Optimality of Zeno Executions in Hybrid Systems William Clark1and Maria Oprea2 Abstract A unique feature of hybrid dynamical systems

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Optimality of Zeno Executions in Hybrid Systems*

William Clark1and Maria Oprea2

Abstract— A unique feature of hybrid dynamical systems

(systems whose evolution is subject to both continuous- and

discrete-time laws) is Zeno trajectories. Usually these trajecto-

ries are avoided as they can cause incorrect numerical results

as the problem becomes ill-conditioned. However, these are

difﬁcult to justiﬁably avoid as determining when and where they

occur is a non-trivial task. It turns out that in optimal control

problems, not only can they not be avoided, but are sometimes

required in synthesizing the solutions. This work explores the

pedagogical example of the bouncing ball to demonstrate the

importance of “Zeno control executions.”

I. INTRODUCTION

A hybrid system is a system whose dynamics are con-

trolled by a mixture of both continuous and discrete transi-

tions. For the purposes of this note, such a dynamical system

will be described via

(˙x=X(x), x ∈M\ S,

x+= ∆(x−), x ∈ S.(1)

The set Swill be referred to as the guard and ∆as the

reset. All of the data will be tacitly assumed to be sufﬁciently

smooth. A unique phenomenon to hybrid systems is that of

Zeno. A trajectory, γ(t), is Zeno if it intersects the guard

inﬁnitely many times in a ﬁnite amount of time. Determining

when/where this occurs is a difﬁcult problem [1], [2], [3] and

is commonly excluded. One way of ensuring that Zeno does

not occur is by requiring that Sand ∆(S)are disjoint and

the set of impact times is closed and discrete [4], [5], [6].

In the case of mechanical systems with impacts, the state-

space is the tangent bundle of the conﬁguration space, M=

T Q, and the guard consists of all outward pointing vectors

at the location of impact, i.e.

S={(x, v)∈T Q :h(x)=0, dhx(v)<0},

where the condition h(x)=0describes the location of im-

pacts. As mechanical impacts reverse the normal component

of velocity,

∆(S) = {(x, v)∈T Q :h(x)=0, dhx(v)>0}.

In this setting, Sand ∆(S)are never disjoint and Zeno

remains a possibility. Although Zeno behaviour will almost

never occur for elastic impacts [7], for inelastic mechanical

impacts, Zeno behavior is to be expected.

*W. Clark was funded by NSF grant DMS-1645643. M. Oprea was

supported by the Army Research Ofﬁce Biomathematics Program Grant

W911NF-18-1-0351.

1William Clark is with the Department of Mathematics, Cornell Univer-

sity, Ithaca, NY, 14580, USA wac76@cornell.edu

2Maria Oprea is with the Center of Applied Mathematics, Cornell

University, Ithaca, NY, 14580, USA mao237@cornell.edu

Fig. 1. An illustration of a Zeno control execution.

The objective of this work is to study solutions to the

following controlled version of (1)

(˙x=X(x, u), x ∈M\ S,

x+= ∆(x−), x ∈ S,(2)

i.e. the ﬂow is controlled but both the guard and reset are

ﬁxed which is a common model for legged locomotion [6],

[8], [9].

For the hybrid control system, (2), we wish to study

solutions to the optimal control problem with cost

J(x0, u(·)) = ZTf

(x(s), u(s)) ds, (3)

subjected to the ﬁxed end-point conditions x(0) = x0

and x(Tf) = xf. A necessary condition for minimizing

(3) subject to (2) is the hybrid maximum principle [10],

[11], [12]. Unfortunately, this procedure breaks down if and

when a Zeno trajectory occurs. However, the hybrid max-

imum principle dictates that the co-state equation satisﬁed

a “Hamiltonian jump condition” and the resulting motion is

analogous of an elastic impact system and, as such, Zeno

will almost never happen [13].

The contribution of this work is to demonstrate that

“almost never Zeno” does not mean “no Zeno.” In the

speciﬁc case of the controlled bouncing ball with dissipation

at impacts, if the terminal time is sufﬁciently large, it is

advantageous to only apply controls around the terminal

time. An illustration of this control procedure is shown in

Fig. 1.

Preliminaries in both hybrid systems and their accom-

panying control systems are outlined in §II. The classical

hybrid maximum principle is presented in §III. A detailed

examination of this problem applied to the case study of

the bouncing ball is performed in §IV where it is explicitly

shown that in certain circumstances the Zeno trajectory

shown in Fig. 1 describes the optimal solution. Finally,

conclusions and future directions are discussed in §V.

arXiv:2210.01056v1 [math.OC] 3 Oct 2022

II. PRELIMINARIES

A hybrid dynamical system is composed of a continuous

ﬂow generated by a vector ﬁeld, and a discrete reset map

which is activated whenever the ﬂow intersects the guard.

The speciﬁc deﬁnition used throughout is given below.

Deﬁnition 1: A hybrid dynamical system is a 4-tuple,

H= (M, S, X, ∆) where

1) Mis a (ﬁnite-dimensional) manifold,

2) S ⊂ Mis an embedded co-dimension 1 submanifold,

3) X:M→T M is a vector-ﬁeld, and

4) ∆ : S → Mis a map.

Throughout this work all of the data will be assumed to be

sufﬁciently smooth. The manifold Mwill be referred to as

the state-space,Sas the guard, and ∆as the reset.

The equations of motion for the hybrid system are char-

acterized by their constituent discrete and continuous parts.

Off the guard S, the hybrid ﬂow will evolve according to the

vector ﬁeld. Whereas on S, the hybrid ﬂow instantaneously

jumps according to the reset map ∆.

(˙x(t) = X(x(t)), x(t)6∈ S,

x(t+) = ∆(x(t−)), x(t−)∈ S.(4)

The major qualitative feature that is unique to hybrid systems

is that of Zeno.

Deﬁnition 2: Let φH

tbe the ﬂow of (4). Then a point

x∈Mhas a Zeno trajectory if there exists an increasing

sequence of times {ti}∞

i=1 such that φH

ti(x)∈ S for all iand

the limit limi→∞ ti=t∞exists and is ﬁnite.

In order to provide a theoretical framework for performing

optimal control in the context of hybrid systems, we extend

the notion of a hybrid dynamical system (Deﬁnition 1) to

include a control variable, denoted by u, to the continuous

component of the dynamics. No control over either the reset

or guard will be assumed.

Deﬁnition 3: A hybrid control system is a 5-tuple, HC =

(M, U,S, X, ∆) where

1) Mis a (ﬁnite-dimensional) manifold,

2) S ⊂ Mis an embedded co-dimension 1 submanifold,

3) U ⊂ Rmis a closed subset of admissible controls,

4) X:M×V→T M is where U ⊂ Vis an open

neighborhood, and

5) ∆ : S → Mis a map.

Again, it will be implicitly assumed that all the data are

sufﬁciently smooth.

The optimal control problem of interest will be the fol-

lowing.

Problem 1: For a given a hybrid control system

(M, U,S, X, ∆), minimize the cost functional:

J:M× U[0,Tf]×[0, Tf]→R,

J(x0, u(·), s) = ZTf

(x(t), u(t)) dt,

subject to the boundary conditions x(s) = x0and x(Tf) =

xf, along with the controlled dynamics

(˙x(t) = X(x(t), u(t)), x(t)6∈ S,

x(t+) = ∆(x(t−)), x(t−)∈ S.

III. HYBRID MAXIMUM PRINCIPLE

As Problem 1 obeys the principle of optimality, it is

amendable to the usual ideas of optimal control theory; there

exists both a hybrid Hamilton-Jacobi-Bellman equation [14]

and a hybrid maximum principle [10], [11], [12]. Although

it will only provide necessary conditions, we will use the hy-

brid maximum principle to study Problem 1. The continuous

part of the optimal trajectory follows the ﬂow of the optimal

Hamiltonian, just as in the classical, non-hybrid, case. For

hybrid control system HC = (M, U,S, X, ∆) with cost

functional Jand running cost , the optimal Hamiltonian,

H:T∗M→R, is given by:

H(x, p) = min

u∈U

H(x, p, u)

= min

u∈U(x, u) + hp, X(x, u)i,

where hp, Xidenotes the natural pairing between a co-vector

and a vector. At impacts, the optimal ﬂow is discontinuous.

As the state variables jump according to the reset map, ∆,

the co-states jump according to the extended reset map,˜

∆,

which satisﬁes:

Id ×˜

∆∗

ϑˆ

H=ι∗ϑˆ

H,(5)

such that the following diagram is commutative

R× S∗R×T∗M

R× S R×M

Id×˜

∆

Id×πMId×πM

Id×∆

where ϑˆ

H=pi·dxi−ˆ

H·dt is the action form, πM:T∗M→

Mis the cotangent projection, and ι:S∗→T∗Mis the

inclusion map of the extended guard,

S∗=(x, p)∈T∗M|S:dhx∂H

∂p >0.

The hybrid maximum principle states that a necessary

condition for a trajectory, x(t), to be a solution to Problem

1, it must have the form x(t) = πM(γ(t)) where γ(t)is an

integral curve of











˙x=∂ˆ

∂p ,˙p=−∂ˆ

∂x , x 6∈ S,

(x+, p+) = ˜

∆(x−, p−), x ∈ S,

such the boundary conditions are satisﬁed: πM(γ(0)) = x0

and πM(γ(Tf)) = xf.

Under the assumption that there exists a unique extended

reset map that satisﬁes the conditions above (one does not

generally expect this to be true; see [13] for a more detailed

analysis on issue), along with some regularity assumptions,

it can be proved that the set of integral curves obeying

the hybrid maximum principle that are Zeno have measure

zero. However, this does not imply that the actual optimal

trajectory will not be Zeno in any speciﬁc example. Indeed,

in the next section we show that the Zeno phenomena is

present even in the simple case of a bouncing ball.

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OptimalityofZenoExecutionsinHybridSystems*WilliamClark1andMariaOprea2AbstractAuniquefeatureofhybriddynamicalsystems(systemswhoseevolutionissubjecttobothcontinuous-anddiscrete-timelaws)isZenotrajectories.Usuallythesetrajecto-riesareavoidedastheycancauseincorrectnumericalresultsastheproblembecomesill...

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Optimality of Zeno Executions in Hybrid Systems William Clark1and Maria Oprea2 Abstract A unique feature of hybrid dynamical systems

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