ERC Advanced Grant
PROACTHIS.

###### Index Terms:

Extended PDS, sweeping process, hybrid integrator,

##  I HIGS

The general form of the hybrid integrator, extending the description of HIGS
in [DeeSha_AUT21a], is mathematically formulated as the scalar-state switched
nonlinear system

$\displaystyle\dot{x}_{h}$ $\displaystyle=f(x_{h},e),\hphantom{k_{h}z}\textup{
if }\left(e,u,\dot{e}\right)\in\mathcal{F}_{1},$ (1a) $\displaystyle{x}_{h}$
$\displaystyle=k_{h}e,\hphantom{f(x_{h},z)}\textup{ if
}\left(e,u,\dot{e}\right)\in\mathcal{F}_{2},$ (1b) $\displaystyle u$
$\displaystyle=x_{h},$ (1c)

with state $x_{h}(t)\in\mathbb{R}$, input $e(t)\in\mathbb{R}$, output
$u(t)\in\mathbb{R}$ at time $t\in\mathbb{R}_{\geq 0}$, and where
$f:\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is a nonlinear function. Here,
$\dot{e}$ denotes the time-derivative. The flow sets $\mathcal{F}_{1}$ and
$\mathcal{F}_{2}$ dictating the active mode in (1) are given by

$\displaystyle\mathcal{F}_{1}$
$\displaystyle:=\left\\{(e,u,\dot{e})\in\mathbb{R}^{3}\mid
eu\geq\frac{u^{2}}{k_{h}}\wedge(e,u,\dot{e})\not\in\mathcal{F}_{2}\right\\},$
(2a) $\displaystyle\mathcal{F}_{2}$
$\displaystyle:=\left\\{(e,u,\dot{e})\in\mathbb{R}^{3}\mid u=k_{h}e\wedge
f(x_{h},e)e>k_{h}\dot{e}e\right\\}$ (2b)

of which the union forms the $[0,k_{h}]$-sector defined as

$\mathcal{F}:=\mathcal{F}_{1}\cup\mathcal{F}_{2}=\left\\{(e,u,\dot{e})\in\mathbb{R}^{3}\mid
eu\geq\frac{u^{2}}{k_{h}},\dot{e}\in\mathbb{R}\right\\}.$ (3)

The sets $\mathcal{F}_{1}$ and $\mathcal{F}_{2}$ in (2) define regions where
$\mathcal{H}$ operates in either a _dynamic mode_ or a _static mode_. The
dynamic mode is referred to as ‘integrator-mode’, since, essentially, the
state value $x_{h}$ is obtained from integration. The static mode is soemtimes
referred to as ‘gain-mode’. Note that the definition of the set $\mathcal{F}$
in (3) shows the input-output pair $(z,u)$ of the hybrid integrator in (1) to
have an equivalent sign at all times, which may benefit transient properties
of a closed-loop system. For a further motivation of the sets in (2), along
with a visualization, the reader is referred to [DeeSha_AUT21a, Section 3].

Note that (1b) is in the form of a (differential) algebraic equation. We could
replace this by an ODE, which then turns (1) into

$\displaystyle\dot{x}_{h}$ $\displaystyle=f(x_{h},e),\hphantom{k_{h}z}\textup{
if }\left(e,u,\dot{e}\right)\in\mathcal{F}_{1},$ (4a)
$\displaystyle\dot{x}_{h}$
$\displaystyle=k_{h}\dot{e},\hphantom{f(x_{h},z)}\textup{ if
}\left(e,u,\dot{e}\right)\in\mathcal{F}_{2},$ (4b) $\displaystyle u$
$\displaystyle=x_{h},$ (4c)

directly revealing also the dependence of the right-hand sides on $\dot{e}$
(in the gain mode).

Concerning the vector field in (1a), the following assumptions are made.

######  Assumption 1.

The function $f$ in (1) satisfies $f(0,0)=0$, and $f(0,e)e\geq 0$ for all
$e\in\mathbb{R}$.

##  II Open-loop formulation of projection-based controllers: HIGS as sweeping
process

###  II-A Sweeping process

Let us first consider the classical HIGS in which $f(x_{h},e)=e$. We can write
this as the sweeping process [REF], where we use that the normal cone for a
given set $K\subseteq\ree^{n}$ at $x\in K$ is defined as

$N_{K}(x)=\\{s\in\ree^{n}\mid\langle s\mid k-x\rangle\leq 0\text{ for all
}k\in K\\}.$ (5)

In case $x\not\in K$ then $N_{K}(x)$ is defined as the empty set.

We assume $e$ is given as a function of time.

$\displaystyle\dot{x}_{h}$ $\displaystyle\in$
$\displaystyle\underbrace{f(x_{h},e(t))}_{\bar{f}(x_{h},t)}-N_{\bar{S}(t)}(x_{h})$
(6a) $\displaystyle x_{h}(t)$ $\displaystyle\in$ $\displaystyle\bar{S}(t)$
(6b)

with

$\bar{S}(t)=\\{x_{h}\in\ree\mid(e(t),x_{h})\in S\\}$ (7)

and $S$ is the sector corresponding to $\mathcal{F}$ defined by

$S=\left\\{(e,u)\in\mathbb{R}^{2}\mid eu\geq\frac{u^{2}}{k_{h}}\right\\}.$ (8)

###  II-B Equivalence of HIGS and SP

Indeed, note that

$\bar{S}(t)=\begin{cases}[0,k_{h}e(t)],&\text{when }e(t)\geq 0,\\\
[k_{h}e(t),0],&\text{when }e(t)\leq 0,\end{cases}$ (9)

which are convex sets (in fact, closed intervals of the real line) for each
$t$.

Note that

$d_{Haus}(\bar{S}(t_{1}),\bar{S}(t_{2}))\leq k_{h}|e(t_{1})-e(t_{2})|$

for all $t_{1},t_{2}$ [check!]. So, by imposing an AC condition on $e$ we get
the required regularity on the set $\bar{S}$.

Solutions are guaranteed now to (6) by [Edmond, Thibault].

Observe that for $x\in[a,b]$ and when $a<b$

$N_{[a,b]}(x)=\begin{cases}\\{0\\},&\text{ when }x\in(a,b,)\\\ \ree_{\leq
0},&\text{ when }x=a,\\\ \ree_{\geq 0},&\text{ when }x=b.\end{cases}$ (10)

Moreover when $a=b$, we get $N_{[a,a]}(x)=\ree$ for $x=a$.

MAURICE: Maybe we should prove existence to the first HIGS formulation and use
uniqueness of solutions of the NC formulation/SP: suppose $f$ is locally
Lipschitz , then take two solutions to SP $x,\tilde{x}$, then

$d/dt|x(t)-\tilde{x}(t)|^{2}\leq 2L|x(t)-\tilde{x}(t)|^{2}$

as normal cone is monotone. So uniqueness of solutions. a solution to HIGS is
a solutions to SP/NC so if we can prove existence of solutions to HIGS – e.g.
with arguments later via Krasovskii??? maybe we can apply them in open -loop,
we show that HIGS is a sweeping process

Maurice we do not get right away that the formuled sweeping process model with
the normal cone is equal to the MODEL of HIGS above – only the “slow/lazy”
solution matches the HIGS formulation - -some work needed to clarify this !!!
Note that also taking the minimal norm/projection on Normal cone of the right-
hand side in (6a) is not helping! There is no way how a $\dot{e}$ gets in this
by just looking at the right-hand sides and doing calculus on that [see some
calculus below], it is really a consequence of the solution satisfying (6b) as
well and hence $\dot{x}_{h}$ somehow has to lie in the tangent cone! This
result on PDS - normal cone equivalence is for constant constraint sets, I
guess, see [BROGLIATO2006, heemelsorl]

Take the case $e(t)>0$ and thus $\bar{S}(t)=[0,k_{h}e(t)]$ and take
$x_{h}=k_{h}e(t)$ (so on the boundary), and $N_{\bar{S}(t)}(x_{h})=\ree_{\geq
0}$ in this case. Hence, (6a) reads $\dot{x}_{h}\in f-\ree_{\geq 0}$. By now
taking the minimal norm solution of the r.h.s. for $f=e$ we do get $\dot{e}$
in. So, this is not a ”static” property of the r.h.s., but it is due to the
fact that (6b) has to hold and thus some viability/tangent cone condition
kicks in due to time-varying nature of $\bar{S}$ , i.e. $\dot{x}_{h}\leq
k_{h}\dot{e}(t)$ has to hold, and this dynamic condition is going to be
important [somehow this is a tangent cone condition to $S$, which appears in
Section III]. I guess a minimal norm solution kicks in under this constraint!
hence, the time-varying nature of $\bar{S}$ (and loosely speaking the
”derivative” of the set, related to $\dot{e}$) plays a role! I will start
digging in the works that you mentioned of Thibault and your observer paper,
etc.

BTW: I do believe the sweeping process formulation (6) and the HIGS-based
formulations (1) or (4) have the same solutions!!!!!!!! only we need to find
the proper setting to connect the two!!

###  II-C Well-posedness

As shown above, first some work needed to embed the HIGS equations (1) or (4)
in an appropriate mathematical (sweeping process framework). If accomplished,
then following questions arise:

  * •

Can we rely on classical results for well-posedness? Locally AC inputs give
local existence and uniqueness?

  * •

global existence of solutions?

  * •

consistency of a time-stepping scheme? how does the particular time-stepping
scheme look if we translate it from classical literature?

  * •

we could also directly make the step in the projection-based controller to
state $x_{h}\in\ree^{n_{h}}$ and put the sector constraint on its input-output
pair! see below

##  III Direct Closed-loop formulations of projection-based controllers: ePDS

Plant (+ smooth controller part) and projection-based controller (with state
$x_{h}$

$\displaystyle\dot{x}_{p}$ $\displaystyle=$ $\displaystyle
f_{p}(x_{p},u,w)=f_{p}(x_{p},g_{h}(x_{h}),w);\ e=g_{p}(x_{p})$
$\displaystyle\dot{x}_{h}$ $\displaystyle=$ $\displaystyle
f_{h}(x_{h},g_{p}(x_{p}));\ u=g_{h}(x_{h})$

We write this as

$\dot{x}=F(x,w)$ (11)

Then

$\dot{x}=\Pi_{S,E}(x,F(x,w))$ (12)

with $n=n_{p}+n_{h}$

$S=\left\\{x\in\mathbb{R}^{n}\mid
e=g_{p}(x_{p}),u=g_{h}(x_{x}),eu\geq\frac{u^{2}}{k_{h}}\right\\}.$ (13)

$E=[0\ I]^{\top}$ (14)

$\Pi_{S,E}(x,v)=\operatorname{argmin}_{w\in T_{S}(x),w-v\in\im E}\|w-v\|$ (15)

This is a mathematical framework that fits and we can show that this leads to
the description (4) [with $f=e$] in closed loop with the plant dynamics. See
[DeeSha_AUT21a] [using a PWL modelling formalism]. Also here we obtain global
existence of solutions w.r.t. bounded piecewise Bohl inputs (including forward
completeness), using lexicographic orderings of derivatives in Taylor
expansions.

Note that with respect to classical PDS there are essential differences:

  * •

projection is not complete in the sense that only some directions (not all)
can be used to keep the state in the constraint set $S$

  * •

using a non-convex constraint set $S$ (note as opposed to setup in Section II,
this is now a constant set, not dependent on time).

In case all the states can be altered/projected, $S$ is convex and no inputs
are present, we recover as a special case of the above framework the classical
PDS.

###  III-A HIGS special case

Maybe first for HIGS or at least the FOPE – with $u=x_{h}$ and $n_{h}=1$

  * •

is the ePDS formulation well-posed in the sense that projection operator is
well defined? see some conditions below that are needed. Assumption 2.

  * •

existence / uniqueness of solutions, see [DeeSha_AUT21a] for piecewise Bohl
inputs.

  * •

MAURICE can this be written as an interconnection of sweeping process and
smooth nonlinear $x_{p}$ dynamics?, i.e. is the ”direct closed-loop
formulation” here ”similar”/same to using the HIGS/FOPE interconnected with
smooth system $\rightarrow$ can we use 2nd order sweeping processes here????
This might work – we typically use a relative degree assumptions, i.e. the
external disturbance $w$ is not directly present in $e=g_{p}(x_{p})$. so, at
least relative degree 1. We can even assume relative degree 2, if this helps!

Regarding the latter item: I would expect something like

$\displaystyle\dot{x}_{p}$ $\displaystyle=$
$\displaystyle=f_{p}(x_{p},g_{h}(x_{h}),w);$ (16a) $\displaystyle\dot{x}_{h}$
$\displaystyle=$ $\displaystyle
f_{h}(x_{h},g_{p}(x_{p}))-N_{\tilde{S}(g_{p}(x_{p}))}(x_{h}),$ (16b)

where

$\tilde{S}(g_{p}(x_{p}))=\\{x_{h}\in\ree\mid(g_{p}(x_{p}),x_{h})\in S\\}$ (17)

with $S$ the sector-like set defined before. Does this fit directly the state-
dependent 2nd order sweeping process formulation?  Does this connect to your
observer design paper with Bernard and Christophe?

Can we use this prove existence (and maybe uniqueness) of solutions for
Lebesgue measurable or AC external disturbance inputs?

And what about time-stepping schemes with consistence?

###  III-B General case

What about the general case ($n_{h}>1$) ? same questions as above.

###  III-C State-dependent norms?

$\Pi_{S,E}(x,v)=\operatorname{argmin}_{w\in T_{S}(x),w-v\in\im
E}\|w-v\|_{P(x)}$ (18)

with $P(x)\succ 0$ or maybe even $E(x)$? See Hauswirth paper where they look
at what they call oblique PDS – full projection but metric $\|\cdot\|_{P(x)}$
in projection operator state-dependent. Connection to that work?

###  III-D Well-posedness

can we prove existence/uniqueness of solutions? what are natural classes of
inputs?

I have been working on a viability/aubin/cellina type of approach, good to
connect to that!

Viability condition for regularized dynamics $\dot{x}=\text{Kras}(\Pi)$ (we
apply regularization to get osc properties) is $\text{Kras}(\Pi)\cap
T_{s}\neq\emptyset$. Moreover, every solution to the regularized dynamics has
to satisfy $\dot{x}=\text{Kras}(\Pi)\cap T_{s}$. To show that this is a
solution to the original (non-regularized) dynamics, it would be sufficient to
show

$\text{Kras}(\Pi)\cap T_{s}=\Pi$

Does this hold?

In general it is not true that

$\text{Kras}(\Pi)=F-N_{S}$

We found examples for $e=x_{h}=u=0$ and $\dot{e}>0$ but fortunately these are
isolated points in time. In fact, this is caused by $x\mapsto T_{S}(x)$ not
being lower semicontinuous!!! maybe nice to add an example illustrating this!

This would lead to the question if $\text{Kras}(\Pi)\cap T_{s}=\Pi$ is
connected to take the slowest solutions in the normal cone DI? same line of
reasoning, would be great to connect those – where Aubin meets Moreau?

##  IV Toulouse 20230203

###  IV-A Time-invariant case

Let $K$ be a convex polyhedral set given and a linear subspace ${\cal E}=\im
E$ where $E\in\ree^{n\times n_{E}}$ is a matrix with full column rank.
Moreover, we assume

######  Assumption 2.

The set $K\subseteq\ree^{n}$ and ${\cal E}$ satisfy

  * •

$K=\\{x\in\ree^{n}\mid Hx\geq b\\}$ for some matrix $H\in\ree{n_{H}\times n}$
MH: rank condition on active constraints? and vector $b\in\ree^{n}$

  * •

${\cal E}+T_{K}(x)=\ree^{n}$ for all $x\in K$ (in case $K$ is pointed convex
cone, then ${\cal E}+K=\ree^{n}$, equivalent - nice to have a more general
condition on $K$)

$\Pi_{K,{\cal E}}(x,v)=\operatorname{argmin}_{w\in T_{K}(x),w-v\in{\cal
E}}\|w-v\|$ (19)

Under Assumption 2 $\\{\w\in T_{K}(x)\mid w-v\in{\cal E}\\}$ is nonempty,
closed and convex. According to the closest point theorem [Luenberger] there
is a unique minimizer, so the projection operator above is well-defined.

$\dot{x}=\Pi_{K,{{\cal E}}}(x,f(x))=:F(x)$ (20)

######  Assumption 3.

$f$ is a continuous mapping in $x$ .

Active constraint index set $J(x):=\\{i\mid H_{i}x=b_{i}\\}$. Then
$T_{K}(x)=\\{v\mid H_{J}v\geq 0\\}$. For convenience we introduce
$T_{J}:=\\{v\mid H_{J}v\geq 0\\}$ for $J\subseteq\\{1,\ldots,n_{N}\\}.$

MH: [we can probably extend all of this $K$ given by mapping $H(x)\geq 0$.]
using $gradH_{i}$

Consider the Krasovskii regularisation of (37) given by

$\dot{x}\in\cap_{\delta>0}\overline{\textup{con}}F(B(x,\delta))=:K_{F}(x)$
(21)

Above, $\overline{\textup{con}}(M)$ denotes the closed convex hull of the set
$M$, in other words, the smallest closed convex set containing $M$.

######  Proposition 4.

If $f$ is continuous and Assumption 2 holds, then the Krasovskii
regularization of $F$ at $x\in K$ is equal to

$K_{F}(x)=\textup{con}\\{P_{T_{J},{\cal E}}(f(x))\mid J\subset J(x)\\}$ (22)

MH: this result reveals the lower semicontinuity of $x\mapsto T_{K}(x)$
through the $J\subset J(x)$ – this property will be crucial in generalizing
the result to general convex or even prox regular sets. However, this requires
to find some useful extension of this proposition characterising the
Krasovskii regularisation

$K_{F}(x)\subseteq(?=?)\overline{\textup{con}}\bigcup\\{P_{T,{\cal
E}}(f(x))\mid T_{K}(x)\subset T,\ T\textup{closed, convex cone}\\}$ (23)

would this hold, at least with $\subseteq$? due to lsc of $T_{K}(x)$ all
nearby $T_{K}(y)$ with $y\in B(x,\delta)$ satisfy $T_{K}(x)\subseteq
T_{K}(y)+eps$ (”loosely speaking”). In fact, this can be used to generalize
all the stuff below!

######  Theorem 5.

Assume that $f$ is continuous and Assumption 2 holds. Then for all $x\in K$ it
holds that

$K_{F}(x)\cap T_{K}(x)=\\{F(x)\\}=\\{\Pi_{K,{{\cal E}}}(x,f(x))\\}$ (24)

###### Proof.

It is obvious that the inclusion $\supseteq$ holds as $\Pi_{K,{{\cal
E}}}(x,f(x))$ clearly belongs to both $T_{K}(x)$ and $K_{F}(x)$ (cf. (50)).
So, let us consider the reversed inclusion $\subseteq$. Thereto, let $v\in
K_{F}(x)\cap T_{K}(x)$. Based on Proposition 4, $v\in K_{F}(x)$ can be written
as

$v=\sum_{J\subseteq J(x)}\lambda_{J}P_{T_{J},{\cal E}}(f(x))$

with $\lambda_{J}\geq 0$, for $J\subset J(x)$, and $\sum_{J\subset
J(x)}\lambda_{J}=1$. Let us consider

$|v-f(x)|=|\sum_{J\subset J(x)}\lambda_{J}P_{T_{J},{\cal E}}(f(x))-f(x)|=\\\
|\sum_{J\subseteq J(x)}\lambda_{J}P_{T_{J},{\cal E}}(f(x))-\sum_{J\subset
J(x)}\lambda_{J}f(x)|\leq\\\ \sum_{J\subseteq J(x)}\lambda_{J}|P_{T_{J},{\cal
E}}(f(x))-f(x)|\leq\\\ \sum_{J\subseteq
J(x)}\lambda_{J}|P_{\underbrace{T_{J(x)}}_{T_{K}(x)},{\cal E}}(f(x))-f(x)|=\\\
|P_{T_{J}(x),{\cal E}}(f(x))-f(x)|=|\Pi_{K,{{\cal E}}}(x,f(x))-f(x)|$ (25)

where we used in the second inequality that $|P_{T_{J},{\cal
E}}(f(x))-f(x)|\leq|P_{T_{J(x)},{\cal E}}(f(x))-f(x)|$ as $T_{J(x)}\subseteq
T_{J}$. Moreover, by definition of $P_{T_{J},{\cal E}}$ it holds that
$P_{T_{J},{\cal E}}(f(x))-f(x)\in{\cal E}$ for each $J\subset J(x)$ and thus
also

$\sum_{J\subseteq J(x)}\lambda_{J}(P_{T_{J},{\cal
E}}(f(x))-f(x))=\underbrace{\sum_{J\subseteq J(x)}\lambda_{J}P_{T_{J},{\cal
E}}(f(x))}_{=v}-f(x)\in{\cal E}$

due to ${\cal E}$ being a linear subspace. Since $v\in T_{K}(x)$ and
$v-f(x)\in{\cal E}$, and $|P_{T_{J(x)},{\cal E}}(f(x))-f(x)|$ is the shortest
distance along ${\cal E}$ between $T_{J}(x)=T_{K}(x)$ and $f(x)$, i.e.,

$\Pi_{K,{\cal E}}(x,f(x))=\operatorname{argmin}_{w\in T_{K}(x),w-f(x)\in{\cal
E}}\|w-f(x)\|$

it must hold that

$|v-f(x)|=|\Pi_{K,{{\cal E}}}(x,f(x))-f(x)|$

and thus $v$ must be the unique closest point in $T_{K}(x)$ to $v$ along
${\cal E}$ and thus $v=\Pi_{K,{{\cal E}}}(x,f(x))$, thereby proving the
result. ∎

######  Theorem 6.

Assume that $f$ is continuous and Assumption 2 holds. Then for every initial
condition $x(t_{0})=x_{0}\in K$ there exists $T>0$ and an AC solution
$x:[0,T]\rightarrow K$ to (37). MH: under boundedness/growth conditions this
should also be extended to global existence of solutions.

###### Proof.

Lemma 5.16 in [Goebel, sanfelice, teel book] show that $K_{F}$ is outer
semicontinous and in case $F$ locally bounded then $K_{F}$ locally bounded.
Moreover, $K_{F}$ takes closed and convex setvalues and for all $x\in K$ there
is an open neighborhood $U$ of $x$ such that for all $y\in U\cap K$ it holds
that $K_{F}(y)\cap T_{K}(y)\neq\emptyset$ (as it contains $\Pi_{K,{\cal
E}}(y,f(y))$). According to Lemma 5.26 (b) [goebel, sanfelice, teel] [maybe
better to refer to original Aubin source to, on viability theory] the
corresponding viability conditions are satisfied implying that the Krasovskii
regularization (50) has a solution $x$. Hence, this solution $x$ satisfies
$x(t)\in K$ for all $t\in[O,T]$, it holds that $\dot{x}(t)\in T_{K}(x(t))$,
almost everywhere, see Lemma 5.26 (a) [goebel, sanfelice, teel]. Hence, it
holds a.e. that

$\dot{x}(t)\in K_{F}(x(t))\cap T_{K}(x(t))$ (26)

Invoking Theorem 5 shows that $x$ is now a solution to (37) as the right-hand
side of (26) is contained of a single element being equal to the right-hand
side of (37) are equal. ∎

MH Interestingly, this is an alternative way to prove existence of solutions
to (ordinary) PDS using Krasovskii regularisation and viability theory
contrasting other proofs building on Skoroghod problem [nagurney and zhang],
plus other proofs – henry? who proved existence of solutions to PDS and how –
of course viability results also build on time-stepping proofs… so maybe deep
down the same?

###  IV-B Time-varying case

$\dot{x}=f(x,t)$

$f$ continuous in $x,t$

$\dot{x}=\Pi_{K,{{\cal E}}}(x,f(x,t))=:F(x,t)$ (27)

can we take $\bar{K}=\\{(x,t)\mid x\in K\\}$ still polyhedral convex set and
$\bar{\cal E}=\bar{E}\times\\{0\\}$ (not allowed to project in $t$ direction).
Define $\bar{f}(x,t)=(f(x,t),1)$). Is, with $\xi=(x,t)$,

$\dot{x}=\Pi_{K,{{\cal E}}}(x,f(x,t))=:F(x,t)$ (28)

”same” as

$\dot{\xi}=\Pi_{\bar{K},\bar{{\cal E}}}(\xi,\bar{f}(\xi))=:\bar{F}(\xi)$ (29)

$\bar{f}$ is continuous in $\xi$. so for continuous inputs we are good and we
can concatenate to go to piecewise continuous.

###  IV-C Further extensions

Possible extensions

  * •

prox regular sets $K$

  * •

what if ${\cal E}={\cal E}(x)$ can depend on $x$ but in a continuous way…

  * •

what if we do not use Euclidean norm $|\cdot|$ but state-dependent norms
$|\cdot|_{P(x)}$ a la Hauswirth, with $P(x)$ positive definite and continuous
in $x$, see details and motivation in Hauswirth’s papers

  * •

time-varying $K(t)$??? maybe the embedding approach with $\dot{t}=1$ solves
this already ;-)

##  V Beyond convexity and prox regularity: What about sector sets

In control theory often sector sets are of interest, see lure… etc etc.

######  Assumption 7.

The set $S\subseteq\ree^{n}$ and $E\in\ree^{n\times n_{E}}$ satisfy

  * •

$S=K\cup-K$ in which $K$ is a convex polyhedral cone given by
$K=\operatorname{pos}G+\im H$ in which the matrix $[G\ H]$ is square and has
rank $n$

  * •

$K\cap-K=\im G$

  * •

$E$ full column rank

  * •

$\im E\cap S=\\{0\\}$ and thus $S+\im E=\ree^{n}$

How can the above ideas be used to prove existence of solutions to

$\dot{x}=\Pi_{S,E}(x,f(x))$ (30)

with

$\Pi_{S,E}(x,v)=\operatorname{argmin}_{w\in T_{S}(x),w-v\in\im E}\|w-v\|$ (31)

for $x\in S$ and $v\in\ree^{n}$.

##  VI Aneel 20230204

MH: I start with simpler setup here, to see if we can make that work and then
let us see if we generalize

suppose after a possible coordinate transformation we get:

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle g(x,u);\ e=x_{n}$
$\displaystyle\dot{u}$ $\displaystyle=$ $\displaystyle h(u,e)$

$\xi=(x,u)$ and then above reads $\dot{\xi}=f(\xi)$.

$S=\left\\{\xi\in\mathbb{R}^{n+1}\mid eu\geq\frac{u^{2}}{k_{h}}\right\\}.$
(32)

Denote for convenience the 2d sector as $\bar{S}$, i.e.

$\bar{S}=\left\\{(e,u)\in\mathbb{R}^{2}\mid
eu\geq\frac{u^{2}}{k_{h}}\right\\}.$ (33)

$T_{S}(\xi)=\ree^{n-1}\times T_{\bar{S}}(e,u)$ (34)

######  Assumption 8.

$h(0,0)=0$

${\cal E}=\textup{Im}[0\ 1]^{\top}$ (35)

$\dot{\xi}=\Pi_{S,{{\cal E}}}(\xi,f(\xi))=:F(\xi)$ (36)

In fact,

$\dot{\xi}=\begin{cases}\Pi_{K,{{\cal E}}}(\xi,f(\xi)),&\text{ when }\xi\in
K\setminus-K\\\ \Pi_{-K,{{\cal E}}}(\xi,f(\xi)),&\text{ when
}\xi\in-K\setminus K\\\ f(\xi),&\text{ when }\xi\in-K\cap K\end{cases}$ (37)

The latter follows from the fact that $f_{h}(0,0)=0$ and therefore $f(\xi)\in
T_{S}(\xi)$. MH: when generalizing to general 2d sectors this will differ

Although $K_{f}\cap T_{S}\neq\Pi$ here, we can still exploit the results in
the preceding section to establish existence of solutions here.

######  Theorem 9.

An AC solutions exists locally for each initial state $\xi_{0}\in S$, i.e. on
$[0,T]$ for some small positive $T$.

###### Proof.

MH: sketch for now Apply Krasovskii regularisation to $\dot{\xi}=F(\xi)$ and
observe that $K_{F}(\xi)\cap T_{S}(\xi)\neq\emptyset$ for all $\xi\in S$.
Hence, the local viability assumption is satisfied and we can establish for
any initial state the existence of an AC solution $\xi:[0,T]\rightarrow S$ to
$\dot{x}\in K_{F}(\xi)$. Clearly, due to the necessity of the viability
condition, we obtain that the solution satisfies a.e.

$\dot{\xi}\in K_{F}(\xi)\cap T_{S}(\xi)$

. Interestingly, for all times $t\in[0,T]$ for which $\xi(t)\not\in K\cap-K$,
we can use the results in the proceeding section that lead for these $t$ to

$K_{F}(\xi(t))\cap T_{S}(\xi(t))=\Pi_{S,{\cal E}}(\xi(t),f(\xi(t)))$ (38)

. In fact, for each such $t$ there is due to closedness of $K\cap-K$ and $\xi$
being continuous, some $\epsilon_{t}>0$ such that this must hold on
$[t,t+\epsilon_{t}]$ Hence, we only have to consider the times where
$\xi(t)\in K\cap-K$, i.e. $(e(t),u(t))=(0,0)$. We will differentiate two
subcases: (i) $\dot{e}(t)\neq 0$ and (ii) $\dot{e}(t)=0$.

maybe consider only times where $\xi$ differentiable

Subcase (i): without loss of generality, let us assume $\dot{e}(t)>0$. Note
that $\dot{e}=g_{n}(x,u)$ which [some properties of $g_{n}$ so that from AC
$x,u$ we can conclude continuous $\dot{e}$]. Hence, we have that there is an
$\epsilon_{t}$ and an $\eta>0$ such that $\dot{e}(\theta)\geq\eta$ for
$\theta\in[t,t+\varepsilon_{t}]$. Hence, we have that $e(t)=0$, but
$e(\theta)\geq\eta(\theta-t)>0$ for $\theta\in[t,t+\epsilon_{t}]$. Hence, the
point where $e$ is zero is an isolated point (measure zero set), and for
$\theta\in(t,t+\epsilon_{t}]$ $\xi(t)\in K\setminus-K$, for which (38) holds
(with $t=\theta$).

Subcase (ii) $\dot{e}(t)=0$. Hence, we have that
$\dot{e}(t)=g_{n}(x(t),u(t))=0$ and $\dot{u}(t)=h(0,0)=0$. Note that
$(\dot{e}(t),\dot{u}(t))\in T_{\bar{S}}(0,0)$. The Krasovskii regularization
eq:kras, recalled here for convenience,

$\dot{x}\in\cap_{\delta>0}\overline{\textup{con}}F(B(x,\delta))=:K_{F}(x)$
(39)

$F(\xi)=(g_{1}(\xi),..g_{n-1}(\xi),\Pi_{[01]^{\top},\bar{S}}((e,u),(g_{n},h))$

Can we express this krasovskii regularization as we did before with polyhedral
sets??? that would be helpful, I guess we can do it!!!

So $K_{F}(\xi)=(g_{1}(\xi),..g_{n-1}(\xi))\times\textup{con}\\{P_{T_{J},{\cal
E}}((g_{n},h))\mid J...)\\}$ where $T_{J}$ are appropriate tangent cones of
points nearby $(e,u)=0$ and $(g_{n},h)=0$. Hence,
$K_{F}(\xi)=\\{(g_{1}(\xi),..g_{n-1}(\xi),0,0)\\}=\Pi_{S,{\cal
E}}(\xi,f(\xi))$.

Hence, in this case we have that for almost all $t\in[0,T]$

$\dot{\xi}(t)=\Pi_{S,{\cal E}}(\xi(t),f(\xi(t)))$

there establishing that $\xi$ is indeed an AC solution to the original pPDS. ∎

MH: let us find a good name for partially projected dynamical systems – is
partial projection a good term? as opposed to oblique projection - used by
hauswirth, although oblique projections as they are classically defined are
possibly closer to partial projection ;-)

######  Corollary 10.

under lipschitz like bounds, we can prove global existence, i.e. on
$[0,\infty)$ solutions

###### Proof.

concatenate, using lipschitz like bounds to avoid finite escape times. ∎

######  Corollary 11.

case of continuous inputs

###### Proof.

embed $t$ in it $\dot{t}=1$ and then for continuous input ∎

######  Corollary 12.

piecewise continuous inputs, make global results

###### Proof.

use previous corollary and concatenate, indicating that at continuities of $w$
you still are in the set $S$! so you can continue with a local solution, and
as long as finite escape times are avoided we should be fine ∎

##  VII Toulouse 20230208

###  VII-A Introduction

###  VII-B Preliminaries on oblique projected dynamical systems

maybe first normal PDS and then e/o PDS

###  VII-C Projection-based controllers

Consider the general nonlinear single-input single-output (SISO) plant given
by

$\displaystyle\dot{x}$ $\displaystyle=$ $\displaystyle g(x,u)$ (40a)
$\displaystyle e$ $\displaystyle=$ $\displaystyle r(x)$ (40b)

with state $x\in\ree^{n}$, control input $u\in\ree$ and output $e\in\ree$.

We will control this by a projection-based controller for which the
unprojected dynamics are given by

$\displaystyle\dot{z}$ $\displaystyle=$ $\displaystyle h(z,e)$ (41a)
$\displaystyle u$ $\displaystyle=$ $\displaystyle z_{1}$ (41b)

with state $z\in\ree^{m}$, controller output $u\in\ree$ and controller input
$e\in\ree$. Note that the controller output equation $u=z_{1}$ can be realizes
under very mild assumptions by a coordinate transformation in case $u=s(z)$
for a function $s:\ree^{m}\rightarrow\ree$. to be added mild condition or
reference! We assume that all the maps $g$, $r$ and $h$ are assumed to be
continuous. Moreover, $r$ is assumed to be continuously differentiable. check
actual needs after proofs!

The projection will take place only on controller states $z$ and the dynamics,
as we cannot change the plant states and dynamics as they adhere to physical
laws, resulting in an oblique or partial projection operation with the goal to
keep the input-output pair $(e,u)$ in a sectorset

$S=\left\\{(e,u)\in\mathbb{R}^{2}\mid(u-k_{1}e)(u-k_{2}e)\leq 0\right\\},$
(42)

where $k_{1},k_{1}\in\ree$ with $k_{1}<k_{2}$. We can write $S$ also as the
union of two polyhedral cones

$S=K\cup-K$ (43)

with

$K=\left\\{(e,u)\in S\mid e\geq 0\right\\}=\left\\{(e,u)\in\ree^{2}\mid u\geq
k_{1}e\text{ and }u\leq k_{2}e\right\\}.$

To write the closed-loop system description we introduce the state
$\xi=(x,z)\in\ree^{n+m}$, the constraint set $S$ on the level of the states
$\xi$ as

${\cal S}=\left\\{\xi=(x,z)\in\mathbb{R}^{n+m}\mid e=r(x),\ u=z_{1},\ (e,u)\in
S\right\\},$ (44)

and the projection subspace as

${\cal E}=\textup{Im}\begin{bmatrix}O_{n}\\\ I_{m}\end{bmatrix}.$ (45)

Note that we can also write ${\mathcal{S}}={\mathcal{K}}\cup-{\mathcal{K}}$
with ${\mathcal{K}}:=\left\\{\xi=(x,z)\in\mathbb{R}^{n+m}\mid(e,u)\in
K\right\\}$.

The closed-loop dynamics can now be written as

$\dot{\xi}=\Pi_{{\cal S},{\cal E}}(\xi,f(\xi)):=F(\xi)$ (46)

with $f$ denoting the unprojected closed-loop dynamics given by

$f(\xi)=(g(x,z_{1}),h(z,r(x))).$ (47)

[for simplicity without external input $w$, but we can aim for embedding time
$\dot{t}=1$ at end again. ]

The oblique projection operator is defined here as maybe better to put this in
preliminaries of paper, where we introduce shortly oblique PDS

$\Pi_{{\cal S},{\cal E}}(x,f)=\operatorname{argmin}_{v\in T_{\cal
S}(x),f-v\in{\cal E}}\|f-v\|$ (48)

This is a mathematical framework that fits and we can show that this leads to
the description (4) [with $f=e$] in closed loop with the plant dynamics. See
[DeeSha_AUT21a] [using a PWL modelling formalism].

###  VII-D Main results

######  Theorem 13.

For each initial state $\xi_{0}\in S$ there exists an AC solutions locally,
i.e., there is a $T>0$ such that $\xi:[0,T]\rightarrow\ree^{n+m}$ with
$\xi(0)=\xi_{0}$ is a solution to (46).

In the proof of Theorem 13, we use the the Krasovskii regularisation of (46)
given by

$\dot{x}\in\cap_{\delta>0}\overline{\textup{con}}F(B(x,\delta))=:K_{F}(x)$
(49)

Above, $\overline{\textup{con}}(M)$ denotes the closed convex hull of the set
$M$, in other words, the smallest closed convex set containing $M$. Hauswirth
had the $\overline{\textup{con}}$ outside of the intersection, might be
easier, but if we go to explicit expressions of Krasovskii regularisations
anyhow, then it does not matter

I think we can generally write under continuity of $f$:

$K_{F}(\xi)=\overline{\textup{con}}P_{\limsup_{\bar{\xi}\rightarrow\xi}T_{{\cal
S}}(\bar{\xi}),{\cal E}}(f(\xi))$ (50) $K_{F}(\xi)=\overline{\rm
con}\limsup_{\bar{\xi}\to\xi}P_{T_{S}(\bar{\xi}),{\mathcal{E}}}(f(\xi))$

or the alike

######  Proposition 14.

The Krasovskii regularization of $F$ at $\xi\in{\cal S}$ is equal to to be
completed:

$K_{F}(x)=\textup{con}\\{P_{T_{J},{\cal E}}(f(x))\mid J\subset J(x)\\}$ (51)

###### Proof.

note that $f$ continuous. ∎

###  VII-E Proof of Theorem 13

MH: sketch for now The Krasovskii regularisation $K_{F}$ is outer
semicontinuous and takes non-empty convex closed setvalues. Moreover, observe
that $K_{F}(\xi)\cap T_{S}(\xi)\neq\emptyset$ for all $\xi\in S$ as
$\Pi_{{\cal S},{\cal E}}(\xi,f(\xi))$ is contained in the intersection. Hence,
the local viability assumption [Aubin, Goebel, sanfelice, teel book] is
satisfied and we can establish for any initial state the existence of an AC
solution $\xi:[0,T]\rightarrow S$ to $\dot{\xi}\in K_{F}(\xi)$ for some $T>0$.
Clearly, due to the necessity of the viability condition, we obtain that the
solution satisfies for almost all times $t\in[0,T]$ that

$\dot{\xi}\in K_{F}(\xi)\cap T_{\cal S}(\xi).$

Interestingly, for all times $t\in[0,T]$ for which $\xi(t)\not\in K\cap-K$, we
can use the results in the proceeding section as

$\Pi_{{\cal S},{\cal E}}(\xi,f(\xi))=\Pi_{{\cal K},{\cal E}}(\xi,f(\xi))$ (52)

in a neighbourhood of $\xi\in{\mathcal{K}}\setminus-{\mathcal{K}}$ (and
similarly for $\xi\in-{\mathcal{K}}\setminus{\mathcal{K}}$). Hence, for almost
all $t$ with $\xi(t)\not\in K\cap-K$ (and due to continuity of $\xi$ and
closedness of $K\cap-K$, there is $\epsilon_{t}$ such that for
$\tau\in[t,t+\epsilon_{t})$, we have $\xi(\tau)\not\in K\cap-K$)

$K_{F}(\xi(t))\cap T_{S}(\xi(t))=\Pi_{S,{\cal E}}(\xi(t),f(\xi(t))).$ (53)

there is a small issue here as the results in previous section are such that
${\cal K}+{\cal E}=\ree^{n}$, which is not the case here … so requires a small
adaptation, but not a serious one. Hence, we only have to consider the times
where $\xi(t)\in K\cap-K$, i.e. $(e(t),u(t))=(0,0)$. We will consider two
cases:

  1. (i)

$\dot{e}(t)\neq 0$, and

  2. (ii)

$\dot{e}(t)=0$

maybe consider only times where $\xi$ differentiable

Case (i): without loss of generality, let us assume $\dot{e}(t)>0$ (when
$\dot{e}(t)<0$ the arguments are similar). Note that
$\dot{e}=\frac{dr}{dx}(x)g(x,z_{1})$, which is a continuous function of time
along solution $\xi$ (as $r$ continuously differentiable and $\xi$
continuous). Hence, there is an $\epsilon_{t}$ and an $\eta>0$ such that
$\dot{e}(\tau)\geq\eta$ for $\tau\in[t,t+\varepsilon_{t}]$. Hence, we have
that $e(t)=0$, but $e(\tau)\geq\eta(\tau-t)>0$ for
$\tau\in[t,t+\epsilon_{t}]$. Hence, the time $t$ where $e$ is zero is an
isolated point (measure zero set) in this case, and for
$\tau\in[t,t+\epsilon_{t}]$ $\xi(t)\in{\mathcal{K}}\setminus-{\mathcal{K}}$,
for which (53) holds (with $t=\tau$).

Case (ii) $\dot{e}(t)=\frac{dr}{dx}(x(t))g(x(t),z_{1}(t))=0$. Note that
$(\dot{e}(t),\dot{u}(t))\in T_{\bar{S}}(0,0)$ [to check: almost everywhere?].

Useful identity: $\xi=(x,z)$ $v=(v_{x},v_{z})$

$(v_{x},v_{z})\in
T_{{\mathcal{S}}}(\xi)\Leftrightarrow(\frac{dr}{dx}(x)v_{x},v_{z,1})\in
T_{S}(r(x),z_{1}).$ (54)

[obvious? proof?]

######  Lemma 15.

The Krasovskii regularization can be written as

$K_{F}(\xi)=\\{g(x,z_{1})\\}\times Z_{1}(\xi)\times\\{\bar{h}(\xi)\\}$ (55)

for some osc mapping $Z_{1}:{\mathcal{S}}\rightarrow\ree$ and
$\bar{h}(\xi):=(h_{2}(z,r(x)),\ldots,h_{m}(z,r(x)))$

###### Proof.

To prove this result, observe that

$\displaystyle\Pi_{{\cal S},{\cal E}}(\xi,f(\xi))=$
$\displaystyle=\operatorname{argmin}_{v\in T_{\cal S}(\xi),f(\xi)-v\in{\cal
E}}\|f(\xi)-v\|$ $\displaystyle=\operatorname{argmin}_{v=(v_{x},v_{z})\in
T_{\cal S}(\xi),v_{x}=g(x,z_{1})}\|h(z,r(x))-v_{z}\|$
$\displaystyle=\operatorname{argmin}_{v=(g(x,z_{1}),v_{z})\in T_{\cal
S}(\xi)}\|h(z,r(x))-v_{z}\|$
$\displaystyle\stackrel{{\scriptstyle\eqref{eq:tangentconelowdim}}}{{=}}\operatorname{argmin}_{v=(g(x,z_{1}),v_{z}),(\frac{dr}{dx}(x)g(x,z_{1}),v_{z,1})\in
T_{S}(r(x),z_{1})}\|h(z,r(x))-v_{z}\|$
$\displaystyle=\operatorname{argmin}_{v=(g(x,z_{1}),v_{z,1},\bar{h}(z,r)),(\frac{dr}{dx}(x)g(x,z_{1}),v_{z,1})\in
T_{S}(r(x),z_{1})}\|h_{1}(z,r(x))-v_{z,1}\|.$

Hence, this shows that

$\Pi_{{\cal S},{\cal E}}(\xi,f(\xi))=(g(x,z_{1}),v_{z,1}^{*},\bar{h}(z,r))$

with

$v_{z,1}^{*}=\operatorname{argmin}_{v_{z,1},(\frac{dr}{dx}(x)g(x,z_{1}),v_{z,1})\in
T_{S}(r(x),z_{1})}\|h_{1}(z,r(x))-v_{z,1}\|.$

From this the result (hopefully ;-)) follows ∎

so lemma above is needed to be able to transfer all reasoning to the 2d
sectorset $S$ instead of the $n+m$d set ${\mathcal{S}}$

We will use this lemma for case (ii), where $(r(x),z_{1})=(0,0)$ and thus
$T_{S}(r(x),z_{1})=S$. Moreover
$\dot{e}(t)=\frac{dr}{dx}(x(t))g(x(t),z_{1}(t))=0$. It follows from the lemma
that

$K_{F}(\xi)=\textup{con}\\{f(\xi),(g(x,z_{1}),0,\bar{h}(z,r))\\}$ (56)

We either have $f(\xi)=(g(x,z_{1}),0,\bar{h}(z,r))$ in which case $K_{F}(\xi)$
is a singleton set and thus must be equal to $\\{\Pi_{{\cal S},{\cal
E}}(\xi,f(\xi))\\}$ (as this one is guaranteed to lie in $K_{F}(\xi)$), or
$f(\xi)\neq(g(x,z_{1}),0,\bar{h}(z,r))$. In the latter case $f(\xi)\not\in
T_{\cal S}(\xi)$ and since $T_{\cal S}(\xi)$ is a cone also $\alpha
f(\xi)\not\in T_{\cal S}(\xi)$ for $\alpha>0$, it follows that

$K_{F}(\xi)\cap T_{\cal
S}(\xi)=\\{(g(x,z_{1}),0,\bar{h}(z,r))\\}=\\{\Pi_{{\cal S},{\cal
E}}(\xi,f(\xi))\\}$ (57)

Hence, in this case we have that for almost all $t\in[0,T]$

$\dot{\xi}(t)=\Pi_{S,{\cal E}}(\xi(t),f(\xi(t)))$

there establishing that $\xi$ is indeed an AC solution to the original pPDS.

Figure 1: Due to the absence of convexity/prox regularity, we do not have that
$K_{F}(\xi)\cap T_{\mathcal{S}}(\xi)=\\{\Pi_{{\mathcal{S}},{\cal
E}}(\xi,f(\xi))\\}$. This is due to the tangent cone lacking a lower
semicontinuity property (in the set ${\mathcal{K}}\cap-{\mathcal{K}}$.)

###  VII-F Corollaries

######  Corollary 16.

under lipschitz like bounds, we can prove global existence, i.e. on
$[0,\infty)$ solutions

###### Proof.

concatenate, using lipschitz like bounds to avoid finite escape times. ∎

######  Corollary 17.

case of continuous inputs

###### Proof.

embed $t$ in it $\dot{t}=1$ and then for continuous input ∎

######  Corollary 18.

piecewise continuous inputs, make global results

###### Proof.

use previous corollary and concatenate, indicating that at continuities of $w$
you still are in the set $S$! so you can continue with a local solution, and
as long as finite escape times are avoided we should be fine ∎

##  VIII Toulouse 20230206

###  VIII-A Extended PDS: Different facets of minimum norm formulation

The objective of this section is to write several analogues of extended PDS
vector field which are similar to some known modeling formalism in the
literature.

For given functions $h_{i}:\mathbb{R}^{n}\to\mathbb{R}$, $i=1,\dots,m$, we
take $S$ to be of the following form:

$S:=\\{x\in\mathbb{R}^{n}\,|\,h_{i}(x)\geq 0\\}.$

The functions $h_{i}$, $i=1,\dots,m$ are smooth and that $\nabla h_{i}$ does
not vanish in a neighborhood of $\\{x\in\mathbb{R}^{n}\,|\,h_{i}(x)=0\\}$. For
all $x\in\mathbb{R}^{n}$, we define the set of active constraints at $x$ by

$J(x)=\\{i\in\\{1,\dots,m\\};h_{i}(x)\leq 0\\}$

The tangent cone to $S$ at $x$ is given by

$T_{S}(x)=\\{v\in R^{n}\,|\,\langle\nabla h_{i}(x),v\rangle\geq 0,i\in
J(x)\\}$

and for $x\in S$, the normal cone is defined as

$N_{S}(x)=\left\\{\sum_{i\in J(x)}\lambda_{i}\nabla h_{i}(x),\lambda_{i}\leq
0\right\\}$

For a vector $f\in\mathbb{R}^{n}$, and a matrix $E$ with ${\mathcal{E}}={\rm
im}\,E$, we consider

$\Pi_{S,E}(x,f)=\operatorname{argmin}_{v\in
T_{S}(x),v-f\in{\mathcal{E}}}\|v-f\|$

We will now rewrite $\Pi_{S,E}(x,f)$ in terms of complementarity relations
involving $T_{S}(x)$ and $N_{S}(x)$.

Figure 2: Orthogonality between $v^{*}=\Pi_{S,E}(x,f)$ and $\eta^{*}$, where
$\eta^{*}$ satisfies $v^{*}-f=E\eta^{*}$.

KintTnotPi Need to show that optimality (of minimum norm problem) implies
cone-complementarity relations.

MAURICE: some stuff below, not relevant now. Just kept it, we might see use of
it - copy paste then

##  IX Preliminaries and classical PDS

###  IX-A Preliminary definitions

polyhedral set : given by the intersection of a finite number of closed half-
spaces

Let $G=[g_{1}\ g_{2}\ldots g_{m}]\in\ree^{n\times m}$ and $g_{i}$,
$i=1,2,\ldots,m$ are the columns of $G$. Then $\operatorname{pos}G$ is the
convex cone consisting of all positive combinations of the columns of $G$
given by

$\operatorname{pos}G=\\{\sum_{i=1}^{m}\alpha_{i}g_{i}\mid\alpha_{i}\geq
0i=1,2,\ldots,m\\}$

$T_{K}(x)$ tangent cone [give definition 5.1.1] when $K$ closed convex cone we
have $T_{K}(x)=\operatorname{cl}\bigcup_{t>0}\frac{K-x}{t}$ (see Remark 5.2.2.
in [Uru] III). In fact, it holds due to convexity that for $x\in K$

$\frac{K-x}{t_{2}}\subseteq\frac{K-x}{t_{1}}\text{ when }0<t_{1}\leq t_{2}$
(58)

projection on a set $S$

$P_{S}(x)=\operatorname{argmin}_{s\in S}\|s-x\|$ (59)

###  IX-B Project dynamical systems

To introduce the “classical” projected dynamical systems (PDSs), consider a
differential equation given by

$\dot{x}(t)=f(x(t))$ (60)

in which $x(t)\in\ree^{n}$ denotes the state at time $t\in\ree_{\geq 0}$.
There is a restriction on the state $x$ in the sense that $x(t)$ has to remain
inside a set $S\subseteq\ree^{n}$, which in PDS is ensured by redirecting the
vector field at the boundary of $S$. Formally, PDS are given for a continuous
vector field $f:\ree^{n}\rightarrow f$ and set $S\subseteq\ree^{n}$ (with
further additional conditions) by

$\dot{x}=\Pi_{S}(x,f(x))$ (61)

with

$\Pi_{S}(x,v)=\operatorname{argmin}_{w\in T_{S}(x)}\|w-v\|$ (62)

for $x\in S$ and $v\in\ree^{n}$.

An equivalent characterisation of $\Pi_{S}(x,v)$, see [Hiriart-Urr, prop.
5.3.5 ], is

$\Pi_{S}(x,v)=\lim_{\delta\downarrow 0}\frac{P_{S}(x+\delta v)-x}{\delta}$
(63)

when $S$ is a closed convex non-empty set. In fact, in many works on PDS [XXX]
this definition is adopted.

##  X Extended projected dynamical systems

###  X-A Model representation

Let set $S\subseteq\ree^{n}$ be given such that $S$ is a non-empty closed
convex set, on which we impose additional conditions later to obtain a well-
posed system. We are interested in dynamical systems

$\dot{x}(t)=f(x(t))$ (64)

in which the state of the system has to reside inside the set $S$. In
classical PDS, as recalled in Subsection IX-B, the latter is ensured by
“projecting” the vector field on the (tangent cone of the) set $S$, cf. (61)
and (62). This projection is along all possible directions of the state in the
sense that it just takes the vector $\Pi_{S}(x,v)\in T_{S}(x)$ that is
“closest” to $v$ irrespective of the direction $\Pi_{S}(x,v)-v$. So, it is
allowed to alter the complete vector field and thus the velocities of all the
states $x$ in (64). Clearly, if (64) is a closed-loop system in the sense of
an interconnection of a physical plant and a controller (and thus the state
$x$ consists of physical plant states $x_{p}$ and controller states $x_{c}$),
one cannot project in all directions. Indeed, the physical state dynamics
cannot be modified by straightforward projection. It is only possible to
”project” the controller ($x_{c}$-)dynamics and possibly a part of the plant
states $x_{p}$-dynamics in which the control input appears explicitly. For an
example on the HIGS, see Section XXX below. Hence, in contrast to PDS, we only
have limited directions in order to “correct ” the vector field $f(x)$ at the
boundary. To formalize, this we model the “correction / projection” direction
by the image of a matrix $E\in\ree^{n\times n_{E}}$, which is assumed to have
full column rank. Formally, we get

$\dot{x}=\Pi_{S,E}(x,f(x))$ (65)

with

$\Pi_{S,E}(x,v)=\operatorname{argmin}_{w\in T_{S}(x),w-v\in\im E}\|w-v\|$ (66)

for $x\in S$ and $v\in\ree^{n}$.

Hence, the projection $\Pi$ projects the direction $v$ at $x\in S$ onto the
set of admissible velocities (tangent cone) along $\im E$ in such a way that
the correction $w-v$ is minimal in norm. For these systems, we coin the term
extended Projected Dynamical Systems (ePDS), as they include the classical PDS
(61) as a special case by taking $\im E=\ree^{n}$.

###  X-B Well-posed projection operator $\Pi_{S,E}$

Clearly, we have to show that the introduced projection $\Pi_{S,E}(x,v)$ is
well-defined in the sense that it provides a unique outcome for each $x\in S$
and each $v\in\ree^{n}$. As in the case of classical PDS, this requires
conditions on the set $S$ and, in this case, also on $E$. Although we envision
that we can work under more general conditions, for the sake of setting the
scene in this paper and inspired by the application of the HIGS, we focus on
the setting below. Future work includes relaxing these conditions.

######  Assumption 19.

The set $S\subseteq\ree^{n}$ and $E\in\ree^{n\times n_{E}}$ satisfy

  * •

$S=K\cup-K$ in which $K$ is a convex polyhedral cone given by
$K=\operatorname{pos}G+\im H$ in which the matrix $[G\ H]$ is square and has
rank $n$

  * •

$K\cap-K=\im G$

  * •

$E$ full column rank

  * •

$\im E\cap S=\\{0\\}$ and thus $S+\im E=\ree^{n}$

Note that $K$ can also be written as

$\\{x\in\ree^{n}\mid Fx\geq 0\\}$ (67)

for some matrix $F\in\ree^{n_{f}\times n}$ (which we assume to be of full
rank)

We are particularly interested in this setup as it can describe sector
conditions as are used in reset controllers and the HIGS, e.g. describing that
input and output of a controller must have the same sign, and also appear in
circle and Popov criteria for the analysis of Lur’e type of systems, see e.g.
[Khalil, Yakubovic]

To prove the well-posedness of (66) observe first that

$T_{S}(x)=\begin{cases}T_{K}(x),&\text{ when }x\in K\setminus-K\\\
K\cup-K,&\text{ when }x\in K\cap-K\\\ -T_{K}(x),&\text{ when }x\in-K\setminus
K\end{cases}$ (68)

[add expression of $T_{K}(x)=\\{w\in\ree^{n}\mid F_{I(x)}w\geq 0\\}$ where
$I(x)=\\{i\in\\{1,2,\ldots,n_{f}\\}\mid F_{i}x=0\\}$ the set of active
constraints at $x$

Note that we can rewrite (66) as

$\Pi_{S,E}(x,v)=v+E\eta^{*}(x,v)$ (69)

with

$\eta^{*}(x,v)=\operatorname{argmin}_{\eta\in\Lambda(x,v)}\|E\eta\|$ (70)

and

$\Lambda(x,v)=\\{\eta\in\ree^{n_{E}}\mid v+E\eta\in T_{S}(x)\\}$ (71)

######  Lemma 20.

Under Assumption 19, it holds for each $x\in S$ and each $v\in\ree^{n}$ that
$\Lambda(x,v)$ is a non-empty closed polyhedral set.

###### Proof.

Let $x\in S$ and $v\in\ree^{n}$ be given. First note that due to $S+\im
E=\ree^{n}$ it follows that $\Lambda(x,v)$ is non-empty.

Clearly, when $x\not\in K\cap-K$ it follows that $T_{S}(x)$, as given in (68),
is a closed polyhedral cone and then also $\Lambda(x,v)$ is a polyhedral set.
So, let us turn our attention to $x\not\in K\cap-K$ in which $T_{S}(x)=K\cup
K$ and thus $\Lambda(x,v)=\\{\eta\in\ree^{n_{E}}\mid v+E\eta\in K\cup-K\\}$.

Claim: $v+E\eta\in K$ and $v+E\bar{\eta}\in-K$ implies that $\eta=\bar{\eta}$.

Note that the Claim would show that if there is $\eta$ with $v+E\eta\in K$
then $\Lambda(x,v)=\\{\eta\in\ree^{n_{E}}\mid v+E\eta\in K\\}$ (as any $\eta$
in $\\{\eta\in\ree^{n_{E}}\mid v+E\eta\in-K\\}$ would also be contained in
$\\{\eta\in\ree^{n_{E}}\mid v+E\eta\in K\\}$, and similarly if there is
$\bar{\eta}$ with $v+E\bar{\eta}\in-K$ then
$\Lambda(x,v)=\\{\eta\in\ree^{n_{E}}\mid v+E\eta\in-K\\}$. As the sets
$\\{\eta\in\ree^{n_{E}}\mid v+E\eta\in K\\}$ and $\\{\eta\in\ree^{n_{E}}\mid
v+E\eta\in-K\\}$ are both closed polyhedral sets, so is $\Lambda(x,v)$.

To prove the claim, observe that due to $K$ being a convex cone and
$-v-E\bar{\eta}\in K$, we get that

$E(\eta-\bar{\eta})=(v+E\eta)-v-E\bar{\eta}\in K$

Since $\im E\cap S=\\{0\\}$ and $E$ full column rank, this shows that
$\eta=\bar{\eta}$ and the result follows. ∎

Due to the constraint set of (70) being a closed polyhedral set, and the
square of the cost function of (69) is $\eta^{\top}E^{\top}E\eta$ is a
quadratic positive definite function (as $E$ full column rank), a unique
minimizer exists, showing the well-posedness of (69) and thus (66).

######  Remark 21.

In case $x(0)\not\in S$, we can use $x(0+)=P_{S,E}(x(0))$ to reset the state
to a state inside $S$. Note that this reset only occurs at the initial time
and not afterwards, as the state never leaves $S$ for time $t>0$.

##  XI Connecting to alternative PDS representations

As already indicated in Subsection LABEL:sec:pds, there is an equivalence
between the definitions (LABEL:piS) and (63) under certain conditions on the
set $S$. In this section, the objective to establish a similar equivalence for
ePDS.

To do so, let us first introduce

$P_{S,E}(x)=\operatorname{argmin}_{s\in S,s-x\in\im E}\|s-x\|$ (72)

Clearly, we can rewrite (66) as

$P_{S,E}(x)=\operatorname{argmin}_{s\in C_{x}}\|s-x\|\text{ with
}C_{x}:=\\{s\in S\mid s-x\in\im E\\}$ (73)

Note that, although this formulation resembles (LABEL:piS), the set
$C_{x}=S\cap(x+\im E)$ is dependent on $x$, which is not the case in
(LABEL:piS). Observe that $C_{x}$ is a non-empty closed and convex set, as can
be seen from a similar reasoning as in the proof of Lemma 20. In fact, this
yields that $C_{x}$ is equal to $\\{s\in K\mid s-x\in\im E\\}$ or
$\\{s\in-K\mid s-x\in\im E\\}$ due to the following implication:

$\left.\begin{array}[]{c}s-x\in\im E,\ s\in K\\\ \bar{s}-x\in\im E,\
\bar{s}\in K\end{array}\right\\}\text{ imply }s=\bar{s}$

Hence, $P_{S,E}(x)=P_{K,E}(x)$ (if $x\in K+\im E$) or $P_{S,E}(x)=P_{-K,E}(x)$
(if $x\in-K+\im E)$ and thus $P_{S,E}(x)$ gives a unique outcome, see, e.g.,
the reasoning on page 116 of [Uru]. Note that based on Theorem 3.3.1 in [Uru]
and using the previous observation, we have that $P_{S,E}(x)$ is characterised
by the following variational inequalities: $s_{x}$ is equal to $P_{S,E}(x)$ if
and only if

$\langle x-s_{x},s-s_{x}\rangle\leq 0\text{ for all }s\in S\cap(x+\im E)$ (74)

In line with (63) for classical PDS, we consider also

$\tilde{\Pi}_{S,E}(x,v)=\lim_{\delta\downarrow 0}\frac{P_{S,E}(x+\delta
v)-x}{\delta}$ (75)

######  Theorem 22.

Assumption 19, it holds that $\tilde{\Pi}_{S,E}(x,v)=\Pi_{S,E}(x,v)$ for all
$x\in S$ and $v\in\ree^{n}$

###### Proof.

Following the arguments of the proof of Prop. 5.3.5 in [Uru], we can obtain
that $\frac{P_{S,E}(x+\delta v)-x}{\delta}$ is equal to
$P_{\frac{S-x}{\delta},E}(v)$. Indeed, using the variational inequalities
characterisation of projections, we have that (74) for
$s_{\delta}:=P_{S,E}(x+\delta v)$ gives that

$\langle x+\delta v-s_{\delta},s-s_{\delta}\rangle\leq 0\text{ for all }s\in
S,\ s-(x+\delta v)\in\im E$

Hence, straightforward algebraic manipulations give for $\delta>0$ that

$\langle
v-\frac{s_{\delta}-x}{\delta},\frac{s-x}{\delta}-\frac{s_{\delta}-x}{\delta}\rangle\leq
0\text{ for all }s\in S,\ s-(x+\delta v)\in\im E$

and thus

$\langle v-\tilde{s},\tilde{s}-\frac{s_{\delta}-x}{\delta}\rangle\leq 0\text{
for all }\tilde{s}\in\frac{S-x}{\delta},\ \tilde{s}-v\in\im E$

where we took $\tilde{s}=\frac{s-x}{\delta}$. From this we conclude indeed
that $\frac{P_{S,E}(x+\delta
v)-x}{\delta}=\frac{s_{\delta}-x}{\delta}=P_{\frac{S-x}{\delta},E}(v)$

Now we use that $T_{S}(x)=\lim_{\delta\downarrow 0}\frac{S-x}{\delta}$ (and
the monotonicity of $\delta\mapsto\frac{S-x}{\delta}$ as in (58)) together
with the fact that $\frac{S-x}{\delta}$ and the limit $T_{S}(x)$ are (the
union of) convex closed sets, that
$\tilde{\Pi}_{S,E}(x,v)=P_{T_{S}(x),E}(v)=\Pi_{S,E}(x,v)$. ∎