# SoK: Cross-Domain MEV

Conor McMenamin Email<EMAIL_ADDRESS>
This project was funded by the Uniswap Foundation Grants Program Department of
Information and Communication Technologies, Universitat Pompeu Fabra,
Barcelona, Spain

###### Abstract

We examine the current state of the cross-domain world, with particular focus
on the protocols being used/planned for use by multi-domain users. We build on
existing frameworks for analyzing how MEV is extracted, while also adding a
new categorization of intrinsic-extractable value vs. time-extractable value
to describe how MEV is generated for an extractor. Together, this provides us
with a framework with which we compare classes of protocols enabling cross-
domain MEV, and the MEV occurring within these classes.

We analyze each protocol class separately and compare current implementations
to an ideal functionality for each. We primarily focus on analyzing the MEV
mitigations that these protocols provide, both now, and into the future. In
each case, we also outline the technical barriers that current protocol
implementations face. With this methodology, we identify sequencers and order-
flow auctions as cross-domain protocols with the greatest potential to
mitigate MEV, but also as protocols with some of the biggest technical
barriers.

##  1 Introduction

The reality of a cross-domain world is fast approaching, and it may already be
upon us. The latency wars that have occurred in other financial domains [55,
20] will come to cross-domain value extraction. To compete for MEV, there is
an ever-increasing need for high-frequency traders to exist and extract on as
many chains and domains as possible. While this is not necessarily a bad thing
if there is a high degree of competition and decentralization, this game is
typically winner takes all [55], which is definitely a bad thing. If one
player controls access and pricing in the system, this player is free to
charge arbitrarily high rents to use the system. As such, there is a need to
understand this cross-domain world, and proactively prepare for this oncoming
reality.

Previous work in the space has tried to define cross-domain MEV [41] and
hypothesize about how MEV changes in a cross-domain world. Despite the
formalism in that work, it is unclear how to apply the provided formal
definitions to draw qualitative or quantitative conclusions about cross-domain
MEV. We note that [41] is a self-proclaimed work-in-progress, and is likely
intended to be thought-provoking rather than a standard for reasoning about
cross-domain MEV in blockchain protocols.

In this work, we systematize prominent cross-domain protocols, focusing on the
types of MEV that occur within each protocol. We extend our analysis to
protocols which can be used to capture and/or mitigate MEV in a cross-domain
world, include protocols like batch auctions (Section 3.8) and encrypted
mempools (Section 3.9). Although not necessarily cross-domain enabling, we see
these protocols as potential solutions to cross-domain MEV. As such, Section 3
analyzes protocols which either directly or indirectly affect cross-domain
MEV.

To do this, in each of the protocols, we breakdown the MEV occurring there by
how the value is extracted, and where the value originates for the extractor.
These categorizations are described in detail in Sections 2.1 and 2.2
respectively. By using a consistent set of MEV categorizations, we distinguish
the types of MEV/methods of MEV extraction occurring in cross-domain enabling
protocols, and what protections these protocols provide. As many of the most
promising cross-domain protocols are still in the early stages of development,
we also discuss the barriers that these protocols face.

Our analysis identifies the censorship-resistant ideal functionalities of
shared sequencers (Section 3.2), multiple single-domain sequencers (Section
3.3), off-chain requesting for quotes (Section 3.12), and batch settlement
protocols (Section 3.8 as viable solutions to cross-domain MEV, and MEV in
general. All of these protocols are not without technical barriers, as
described in the respective sections, but stand as areas of particular
interest for further research.

##  2 Cross-Domain MEV $\subseteq$ MEV

There are many potential categorizations for, and components of, MEV. We can
consider extractable value based on how the value is extracted (e.g.
$EV_{ordering}$ vs $EV_{signal}$ [48]), where the value originates (Section
2.2), who controls access the extractable value (e.g. Mafia vs. Monarch vs.
Moloch [51]), where the extraction occurs, how the extracted value can be
isolated (e.g. proposer-builder separation (PBS, see Appendix A), order-flow
auctions (see Section 3.4), etc.), who the extracted value can be
redistributed to (users vs. builders vs protocols vs. everyone), to name but a
few. Each of these distinctions has their own benefits and insights for each
type of user within the system, be they protocol designers, searchers, users,
block builders or validators.

In this work, we focus on the initial two areas; how the MEV is extracted and
where the extractable value is originating. Using these categories and
components, we provide insight into the cross-domain MEV landscape, especially
within the protocols that we introduce in Section 3. We consider extractable
value from the extractor’s perspective. We note that extractor value may vary
depending on ability to propose blocks, capital, latency, etc.. We generalize
the notion of extractor to be the extractor with the most value to gain, as
this is typically the extractor that wins the right to perform the extraction
(at least in the PBS model). Furthermore, extractor value vs. user/protocol
value is not necessarily zero-sum, because of the utility that users and
protocols get from performing blockchain actions, making the sum of utilities
positive.

###  2.1 How Value is Extracted

This categorization has already been developed through a series of blog-posts
advocating it [48]. To summarize, MEV is either extracted as a result of
ordering or signal. Extraction due to ordering involves strategically
inserting and sequencing a set of transactions to be executed, where the value
of tokens at the start of the sequence is greater than the value at the end of
the sequence, using on-chain data only. The clearest example of this is
starting the sequence with some token $X$, and ending the sequence with a
greater balance of token $X$, although it can be extended to tokens with the
same peg (although this has its limitations [14, 21]). Ordering extraction was
popularized in the original Flashbots paper [20] (signal extraction was not),
and quantified here [26].

Signal extraction is based on information occurring off-chain which creates a
profitable opportunity to submit a set of transactions on-chain. Signals can
occur in the mempool, on other blockchains, centralized exchanges, in news
headlines, to name but a few. A signal-based extractable value opportunity
occurs when the value to act on some transaction or on-chain state for an
extractor becomes higher as a result of new off-chain information.

With a knowledge of underlying token valuations and/or the existence of
hedging opportunities off-chain, signal opportunities involve the extractor
starting and ending the on-chain extraction sequence in different tokens. It
is with these valuations/off-chain opportunities that the starting token
balance is worth less than the final balance to the extractor.

A common example of signal extraction is loss-versus-rebalancing [38], when
the price of a token swap on a centralized exchange moves sufficiently far
away from an on-chain AMM price, creating a profitable opportunity for an
extractor to realign the AMM with the centralized exchange price. This signal
opportunity is commonly known as CEX-DEX arbitrage.

Both signal and ordering extraction techniques are used in blockchains. The
profit from ordering for extractors, after paying for the right to extract, is
almost 0 due to its risk-free nature (fork attacks aside) and its
deterministic valuation (approximately the same profit for all extractors).
Profit from signal extraction is much more subjective, depending on factors
like risk-appetite, access and costs to trade against centralized exchanges,
signal latency, etc.

###  2.2 Where the Extractable Value Originates

Figure 1: A representation of how the expected extractable value of
transaction/protocol states with the same intrinsic-extractable value at
expiration is increasing in the time until expiration. This is analogous to
the time-value of options in traditional finance [5]. The exact expected
extractable value function depends on the transaction/protocol, and the
optionality that the extractor has.

This categorization has likely been hinted at in other works, although we have
not seen it explicitly used. We propose that the expected extractable value of
a transaction can be be split into mutually exclusive intrinsic-extractable
value and time-extractable value components. This is exactly the same way
options are valued [36]. We note that due to extractor competition, the time
at which positive extractable value opportunities can be acted on typically
corresponds to the time at which the extractable value opportunity must be
acted on. There are exceptions to this, particularly related to monopolization
of inclusion/censorship powers. In what follows, we assume that the first time
at which a state/transaction creates a positive extractable value opportunity
corresponds to time when the state/transaction must be acted upon. We refer to
this time as expiry/expiration.111When censorship is possible, time-
extractable value is bounded below by time-extractable value between now and
the first time at which the state can be acted on. However, expected-
extractable value can decrease between state-update times (e.g. decreasing
variations of the fee escalators introduced here [46]). As such, generalized
statements about the expected extractable value of a state increasing for an
extractor over many state-update times are not possible.

  * •

Intrinsic-extractable value is the expected value to the extractor at the time
when the blockchain state or transaction must be acted on. For a Uniswap
order, this can be approximated by the maximum of the expected value of all of
the front-running and back-running opportunities. For a Uniswap pool, it is
the expected extractable value from moving the price up or down at the instant
when orders are to be included in the blockchain (typically arbitraging the
price in line with all other on- and off-chain liquidity). Intrinsic-
extractable value can be realized when the time to act on the order of
blockchain state is 0 (the order/state must be acted on now).

  * •

Time-extractable value is slightly more complex. With respect to a blockchain
protocol/transaction state on which the extractor can act on, the time-
extractable value can be derived in a similar way to that of an option. The
extractor has the length of time between confirmation times/blocks to decide
whether or not to act on the blockchain state in question. The time-
extractable value to the extractor of this optionality is the sum of all paths
with a positive extractable value at expiration, times the probability of that
path happening.
For user transactions, time-extractable value is similar. For a transaction,
the time value to an extractor is the expected value that can be extracted
from the transaction between seeing the transaction and including the
transaction. This is again analogous to options pricing. 222The intuition
behind why time to expiration has a positive value for the extractor is the
fact that the extractor has no obligation to include transactions at
expiration. Furthermore, the extractor only includes transactions if the
transactions’ value to the extractor becomes positive at expiration. This
creates a positive drift in the state’s value to the extractor, with all
negative paths removed from the sum, thanks to the optionality that the
extractor receives. More information on why increasing the length of time to
expiration increases the expected extractable value for an extractor, the
owner of the option to act on the transaction/state, can be found here [5].

####  2.2.1 Examples of MEV Opportunities Developing with Time

A basic example of the value of time until expiration is the case of the
prospector and the gold mine. A prospector is told by a surveyor that a mine
has a $20\%$ chance to contain gold and be worth $\$1$M, or an $80\%$ chance
to contain no gold and be worth $\$0$. If the prospector is allowed to check
the mine tomorrow and find out if it contains gold, how much would the miner
pay for the chance to buy the mine tomorrow after inspecting it for $\$100$k?
To calculate the value of the option, we can see that there is a $20\%$ chance
that the miner makes $\$900$k (buys the mine), and an $80\%$ chance that the
miner loses nothing, as there is no obligation to buy the mine. As such, this
option is worth

$0.2*\$900\text{k}+0.8*\$0=\$180\text{k}.$ (1)

In fact, as long as the cost to buy the mine is less than the value of the
mine if it contains gold, the option to buy the mine has value. In this case,
the current owner of the mine must charge for the option, or give away value
to the prospector. With respect to blockchain protocols and transactions, this
option is typically given away for fee. Although payoffs in the real-world are
not so clear-cut, the same intuition applies. In the time between an extractor
seeing an mempool transaction/protocol state and acting on it at expiration,
the value of acting on the state changes333potentially millions of mini
gold/no gold discoveries are made every second, in a phenomenon described by a
Wiener process [56]), but the cost for the extractor typically does not.

A popular form of this cost which is actively and consistently being incurred
by AMMs like Uniswap is loss-versus-rebalancing [38].444We note that censoring
a protocol can also cause the value of the protocol to drop, potentially
transferring the value to a competing protocol. Although this is an example of
value being extracted due to time, it is not particularly useful for the
analyses in this work. This effect is also felt by transactions in the
mempool. Extractors and builders typically have no obligation to include
transactions as they are submitted. Waiting the maximum amount of time at each
blockchain update before deciding on whether or not to include a transaction
allows extractors to maximize the expected extractable value from the
transaction. This can be seen in the case where in the seconds before
submitting a bundle/block, a significant event occurs in the real world which
affects the underlying price of a token pair (like finding or not finding gold
in the mine). By waiting until block-building time to decide on whether or not
to include the transaction, the builder can choose to front-run the swap
(profitable at block confirmation time) instead of back-running it (profitable
when the transaction was first seen, but not profitable at block confirmation
time).

####  2.2.2 Using Intrinsic- vs. Time-Extractable Value to Analyze MEV

When examining the MEV occurring in each of the protocols in the proceeding
section, we consider the effects that these protocols have on both intrinsic-
and time-extractable value. Intrinsic-extractable value can be reduced by
things like reduced slippage requirements, batch execution, and/or encrypting
some or all of a transaction’s data. These are all things which restrict the
price, execution capabilities, and/or information with which an extractor can
act on transactions respectively. Users over-paying for a swap, execution
freedom, and informational advantages are all things which allow an extractor
to extract value from user transactions as soon as the transactions are seen;
what we consider to be a transaction’s intrinsic value.

Reducing time-extractable value is simpler in theory: reduce the time
extractors have to act on a state, the time until expiration. As we will see
in the proceeding section, this is fraught with complications. Another way to
reduce the net time-extractable value paid by a user or protocol is charging
for the time-extractable value that protocols and users give up (see Section
3.6), as is done in options trading.

From a recent result [37], the time cost for AMM pools grows proportionally to
the square root of block time, which stands as the expiration time for the
option to act on the AMM pool. This holds true in standard options pricing
too, where the time value of options typically grows proportionally to the
square root of time to expiration. For the purpose of this document, it
suffices to understand that the time-extractable value for a
protocol/transaction grows with respect to time until expiration, as in Figure
1. As the time with which extractors have to act on a protocol or transaction
approaches 0, so too does the time-extractable value, collapsing to the
intrinsic-extractable value.

##  3 Cross-Domain Protocols and MEV

This section introduces a list of protocols at the core of cross-domain
conversation. All of these protocols play some role in cross-domain MEV, with
some protocols explicitly intended to provide some sort of MEV mitigation. In
each of the protocols we discuss both the type of value extraction taking
place (Section 2.1), and where the extractable value typically originates
(Section 2.2). With each protocol we describe the ideal protocol
functionality, current implementations, and open questions that exist towards
implementing the ideal functionality specified. We also describe the state of
MEV in each, including, but not limited to, cross-domain MEV.

In the analysis that follows, we consider the centralization of power at any
point in the transaction/block-building/MEV supply-chain, whether relaying,
building, running an auction, block proposing, as negative for protocols. If
protocols were to centralize power to a truly trusted player/set of players,
this would fine. However, the existence of such players ad infinitum is
unrealistic, and cannot be guaranteed in practice without strict rules and
incentives. These rules and incentives do not exist as a given, and require
careful construction. In-keeping with the core decentralization values of
blockchains, we highlight centralizing effects that protocols have.
Furthermore, in each such protocol, we leave the “Can the protocol protect
against such effects?” as an important open question.

|  | Current Implementations | Ideal Functionality
---|---|---|---
|  | Origin | How Extracted | Origin | How Extracted
Protocol | Sec. | Intrinsic | Time | Ordering | Signal | Intrinsic | Time | Ordering | Signal
Multi-domain Sequencers | 3.2 |  |  |  |  |  |  |  |
Single-domain Sequencers | 3.3 |  |  |  |  |  |  |  |
Order-flow auctions | 3.4 |  |  |  |  |  |  |  |
App-chains | 3.5 |  |  |  |  |  |  |  |
Priority Auction | 3.6 |  |  |  |  |  |  |  |
Slot Auctions | 3.7 |  |  |  |  |  |  |  |
Batch Settlement | 3.8 |  |  |  |  |  |  |  |
Encrypted Mempool | 3.9 |  |  |  |  |  |  |  |
Reduced Block Times | 3.10 |  |  |  |  |  |  |  |

Table 1: A breakdown of expected extractable value from users and LPs within
an AMM protocol, when order submission to the protocol is controlled by the
respective cross-domain protocols. The included protocols are not necessarily
exclusive, and may be composed. The following labels* indicate that the
corresponding MEV component:
/: Is effectively eliminated in the corresponding protocol for users/LPs.
/: Exists in the corresponding best-in-class protocol for users/LPs.
/: Is theoretically eliminated for users/LPs, but current implementations have
seen limited deployment, depend on centralized trust, or significant technical
barriers remain.
*Readers should consult the respective subsections for full details on the extent of these indicative labels, and the assumptions made related to the ideal functionalities.

###  3.1 Framework for Analysis

We consider blockchains as state machines555A state machine consists of set of
variables, and sequence of commands/updates on those variables, producing some
output.. Blockchains can be represented as a directed acyclic graph, with each
block containing state machine updates. The updates in a block $B$ act on the
state achieved by applying all blocks in the directed sub-graph reachable from
$B$. Where multiple competing sub-graphs exist, blockchain protocols have tie-
breaking rules for deciding which sub-graph to choose (most blocks, greatest
height, etc.). Unless otherwise stated, blocks contain a sequence of
transactions (state machine updates) which must be applied in
order666Transactions can also be batch executed as in Section 3.8., with this
sequence chosen by a single elected block proposer for each block777Protocols
can force proposers to out-source sequencing to a dedicated sequencer (see
Sections 3.2 and 3.3), or incentivize proposers to auction the right to
sequence blocks (see Section 3.7 and Appendix A).. Blocks are added to the
blockchain at discrete time intervals.

Where appropriate, our analysis focuses on the MEV occurring in a an AMM,
where order submission is controlled by the respective protocols. We isolate
the MEV being extracted from the users submitting orders for inclusion, and
liquidity providers (LPs) deploying liquidity, in the case of AMM pools. This
intended to simplify comparisons, as the action space is restricted to swap
orders, and the value being extracted is intuitive. A summary of this analysis
is provided in Table 1.

###  3.2 Shared Sequencer

####  3.2.1 Ideal Functionality

A single entity, the shared sequencer, accepts transactions for 2 or more
blockchains, committing to include valid transactions in the order in which
they are sequenced. Commitments can take place at any time (independently to
block production time on the sequenced chains). The validity condition for
transactions can range from fee payment, at a minimum, to ‘the transaction is
executable in the given sequence’.

In an ideal implementation of a shared sequencer, the delay between valid
transaction submission from users, and inclusion in the sequence of
transactions is negligible. Furthermore, all builders are incentivized to obey
the shared sequencer (for this to be true, builders would need to face
economic punishment for disobeying the shared sequencer), and there is some
way for builders to charge and control fees in the shared sequencer. Moreover,
an ideal shared sequencer allows for atomic cross-chain transactions, which
are discussed in Appendix B.

All of this practically removes extractable value opportunities. Intrinsic
extractable value is prevented as no processing of transactions is possible,
removing an extractor’s ability to strategically bundle transactions. Time-
extractable value is effectively eliminated as a result of the negligible
confirmation times for transactions, which means no time for extractors to
wait for more profitable opportunities. As soon as an extractable value
opportunity exists with respect to a protocol, an extractor must take it, or
miss the opportunity.

Shared sequencing is intended to shift at least some of the signal MEV
extraction to ordering MEV extraction (signals that previously existed across
domains would exist on the same domain in the shared sequencer). If sequencing
is first-come first-served, ordering MEV would be removed. As time-extractable
value approaches 0, so too does the profitability of signal MEV.

####  3.2.2 Implementations and Discussion

Prominent shared sequencer implementations include Espresso [23], Astria [4],
and Zenith [59]. Detailed write-ups on Espresso, Astria, and shared sequencers
in general can be found here [10, 9]. All of these implementations are
currently focused on sequencing layer-2s.

####  3.2.3 Open Questions

Unfortunately, shared sequencers are highly complex and depend on strong
assumptions that are unlikely to hold without significant technological and
incentivization advancements. Specifically, there is currently no protocol to
guarantee that proposers and builders on sequenced chains obey the sequencer.
This may be related to the difficulties in provably ensuring availability of
data for all sequenced domains. Without proof that a sequence of transactions
was delivered, punishments for not adhering to the sequence are limited. Data
availability proofs on Celestia [8], a leading data availability protocol,
take upwards of 10 seconds, slower than almost all block production times. In
the PBS paradigm that exists in Ethereum Proof-of-stake, relayers collect
built block information, and are responsible for providing that information to
the proposer and validators when a block is selected for inclusion. If
proposers on sequenced chains must build blocks according to sequencer
restrictions, the entire sequence needs to be available to the proposer before
blocks are built.

Related to centralization of power, the end-game for shared sequencers may be
to also build the blocks for sequenced domains, much in the same way as
proposed in SUAVE [27](discussed in Section 3.4). This potentially shifts all
of the MEV/transaction fee revenue control to the shared-sequencer, which does
not align with the current paradigm of proposers on the individual chains
controlling this revenue. A rational sequencer who can induce even small
sequencing delays can extract all of the intrinsic extractable value from
transactions. With centralization of the sequencer, such attack vectors are
likely possible and undetectable. That said, time-extractable value is likely
to be negligible in any sequencer implementation.

Existing shared sequencer solutions do not require transaction validity
checks, which is problematic for a shared sequencer. If a bundle of cross-
domain transactions are intended to be all-or-nothing (all transactions are
executed, or none), rational/malicious players only need to front-run one of
the domains (on the shared sequencer) to invalidate the bundle. This in turn
may invalidate all proceeding transactions in that domain. For a single domain
where all-or-nothing restrictions are enforced more naturally, this is much
less of an issue, if an issue at all.

There likely needs to be pre-processing or post-processing on shared sequencer
transactions before the block is confirmed in which sequenced transactions are
promised to be executable by one or more watchtowers. Incorrect (in)validation
by a watchtower could then carry a severe penalty. These watchtowers can be
executors or simply state-update checkers. There is a natural requirement
therein to validate these transactions in the order they are sequenced.
Although this is centralizing, a decentralized sequencer who can enforce
punishments on these watchtowers neutralizes the bad effects it has (although
if the validation is done before sequencing, we need competition. As such,
post-processing is preferred, and can be done by even a single player,
although we then need DOS protection).

###  3.3 Multiple Single-Domain Sequencers

####  3.3.1 Ideal Functionality

Each blockchain runs a sequencer, who commits to including valid sequenced
transactions as they are sequenced. As in shared sequencers, this validity
condition can range from fee payment, to executability. The delay between
valid transaction submission from users, and inclusion in the sequence of
transaction is negligible, with builders incentivized to obey the sequencer
(for this to be true, builders would need to face economic punishment for
disobeying the shared sequencer), and there is some way for builders to charge
and control fees in the shared sequencer. Extractable value is eradicated in
this ideal functionality, for the same reasons as the ideal functionality of
the shared sequencer. Single-domain sequencers would not affect the signals
existing outside of the chain being sequenced, although the signal MEV
opportunities would reduce in line with the reduction in time-extractable
value.

####  3.3.2 Implementations and Discussion

A single-domain sequencer is exactly that; a sequencer in charge of a single
blockchain. Arbitrum [3] and zkSync Era [60] are examples of blockchains using
dedicated single-domain sequencers. We have seen that shared-sequencer
implementations are fraught with unsolved issues and considerations. In the
short term, it seems we would be happier to accept the benefits of sequencing
in a single domain at the expense of this added functionality.

Furthermore, for a given transaction to be split across multiple domains, with
single-domain sequencers, we can practically achieve much of the same
functionality given the existence of cross-domain agents. Actions/intents
requiring multiple domains can have conditional execution constraints
controlled by the user, or delegated cross-domain executors. As users can
interact with directly with these cross-domain agents, the trust required is
significantly less (pairwise economic/social/legal contracts can be
established).

####  3.3.3 Open Questions

Current implementations of single-domain sequencers are centralized.
Decentralizing the sequencer without degrading performance and usability is an
active area of research. However, single-domain sequencers have significantly
less technical barriers than shared sequencers, with connectivity and the
alignment of incentives simplified in the single domain case. The trade-off
here is the ability to execute cross-domain transactions atomically. As
discussed above, this may not be such a big barrier.

###  3.4 Order-flow Auctions

####  3.4.1 Ideal Functionality

Users are able to auction their orders/order information among a set of
searchers. The proceeds of these auctions are returned to the user. An ideal
cross-domain order-flow auction eliminates both intrinsic and time-extractable
value for the user, assuming the order-flow auctions are indeed revenue
maximizing. The revenue from such an auction should equal to total extractable
value for the extractor.

Protocol-level value extraction (against the LPs) still remains. Multi-domain
order flow auctions (like the proposed SUAVE protocol) will shift some of the
signal MEV to ordering MEV in a similar fashion to shared sequencers.

####  3.4.2 Implementations and Discussion

The heavyweight contender in this domain is the Flashbots-proposed SUAVE
(Single Unified Auction for Value Expression) chain [27]. Here, users can post
transactions/intents to be executed across potentially many domains
participating in the SUAVE domain. The general idea is for searchers
(executors in the SUAVE notation) to compete via an auction to execute these
intents, with some or all of the proceeds from this competition capturable by
the user. The competition among searchers in SUAVE is intended to be based on
partially encrypted transactions, with hints provided by users revealing
enough information to searchers to optimally bundle transactions, while
retaining some form of pre-execution privacy, and the value that comes with it
(extractable value that is not given to the searchers).

Searchers bundle transactions for inclusion in the SUAVE chain. The SUAVE
chain logic then selects and orders the bundles maximizing the (extractable)
value of the set of bundles according to the encoded valuation functions of
the chain. These sets of bundles represent blocks to be added to participating
blockchains. With enough participation, SUAVE is intended to act as a one-
stop-shop for transaction submission across many blockchains.

Uniswap have announced UniswapX [2], an order-flow auction (Dutch Auction) to
assist with routing in the proposed Uniswap V4 [1]. Due to the potential for
thousands of liquidity pools for the same token pair in Uniswap V4, some off-
chain routing is likely needed to ensure users receive the best possible
pricing. Emulating a Dutch Auction among searchers to fill user orders should
help in this regard. However, the UniswapX proposal also mentions the ability
for auction winners to use their own private inventory/other on-chain
liquidity, which might have negative implications for on-chain LP providers. A
legacy order-flow auction protocol that appears to have shutdown in recent
months is Rook [50].

####  3.4.3 Open Questions

SUAVE depends on a centralized auctioneer [28] (Flashbots or an additional
trusted third party). The Flashbots hope is that this auctioneer can
eventually be replicated by some form of trusted execution environment, and
eventually a transparent decentralized system. The path to such an end-goal
remains unsolved and a contentious point. Trust in a trusted auctioneer
appears to be widely accepted, at least in the case of Flashbots as the
trusted auctioneer, evidenced by the amount of blocks produced by MEV-boost
[29], which also depends on trusting Flashbots as an auctioneer.

The removal of extractable value through revenue-maximizing off-chain auctions
depends on many factors, including searcher competition and auctioneer trust,
both of which need further investigation. Without atomicity guarantees, most
searchers in these domains must charge fees to off-set the cost and
probability of not executing intended orders on dependent domains.
Specifically, if a searcher agrees to sell some amount of tokens to a user in
the order-flow auction with the intention of/assuming that they will be buying
those tokens back on Ethereum, the searcher must increase their fee
proportionally to the cost of not executing the swap on Ethereum. Sequencers
would practically eliminate this cost. With such costs, searcher competition
likely diminishes, reintroducing extractable value opportunities for dominant
searchers.

The logic around maximizing extractable value through re-ordering bundles is
intended to be run by trusted execution environments [39], such as trusted
off-chain hardware or cloud infrastructure. Centralizing power off-chain to
trusted execution environments requires significant further study, with
guarantees/limitations of such solutions clearly stated for the community.
Another concern for order-flow auctions is the ability to auction bundles of
orders in a revenue-maximizing way for users generating the orders. The
concern here is the creation of incentives to spam the auction and fill
bundles with fake orders, paying less revenue to the affected users. This is
discussed in detail here [34].

With respect to UniswapX, questions remain about the effect that improved user
execution has for on-chain LPs. Given a particular Uniswap V4 pool is trading
with a user, this pool must be offering a better price than any other on- or
off-chain liquidity source, which also means the LP’s price is likely mis-
priced. To this end, some defragmentation of liquidity (restrict the execution
of orders to Uniswap pools only) might be required to remove this potentially
toxic element of efficient routing through off-chain auctions.

###  3.5 App-Chains

####  3.5.1 Ideal Functionality

A single instance of the protocol is implemented on a dedicated blockchain,
with users required to bridge their assets to and from the chain. App-chains
likely eliminate the intrinsic extractable value of orders, but their effect
on time-extractable value is not clear. Like previous protocols merging
users/order-flow from multiple domains, app-chains will also shift some of the
MEV from signal to ordering, although signal opportunities likely still remain
in the discrete block time, single proposer model. Based on the thesis of
Osmosis [43], ordering extractable will be fully redistributable to both users
and LPs.

####  3.5.2 Implementations and Discussion

Related to shared sequencers is the idea of app-chains. An app-chain is
exactly that, a separate blockchain/side-chain on which all app-specific
actions are carried out. For a protocol such as Uniswap with fragmented
liquidity, isolated/unused domains are exposed to increased time-extractable
value, in addition to the reduced fees from not being actively used. An app-
chain removes this inter-protocol staleness. An app-chain remains exposed to
$t$-staleness where $t$ is the time between confirmations on the chain. For a
protocol like Uniswap, the hope would be the app-chain would adapt to the core
needs of the protocol, and favour short confirmation times to address the
costs associated with time-extractable value. App-chains also have the
potential to reduce intrinsic-extractable value existing in the 1 app per
chain paradigm by defragmenting liquidity, and the reduced slippage/higher
indicative price accuracy this would bring.

Examples of app-chains include Osmosis [43], dydx [22], and ThorChain [54]. A
detailed discussion on the value proposition of app-chains is provided here
[32].

An interesting feature of Osmosis, one of the emerging app-chains in the DEX
space, is their prospective ProtoRev feature [45]. ProtoRev effectively tries
to realign all on-chain liquidity sources after each transaction, back-running
every transaction with an attempted arbitrage of on-chain liquidity pools
against each other. The surplus from this process (there can only be a surplus
under current conditions) is shared among the DAO controlling the Osmosis
chain. Arbitrary post-processing is not considered plausible in general
purpose L1 chains, due to the added computational cost. That being said,
ProtoRev demonstrates the power that app-chains have given an ability to
tailor consensus and execution specifically to the needs of the dedicated
application.

####  3.5.3 Open Questions

The main problem that app-chains face is the need to bridge assets to and from
the app-chain (which may have prohibitive costs, including additional order
staleness, depending on the bridge), and worse user experience as a result.
App-chains on their own do not necessarily affect the timing issues that exist
on normal blockchains. Therefore, it is important to understand the timing,
trust and security specifications of any proposed app-chain when considering
its use. It has been demonstrated that app-chains can potentially provide
meaningful ordering protections where multi-purpose blockchains cannot [12].
This is in line with the promises of Osmosis. The extent to which these
protections can be provided for each application stands as an open question.

###  3.6 Priority Auction to Interact with Application First

####  3.6.1 Ideal Functionality

The ability to interact with the protocol is auctioned off, with proceeds paid
to the protocol. This eliminates extractable value against LPs in expectancy,
with the proceeds of both LP- and user-extractable value paid to the protocol
itself. By explicitly capturing some of the MEV proceeds through the auction,
net profitability is reduced, however, the gross MEV from ordering and signal
techniques should remain largely unchanged.

####  3.6.2 Implementations and Discussion

A proposed implementation of this is MEV-capturing AMMs (McAMMs) [31], which
proposes the protocol itself auctioning off the right to send the first swap
to an Mc-AMM ahead of time, with reduced protocol performance if the
transaction is not included. The auction proceeds should reflect the expected
extractable value to be extracted from the pool within the block, both from
the intrinsic extractable value of user orders, and the time-extractable value
of both orders and the pool pricing function between blocks. In this regard,
McAMMs have many similarities to slot auctions, although here the proceeds are
more clearly exploited by the protocols themselves.

Diamond [35], a variation of a priority auction which shifts this auction to
the builders, has both been described as a prospective Uniswap V4 pool [53],
following the release of Uniswap V4 specifications [1]. Builders must repay
some percentage of the net tokens removed in the pool within the block. The
final pool price in a block is claimed to correspond to the builders best
guess of the true underlying price of the pool, with this net amount of tokens
removed corresponding to the MEV as a result of time-value extraction.
Comparing these two solutions, Diamond does not restrict access to any
individual ahead of block production time, removing any possibility of denial-
of-service attacks either on, or by, the player with the exclusive right to
execute the first swap.

####  3.6.3 Open Questions

Both of these solutions effectively require the builder/searchers to arbitrage
the pool against this true price, and then provide liquidity to the user
orders. In these solutions, the pool reserves play a limited role beyond
providing users with an estimate of how their orders will be executed. On the
other hand, the end game in PBS also appears to be for builders to implicitly
provide liquidity to user orders, taking on inventory risk in order to extract
all available value from a block. Importantly, McAMMs and Diamond return the
value to where it is being extracted, as opposed to the block proposers in
PBS, so has clear advantages over existing solutions in this regard.

Priority auctions likely incentivize private order-flow. By ensuring a
significant share of order-flow, an extractor can better price priority
auctions, allowing such an extractor to minimally increment the second best
auction bid, guaranteeing priority. This creates a cycle, attracting more
private flow. This in turn allows the extractor to provide private order-flow
extracting arbitrarily high rents as the only player in town with access to
the protocols offering priority. Such negative centralizing externalities need
to be properly addressed.

Additionally, in the case where auctions must be carried out by a trusted
auctioneer, such as in McAMMs, it is not clear if the incentives of the
auctioneer are aligned with that of protocol. A clear specification and
analysis of the role of the auctioneer is required.

###  3.7 Slot Auctions

####  3.7.1 Ideal Functionality

Block proposers auction off the right to propose a block ahead of time. For a
player winning multiple slot auctions across several chains, this gives the
ability to simultaneously propose a block on these chains. This unlocks the
ability to provide users with atomic cross-chain transactions (Appendix B).
Atomic order-inclusion confirmation has the potential to eliminate time-
extractable value if these confirmations are provided immediately, with
sequence guaranteed. Retaining the ability to sequence transactions provides
these slot auction winners with some time-extractable value (although less
than if the transactions were not confirmed to be included). By simultaneously
executing cross-chain transactions, signal MEV is reduced, while ordering MEV
opportunities stand to increase, due to the increase in possible orderings for
proposers/searchers controlling multiple simultaneous blocks.

####  3.7.2 Implementations and Discussion

Related to shared sequencing is the idea of slot auctions, where block
proposers auction off the right to build the block ahead of time. This idea
was discussed in a recent blog-post [33], with the Interchain Scheduler [52]
an initial proposed solution.

####  3.7.3 Open Questions

The exact implementation of such a system is yet to be released. As with any
solution requiring builders to control and coordinate themselves across
multiple domains, slot auctions centralize the builder set across domains.
This is likely exacerbated with slot auctions due to the need to place bids
ahead of time. Making the right to build in a particular slot tradeable via a
slot market may mitigate this effect, allowing builders to adjust to
information closer to proposal time. How this could be done is beyond the
scope of this work.

Incentivizing multi-chain searchers to win simultaneous slots across multiple
chains creates a single point of failure shared by each of the chains.
Requiring searchers to pay for blocks before receiving the transactions
creates unstudied incentives related to private order-flow. Players with more
private order-flow are better positioned to predict the value of a slot for
each chain and able to pay the minimum amount to guarantee control of those
slots. This centralizing vector requires further research.

###  3.8 Batch Auctions/Batch Settlement

####  3.8.1 Ideal Functionality

Orders are executed at a common price. In an ideal functionality with no
censorship, extractable value opportunities are effectively eliminated.

####  3.8.2 Implementations and Discussion

The most common example of such a batch settlement protocol is CowSwap [18].
In CowSwap, orders are exposed to time-extractable value due to the previously
discussed power of inclusion that solvers in the protocol have when deciding
which orders to execute in a batch. It can be argued that intrinsic value is
reduced, especially when some trust exists in the solvers, as orders are less
likely to be maximally extracted against in a batch. Penumbra [44] also
proposes batch settlement, with orders there encrypted, mitigating extractable
value. In Penumbra, trust is placed in the threshold-decryption committee to
not decrypt information prior to inclusion in the batch.

####  3.8.3 Open Questions

Transaction censorship in batch auctions diminishes the benefits of such
protocols, exposing users orders to both intrinsic- and time-extractable
value. Furthermore, searchers and/or block producers in charge of order
inclusion can estimate the imbalance of an auction, either from unencrypted
order information or from the implicit information leakage through identity-
reveal, timing, etc.. These players can then use their ’last-look’
capabilities to enter an auction if the implicit settlement price of a batch
is profitable. This is time-extractable value, albeit diluted amongst orders
in a batch. Such value can be seen as a combination of both ordering and
signal MEV, although we consider it more of a signal opportunity. How to
reduce trust/censorship capabilities of searcher/solvers in current batch
protocols is an important open question.

###  3.9 Encrypted Mempool

####  3.9.1 Ideal Functionality

Orders are committed to the blockchain before order information is revealed to
the block producer. Orders are decrypted and executed at some point in the
future. User-level extractable value is removed when no information about a
user’s transaction can be learned before it is executed. The effect on LP MEV
is unclear. Although we would expect protocols to adjust to an encrypted
mempool paradigm, in isolation (without batch execution for example) LPs
remain exposed to value-extraction attacks, such as losses-versus-rebalancing.

####  3.9.2 Implementations and Discussion

Secret Network [40] and Penumbra [44] are examples of protocols using
encrypted mempools. For detailed discussions on the effects of encrypted
mempools on MEV see [49, 11].

####  3.9.3 Open Questions

As mentioned previously, trust is currently placed in decryption committees
not to decrypt transactions before committing transactions to the chain.
Solutions without a committee [6] are promising, although significant work is
still required to make these techniques feasible at scale.

When generic state updates are encrypted, their executability and
computational requirements are not necessarily known. This leads solutions
like Penumbra and Osmosis to restrict the action space of transactions.
Implementing encrypted mempools outside of application-specific chains like
the ones mentioned remains an area requiring further research.

###  3.10 Reduced Block times

####  3.10.1 Ideal Functionality

The consensus protocol is changed to allow for quicker block times, reducing
both time- and signal-extractable value for LPs. In an ideal functionality
without censorship, this also holds true for users. Intrinsic- and ordering-
extractable value opportunities still remain for any delay between block
times, which we assume to be the case.

####  3.10.2 Implementations and Discussion

There is a nuanced, yet crucial difference between a sequencer that can
provide fast confirmations, and increasing the throughput of a system through
faster block times. The former is typically intended to provide
cryptographic/economic guarantees between sequencer and the
proposer/validators that when a block is added, the block will contain the
transactions included in the sequencer. It is only the block contents that are
applied to the state machine, so all of the security still rests on the
proposer/validator.

Faster block-confirmation times are being seen across many L2 chains, where
the security lost to increased speed/centralization can be off-set be the
security and oversight of a slower L1 blockchain.

####  3.10.3 Open Questions

Consensus layer changes must be carefully considered. Reduced block times come
at the expense of security, with faster blocks increasing centralization, and
incentivizing attacks such as forking the blockchain [7]. Faster confirmation
at the layer-1 level has also lead to significant liveness issues [15, 16].

###  3.11 Bridges

####  3.11.1 Ideal Functionality

Allows for the passing of messages between two blockchains. Messages on both
the sending and receiving chain are subject to the constraints of the
respective blockchains. Bridges are exposed to vulnerability attacks, which we
deem to be intrinsic value to extractors, extracted through ordering attacks.

####  3.11.2 Implementations and Discussion

Bridges alone are not necessarily responsible for MEV related to decentralized
exchange of tokens. They do however play an important role in cross-domain
MEV. Bridges facilitate swaps that otherwise would not be possible, allowing
users to transfer value on one chain to another. A basic example of this is
locking tokens in a smart contract (bridge contract) on the source chain. This
information is passed to the destination chain through a corresponding bridge
contract, which creates/unlocks tokens tied to the tokens locked up on the
source chain. Eventually, if users on the destination chain want to unlock the
tokens in the source-chain bridge contract, the tokens must be
locked/destroyed on the destination chain. When this happens, and is
communicated back to the source chain, the source chain tokens can be
unlocked. Implementations of bridges can be found in all Ethereum-native
layer-2 protocols (Arbitrum, Optimism, zkSync, etc. [17]), with cross-chain
bridges summarized here [19].

####  3.11.3 Open Questions

An important result related to bridges is the fact that all messages (the
proof-of-lock/-unlock in the toy example) between separate blockchains must be
sent by a trusted third party/set of parties [58]. Done naively, this is
highly centralizing. These bridge contracts/third parties are the key-holders
for the locked tokens on the source chain. By compromising the key-
holders/vulnerabilities in how the key-holders interact with the bridge,
extractors can drain bridge contracts. This creates a vector for MEV that does
not exist elsewhere, which effects almost all of the proceeding protocols that
use off-chain data. Examples of bridge attacks can be found here [13].

###  3.12 Off-Chain User-Liquidity Provider Interactions (RFQs)

####  3.12.1 Ideal Functionality

The user requests a price/market in a particular set of tokens from a set of
liquidity providers. The liquidity providers respond with the required prices,
to which the user can (or must) respond, typically within a certain time. In
this ideal functionality, extractable value is removed.

####  3.12.2 Implementations and Discussion

Most importantly, this protocol is not applicable for automated MMs, requiring
active MMs who respond in real time to users. As a result, we relegate RFQs to
the end of the section and exclude RFQs from Table 1, which tries to isolate
protocol effects on automated MMs. Despite this critical difference, requests
for quotes (RFQs) allow liquidity providers to eliminate intrinsic extractable
value, and account for time-extractable value by charging a dynamic fee.
Implementations include ODOS [42], Hashflow [30], Hop Exchange [25], and WOOFi
[57]. Correctly implemented RFQs remove time-extractable value for the user by
allowing users to instantly trade with quoted markets. Existing solutions
side-step this with uncompetitive quotes/high fees, or by giving MMs the final
say on whether to include a user transaction or not.

####  3.12.3 Open Questions

Can RFQ-style solutions be implemented on a blockchains to allow for near-
instantaneous trades after quotes are provided? Sequencers, dedicated RFQ-
chains and/or SUAVE may help in this regard.

##  4 Conclusion

We provide a summary of cross-domain protocols, not only as a guide to
understanding cross-domain MEV, but also as a resource for understanding the
cross-domain world in general. Some of the most promising protocols for
enabling a cross-domain world, such as shared sequencers and generalized
order-flow auctions, are also the protocols with the most open questions. This
highlights that the cross-domain world of today is still under construction,
and will likely be very different to that of the future. This future will be
shaped by the answers to the open questions that we have posed among others.
As value is created in the cross-domain world, so to will value extraction
opportunities. That being said, many of the ideal protocol functionalities of
Section 3 effectively eliminate MEV, or allow for its redistribution.
Censorship and centralization are common barriers for many of the protocols
that we discuss. These stand as areas that warrant particular focus from the
research community.

##  5 Acknowledgements

Thanks to Ankit Chiplunkar, Alana Levin, Alex Obadia, Bruno Mazorra, Federico
Landini, Quintus Kilbourn, and Xin Wan for their insightful reviews and
comments.

## References

  * [1] Hayden Adams.  Our vision for uniswap v4.  https://blog.uniswap.org/uniswap-v4.  Accessed: 19/07/2023.
  * [2] Hayden Adams, Noah Zinsmeister, Mark Toda, Emily Williams, Xin Wan, Matteo Leibowitz, Will Pote, Allen Lin, Eric Zhong, Zhiyuan Yang, Riley Campbell, Alex Karys, and Dan Robinson.  Uniswapx.  https://uniswap.org/whitepaper-uniswapx.pdf.  Accessed: 21/07/2023.
  * [3] Arbitrum.  The sequencer and censorship resistance.  https://developer.arbitrum.io/sequencer.  Accessed: 25/07/2023.
  * [4] Astria.  https://www.astria.org/.  Accessed: 19/07/2023.
  * [5] Caroline Banton.  The importance of time value in options trading.  https://www.investopedia.com/articles/optioninvestor/02/021302.asp.  Accessed: 23/07/2023.
  * [6] Jeffrey Burdges and Luca De Feo.  Delay encryption.  In Anne Canteaut and François-Xavier Standaert, editors, Advances in Cryptology – EUROCRYPT 2021, pages 302–326, Cham, 2021. Springer International Publishing.
  * [7] Vitalik Buterin.  Toward a 12-second block time.  https://blog.ethereum.org/2014/07/11/toward-a-12-second-block-time.  Accessed: 19/07/2023.
  * [8] Celestia.  Data availability layer faqs.  https://docs.celestia.org/concepts/how-celestia-works/data-availability-layer/.  Accessed: 19/07/2023.
  * [9] Jon Charbonneau.  Rollups aren’t real: Decentralizing, shared sequencers, x-chain atomicity, suave, & mev.  https://joncharbonneau.substack.com/p/rollups-arent-real.  Accessed: 19/07/2023.
  * [10] Jon Charbonneau.  Suave, anoma, shared sequencers, & super builders.  https://dba.mirror.xyz/NTg5FSq1o_YiL_KJrKBOsOkyeiNUPobvZUrLBGceagg.  Accessed: 19/07/2023.
  * [11] Jon Charbonneau.  Threshold encrypted mempools: Limitations and considerations.  https://joncharbonneau.substack.com/p/encrypted-mempools.  Accessed: 21/07/2023.
  * [12] Tarun Chitra.  Towards a theory of maximal extractable value ii: Uncertainty, 2023.
  * [13] CoinDesk.  Blockchain bridges keep getting attacked. here’s how to prevent it.  https://www.coindesk.com/layer2/2022/10/14/blockchain-bridges-keep-getting-attacked-heres-how-to-prevent-it/.  Accessed: 19/07/2023.
  * [14] CoinDesk.  The Fall of Terra: A Timeline of the Meteoric Rise and Crash of UST and LUNA.  https://www.coindesk.com/learn/the-fall-of-terra-a-timeline-of-the-meteoric-rise-and-crash-of-ust-and-luna/.  Accessed: 21/07/2023.
  * [15] CoinTelegraph.  Solana attributes major outage to denial-of-service attack targeting dex offering.  https://cointelegraph.com/news/solana-attributes-major-outage-to-denial-of-service-attack-targeting-dex-offering.  Accessed: 19/07/2023.
  * [16] CoinTelegraph.  Solana outage triggers ballistic reaction from the crypto community.  Solanaoutagetriggersballisticreactionfromthecryptocommunity.  Accessed: 19/07/2023.
  * [17] Ethereum Community.  Summary of layer-2 protocols.  https://ethereum.org/en/layer-2/.  Accessed: 19/07/2023.
  * [18] CoW Protocol.  https://docs.cow.fi/.  Accessed: 11/10/2022.
  * [19] CrossChainBridges.  Cross-chain bridge & interoperability protocols.  https://crosschainbridges.blog/blockchain-bridges-interoperability-protocols/.  Accessed: 19/07/2023.
  * [20] Philip Daian, Steven Goldfeder, Tyler Kell, Yunqi Li, Xueyuan Zhao, Iddo Bentov, Lorenz Breidenbach, and Ari Juels.  Flash Boys 2.0: Frontrunning, Transaction Reordering, and Consensus Instability in Decentralized Exchanges.  https://arxiv.org/abs/1904.05234, 2019.  Accessed: 19/01/2022.
  * [21] Decrypt.  USDC Stablecoin Falls to 87 Cents After Circle Discloses Exposure to Silicon Valley Bank.  https://decrypt.co/123211/usdc-stablecoin-depegs-90-cents-circle-exposure-silicon-valley-bank.  Accessed: 21/07/2023.
  * [22] dydx.  https://dydx.exchange/.  Accessed: 19/07/2023.
  * [23] Espresso.  https://www.espressosys.com/sequencer.  Accessed: 19/07/2023.
  * [24] Ethereum.  Proposer-builder separation.  https://ethereum.org/nl/roadmap/pbs/.  Accessed: 20/07/2023.
  * [25] HOP Exchange.  https://hop.exchange/.  Accessed: 19/07/2023.
  * [26] Flashbots.  https://explore.flashbots.net.  Accessed: 21/07/2023.
  * [27] Flashbots.  The future of mev is suave.  https://writings.flashbots.net/the-future-of-mev-is-suave/.  Accessed: 19/07/2023.
  * [28] Flashbots.  The future of mev is suave.  https://writings.flashbots.net/mevm-suave-centauri-and-beyond.  Accessed: 19/07/2023.
  * [29] Flashbots.  Percentage of blocks proposed by mev-boost relay.  https://www.theblock.co/data/on-chain-metrics/ethereum/percentage-of-blocks-proposed-by-mev-boost-relay.  Accessed: 19/07/2023.
  * [30] Hashflow.  https://docs.hashflow.com/hashflow/product/bridgeless-cross-chain-swaps.  Accessed: 19/07/2023.
  * [31] Josojo.  MEV capturing AMMs.  https://ethresear.ch/t/mev-capturing-amm-mcamm/13336, 2022.  Accessed: 10/02/2023.
  * [32] Alana Levin.  App-specific rollups: A trade-off between connectivity and control.  https://variant.fund/articles/app-specific-rollups-trade-off-connectivity-control/.  Accessed: 19/07/2023.
  * [33] Alana Levin.  Cross-chain order flow auctions.  https://variant.fund/articles/cross-chain-order-flow-auctions/.  Accessed: 21/06/2023.
  * [34] Bruno Mazorra and Nicolas Della Penna.  Towards optimal prior-free permissionless rebate mechanisms, with applications to automated market makers & combinatorial orderflow auctions.  https://arxiv.org/abs/2306.17024, 2023.  Accessed: 26/07/2023.
  * [35] Conor McMenamin, Vanesa Daza, and Bruno Mazorra.  Diamonds are Forever, Loss-Versus-Rebalancing is Not.  https://arxiv.org/abs/2210.10601, 2022.  Accessed: 04/01/2023.
  * [36] Merrill.  Options pricing.  https://www.merrilledge.com/investment-products/options/options-pricing-valuation.  Accessed: 19/07/2023.
  * [37] Jason Milionis, Ciamac C. Moallemi, and Tim Roughgarden.  Automated market making and arbitrage profits in the presence of fees.  https://arxiv.org/abs/2305.14604.  Accessed: 17/07/2023.
  * [38] Jason Milionis, Ciamac C. Moallemi, Tim Roughgarden, and Anthony Lee Zhang.  Quantifying Loss in Automated Market Makers.  In Fan Zhang and Patrick McCorry, editors, Proceedings of the 2022 ACM CCS Workshop on Decentralized Finance and Security. ACM, 2022.
  * [39] Andrew Miller.  Suave smart contract programming model: Tee-based smart contracts for block building.  https://www.youtube.com/watch?v=DhsDFKnHPa0.  Accessed: 19/07/2023.
  * [40] Secret Network.  https://scrt.network/.  Accessed: 19/07/2023.
  * [41] Alexandre Obadia, Alejo Salles, Lakshman Sankar, Tarun Chitra, Vaibhav Chellani, and Philip Daian.  Unity is strength: A formalization of cross-domain maximal extractable value, 2021.
  * [42] ODOS.  https://www.odos.xyz/arbitrage.  Accessed: 19/07/2023.
  * [43] Osmosis.  https://osmosis.zone/.  Accessed: 19/07/2023.
  * [44] Penumbra.  https://penumbra.zone/.  Accessed: 23/01/2023.
  * [45] Skip Protocol.  Protorev.  https://osmosis.zone/blog/osmosis-protorev-by-skip-protocol-on-chain-app-directed-arbitrage.  Accessed: 19/07/2023.
  * [46] Quintus.  Order flow, auctions and centralisation ii: order flow auctions.  https://collective.flashbots.net/t/order-flow-auctions-and-centralisation-ii-order-flow-auctions/284.  Accessed: 21/07/2023.
  * [47] Quintus.  Pbs (and alternatives) wiki.  https://collective.flashbots.net/t/pbs-and-alternatives-wiki/2034.  Accessed: 26/07/2023.
  * [48] Frontier Research.  A tale of two arbitrages.  https://frontier.tech/a-tale-of-two-arbitrages.  Accessed: 19/07/2023.
  * [49] Antoine Rondelet and Quintus Kilbourn.  Threshold encrypted mempools: Limitations and considerations.  https://arxiv.org/abs/2307.10878, 2023.  Accessed: 21/07/2023.
  * [50] Rook.  https://www.rook.fi/about.  Accessed: 20/07/2023.
  * [51] sxysun.  Short note on Monarch, Moloch, Mafia extractable value.  https://hackmd.io/@sxysun/short-note-ext.  Accessed: 19/07/2023.
  * [52] Informal Systems.  Interchain scheduler design update.  https://informal.systems/2022/10/24/interchain-scheduler-design-update.  Accessed: 19/07/2023.
  * [53] The_CTra1n.  Lvr-minimization in uniswap v4.  https://ethresear.ch/t/lvr-minimization-in-uniswap-v4/15900.  Accessed: 19/07/2023.
  * [54] ThorChain.  https://thorchain.org/.  Accessed: 19/07/2023.
  * [55] Wikipedia.  Flash boys.  https://en.wikipedia.org/wiki/Flash_Boys.  Accessed: 19/07/2023.
  * [56] Wikipedia.  Wiener process.  https://en.wikipedia.org/wiki/Wiener_process.  Accessed: 24/07/2023.
  * [57] WOOFi.  https://fi.woo.org/.  Accessed: 19/07/2023.
  * [58] Alexei Zamyatin, Mustafa Al-Bassam, Dionysis Zindros, Eleftherios Kokoris-Kogias, Pedro Moreno-Sanchez, Aggelos Kiayias, and William J. Knottenbelt.  Sok: Communication across distributed ledgers.  Cryptology ePrint Archive, Paper 2019/1128, 2019.  https://eprint.iacr.org/2019/1128.
  * [59] Zenith.  https://meka.tech/zenith.  Accessed: 19/07/2023.
  * [60] zkSync Era.  Sequencing transactions.  https://era.zksync.io/docs/reference/concepts/hyperscaling.html#sovereignty.  Accessed: 25/07/2023.

##  Appendix A Proposer-Builder Separation

Proposer-Builder separation (PBS) [24], popularized by Flashbots’ MEV-Boost
[29], involves the proposers of blocks, as elected by the blockchain consensus
mechanism, auctioning off the right to build the block to a set of competing
builders. PBS is simultaneously intended to remove the barrier to entry for
block proposing, and censorship of transactions.

In Ethereum proof-of-work, proposing and building were done by the same
individual. Block-building has always been computationally intensive due to
the need to store the entire blockchain’s state, and simulate state over many
combinations and permutations of prospective transactions. More recently, it
has become extremely competitive due to the need to maximize extractable value
from included transactions. By allowing proposers to out-source this work
while still profiting from block-building due to the auction process, everyone
can theoretically be a proposer. Furthermore, although block building is
competitive and centralizing, block proposing remains relatively
decentralized. By involving two unique individuals in the block proposing
mechanism, the hope is that one of these individuals (typically the proposer)
will prevent transaction censorship. From the proposers perspective, this can
be done by adding constraints on blocks that the proposer will accept from
builders. A comprehensive list of resources on PBS can be found here [47].

##  Appendix B Atomic Cross-chain Transactions

We briefly outline atomic cross-chain transactions, a prospective advancement
with the potential to add significant utility (and MEV opportunities) to the
cross-domain MEV space. An atomic cross-chain transaction allows for the
simultaneous execution of transactions across multiple blockchains. More than
this, atomic cross-chain transactions may even allow for the moving of assets
among multiple blockchains. This would allow for instantaneously bridging
assets and using them on destination chains, atomic cross-chain swaps, and
potentially even flash-loans. Such functionalities would unlock a world of
use-cases and value for users.

That being said, any atomic cross-chain transaction solution requires each
participating blockchain to trust in a third party external to the blockchain
itself [58]. This is a contentious issue, especially with regard to
decentralization. To the best of our knowledge, no satisfactory solution to
these strict trust requirements exists. However, the promise and potential of
atomic cross-chain transactions, especially within protocols like shared
sequencers, warrants some consideration. Protocols potentially allowing for
atomic cross-chain transactions are discussed in Section 3.


# On the Calculation of Differential Parametrizations for the Feedforward
Control of an Euler-Bernoulli Beam

Bernd Kolar , Nicole Gehring and Markus Schöberl E-mail<EMAIL_ADDRESS>
###### Abstract

This contribution is concerned with the motion planning for underactuated
Euler-Bernoulli beams. The design of the feedforward control is based on a
differential parametrization of the beam, where all system variables are
expressed in terms of a free time function and its infinitely many
derivatives. In the paper, we derive an advantageous representation of the set
of all formal differential parametrizations of the beam. Based on this
representation, we identify a well-known parametrization, for the first time
without the use of operational calculus. This parametrization is a flat one,
as the corresponding series representations of the system variables converge.
Furthermore, we discuss a formal differential parametrization where the free
time function allows a physical interpretation as the bending moment at the
unactuated boundary. Even though the corresponding series do not converge, a
numerical simulation using a least term summation illustrates the usefulness
of this formal differential parametrization for the motion planning.

Institute of Automatic Control and Control Systems Technology, Johannes Kepler
University Linz, Altenbergerstraße 66, 4040 Linz, Austria

##  1 Introduction

The feedforward control for boundary actuated Euler-Bernoulli beams is usually
concerned with the question of how an input has to be chosen in order to
transition the beam from one steady state to another. The flatness-based
approach, originally introduced for lumped-parameter systems, has proven very
valuable for the motion planning in distributed-parameter systems (e.g. [1],
[2], [3], [4]). It relies on a differential parametrization of the spatially-
dependent system solution in terms of a parametrizing (boundary) output, and,
in the case of the Euler-Bernoulli beam or parabolic systems like the heat
equation, involves infinite series. As these series comprise infinitely many
time derivatives of the parametrizing output, in order to transition the
system between two steady states and have the series converge, the desired
trajectory of the parametrizing output has to be a non-analytic, smooth
function of appropriate Gevrey class (e.g. [5]). If a Gevrey class ensuring
convergence exists, the parametrizing output is called a flat output and the
differential parametrization is a flat one (e.g. [2]). On the other hand, if
no such Gevrey class exists, the differential parametrization is specified by
the prefix formal. However, even in this case, a flatness-based motion
planning may still be possible (e.g. [5], [6]).

The main challenge of the flatness-based approach lies in finding a
differential parametrization. For the heat conduction problem this is either
based on the ansatz of a power series in the spatial variable (e.g. [5]), the
ansatz in [7] which generalizes the Brunovsky decomposition, or some kind of
operational calculus (e.g. [8], [3]). All these ideas could be applied in a
straightforward way to the fully actuated Euler-Bernoulli beam, i.e.
configurations with two boundary inputs at one end. In contrast, in the more
difficult case with only one input, considered here, all differential
parametrizations found in the literature solely rely on operational calculus
(e.g. [9], [3], [10], [11]), to the best of the authors’ knowledge.

In this paper, we demonstrate that a time-domain approach similar to [7] is
also applicable to an Euler-Bernoulli beam with one boundary input. One of our
main results is an advantageous representation of the set of all formal
differential parametrizations of the beam. Based on this representation, it is
easy to identify a distinguished differential parametrization that is a flat
one and well-known from the literature (see e.g. [3]). As no physical
interpretation is known for this flat output, we discuss an interesting formal
differential parametrization with a parametrizing output that allows an
interpretation as the bending moment at the unactuated boundary. Although the
corresponding series do not converge (for any non-analytic function), inspired
by the computations with divergent series in [5], simulation results based on
a least term summation illustrate that this formal differential
parametrization can still be useful for motion planning.

#### Notation

We use the convention that the set of natural numbers $\mathbb{N}$ includes
$0$. Thus, $(a_{k})_{k\in\mathbb{N}}$ denotes the infinite sequence
$(a_{0},a_{1},a_{2},\ldots)$. By $(a_{0},a_{1},\ldots,a_{k})$ we denote a
finite sequence with the last element $a_{k}$ for some fixed $k$.

##  2  A Series Ansatz for Solutions of the Euler-Bernoulli Beam

In this contribution, we consider an Euler-Bernoulli beam with a clamped end
at $z=0$ and a free end at $z=1$, with the bending moment serving as the
control input $u(t)$ at the latter. The deflection $w(z,t)$ satisfies the
(normalized) beam equation

$\partial_{t}^{2}w(z,t)=-\partial_{z}^{4}w(z,t)\,,\quad 0\leq z\leq
1\,,\,t\geq 0$ (1a) and the boundary conditions
$\begin{array}[]{rclcrcl}w(0,t)&=&0\,,&&\partial_{z}^{2}w(1,t)&=&u(t)\\\
\partial_{z}w(0,t)&=&0\,,&&\partial_{z}^{3}w(1,t)&=&0\,.\end{array}$ (1b)

This is a classical beam configuration that has been studied in the context of
flatness-based feedforward control, e.g., in [3] and [12].111The general
constant-coefficient case
$\mu\partial_{t}^{2}w(z,t)=-EI\partial_{z}^{4}w(z,t)\,,\quad 0\leq z\leq
L\,,\,t\geq 0$ with the linear mass density $\mu>0$, the flexural rigidity
$EI>0$, and a spatial domain $[0,L]$ can always be traced back to the
normalized case (1) by transformations of the independent variables $z$ and
$t$.

The objective of the present paper is the construction of formal differential
parametrizations for the Euler-Bernoulli beam (1). We call a representation

$\displaystyle w(z,t)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\alpha_{k}(z)y^{(k)}(t)$ (2a) $\displaystyle
u(t)$ $\displaystyle=$ $\displaystyle\sum_{k=0}^{\infty}\beta_{k}y^{(k)}(t)$
(2b)

of the deflection $w(z,t)$ and the input $u(t)$ by a parametrizing output
$y(t)$ and its time derivatives a formal differential parametrization of the
system (1), if the series (2a) and (2b) formally satisfy the PDE (1a) and the
boundary conditions (1b) for arbitrary smooth functions $y(t)$. It is
important to emphasize the word formal, since we are dealing with formal
solutions222The series (2a) and (2b) are formal solutions if they satisfy (1a)
and (1b) after formally interchanging differentiation and summation, even if
they do not converge. and discuss the problem of finding parametrizations (2)
separately from the question of convergence, see also [6] or [5]. For a
convergence analysis, $y(t)$ has to be restricted to so-called Gevrey
functions with appropriately bounded derivatives, see e.g. [5].

######  Definition 1.

A smooth function $y(t)$ defined on $\mathbb{R}^{+}$ is Gevrey of order
$\gamma$ if there exist constants $M,R>0$ such that

$\sup_{t\in\mathbb{R}^{+}}\left|y^{(m)}(t)\right|\leq
M\tfrac{(m!)^{\gamma}}{R^{m}}\,,\quad\forall m\in\mathbb{N}\,.$ (3)

Gevrey functions of order $\gamma=1$ are analytic. For planning transitions
between equilibria of the system (1), which are characterized by constant
values of $y(t)$, they cannot be used. The reason is that any analytic
function that is constant on an open subset of $\mathbb{R}^{+}$ is constant
everywhere. In contrast, this is no longer true for Gevrey functions $y(t)$ of
order $\gamma>1$. If the convergence of the series (2a) and (2b) can be shown
for a Gevrey class with $\gamma>1$, the parametrizing output $y(t)$ is called
a flat output.

######  Remark 2.

In the context of motion planning, the question arises whether the span over
the set $\\{\alpha_{k}(z)$, $k\in\mathbb{N}\\}$ of functions of a given
parametrization (2a) is dense in the state space, see also [7]. However, this
question is not within the scope of the present paper.

In order to derive conditions on the functions $\alpha_{k}(z)$ and
coefficients $\beta_{k}$, let us plug the ansatz (2) into the PDE (1a) and the
boundary conditions (1b). Since the resulting equations must hold for
arbitrary smooth functions $y(t)$, after formally interchanging
differentiation and summation, the factors of all time derivatives of $y(t)$
have to vanish. Hence

$\displaystyle\alpha_{0}^{\prime\prime\prime\prime}(z)$ $\displaystyle=$
$\displaystyle 0$ $\displaystyle\alpha_{1}^{\prime\prime\prime\prime}(z)$
$\displaystyle=$ $\displaystyle 0$
$\displaystyle\alpha_{k}^{\prime\prime\prime\prime}(z)$ $\displaystyle=$
$\displaystyle-\alpha_{k-2}(z)\,,\quad k\geq 2$ (4a) and
$\left.\begin{array}[]{cclcccl}\alpha_{k}(0)&=&0\,,&&\alpha_{k}^{\prime\prime}(1)&=&\beta_{k}\\\
\alpha_{k}^{\prime}(0)&=&0\,,&&\alpha_{k}^{\prime\prime\prime}(1)&=&0\end{array}\quad\right\\}\quad
k\geq 0\,.$ (4b)

Therein, ′ denotes the differentiation with respect to the spatial variable
$z$. With (4) we have to solve a sequence of boundary value problems in the
independent variable $z$. Obviously, for all $k\geq 2$, the function
$\alpha_{k}(z)$ follows from a fourfold integration of $-\alpha_{k-2}(z)$. The
integration constants are determined by the boundary values (4b).
Consequently, the functions $\alpha_{k}(z)$ of the series representation (2a)
of the deflection $w(z,t)$ are uniquely determined by the coefficients
$(\beta_{0},\beta_{1},\ldots,\beta_{k})$ of the series representation (2b) of
the input $u(t)$. More precisely, the functions $\alpha_{k}(z)$ with even $k$
are determined by the coefficients $(\beta_{0},\beta_{2},\ldots,\beta_{k})$
with even indices, the functions $\alpha_{k}(z)$ with odd $k$ by the
coefficients $(\beta_{1},\beta_{3},\ldots,\beta_{k})$ with odd indices. Hence,
the functions $\alpha_{k}(z)$ with even and odd indices can be calculated
independently. In particular, setting all $\beta_{k}$ with odd $k$ to zero
yields a parametrization of the form

$\displaystyle w(z,t)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\alpha_{2k}(z)y^{(2k)}(t)$ (5a)
$\displaystyle u(t)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\beta_{2k}y^{(2k)}(t)\,,$ (5b)

where only time derivatives of even order occur. In the remainder of the
paper, for simplicity, we restrict ourselves to this special case. The
calculations for the case with odd indices could be performed in an analogous
way.

Integrating $\alpha_{0}^{\prime\prime\prime\prime}(z)=0$ four times and taking
account of (4b) yields the first function

$\alpha_{0}(z)=\tfrac{1}{2}\beta_{0}z^{2}$ (6)

of the series (5a). Note that $\alpha_{0}(z)$ corresponds to an equilibrium
profile of the beam, since all time derivatives of $y(t)$ vanish in steady
state. Continuing by solving
$\alpha_{2}^{\prime\prime\prime\prime}(z)=-\alpha_{0}(z)$ and
$\alpha_{4}^{\prime\prime\prime\prime}(z)=-\alpha_{2}(z)$, it can be shown
that the functions $\alpha_{2k}(z)$ are polynomials of the form

$\alpha_{2k}(z)=\sum_{i=0}^{k}\left(c_{k,i}z^{4i+2}+d_{k,i}z^{4i+3}\right)$
(7)

with $d_{k,k}=0$, where $c_{k,i}$ denotes the coefficients of the even powers
$z^{4i+2}$, and $d_{k,i}$ denotes the coefficients of the odd powers
$z^{4i+3}$.

In principle, by successive integration we could calculate all functions
$\alpha_{2k}(z)$ – up to some arbitrary index $k$ – in terms of the sequence
$(\beta_{0},\beta_{2},\ldots,\beta_{2k})$. However, this is cumbersome and
does not answer the important question how one has to choose the sequence
$(\beta_{2k})_{k\in\mathbb{N}}$ in order for the series (5a) and (5b) to
converge for a sufficiently large class of functions $y(t)$. Therefore, in the
following section, we study the connections between the sequence
$(\beta_{2k})_{k\in\mathbb{N}}$ and the coefficients of the polynomials (7) in
more detail.

##  3 Derivation of Formal Differential Parametrizations

First, we determine how the coefficients of (7) depend on the coefficients of

$\alpha_{2(k-1)}(z)=\sum_{i=0}^{k-1}\left(c_{k-1,i}z^{4i+2}+d_{k-1,i}z^{4i+3}\right)\,.$
(8)

Integrating $-\alpha_{2(k-1)}(z)$ four times and taking account of (4b), a
comparison with (7) shows that the coefficients of (7) can be calculated from
the coefficients of (8) according to

$\displaystyle c_{k,0}$ $\displaystyle=$
$\displaystyle\tfrac{1}{2}\beta_{2k}-\tfrac{1}{2}\sum_{i=0}^{k-1}\left(c_{k-1,i}\tfrac{1}{4i+4}+d_{k-1,i}\tfrac{1}{4i+5}\right)$
(9) $\displaystyle d_{k,0}$ $\displaystyle=$
$\displaystyle\tfrac{1}{6}\sum_{i=0}^{k-1}\left(c_{k-1,i}\tfrac{1}{4i+3}+d_{k-1,i}\tfrac{1}{4i+4}\right)$
(10)

and

$\left.\begin{array}[]{ccc}c_{k,i}&=&-c_{k-1,i-1}\frac{(4i-2)!}{(4i+2)!}\\\
d_{k,i}&=&-d_{k-1,i-1}\frac{(4i-1)!}{(4i+3)!}\end{array}\quad\right\\}\quad
1\leq i\leq k\,.$ (11)

It can be observed that, according to (9) and (10), the coefficients $c_{k,0}$
and $d_{k,0}$ of the powers $z^{2}$ and $z^{3}$ in (7) depend on all
coefficients of (8) and on $\beta_{2k}$. In contrast, the coefficients of the
higher powers of $z$ in (7) depend only on one coefficient of (8) each (cf.
(11)).

Next, it is straightforward to show that the coefficients $c_{k,0}$ and
$d_{k,0}$ of (7) can alternatively be expressed by the sequences
$(c_{0,0},c_{1,0},\ldots,c_{k-1,0})$ and $(d_{0,0},d_{1,0},\ldots,d_{k-1,0})$,
i.e. the coefficients of $z^{2}$ and $z^{3}$ of the polynomials
$\alpha_{0}(z),\ldots,\alpha_{2(k-1)}(z)$, as well as $\beta_{2k}$: By a
repeated application of (11) we immediately get

$\left.\begin{array}[]{ccc}c_{k,i}&=&(-1)^{i}\frac{2!}{(4i+2)!}c_{k-i,0}\\\
d_{k,i}&=&(-1)^{i}\frac{3!}{(4i+3)!}d_{k-i,0}\end{array}\quad\right\\}\quad
0\leq i\leq k\,,$ (12)

and plugging (12) into (9) and (10) results in

$\displaystyle c_{k,0}$
$\displaystyle=\tfrac{1}{2}\beta_{2k}-\tfrac{1}{2}\sum_{i=0}^{k-1}(-1)^{i}\left(c_{k-1-i,0}\tfrac{2(4i+3)}{(4i+4)!}+d_{k-1-i,0}\tfrac{6(4i+4)}{(4i+5)!}\right)$
(13) $\displaystyle d_{k,0}$
$\displaystyle=\tfrac{1}{6}\sum_{i=0}^{k-1}(-1)^{i}\left(c_{k-1-i,0}\tfrac{2}{(4i+3)!}+d_{k-1-i,0}\tfrac{6}{(4i+4)!}\right)\,.$
(14)

Based on (12), (13), and (14), we can now formulate one of the key results of
the paper.

######  Theorem 3.

The functions $\alpha_{2k}(z)$ and coefficients $\beta_{2k}$ of the formal
differential parametrization (5) are uniquely determined by the sequence
$(c_{k,0})_{k\in\mathbb{N}}$.

###### Proof.

If we fix a sequence $(c_{k,0})_{k\in\mathbb{N}}$, (14) allows us to calculate
step by step the corresponding sequence $(d_{k,0})_{k\in\mathbb{N}}$, starting
with $d_{0,0}=0$. Subsequently, by (12), the remaining coefficients of the
functions $\alpha_{2k}(z)$ of (5a) can be determined from the sequences
$(c_{k,0})_{k\in\mathbb{N}}$ and $(d_{k,0})_{k\in\mathbb{N}}$. Finally,
solving (13) for $\beta_{2k}$ yields the coefficients of (5b). ∎

Based on the fact that the sequence $(c_{k,0})_{k\in\mathbb{N}}$ can be chosen
arbitrarily, the following corollary can be formulated.

######  Corollary 4.

There is a one-to-one correspondence between arbitrary sequences
$(c_{k,0})_{k\in\mathbb{N}}$ and formal differential parametrizations (5) of
the Euler-Bernoulli beam.

###### Proof.

By Theorem 3, every sequence $(c_{k,0})_{k\in\mathbb{N}}$ determines a unique
formal differential parametrization (5). Conversely, since the sequence
$(c_{k,0})_{k\in\mathbb{N}}$ consists of the coefficients of $z^{2}$ in the
functions $\alpha_{2k}(z)$ of (5a), every formal differential parametrization
(5) determines a unique sequence $(c_{k,0})_{k\in\mathbb{N}}$. ∎

As already mentioned in Section 2, the functions $\alpha_{2k}(z)$ of (5a) are
uniquely determined by the coefficients
$(\beta_{0},\beta_{2},\ldots,\beta_{2k})$ of (5b). Hence, the formal
differential parametrization (5) is also uniquely determined by the sequence
$(\beta_{2k})_{k\in\mathbb{N}}$. However, a representation of all formal
differential parametrizations of the Euler-Bernoulli beam using the sequence
$(c_{k,0})_{k\in\mathbb{N}}$ as the free design parameter is advantageous.
This becomes apparent when we want to find flat parametrizations, where the
series (5a) and (5b) converge for functions $y(t)$ of appropriate Gevrey
order: Since the input $u(t)$ is the bending moment at the free boundary,
which follows from the deflection $w(z,t)$ as $u(t)=\partial_{z}^{2}w(1,t)$,
the convergence of the series (5a) implies the convergence of the series (5b),
under the assumption that $w(z,t)$ is twice differentiable with respect to $z$
at $z=1$. Thus, it is sufficient to check the convergence of the series
representation (5a) of the deflection $w(z,t)$. In (5a), the elements of the
sequence $(c_{k,0})_{k\in\mathbb{N}}$ appear as coefficients of the
polynomials $\alpha_{2k}(z)$. This facilitates the construction of
differential parametrizations, as will be illustrated by means of the examples
in Section 4.

In the following, we derive explicit expressions for the sequences
$(d_{k,0})_{k\in\mathbb{N}}$ and $(\beta_{2k})_{k\in\mathbb{N}}$ in terms of
the sequence $(c_{k,0})_{k\in\mathbb{N}}$.

######  Lemma 5.

The sequence $(d_{k,0})_{k\in\mathbb{N}}$ is generated by the discrete
convolution

$d_{k,0}=\sum_{i=0}^{k}\eta_{k-i}c_{i,0}$ (15)

of the sequence $(c_{k,0})_{k\in\mathbb{N}}$ with the sequence
$(\eta_{k})_{k\in\mathbb{N}}$, which is defined recursively by $\eta_{0}=0$
and

$\eta_{k}=-\tfrac{(-1)^{k}}{3(4k-1)!}-\sum_{i=1}^{k}\eta_{k-i}\tfrac{(-1)^{i}}{(4i)!}\,,\quad
k\geq 1\,.$ (16)

###### Proof.

First, it should be noted that $d_{k,0}$ could be calculated from (14) by
eliminating successively $d_{k-1,0},d_{k-2,0},\ldots$ with shifted versions of
(14). This shows that $d_{k,0}$ is a linear combination of the coefficients of
the sequence $(c_{0,0},c_{1,0},\ldots,c_{k,0})$, justifying an ansatz of the
form (15). Due to $d_{0,0}=0$, from (15) we immediately get $\eta_{0}=0$. In
order to determine $\eta_{k}$, $k\geq 1$, we plug the ansatz (15) into the
relation (14). After some simplifications this yields the equation

$\sum_{i=0}^{k}\eta_{k-i}c_{i,0}=\tfrac{1}{3}\sum_{n=0}^{k-1}(-1)^{k-1-n}\tfrac{1}{(4(k-n)-1)!}c_{n,0}+\sum_{n=0}^{k-1}(-1)^{k-1-n}\sum_{j=0}^{n}\eta_{n-j}\tfrac{1}{(4(k-n))!}c_{j,0}\,,$
(17)

which must hold for every choice of the sequence
$(c_{0,0},c_{1,0},\ldots,c_{k,0})$. By the special choice $(1,0,\ldots,0)$,
the relation (17) can be simplified to (16). ∎

In a similar way, the sequence $(\beta_{2k})_{k\in\mathbb{N}}$ can be
calculated from the sequence $(c_{k,0})_{k\in\mathbb{N}}$.

######  Lemma 6.

The sequence $(\beta_{2k})_{k\in\mathbb{N}}$ is generated by the discrete
convolution

$\beta_{2k}=\sum_{i=0}^{k}\mu_{k-i}c_{i,0}$ (18)

of the sequence $(c_{k,0})_{k\in\mathbb{N}}$ with the sequence
$(\mu_{k})_{k\in\mathbb{N}}$, which is defined recursively by $\mu_{0}=2$ and

$\mu_{k}=\tfrac{4^{k}}{(4k)!}-\sum_{i=1}^{k}\mu_{k-i}\tfrac{(-1)^{i}}{(4i)!}\,,\quad
k\geq 1\,.$ (19)

###### Proof.

In principle, $\beta_{2k}$ can be calculated by solving (13) for $\beta_{2k}$
and replacing $d_{k-1,0},d_{k-2,0},\ldots$ successively with shifted versions
of (14). This shows that $\beta_{2k}$ is a linear combination of the
coefficients of the sequence $(c_{0,0},c_{1,0},\ldots,c_{k,0})$, justifying an
ansatz of the form (18). A comparison of (6) and (7) shows that
$c_{0,0}=\frac{1}{2}\beta_{0}$, and from (18) with $k=0$ we immediately get
$\mu_{0}=2$. In order to determine $\mu_{k}$, $k\geq 1$, we solve (13) for
$\beta_{2k}$ and insert the ansatz (18), which yields

$\sum_{i=0}^{k}\mu_{k-i}c_{i,0}=2c_{k,0}+\sum_{i=0}^{k-1}(-1)^{i}\left(c_{k-1-i,0}\tfrac{2(4i+3)}{(4i+4)!}+d_{k-1-i,0}\tfrac{6(4i+4)}{(4i+5)!}\right)\,.$
(20)

To get rid of the coefficients $c_{i,0}$ and $d_{i,0}$ in (20), we can use the
fact that the particular sequence $c_{k,0}=\frac{(-1)^{k}}{(4k)!}$ determines
the sequence $d_{k,0}=-\frac{4k(-1)^{k}}{3(4k)!}$. This follows rather easily
from (15) and (16) and will be shown in Section 4.1. After inserting these
particular sequences, (20) can be simplified to (19). ∎

A numerical evaluation of the recursively defined sequences (16) and (19)
strongly suggests, that $\eta_{k}$ and $\mu_{k}$ are defined by

$\eta_{k}=\tfrac{4^{k+1}(1-16^{k})}{6(4k)!}B_{4k}\,,\quad k\geq 0$

and

$\mu_{k}=\tfrac{2}{4^{k}(4k)!}\sum_{i=0}^{2k}(-1)^{i}E_{2i}\tbinom{4k}{2i}\,,\quad
k\geq 0\,,$

where $B_{i}$ denotes the Bernoulli numbers and $E_{i}$ the Euler numbers.
This explicit representation may be advantageous for a convergence analysis of
the parametrizations (5a) and (5b).

##  4 Two Notable Differential Parametrizations

In the previous section, we have shown that the sequence
$(c_{k,0})_{k\in\mathbb{N}}$ can be used as a design parameter for the formal
differential parametrizations (5) of the Euler-Bernoulli beam. Based on the
representations (15), (16) and (18), (19) of the sequences
$(d_{k,0})_{k\in\mathbb{N}}$ and $(\beta_{2k})_{k\in\mathbb{N}}$, we show that
there is a natural choice for the sequence $(c_{k,0})_{k\in\mathbb{N}}$ that
leads to a flat parametrization, which is well-known from the literature (see
e.g. [3]). Subsequently, we discuss a special formal differential
parametrization where the parametrizing output $y(t)$ allows a physical
interpretation.

###  4.1 A Natural Choice

Consider the sum (15). If we split off the term with $i=0$ and substitute (16)
for $\eta_{k}$, we get

$d_{k,0}=-\tfrac{(-1)^{k}}{3(4k-1)!}c_{0,0}+\sum_{i=1}^{k}\eta_{k-i}\left(c_{i,0}-\tfrac{(-1)^{i}}{(4i)!}c_{0,0}\right)\,.$
(21)

With the special choice

$c_{k,0}=\tfrac{(-1)^{k}}{(4k)!}c_{0,0}\,,\quad k\geq 0$ (22)

for the sequence $(c_{k,0})_{k\in\mathbb{N}}$, the sum in (21) vanishes, and
therefore (21) simplifies to

$d_{k,0}=-\tfrac{(-1)^{k}}{3(4k-1)!}c_{0,0}\,,\quad k\geq 1\,.$

Expanding with $4k$ finally yields the relation

$d_{k,0}=-\tfrac{4k(-1)^{k}}{3(4k)!}c_{0,0}\,,\quad k\geq 0\,,$ (23)

which also includes the case $k=0$ with $d_{0,0}=0$.

Analogously, splitting off the term with $i=0$ in the sum (18) and
substituting (19) for $\mu_{k}$ yields

$\beta_{2k}=\tfrac{4^{k}}{(4k)!}c_{0,0}+\sum_{i=1}^{k}\mu_{k-i}\left(c_{i,0}-\tfrac{(-1)^{i}}{(4i)!}c_{0,0}\right)\,.$
(24)

By the choice (22) for the sequence $(c_{k,0})_{k\in\mathbb{N}}$, it is
evident that the sum vanishes again, and (24) simplifies to

$\beta_{2k}=\tfrac{4^{k}}{(4k)!}c_{0,0}\,,\quad k\geq 1\,.$

For $k=0$ we have $\beta_{0}=2c_{0,0}$, and therefore the complete sequence is
given by

$\beta_{2k}=\begin{cases}2c_{0,0}\,,&k=0\\\
\tfrac{4^{k}}{(4k)!}c_{0,0}\,,&k\geq 1\,.\end{cases}$ (25)

Finally, plugging (22) and (23) into (12) results in the complete set of
coefficients

$\displaystyle c_{k,i}$
$\displaystyle=\tfrac{4(-1)^{k}}{(4i+2)!(4(k-i))!}c_{0,0}\,,$ $\displaystyle
0\leq i\leq k\,,\,k\geq 0$ (26) $\displaystyle d_{k,i}$
$\displaystyle=-\tfrac{16(-1)^{k}(k-i)}{(4i+3)!(4(k-i))!}c_{0,0}\,,$
$\displaystyle 0\leq i\leq k\,,\,k\geq 0$ (27)

of the polynomials (7). If we set the scaling factor in (22) to $c_{0,0}=2$,
we get the same differential parametrization

$\displaystyle w(z,t)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}4(-1)^{k}\sum_{i=0}^{k}\left(\tfrac{1}{(4i+2)!(4(k-i))!}z^{4i+2}-\tfrac{4(k-i)}{(4i+3)!(4(k-i))!}z^{4i+3}\right)y^{(2k)}(t)$
(28a) $\displaystyle u(t)$ $\displaystyle=$ $\displaystyle
4y(t)+\sum_{k=1}^{\infty}2\tfrac{4^{k}}{(4k)!}y^{(2k)}(t)$ (28b)

that was derived, e.g., in [3] by means of operational calculus.333In contrast
to our double sum representation (28a), in [3], the parametrization of the
deflection $w(z,t)$ is expressed by a single sum of real and imaginary parts
of powers of complex numbers. Also, the input used in [3] has the opposite
sign as compared to our input $u(t)$. As shown in [3], the series (28a) and
(28b) converge for all trajectories $y(t)$ of Gevrey class $\gamma<2$. Hence,
this parametrizing output $y(t)$ is a flat output.

###  4.2 Formal Differential Parametrization by the Bending Moment at the
Clamped Boundary

In motion planning, one is often interested in a physical interpretation of
the parametrizing output $y(t)$ in terms of a boundary value of the system.
For the flat output $y(t)$ of the differential parametrization (28), no such
physical interpretation is known. In contrast, for the fully-actuated Euler-
Bernoulli beam, with both the bending moment and the shear force at the free
end as inputs, it is not difficult to show that the bending moment and the
shear force at the clamped end form a flat output. For our underactuated
configuration (1), we show that there exists at least one (useful) formal
differential parametrization (5) where the parametrizing output $y(t)$ allows
a physical interpretation.

First, let us restate the formal differential parametrization of the
deflection $w(z,t)$ by plugging (7) into (5a):

$w(z,t)=\sum_{k=0}^{\infty}\sum_{i=0}^{k}\left(c_{k,i}z^{4i+2}+d_{k,i}z^{4i+3}\right)y^{(2k)}(t)\,.$

Evaluating its second spatial derivative at $z=0$ yields the parametrization
of the bending moment

$\partial_{z}^{2}w(0,t)=\sum_{k=0}^{\infty}2c_{k,0}y^{(2k)}(t)$ (29)

at the clamped boundary. If we choose the sequence
$(c_{k,0})_{k\in\mathbb{N}}$ as

$(c_{0,0},c_{1,0},c_{2,0},\ldots)=(\tfrac{1}{2},0,0,\ldots)\,,$ (30)

only the first term of the sum is left, and (29) simplifies to

$y(t)=\partial_{z}^{2}w(0,t)\,.$ (31)

Thus, in this case, the function $y(t)$ is simply the bending moment at the
clamped boundary. By the choice (30) the discrete convolutions (15) and (18)
simplify to $d_{k,0}=\tfrac{1}{2}\eta_{k}\,,\,k\geq 0$ and
$\beta_{2k}=\tfrac{1}{2}\mu_{k}\,,\,k\geq 0$. Then with (12) we get the formal
differential parametrization

$\displaystyle w(z,t)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\left((-1)^{k}\tfrac{1}{(4k+2)!}z^{4k+2}+\sum_{i=0}^{k-1}(-1)^{i}\tfrac{3}{(4i+3)!}\eta_{k-i}z^{4i+3}\right)y^{(2k)}(t)$
(32a) $\displaystyle u(t)$ $\displaystyle=$
$\displaystyle\sum_{k=0}^{\infty}\mu_{k}y^{(2k)}(t)\,.$ (32b)

However, since the sequence $(\mu_{k})_{k\in\mathbb{N}}$ does not go to zero
fast enough, (32b) cannot converge for any non-analytic function $y(t)$ of
Gevrey order $\gamma>1$. Hence, neither does (32a). A numerical evaluation of
the elements $\mu_{k}$ reveals that
$\frac{\mu_{k+1}}{\mu_{k}}\approx\frac{1}{24}$, for $k\geq 2$. Since
$24^{k}<(2k)!$ for large $k$, the bound
$\sup_{t\in\mathbb{R}^{+}}\left|y^{(2k)}(t)\right|$ guaranteed by (3) grows
much faster than the coefficients $\mu_{k}$ converge to zero. Consequently,
the product $\mu_{k}y^{(2k)}(t)$ diverges and the parametrizing output $y(t)$
is not a flat one.

However, a flatness-based transition between two equilibria is still possible
using the divergent series (32b). In [5], simulation results for a heat
conduction problem showed that a least term summation allows feedforward
control even based on divergent series. More precisely, the divergent series
discussed in [5] first converges very fast and then diverges very fast. A
similar effect can be observed for our beam parametrization (32b), for
suitable trajectories $y(t)$. Now, the idea of the least term summation in [5]
is to take into account only the convergent part for each time $t$. Hence,
instead of the series (32b), the feedforward control $u(t)$ is calculated by

$u(t)=\sum_{k=0}^{n_{t}}\mu_{k}y^{(2k)}(t)\,,$ (33)

with $n_{t}$ defined for each $t$ as the smallest integer greater than $1$
that meets

$\left|\mu_{n_{t}+1}y^{(2(n_{t}+1))}(t)\right|>\left|\mu_{n_{t}}y^{(2n_{t})}(t)\right|\,.$

Simulation studies were performed in order to illustrate the usefulness of
this approach. Here, we defined the desired trajectory for $y(t)$ based on a
function

$\Phi_{\sigma}(t)=\begin{cases}0&t\leq 0\\\
\tfrac{\int_{0}^{t}\phi_{\sigma}(\tau)\mathrm{d}\tau}{\int_{0}^{1}\phi_{\sigma}(\tau)\mathrm{d}\tau}&t\in(0,1)\\\
1&t\geq 1\,,\end{cases}$

with $\phi_{\sigma}(t)=\exp\left(-1/((1-t)t)^{\sigma}\right)$, scaled for a
transition in $1\,\mathrm{s}$. This function is Gevrey of order
$1+\tfrac{1}{\sigma}$ for $\sigma>0$. In order to transition the beam between
the equilibria $w(z,0)=0$ and $w(z,T)=\tfrac{1}{2}z^{2}$ in finite time $T$,
based on the definition of the parametrizing output $y(t)$ in (31), the
corresponding initial and final value of $y(t)$ are $y(0)=0$ and $y(T)=1$. By
setting $T=5\,\mathrm{s}$ and $\sigma=1.1$, we use the same reference
trajectory

$y(t)=\Phi_{\sigma}(\tfrac{t}{T})$ (34)

that was used in [12] for the flat output in (28).

######  Remark 7.

In [12], the transition time is $11.5\,\mathrm{ms}$. As our normalized beam
and the one with physical parameters in [12] are related by a scaling of the
spatial variable and a time-scaling with the factor $\tfrac{1}{2.3}10^{3}$,
this corresponds to the transition time $T=5\,\mathrm{s}$ used in our
simulation.

Fig. 1 and Fig. 2 give simulation results for the feedforward control (33)
based on the reference trajectory (34). The distributed beam deflection in
Fig. 1 shows that the desired transition is achieved. In Fig. 2, the
(simulated) bending moment at the clamped boundary, i.e. the parametrizing
output $y(t)$, matches the desired reference trajectory (34), apart from some
small deviations. Thus, using only the formal differential parametrization
(32), we managed to implement the same transition considered in [12] on the
basis of the flat differential parametrization (28). However, in contrast to
the flat output used in [12], our parametrizing output offers a physical
interpretation. This is particularly interesting for applications where the
bending moment must not exceed certain bounds, since we can take account of
these bounds in the design of the desired trajectory $y(t)$ directly.

Nevertheless, it is important to mention that choosing the transition time $T$
too small means that the series (32b) diverges too soon. Consequently, a least
term summation would not make sense any more. In contrast, increasing the
transition time improves the results, in particular with respect to the small
deviations visible in Fig. 2. For instance, with $T=10\,\mathrm{s}$ the
simulated bending moment at the clamped boundary almost perfectly matches the
reference (34). We also expect that evaluating the divergent series (32b) with
the more sophisticated summation methods used in [6] and [13] should further
improve the results.

Figure 1: Deflection $w(z,t)$ of the beam Figure 2: Bending moment $y(t)$ at
the clamped boundary

#### Acknowledgements

This work has been supported by the Austrian Science Fund (FWF) under grant
number P 29964-N32 and the Pro2Future competence center in the framework of
the Austrian COMET-K1 programme under contract no. 854184.

## References

  * [1] M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph, “Controllability and motion planning for linear delay systems with an application to a flexible rod,” in _Proceedings 34th IEEE Conference on Decision and Control (CDC)_ , 1995, pp. 2046–2051.
  * [2] A. Lynch and J. Rudolph, “Flatness-based boundary control of a class of quasilinear parabolic distributed parameter systems,” _International Journal of Control_ , vol. 75, no. 15, pp. 1219–1230, 2002.
  * [3] J. Rudolph, _Flatness Based Control of Distributed Parameter Systems_. Aachen: Shaker Verlag, 2003.
  * [4] T. Knüppel and F. Woittennek, “Control design for quasi-linear hyperbolic systems with an application to the heavy rope,” _IEEE Transactions on Automatic Control_ , vol. 60, no. 1, pp. 5–18, 2015.
  * [5] B. Laroche, P. Martin, and P. Rouchon, “Motion planning for the heat equation,” _International Journal of Robust and Nonlinear Control_ , vol. 10, pp. 629–643, 2000.
  * [6] M. Wagner, T. Meurer, and M. Zeitz, “K-summable power series as a design tool for feedforward control of diffusion-convection-reaction systems,” in _Proceedings 6th IFAC Symposium on Nonlinear Control Systems (NOLCOS)_ , 2004, pp. 147–152.
  * [7] B. Laroche and P. Martin, “Motion planning for a 1-D diffusion equation using a Brunovsky-like decomposition,” in _Proceedings 14th International Symposium on Mathematical Theory of Networks and Systems (MTNS)_ , 2000.
  * [8] P. Martin, R. Murray, and P. Rouchon, “Flat systems,” in _Plenary Lectures and Mini-Courses, Proceedings 4th European Control Conference (ECC)_ , G. Bastin and M. Gevers, Eds., 1997, pp. 211–264.
  * [9] M. Fliess, H. Mounier, P. Rouchon, and J. Rudolph, “Systèmes linéaires sur les opérateurs de Mikusiński et commande d’une poutre flexible,” _ESAIM: Proc._ , vol. 2, pp. 183–193, 1997.
  * [10] J. Rudolph and F. Woittennek, “Motion planning for Euler-Bernoulli beams,” in _Algebraic Methods in Flatness, Signal Processing and State Estimation_ , H. Sira-Ramirez and G. Silva-Navarro, Eds. Innovacion Editorial Lagares Mexico, 2003, ch. 8, pp. 131–148.
  * [11] T. Meurer, J. Schröck, and A. Kugi, “Motion planning for a damped Euler-Bernoulli beam,” in _Proceedings 49th IEEE Conference on Decision and Control (CDC)_ , 2010, pp. 2566–2571.
  * [12] W. Haas and J. Rudolph, “Steering the deflection of a piezoelectric bender,” in _Proceedings 5th European Control Conference_ , 1999, pp. 3238–3243.
  * [13] M. Wagner, T. Meurer, and A. Kugi, “Feedforward control design for the inviscid burger equation using formal power series and summation methods,” in _Proceedings 17th IFAC World Congress_ , 2008, pp. 8743–8748.