TOWARDS SCALABLE SYNTHESIS OF STOCHASTIC CONTROL SYSTEMS

arXiv:1602.01358v1 [] 3 Feb 2016

MAJID ZAMANI1 , ILYA TKACHEV2 , AND ALESSANDRO ABATE3

Abstract. Formal control synthesis approaches over stochastic systems have received significant attention
in the past few years, in view of their ability to provide provably correct controllers for complex logical
specifications in an automated fashion. Examples of complex specifications of interest include properties
expressed as formulae in linear temporal logic (LTL) or as automata on infinite strings. A general methodology
to synthesize controllers for such properties resorts to symbolic abstractions of the given stochastic systems.
Symbolic models are discrete abstractions of the given concrete systems with the property that a controller
designed on the abstraction can be refined (or implemented) into a controller on the original system. Although
the recent development of techniques for the construction of symbolic models has been quite encouraging, the
general goal of formal synthesis over stochastic control systems is by no means solved. A fundamental issue
with the existing techniques is the known “curse of dimensionality,” which is due to the need to discretize state
and input sets and that results in an exponential complexity over the number of state and input variables in
the concrete system. In this work we propose a novel abstraction technique for incrementally stable stochastic
control systems, which does not require state-space discretization but only input set discretization, and that
can be potentially more efficient (and thus scalable) than existing approaches. We elucidate the effectiveness
of the proposed approach by synthesizing a schedule for the coordination of two traffic lights under some safety
and fairness requirements for a road traffic model. Further we argue that this 5-dimensional linear stochastic
control system cannot be studied with existing approaches based on state-space discretization due to the very
large number of generated discrete states.

1. Introduction
In the last decade many techniques have been developed providing controllers for control systems (both
deterministic and, more recently, stochastic) in a formal and automated fashion against some complex logical
specifications. Examples of such specifications include properties expressed as formulae in linear temporal logic
(LTL) or as automata on infinite strings [BK08], and as such they are not tractable by classical techniques for
control systems. A general scheme for providing such controllers is by leveraging symbolic models of original
concrete systems. Symbolic models are discrete abstractions of the original systems in which each symbol
represents an aggregate of continuous variables. When such symbolic models exist for the concrete systems,
one can leverage the algorithmic machinery for automated synthesis of discrete models [dAHM01, MNA03] to
automatically synthesize discrete controllers which can be refined to hybrid controllers for the original systems.
The construction of symbolic models for continuous-time non-probabilistic systems has been thoroughly investigated in the past few years. This includes results on the construction of approximately bisimilar symbolic
models for incrementally stable control systems [MZ12, PGT08], switched systems [GPT09], and control systems with disturbances [PT09], non-uniform abstractions of nonlinear systems over a finite-time horizon [TI09],
as well as the construction of sound abstractions based on the convexity of reachable sets [Rei11], feedback refinement relations [RWR15], robustness margins [LO14], and for unstable control systems [ZPJT12]. Recently,
there have been some results on the construction of symbolic models for continuous-time stochastic systems,
including the construction of finite Markov decision process approximations of linear stochastic control system,
however without providing a quantitative relationship between abstract and concrete model [LAB09], approximately bisimilar symbolic models for incrementally stable stochastic control systems [ZEM+ 14], stochastic
switched systems [ZAG15], and randomly switched stochastic systems [ZA14], as well as sound abstractions
for unstable stochastic control systems [ZEAL13].
1

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M. ZAMANI, I. TKACHEV, AND A. ABATE

Note that all the techniques provided in [MZ12, PGT08, GPT09, PT09, TI09, Rei11, RWR15, LO14, ZPJT12,
LAB09, ZEM+ 14, ZA14, ZEAL13] are fundamentally based on the discretization of continuous states. Therefore, they suffer severely from the curse of dimensionality due to gridding those sets, which is especially
irritating for models with high-dimensional state sets. In this work we propose a novel approach for the
construction of approximately bisimilar symbolic models for incrementally stable stochastic control systems
not requiring any state set discretization but only input set discretization. Therefore, it can be potentially
more efficient than the proposed approaches in [ZEM+ 14] when dealing with higher dimensional stochastic
control systems. We provide a theoretical comparison with the approach in [ZEM+ 14] and a simple criterion
that helps choosing the most suitable among two approaches (in terms of the sizes of the symbolic models)
for a given stochastic control system. Another advantage of the technique proposed here is that it allows
us to construct symbolic models with probabilistic output values, resulting in less conservative symbolic abstractions than those proposed in [ZEM+ 14, ZA14, ZEAL13] that allow for non-probabilistic output values
exclusively. We then explain how the proposed symbolic models with probabilistic output values can be used
for synthesizing hybrid controllers enforcing logic specifications. The proposed approaches in [ZAG15] also
provide symbolic models with probabilistic output values and without any state set discretization. However,
the results in [ZAG15] are for stochastic switched systems rather than stochastic control systems as in this
work and they do not provide any intuition behind the control synthesis over symbolic models with probabilistic output values. The effectiveness of the proposed results is illustrated by synthesizing a schedule for
the coordination of two traffic lights under some safety and fairness requirements for a model of road traffic
which is a 5-dimensional linear stochastic control system. We also show that this example is not amenable to
be dealt with the approaches proposed in [ZEM+ 14]. Although the main proposed results in this work are for
incrementally stable stochastic control systems, the similar results for incrementally stable non-probabilistic
control systems can be recovered in the same framework by simply setting the diffusion term to zero.
Alongside the relationship with and extension of [ZAG15, ZEM+ 14], this paper provides a detailed and extended elaboration of the results first announced in [ZTA14], including the proofs of the main results, a detailed
discussion on how to deal with probabilistic output values and a generalization of the corresponding result
with no requirement on compactness, and finally discussing a new case study on road traffic control.

2. Stochastic Control Systems
2.1. Notation. The identity map on a set A is denoted by 1A . The symbols N, N0 , Z, R, R+ , and R+
0 denote
the set of natural, nonnegative integer, integer, real, positive, and nonnegative real numbers, respectively. The
symbols In , 0n , and 0n×m denote the identity matrix, the zero vector, and the zero matrix in Rn×n , Rn , and
Rn×m , respectively. Given a vector x ∈ Rn , we denote by xi the i–th element of x, and by kxk the infinity
norm of x, namely, kxk = max{|x1 |, |x2 |, ...,P
|xn |}, where |xi | denotes the absolute value of xi . Given a matrix
n
P = {pij } ∈ Rn×n , we denote by Tr(P ) = i=1 pii the trace of P . We denote by λmin (A) and λmax (A) the
minimum and maximum eigenvalues of a symmetric matrix A, respectively. The diagonal set ∆ ⊂ Rn × Rn is
defined as: ∆ = {(x, x) | x ∈ Rn }.
The closed ball centered at x ∈QRm with radius λ is defined by Bλ (x) = {y ∈ Rm | kx − yk ≤ λ}. A set
m
B ⊆ Rm is called a box if B = i=1 [ci , di ], where ci , di ∈ R with ci < di for each i ∈ {1, . . . , m}. The span
of a box B is defined as span(B) = min{|di − ci | | i = 1, . . . , m}. For a box B ⊆ Rm and µ ≤ span(B),
define the µ-approximation [B]µ = [Rm ]µ ∩ B, where [Rm ]µ = {a ∈ Rm | ai = ki µ, ki ∈ Z, i = 1, . . . , m}.
Note that [B]µ 6= ∅ for any µ ≤ span(B). Geometrically, for any µS∈ R+ with µ ≤ span(B) and λ ≥ µ, the
collection of sets {Bλ (p)}p∈[B]µ is a finite covering of B, i.e. B ⊆ p∈[B]µ Bλ (p). We extend the notions of
SM
span and approximation to finite unions of boxes as follows. Let A = j=1 Aj , where each Aj is a box. Define
SM
span(A) = min{span(Aj ) | j = 1, . . . , M }, and for any µ ≤ span(A), define [A]µ = j=1 [Aj ]µ .
n
Given a measurable function f : R+
0 → R , the supremum of f is denoted by kf k∞ := (ess)sup{kf (t)k, t ≥ 0}.
+
+
A continuous function γ : R0 → R0 is said to belong to class K if it is strictly increasing and γ(0) = 0; γ is

TOWARDS SCALABLE SYNTHESIS OF STOCHASTIC CONTROL SYSTEMS

3

+
+
said to belong to class K∞ if γ ∈ K and γ(r) → ∞ as r → ∞. A continuous function β : R+
0 × R0 → R0 is
said to belong to class KL if, for each fixed s, the map β(r, s) belongs to class K with respect to r and, for
each fixed nonzero r, the map β(r, s) is decreasing with respect to s and β(r, s) → 0 as s → ∞. We identify a
relation R ⊆ A × B with the map R : A → 2B defined by b ∈ R(a) iff (a, b) ∈ R. Given a relation R ⊆ A × B,
R−1 denotes the inverse relation defined by R−1 = {(b, a) ∈ B × A : (a, b) ∈ R}. Given a finite sequence S,
we denote by σ := (S)ω the infinite sequence generated by repeating S infinitely, i.e. σ := SSSSS . . ..

2.2. Stochastic control systems. Let (Ω, F, P) be a probability space endowed with a filtration F = (Fs )s≥0
satisfying the usual conditions of completeness and right continuity [KS91, p. 48]. Let (Ws )s≥0 be a pdimensional F-adapted Brownian motion.
Definition 2.1. A stochastic control system Σ is a tuple Σ = (Rn , U, U, f, σ), where
Rn is the state space;
U ⊆ Rm is a bounded input set;
U is a subset of the set of all measurable functions of time from R+
0 to U;
f : Rn × U → Rn satisfies the following Lipschitz assumption: there exist constants Lx , Lu ∈ R+ such
that: kf (x, u) − f (x0 , u0 )k ≤ Lx kx − x0 k + Lu ku − u0 k for all x, x0 ∈ Rn and all u, u0 ∈ U;
• σ : Rn → Rn×p satisfies the following Lipschitz assumption: there exists a constant Z ∈ R+ such that:
kσ(x) − σ(x0 )k ≤ Zkx − x0 k for all x, x0 ∈ Rn .

•
•
•
•

n
A continuous-time stochastic process ξ : Ω × R+
0 → R is said to be a solution process of Σ if there exists
υ ∈ U satisfying the following stochastic differential equation (SDE) P-almost surely (P-a.s.)

d ξ = f (ξ, υ) d t + σ(ξ) d Wt ,

(2.1)

where f is known as the drift and σ as the diffusion. We also write ξaυ (t) to denote the value of the solution
process at time t ∈ R+
0 under the input curve υ from initial condition ξaυ (0) = a P-a.s., in which a is a random
variable that is measurable in F0 . Let us emphasize that the solution process is unambiguously determined,
since the assumptions on f and σ ensure its existence and uniqueness [Oks02, Theorem 5.2.1, p. 68].
3. Incremental Stability
We recall a stability notion for stochastic control systems, introduced in [ZEM+ 14], on which the main results
presented in this work rely.
Definition 3.1. A stochastic control system Σ is incrementally input-to-state stable in the qth moment (δISS-Mq ), where q ≥ 1, if there exist a KL function β and a K∞ function γ such that for any t ∈ R+
0 , any
Rn -valued random variables a and a0 that are measurable in F0 , and any υ, υ 0 ∈ U, the following condition is
satisfied:

q 
q
(3.1)
E [kξaυ (t) − ξa0 υ0 (t)k ] ≤ β E ka − a0 k , t + γ (kυ − υ 0 k∞ ) .
It can be easily verified that a δ-ISS-Mq stochastic control system Σ is δ-ISS [Ang02] in the absence of any
noise as in the following:
kξaυ (t) − ξa0 υ0 (t)k ≤ β (ka − a0 k , t) + γ (kυ − υ 0 k∞ ) ,
0

(3.2)

n

for a, a ∈ R , some β ∈ KL, and some γ ∈ K∞ .
Similar to the characterization of δ-ISS in terms of the existence of so-called δ-ISS Lyapunov functions in
[Ang02], one can describe δ-ISS-Mq in terms of the existence of so-called δ-ISS-Mq Lyapunov functions, as
shown in [ZEM+ 14] and defined next.
Definition 3.2. Consider a stochastic control system Σ and a continuous function V : Rn × Rn → R+
0 that
is twice continuously differentiable on {Rn × Rn }\∆. The function V is called a δ-ISS-Mq Lyapunov function
for Σ, where q ≥ 1, if there exist K∞ functions α, α, ρ, and a constant κ ∈ R+ , such that

4

M. ZAMANI, I. TKACHEV, AND A. ABATE

(i) α (resp. α) is a convex (resp. concave) function;
(ii) for any x, x0 ∈ Rn , α (kx − x0 kq ) ≤ V (x, x0 ) ≤ α (kx − x0 kq );
(iii) for any x, x0 ∈ Rn , x 6= x0 , and for any u, u0 ∈ U,





 ∂x,x V
1
σ(x)  T
f (x, u)
u,u0
0
T
0
L
V (x, x ) := [∂x V ∂x0 V ]
σ (x) σ (x )
+ Tr
σ(x0 )
f (x0 , u0 )
∂x0 ,x V
2
0
0
≤ − κV (x, x ) + ρ(ku − u k),

∂x,x0 V
∂x0 ,x0 V



0

where Lu,u is the infinitesimal generator associated to the process V (ξ, ξ 0 ), and where ξ and ξ 0 are solution
processes of the SDE (2.1) [Oks02, Section 7.3]. The symbols ∂x and ∂x,x0 denote first- and second-order
partial derivatives with respect to x and (x, x0 ), respectively.

Although condition (ii) in the above definition implies that the growth rate of functions α and α is linear,
this requirement does not restrict the behavior of α and α to be linear on a compact subset of Rn . Note that
condition (i) is not required in the context of non-probabilistic control systems for the corresponding δ-ISS
Lyapunov functions [Ang02]. The following theorem, borrowed from [ZEM+ 14], describes δ-ISS-Mq in terms
of the existence of δ-ISS-Mq Lyapunov functions.
Theorem 3.3. A stochastic control system Σ is δ-ISS-Mq if it admits a δ-ISS-Mq Lyapunov function.



One can resort to available software tools, such as SOSTOOLS [PAV+ 13], to search for appropriate δ-ISSMq Lyapunov functions for systems Σ of polynomial type. We refer the interested readers to the results
in [ZEM+ 14] for the discussion of special instances where these functions can be easily computed, and limit
ourselves to mention that, as an example, for linear stochastic control systems Σ (with linear drift and diffusion
terms), one can search for appropriate δ-ISS-Mq Lyapunov functions by solving a linear matrix inequality
(LMI).
3.1. Noisy and noise-free trajectories. In order to introduce the symbolic models in Subsection 5.2 (Theorems 5.7 and 5.9) for a stochastic control system, we need the following technical result, borrowed from
[ZEM+ 14], which provides an upper bound on the distance (in the qth moment) between the solution process
of Σ and the solution of a derived non-probabilistic control system Σ obtained by disregarding the diffusion
term σ. From now on, we use the notation ξ xυ to denote the solution of Σ = (Rn , U, U, f, 0n×p )1, starting from
the non-probabilistic initial condition x and under the input curve υ, which satisfies the ordinary differential
equation (ODE) ξ˙ xυ = f (ξ xυ , υ).
Lemma 3.4. Consider a stochastic control system Σ such that f (0n , 0m ) = 0n and σ(0n ) = 0n×p . Suppose
that q ≥ 2 and that there exists a δ-ISS-Mq Lyapunov function V for Σ such that its Hessian is a positive
semidefinite matrix in R2n×2n and ∂x,x V (x, x0 ) ≤ P , for any x, x0 ∈ Rn , and some positive semidefinite matrix
P ∈ Rn×n . Then for any x ∈ Rn and any υ ∈ U, we have
h
i
q
E ξxυ (t) − ξ xυ (t)
≤ hx (t),
(3.3)
where
hx (t) = α−1

1 √
P
2

2


 q2 !
Z t
q
n min{n, p}Z 2 e−κt
β (kxk , s) + γ sup {kuk}
ds ,
0

u∈U

and where Z is the Lipschitz constant, introduced in Definition 2.1, and β is the KL function appearing in
(3.1).

It can be readily seen that the nonnegative-valued function hx tends to zero as t → 0, t → +∞, or as Z → 0,
and is identically zero if the diffusion term is identically zero (i.e. Z = 0, which is the case for Σ). The
interested readers are referred to [ZEM+ 14], which provides results in line with that of Lemma 3.4 for (linear)
stochastic control systems Σ admitting a specific type of δ-ISS-Mq Lyapunov functions.
1Here, we have abused notation by identifying 0
n
n×p with the map σ : x → 0p ∀x ∈ R .

TOWARDS SCALABLE SYNTHESIS OF STOCHASTIC CONTROL SYSTEMS

5

4. Systems and Approximate Equivalence Relations
4.1. Systems. We employ the abstract and general notion of “system,” as introduced in [Tab09], to describe
both stochastic control systems and their symbolic models.
- , Y, H), where X is a set of states (possibly
Definition 4.1. A system S is a tuple S = (X, X0 , U,
- ⊆
infinite), X0 ⊆ X is a set of initial states (possibly infinite), U is a set of inputs (possibly infinite),
X × U × X is a transition relation, Y is a set of outputs, and H : X → Y is an output map.

- is also denoted by x u- x0 . For a transition x u- x0 , state x0 is called a
A transition (x, u, x0 ) ∈
u-successor, or simply a successor, of state x. We denote by Postu (x) the set of all u-successors of a state x.
For technical reasons, we assume that for any x ∈ X, there exists some u-successor of x, for some u ∈ U —
let us remark that this is always the case for the considered systems later in this paper.
A system S is said to be
• metric, if the output set Y is equipped with a metric d : Y × Y → R+
0;
• finite (or symbolic), if X and U are finite sets;
• deterministic, if for any state x ∈ X and any input u ∈ U , |Postu (x)| ≤ 1.
- , Y, H) and given any initial state x0 ∈ X0 , a finite state run generated
For a system S = (X, X0 , U,
from x0 is a finite sequence of transitions:
x0

u0

- x1

u1

- ···

un−2

- xn−1

un−1

- xn ,

(4.1)

ui

such that xi - xi+1 for all 0 ≤ i < n. A finite state run can be directly extended to an infinite state run
as well. A finite output run is a sequence {y0 , y1 , . . . , yn } such that there exists a finite state run of the form
(4.1) with yi = H(xi ), for i = 0, . . . , n. A finite output run can also be directly extended to an infinite output
run as well.
4.2. Relations among systems. We recall the notion of approximate (bi)simulation relation, introduced in
[GP07], which is cruicial when analyzing or synthesizing controllers for deterministic systems.
Definition 4.2. Let Sa = (Xa , Xa0 , Ua ,

- , Ya , Ha ) and Sb = (Xb , Xb0 , Ub ,

a

- , Yb , Hb ) be metric sys-

b

tems with the same output sets Ya = Yb and metric d. For ε ∈ R+
0 , a relation R ⊆ Xa × Xb is said to be
an ε-approximate simulation relation from Sa to Sb if, for all (xa , xb ) ∈ R, the following two conditions are
satisfied:
(i) d(Ha (xa ), Hb (xb )) ≤ ε;
ua
- x0a in Sa implies the existence of xb

(ii) xa

a

ub

- x0 in Sb satisfying (x0a , x0 ) ∈ R.
b
b

b

A relation R ⊆ Xa × Xb is said to be an ε-approximate bisimulation relation between Sa and Sb if R is an
ε-approximate simulation relation from Sa to Sb and R−1 is an ε-approximate simulation relation from Sb to
Sa .
System Sa is ε-approximately simulated by Sb , or Sb ε-approximately simulates Sa , denoted by Sa εS Sb , if
there exists an ε-approximate simulation relation R from Sa to Sb such that:
• for every xa0 ∈ Xa0 , there exists xb0 ∈ Xb0 with (xa0 , xb0 ) ∈ R.
System Sa is ε-approximately bisimilar to Sb , denoted by Sa ∼
=εS Sb , if there exists an ε-approximate bisimulation relation R between Sa and Sb such that:
• for every xa0 ∈ Xa0 , there exists xb0 ∈ Xb0 with (xa0 , xb0 ) ∈ R;
• for every xb0 ∈ Xb0 , there exists xa0 ∈ Xa0 with (xa0 , xb0 ) ∈ R.



6

M. ZAMANI, I. TKACHEV, AND A. ABATE

5. Symbolic Models for Stochastic Control Systems
5.1. Describing stochastic control systems as metric systems. In order to show the main results
of the paper, we use the notion of system introduced above to abstractly represent a stochastic control
system. More precisely, given a stochastic control system Σ, we define an associated metric system S(Σ) =
- , Y, H), where:
(X, X0 , U,
X is the set of all Rn -valued random variables defined on the probability space (Ω, F, P);
X0 is a subset of the set of Rn -valued random variables that are measurable over F0 ;
U = U;
υ
+
x - x0 if x and x0 are measurable in Ft and Ft+τ , respectively, for some t ∈ R+
0 and τ ∈ R , and
+
n
0
there exists a solution process ξ : Ω × R0 → R of Σ satisfying ξ(t) = x and ξxυ (τ ) = x P-a.s.;
• Y = X;
• H = 1X .
1

q 
We assume that the output set Y is equipped with the metric d(y, y 0 ) = E ky − y 0 k q , for any y, y 0 ∈ Y
and some q ≥ 1. Let us remark that the set of states and inputs of S(Σ) are uncountable and that S(Σ)
is a deterministic system in the sense of Definition 4.1, since (cf. Subsection 2.2) the solution process of
Σ is uniquely determined. Note that for the case of non-probabilistic control system Σ, one obtains S(Σ) =
υ
(X, X0 , U, - , Y, H), where X = Rn , X0 is a subset of Rn , U = U, x - x0 iff x0 = ξ xυ (τ ) for some τ ∈ R+ ,
Y = X, H = 1X , and the metric on the output set reduces to the natural Euclidean one: d(y, y 0 ) = ky − y 0 k,
for any y, y 0 ∈ Y .
•
•
•
•

Notice that, since the concrete system S(Σ) is uncountably infinite, it does not allow for a straightforward
discrete controller synthesis with the techniques in the literature [dAHM01, MNA03]. We are thus interested in
finding a finite abstract system that is (bi)similar to the concrete system S(Σ). In order to discuss approximate
(bi)simulation relations between two metric systems, they have to share the output space (cf. Definition 4.2).
System S(Σ) inherits a classical trace-based semantics (cf. definition of output run after (4.1)) [BK08], however
the outputs of S(Σ) (and necessarily those of any approximately (bi)similar one) are random variables. This
fact is especially important due to the metric d that the output set is endowed with: for any non-probabilistic
point one can always find a non-degenerate random variable that is as close as desired to the original point in
the metric d.
To further elaborate the discussion in the previous paragraph, let us consider the following example. Let
A ⊂ Rn be a set (of non-probabilistic points). Consider a safety problem, formulated as the satisfaction of
the LTL formula2 ϕA , where ϕA is a label (or proposition) characterising the set A. Suppose that over the
abstract system we are able to synthesize a control strategy that makes an output run of the abstraction satisfy
ϕA . Although the run would in general be consisting of random variables y, the fact that y ∈ A means that
y has a Dirac probability distribution centered at y, that is y ∈ Y is a degenerate random variable that can
be identified with a point in A ⊂ Rn ⊂ Y . Note that since any non-probabilistic point can be regarded as
a random variable with a Dirac probability distribution centered at that point, Rn can be embedded in Y ,
which we denote as Rn ⊂ Y with a slight abuse of notation. As a result, satisfying ϕA precisely means
that the output run of the abstraction indeed stays in the set A ⊂ Rn forever. On the other hand, suppose
that the original system is ε-approximate bisimilar to the abstraction. If we want to interpret the result ϕA
obtained over the abstraction, we can guarantee that the corresponding output run of the original system
satisfies ϕAε , that is any output y of the run of the original system is within ε d-distance from the set
A: d(y, A) = inf a∈A d(y, a) ≤ ε. Note that although the original set A ⊂ Y is a subset of Rn ⊂ Y , its
ε-inflation Aε = {y ∈ Y : d(y, A) ≤ ε} is not a subset of Rn anymore and hence contains non-degenerate
random variables. In particular, Aε 6= {y ∈ Rn : inf a∈A ky − ak ≤ ε} and is in fact bigger than the latter set
of non-probabilistic points. As a result, although satisfying ϕAε does not necessarily mean that a trajectory
2We refer the interested readers to [BK08] for the formal semantic of the temporal formula ϕ expressing the safety property
A

over set A.

TOWARDS SCALABLE SYNTHESIS OF STOCHASTIC CONTROL SYSTEMS

7

of Σ always stays within some non-probabilistic set, it means that the associated random variables always
belong to Aε and, hence, are close to the non-probabilistic set A with respect to the qth moment metric.
We are now able to provide two versions of finite abstractions: one whose outputs are always non-probabilistic
points – that is degenerate random variables, elements of Rn ⊂ Y , and one whose outputs can be nondegenerate random variables. Recall, however, that in both cases the output set is still the whole Y and the
semantics is the same as for the original system S(Σ).
5.2. Main results. This subsection contains the main contributions of the paper. We show that for any
δ-ISS-Mq (resp. δ-ISS) stochastic control system Σ (resp. non-probabilistic control system Σ), and for any
precision level ε ∈ R+ , we can construct a finite system that is ε-approximate bisimilar to Σ (resp. Σ) without
any state set discretization. The results in this subsection rely on additional assumptions on the model Σ
that are described next. We restrict our attention to stochastic control systems Σ with input sets U that are
assumed to be finite unions of boxes (cf. Subsection 2.1). We further restrict our attention to sampled-data
stochastic control systems, where input curves belong to set Uτ , which contains exclusively curves that are
constant over intervals of length τ ∈ R+ , i.e.
n
o
Uτ = υ ∈ U | υ(t) = υ((k − 1)τ ), t ∈ [(k − 1)τ, kτ [, k ∈ N .
Let us denote by Sτ (Σ) a sub-system of S(Σ) obtained by selecting those transitions of S(Σ) corresponding
to solution processes of duration τ and to control inputs in Uτ . This can be seen as the time discretization of
Σ. More precisely, given a stochastic control system Σ and the corresponding metric system S(Σ), we define
a new associated metric system


- , Yτ , Hτ ,
Sτ (Σ) = Xτ , Xτ 0 , Uτ ,
τ

where Xτ = X, Xτ 0 = X0 , Uτ = Uτ , Yτ = Y , Hτ = H, and
• xτ

υτ

- x0τ if xτ and x0τ are measurable, respectively, in Fkτ and F(k+1)τ for some k ∈ N0 , and there

τ

n
0
exists a solution process ξ : Ω × R+
0 → R of Σ satisfying ξ(kτ ) = xτ and ξxτ υτ (τ ) = xτ P-a.s..

Similarly, one can define Sτ (Σ) as the time discretization of Σ. Notice that a finite state run
x0

υ0

υ1

υN −1

τ

τ

τ

- x1

- ···

- xN

of Sτ (Σ), where υi−1 ∈ Uτ and xi = ξxi−1 υi−1 (τ ) P-a.s. for i = 1, . . . , N , captures the solution process of Σ at
times t = 0, τ, . . . , N τ , started from the initial condition x0 and resulting from a control
 input υ obtained by
the concatenation of the input curves υi−1 i.e. υ(t) = υi−1 (t) for any t ∈ [(i − 1)τ, i τ [ , for i = 1, . . . , N .
Let us proceed introducing two fully symbolic systems for the concrete model Σ. Consider a stochastic control
system Σ and a tuple q = (τ, µ, N, xs ) of parameters, where τ is the sampling time, µ is the input set
quantization, N ∈ N is a temporal horizon, and xs ∈ Rn is a source state. Given Σ and q, let us introduce the
following two symbolic systems:
- , Yq , Hq ),
Sq (Σ) = (Xq , Xq0 , Uq ,
q

S q (Σ) = (Xq , Xq0 , Uq ,

- , Yq , H q ),

q

consisting of:
N times

•
•
•
•

z
}|
{
Xq = (u1 , . . . , uN ) ∈ [U]µ × · · · × [U]µ ;
Xq0 = Xq ;
Uq = [U]µ ;
uq
xq - x0q , where xq = (u1 , u2 , . . . , uN ), if and only if x0q = (u2 , . . . , uN , uq );


q

8

M. ZAMANI, I. TKACHEV, AND A. ABATE

• Yq is the set of all Rn-valued random variables
defined on the probability space (Ω, F, P);

• Hq (xq ) = ξxs xq (N τ ) H q (xq ) = ξ xs xq (N τ ) .
Note that the transition relation in Sq (Σ) admits a compact representation in the form of a shift operator. We
have abused notation by identifying uq ∈ [U]µ with the constant input curve with domain [0, τ [ and value uq ,
and by identifying xq ∈ [U]N
xq (t) = ui for any
µ with the concatenation of N control inputs ui ∈ [U]µ i.e. 

t ∈ [(i − 1)τ, iτ [ for i = 1, . . . , N . Notice that the proposed abstraction Sq (Σ) resp. S q (Σ) is a deterministic
system in the sense of Definition 4.1. Note that Hq and H q are mappings from a non-probabilistic point xq
to the random variable ξxs xq (N τ ) and to the one with a Dirac probability distribution centered at ξ xs xq (N τ ),
respectively. Finally, note that in the case of a non-probabilistic control system Σ, one obtains the symbolic
- , Yq , H q ), where Xq , Xq0 , Uq ,
- , and H q are the same as before, but
system S q (Σ) = (Xq , Xq0 , Uq ,
q
q
where the output set reduces to Yq = Rn .
Note that the idea behind the definitions of symbolic models Sq (Σ) and S q (Σ) hinges on the δ-ISS-Mq property.
Given an input υ ∈ U, all solution processes of Σ under the input υ forget the mismatch between their initial
conditions and converge to each other with respect to the qth moment metric. Therefore, the longer the
applied inputs are, the less relevant is the mismatch between initial conditions. Then, the fundamental idea
of the introduced abstractions consists in taking the N applied inputs as the state of the symbolic model.
The control synthesis over S q (Σ) (resp. S q (Σ)) is simple as the outputs are non-probabilistic points, whereas
for Sq (Σ) it is perhaps less intuitive. Hence, we discuss it in more details later in Subsection 5.3.
Example 5.1. An example of an abstraction Sq (Σ) with N = 3 and Uq = {0, 1} is depicted in Figure 1, where
the initial states are shown as targets of sourceless arrows. Note that, regardless of the size of the state set
and of its dimension, Sq (Σ) only has eight possible states, namely:
Xq = {(0, 0, 0), (0, 0, 1), (0, 1, 0), (0, 1, 1), (1, 0, 0), (1, 0, 1), (1, 1, 0), (1, 1, 1)}.
In order to obtain some of the main results of this work, we raise an assumption on the δ-ISS-Mq Lyapunov
function V we will work with, as follows:
|V (x, y) − V (x, z)| ≤ γ
b(ky − zk),

(5.1)

for any x, y, z ∈ Rn , and for some K∞ and concave function γ
b. As long as one is interested to work in a
compact subset of Rn , the function γ
b in (5.1) can be readily computed. Indeed, for all x, y, z ∈ D, where
D ⊂ Rn is compact, one can readily apply the mean value theorem to the function y → V (x, y) to get


∂V (x, y)
|V (x, y) − V (x, z)| ≤ γ
b (ky − zk) , where γ
b(r) =
max
r.
∂y
x,y∈D\∆
q
T
In particular, for the δ-ISS-M1 Lyapunov function V (x, x0 ) := (x − x0 ) P (x − x0 ), for some positive definite
matrix P ∈ Rn×n and for all x, x0 ∈ Rn , one obtains γ
b(r) = √λmax (P ) r [Tab09, Proposition 10.5], which satisfies
λmin (P )

(5.1) globally on Rn . Note that for non-probabilistic control systems, the concavity assumption of γ
b is not
required.
Before providing the main results of the paper, we need the following technical lemmas.
Lemma 5.2. Consider a stochastic control system Σ, admitting a δ-ISS-Mq Lyapunov function V , and consider its corresponding symbolic model S q (Σ). We have that

η≤ α

−1


e

−κN τ



max V ξ xs uq (τ ), xs

uq ∈Uq

1/q

,

(5.2)

TOWARDS SCALABLE SYNTHESIS OF STOCHASTIC CONTROL SYSTEMS

0

1

(0, 0, 0)

(0, 0, 1)

1

ξxs (0,0,0) (3τ )

0

9

1

ξxs (0,0,1) (3τ )
1

(0, 1, 1)

(1, 1, 1)

1

ξxs (1,1,1) (3τ )

ξxs (0,1,1) (3τ )

0

0

1

0

1
(1, 0, 0)
ξxs (1,0,0) (3τ )

(0, 1, 0)
0

(1, 0, 1)
0

ξxs (0,1,0) (3τ )

(1, 1, 0)
1

ξxs (1,0,1) (3τ )

ξxs (1,1,0) (3τ )

0
Figure 1. Example of abstraction Sq (Σ) with N = 3 and Uq = {0, 1}. The lower part of the
states are intended as labels, corresponding to their output values. Initial states are targets
of sourceless arrows.
where
η :=

max

uq ∈Uq ,xq ∈Xq
x0q ∈Postuq (xq )

ξ H q (xq )uq (τ ) − H q x0q



.

(5.3)

The proof of Lemma 5.2 is provided in the Appendix. The next lemma provides similar result as the one in
Lemma 5.2, but without explicitly using any Lyapunov function.
Lemma 5.3. Consider a δ-ISS-Mq stochastic control system Σ and its corresponding symbolic model S q (Σ).
We have:
 
1/q
q
η ≤ β max ξ xs uq (τ ) − xs , N τ
,
(5.4)
uq ∈Uq

where η is given in (5.3) and β is the KL function appearing in (3.1).



The proof of Lemma 5.3 is provided in the Appendix. The next two lemmas provide similar results as Lemmas
5.2 and 5.3, but by using the symbolic model Sq (Σ) with probabilistic output values rather than S q (Σ) with
non-probabilistic output values.
Lemma 5.4. Consider a stochastic control system Σ, admitting a δ-ISS-Mq Lyapunov function V , and consider its corresponding symbolic model Sq (Σ). One has:




 1/q
−1
−κN τ
ηb ≤ α
e
max E V ξxs uq (τ ), xs
,
(5.5)
uq ∈Uq

10

M. ZAMANI, I. TKACHEV, AND A. ABATE

where
ηb :=

max

uq ∈Uq ,xq ∈Xq
x0q ∈Postuq (xq )

E



ξHq (xq )uq (τ ) − Hq x0q

 

.

(5.6)

Proof. The proof is similar to the one of Lemma 5.2 and can be shown by using convexity of α and Jensen
inequality [Oks02].


Lemma 5.5. Consider a δ-ISS-Mq stochastic control system Σ and its corresponding symbolic model Sq (Σ).
We have:
 
1/q

q
ηb ≤ β max E ξxs uq (τ ) − xs
, Nτ
,
(5.7)
uq ∈Uq

where ηb is given in (5.6) and β is the KL function appearing in (3.1).



Proof. The proof is similar to the one of Lemma 5.3 and can be shown by using Jensen inequality [Oks02]. 
Remark 5.6. It can be readily verified that by choosing N sufficiently large, η and ηb can be made arbitrarily
small. One can as well try to reduce the upper bound for η (in (5.2) for example) by selecting the initial point
xs as follows:


(5.8)
xs = arg minn max V ξ xuq (τ ), x .
x∈R uq ∈Uq

We can now present the first main result of the paper, which relates the existence of a δ-ISS-Mq Lyapunov
function to the construction of an approximately bisimilar symbolic model.
Theorem 5.7. Consider a stochastic control system Σ with f (0n , 0m ) = 0n and σ(0n ) = 0n×p , admitting
a δ-ISS-Mq Lyapunov function V , of the form of the one explained in Lemma 3.4, such that (5.1) holds for
some concave γ
b ∈ K∞ . Let η be given by (5.3). For any ε ∈ R+ and any tuple q = (τ, µ, N, xs ) of parameters
satisfying µ ≤ span(U) and


1
1
(5.9)
b (hxs ((N + 1)τ )) q + η ≤ α (εq ) ,
e−κτ α (εq ) + ρ(µ) + γ
eκ
the relation (cf. Definition 4.2)



R = (xτ , xq ) ∈ Xτ × Xq | E V xτ , H q (xq ) ≤ α (εq )
is an ε-approximate bisimulation relation between S q (Σ) and Sτ (Σ).



The proof can be found in the Appendix. By choosing N sufficiently large and using the results in Lemmas
3.4 and 5.2, one can enforce hxs ((N + 1)τ ) and η in (5.9) to be sufficiently small. Hence, it can be readily
seen that for a given precision ε, there always exists a sufficiently small value of µ and a large value of N , such
that the condition in (5.9) is satisfied. A result similar as that in Theorem 5.7 can be recovered for a δ-ISS
non-probabilistic control system Σ, as provided in the following corollary.
Corollary 5.8. Consider a non-probabilistic control system Σ admitting a δ-ISS Lyapunov function V such
that (5.1) holds for some γ
b ∈ K∞ . Let η be given by (5.3). For any ε ∈ R+ and any tuple q = (τ, µ, N, xs ) of
parameters satisfying µ ≤ span(U) and
e−κτ α (ε) +

1
ρ(µ) + γ
b (η) ≤ α (ε) ,
eκ

(5.10)

the relation


R = (xτ , xq ) ∈ Xτ × Xq | V xτ , H q (xq ) ≤ α (ε)
is an ε-approximate bisimulation relation between S q (Σ) and Sτ (Σ).



TOWARDS SCALABLE SYNTHESIS OF STOCHASTIC CONTROL SYSTEMS

11

The proof is similar to the one of Theorem 5.7. In order to mitigate the conservativeness that might result
from using Lyapunov functions, the next theorem provides a result that is similar to the one of Theorem 5.7,
which is however not obtained by explicit use of δ-ISS-Mq Lyapunov functions, but by using functions β and
γ as in (3.1).
Theorem 5.9. Consider a δ-ISS-Mq stochastic control system Σ, satisfying the result of Lemma 3.4. Let η
be given by (5.3). For any ε ∈ R+ , and any tuple q = (τ, µ, N, xs ) of parameters satisfying µ ≤ span(U) and
1

1

(β (εq , τ ) + γ(µ)) q + (hxs ((N + 1)τ )) q + η ≤ ε,

(5.11)

the relation

R=


 h
i q1
q
(xτ , xq ) ∈ Xτ × Xq | E xτ − H q (xq )
≤ε

is an ε-approximate bisimulation relation between S q (Σ) and Sτ (Σ).



The proof can be found in the Appendix. By choosing N sufficiently large and using the results in Lemmas
3.4 and 5.3, one can force hxs ((N + 1)τ ) and η in (5.11) to be sufficiently small. Hence, it can be readily seen
that for a given precision ε, there always exist a sufficiently large value of τ and N and a small enough value of
µ such that the condition in (5.11) is satisfied. However, unlike the result in Theorem 5.7, notice that here for
a given fixed sampling time τ , one may not find any values of N and µ satisfying (5.11) because the quantity
1
(β (εq , τ )) q may be larger than ε. The symbolic model S q (Σ), computed using the parameter q provided in
Theorem 5.9 (whenever existing), is likely to have fewer states than the model computed using the parameter
q provided in Theorem 5.7 – a similar fact has been experienced in the first example in [ZEM+ 14]. A result
similar to the one in Theorem 5.9 can be fully recovered for a δ-ISS non-probabilistic control system Σ, as
provided in the following corollary.
Corollary 5.10. Consider a δ-ISS non-probabilistic control system Σ. Let η be given by (5.3). For any
ε ∈ R+ , and any tuple q = (τ, µ, N, xs ) of parameters satisfying µ ≤ span(U) and3
β (ε, τ ) + γ(µ) + η ≤ ε,

(5.12)

the relation

R = (xτ , xq ) ∈ Xτ × Xq |

xτ − H q (xq ) ≤ ε

is an ε-approximate bisimulation relation between S q (Σ) and Sτ (Σ).



The proof is similar to the one of Theorem 5.9. The next theorems provide results that are similar to those
of Theorems 5.7 and 5.9, but by using the symbolic model Sq (Σ) with probabilistic output values rather than
S q (Σ) with non-probabilistic output values.
Theorem 5.11. Consider a stochastic control system Σ, admitting a δ-ISS-Mq Lyapunov function V such that
(5.1) holds for some concave γ
b ∈ K∞ . Let ηb be given by (5.6). For any ε ∈ R+ and any tuple q = (τ, µ, N, xs )
of parameters satisfying µ ≤ span(U) and
e−κτ α (εq ) +

1
ρ(µ) + γ
b (b
η ) ≤ α (εq ) ,
eκ

(5.13)

the relation
R = {(xτ , xq ) ∈ Xτ × Xq | E [V (xτ , Hq (xq ))] ≤ α (εq )}
is an ε-approximate bisimulation relation between Sq (Σ) and Sτ (Σ).
The proof is similar to the one of Theorem 5.7.
3Here, β and γ are the KL and K
∞ functions, respectively, appearing in (3.2).



12

M. ZAMANI, I. TKACHEV, AND A. ABATE

Theorem 5.12. Consider a δ-ISS-Mq stochastic control system Σ. Let ηb be given by (5.6). For any ε ∈ R+ ,
and any tuple q = (τ, µ, N, xs ) of parameters satisfying µ ≤ span(U) and
1

(β (εq , τ ) + γ(µ)) q + ηb ≤ ε,
the relation

(5.14)

o
n
q 1
R = (xτ , xq ) ∈ Xτ × Xq | (E [kxτ − Hq (xq )k ]) q ≤ ε

is an ε-approximate bisimulation relation between Sq (Σ) and Sτ (Σ).



The proof is similar to the one of Theorem 5.9.
Remark 5.13. The symbolic model Sq (Σ), computed using the parameter q provided in Theorem 5.11 (resp.
Theorem 5.12), has fewer (or at most equal number of ) states than the symbolic model S q (Σ), computed by
using the parameter q provided in Theorem 5.7 (resp. Theorem 5.9) while having the same precision. However,
the symbolic model Sq (Σ) has states with probabilistic output values, rather than non-probabilistic ones, which
is likely to require more involved control synthesis procedures (cf. Subsection 5.3).

Remark 5.14. Although we assume that the set U is infinite, Theorems 5.7, 5.9, 5.11, and 5.12 and Corollaries
5.8 and 5.10 still hold when the set U is finite, with the following modifications. First, the systems Σ and Σ
are required to satisfy the properties (3.1) and (3.2), respectively, for υ = υ 0 . Second, take Uq = U in the
definitions of S q (Σ) (resp. Sq (Σ)) and S q (Σ). Finally, in the conditions (5.9), (5.10), (5.11), (5.12), (5.13),
and (5.14) set µ = 0.

Finally, we establish the results on the existence of symbolic model S q (Σ) (resp. Sq (Σ)) such that S q (Σ) ∼
=εS Sτ (Σ)
ε
ε
(resp. Sq (Σ) ∼
=S Sτ (Σ)) and S q (Σ) ∼
=S Sτ (Σ).
Theorem 5.15. Consider the results in Theorem 5.7. If we select
n
o
1
Xτ 0 = x ∈ Rn | x − H q (xq0 ) ≤ α−1 (α (εq )) q , ∃xq0 ∈ Xq0 ,
then we have S q (Σ) ∼
=εS Sτ (Σ).



Proof. We start by proving that Sτ (Σ) εS S q (Σ). For every xτ 0 ∈ Xτ 0 there always exists xq0 ∈ Xq0 such
1
that kxτ 0 − H q (xq0 )k ≤ α−1 (α (εq )) q . Then,



E V xτ 0 , H q (xq0 ) = V xτ 0 , H q (xq0 ) ≤ α(kxτ 0 − H q (xq0 )kq ) ≤ α (εq ) ,
since α is a K∞ function. Hence, (xτ 0 , xq0 ) ∈ R implying that Sτ (Σ) εS S q (Σ). In a similar way, we can show
that S q (Σ) εS Sτ (Σ) which completes the proof.

The next theorem provides a similar result in line with the one of previous theorem, but by using a different
relation.
Theorem 5.16. Consider the results in Theorem 5.9. If we select

Xτ 0 = x ∈ Rn | x − H q (xq0 ) ≤ ε, ∃xq0 ∈ Xq0 ,
then we have S q (Σ) ∼
=εS Sτ (Σ).



Proof. We start by proving that Sτ (Σ) εS S q (Σ). For every xτ 0 ∈ Xτ 0 there always exists xq0 ∈ Xq0 such that
 h
i q1
q
kxτ 0 −H q (xq0 )k ≤ ε and E xτ 0 − H q (xq0 )
≤ ε. Hence, (xτ 0 , xq0 ) ∈ R implying that Sτ (Σ) εS S q (Σ).
In a similar way, we can show that S q (Σ) εS Sτ (Σ) which completes the proof.



The next two corollaries provide similar results as the ones of Theorems 5.15 and 5.16, but for non-probabilistic
control systems Σ.

TOWARDS SCALABLE SYNTHESIS OF STOCHASTIC CONTROL SYSTEMS

Corollary 5.17. Consider the results in Corollary 5.8. If we select


Xτ 0 = x ∈ Rn | x − H q (xq0 ) ≤ α−1 (α (ε)) , ∃xq0 ∈ Xq0 ,
then we have S q (Σ) ∼
=ε Sτ (Σ).
S

13



The proof is similar to the one of Theorem 5.15.
Corollary 5.18. Consider the results in Corollary 5.10. If we select

Xτ 0 = x ∈ Rn | x − H q (xq0 ) ≤ ε, ∃xq0 ∈ Xq0 ,
then we have S q (Σ) ∼
=ε Sτ (Σ).
S



The proof is similar to the one of Theorem 5.16. The next two theorems provide similar results as the ones of
Theorems 5.15 and 5.16, but by using the symbolic model Sq (Σ).
Theorem 5.19. Consider the results in Theorem 5.11. Let A denote the set of all Rn -valued random variables,
measurable over F0 . If we select
n
o
1
q 1
Xτ 0 = a ∈ A| (E [ka − Hq (xq0 )k ]) q ≤ α−1 (α (εq )) q , ∃xq0 ∈ Xq0 ,
then we have Sq (Σ) ∼
=εS Sτ (Σ).



The proof is similar to the one of Theorem 5.15.
Theorem 5.20. Consider the results in Theorem 5.12. Let A denote the set of all Rn -valued random variables,
measurable over F0 . If we select
n
o
q 1
Xτ 0 = a ∈ A | (E [ka − Hq (xq0 )k ]) q ≤ ε, ∃xq0 ∈ Xq0 ,
then we have Sq (Σ) ∼
=εS Sτ (Σ).



The proof is similar to the one of Theorem 5.16.
5.3. Control synthesis over Sq (Σ). Note that both S q (Σ) and Sq (Σ) are finite systems. The only difference
is that the outputs of the former system are always non-probabilistic points, whereas those of the latter can
be non-degenerate random variables. Let us describe the control synthesis for these systems over quantitative
specifications, and for example over the safety formula ϕA , for A ⊂ Rn ⊂ Y (as already been used in
Subsection 5.1). Clearly, since the original system Sτ (Σ) is stochastic in the sense that its outputs are nondegenerate random variables similarly to Sq (Σ), it would be too conservative to require that it satisfies the
formula exactly. Thus, we are rather interested in an input policy that makes Sτ (Σ) satisfy ϕAε with some
ε > 0: recall from Subsection 5.1 that the latter LTL formula can be satisfied by non-degenerate random
variables, in contrast to ϕA . Let us recap how to use abstractions for this task, and let us start with S q (Σ)
belonging to a more familiar type of systems whose outputs are non-probabilistic.
We label a state xq of S q (Σ) with A if H q (xq ) ∈ A and, say, with B otherwise. As a result, we obtain a
transition system with labels over the states and can synthesize a control strategy by solving a safety game
[Tab09] that makes an output run of S q (Σ) satisfy ϕA . After that, we can exploit ε-approximate bisimilarity
to guarantee that the refined input policy makes the corresponding output run of the original system satisfy
ϕAε .
The main subtlety in the case of Sq (Σ) is how to label its states. We cannot do this as for S q (Σ), since Hq (xq )
may never be an element of A for any xq ∈ Xq : indeed, the latter is a set of non-probabilistic points, whereas
all the outputs of Sq (Σ) can happen to be non-degenerate random variables. In order to cope with this issue,
we propose to relax the original problem and at the same time to strengthen the quality of the abstraction.
Namely, we can consider a relaxed problem ϕAδ over the abstraction Sq (Σ), for some δ ∈]0 ε[, where the
latter is now required to be (ε − δ)-approximate (rather than just ε-approximate) bisimilar to the original

14

M. ZAMANI, I. TKACHEV, AND A. ABATE

system. Clearly (Aδ )ε−δ ⊆ Aε , so that whenever the control policy for ϕAδ is synthesized over Sq (Σ), its
refined version is guaranteed to enforce ϕAε over the original system. Thanks to the fact that Aδ contains
non-degenerate random variables, we eliminate the conservativeness presented before in the sense that it is
likely that there are now points xq ∈ Xq in Sq (Σ) such that Hq (xq ) ∈ Aδ . The only remaining question is how
to check whether Hq (xq ) ∈ Aδ . To answer this question, we check that the distance
1/q
d (Hq (xq ), A) = inf Ekξxs xq (N τ ) − akq
(5.15)
a∈A

is smaller than δ, which involves both computing the expectation over the solution of the SDE, and optimizing
the value of this expectation. Clearly, such a computation in general cannot be done analytically, and the
evaluation of the expectation itself is a highly non-trivial task unless the SDE has a very special form.
We propose a Monte Carlo approach to compute an approximation of the quantity in (5.15) by means of
empirical expectations. Using such an approach, we can estimate d (Hq (xq ), A) only up to some precision, say
θ. If the estimated distance is less than δ −θ, we are safe to label xq with A, whereas all other states are labeled
by B. Furthermore, since this result is based on a Monte Carlo method, it holds true only with a certain
confidence level 1 − π where π ∈ [0 1]. The benefit of our approach is that it is not only valid asymptotically
(as the number of samples grows to infinity), but we are also able to provide a number of simulations that is
sufficient to estimate d (Hq (xq ), A) with any given precision θ and with any given confidence 1 − π. This can
be considered as an extension of the well-known Hoeffding’s inequality [Hoe63] to the case when one has to
deal with an optimization problem. Note that regardless of the specification of interest, the main task over
Sq (Σ) is always to compute some distance as in (5.15) for any set that appears in the specification, so the
method below applies not only to the safety formula ϕA , but also to more general formulae, which are left
as object of the future research.
n
r
n
Suppose
S that A as in (5.15) is a compact subset of R , and let A be the smallest subset of [R ]r such that
A ⊆ p∈Ar B r2 (p). Let M be the number of samples and let

drM

:= minr
a∈A

M
1 X i
ξ
(N τ ) − a
M i=1 xs xq

q

! q1
,

where the superscript i denotes the index of samples. Now, we have the following theorem.
Theorem 5.21. For any stochastic control system Σ one has |d (Hq (xq ), A) − drM | ≤ θ with confidence of at
least 1 − π, given that r < 2θ and that
|Ar |b(a∗ , 2q)
M≥
,
π(θ − r/2)2q
where b(a, p) := (1 + |xs − a|p )ep(p+1) max{Lx ,Z}N τ and a∗ ∈ arg maxa0 ∈Ar kxs − a0 k.
The proof can be found in the Appendix. Let us make some comments on Theorem 5.21. First of all, no
matter how many distances one has to evaluate, one can always use the same samples ξ i and there is no need
to generate new samples. Second, to the best of our knowledge, logarithmic bounds on M (as per [KP92]) are
not available in this general case due to the fact that we deal with an unbounded state space.
5.4. Relationship with existing results in the literature. Note that given any precision ε and sampling
time τ , one can always use the results in Theorem 5.15 to construct a symbolic model S q (Σ) that is εapproximate bisimilar to Sτ (Σ) without any state set discretization. Note that the results in Theorem 5.1
in [ZEM+ 14] also provide symbolic models that are ε-approximate bisimilar to Sτ (Σ). However, the results
in [ZEM+ 14] require state set discretization and cannot be applied for any sampling time τ if the precision
ε is lower than the thresholds introduced in inequality (5.5) in [ZEM+ 14]. Furthermore, while the results in
[ZEM+ 14] only provide symbolic models with non-probabilistic output values, the ones in this work provide
symbolic models with probabilistic output values as well, which can result in less conservative symbolic models
(cf. Remark 5.13).

TOWARDS SCALABLE SYNTHESIS OF STOCHASTIC CONTROL SYSTEMS

15

One can compare the results provided in Theorems 5.7 (corr. 5.15) and 5.9 (corr. 5.16) with the results
provided in Theorems 5.1 and 5.3 in [ZEM+ 14] in terms of the size of the generated symbolic models.
One can readily verify that the precisions of the symbolic models proposed here and the ones proposed
in [ZEM+ 14] are approximately the same as long as both use the same input set quantization parameter µ
and the state space quantization parameter, called ν, in [ZEM+14] is equal to the parameter η in (5.3), i.e.
1/q
, where η0 = maxuq ∈Uq V ξ xs uq (τ ), xs . The reason their precisions are approxiν ≤ α−1 e−κN τ η0
mately (rather than exactly) the same is because we use hxs (σ, (N + 1)τ ) in conditions (5.9) and (5.11) in
this paper rather than h(τ ) = supx∈D hx (τ ) that is being used in conditions 5.4 and 5.14 in [ZEM+ 14] for
1
1
a compact set D ⊂ Rn . By assuming that hxs (σ, (N + 1)τ ) q and h(τ ) q are much smaller than η and ν,
respectively, or hxs (σ, (N + 1)τ ) ≈ h(τ ), one should expect to obtain the same precisions for the symbolic
models provided here and those provided in [ZEM+ 14] under the aforementioned conditions.
N

The number of states of the proposed symbolic model in this paper is [U]µ . Assume that we are interested
in the dynamics of Σ on a compact set D ⊂ Rn . Since the set of states of the proposed symbolic model
in [ZEM+ 14] is [D]ν , its size is |[D]ν | = νKn , where K is a positive constant proportional to the volume of
D. Hence, it is more convenient to use the proposed symbolic model here rather than the one proposed in
[ZEM+ 14] as long as:
K

N

[U]µ

≤

(α−1

(e−κN τ η

n/q
0 ))

.

Without loss of generality, one can assume that α(r) = r for any r ∈ R+
0 . Hence, for sufficiently large value
of N , it is more convenient to use the proposed symbolic model here in comparison with the one proposed in
[ZEM+ 14] as long as:
[U]µ e

−κτ n
q

≤ 1.

(5.16)

Note that the methodology proposed in this paper allows us to construct less conservative symbolic models
with probabilistic output values while the proposed one in [ZEM+ 14] only provides conservative symbolic
models with non-probabilistic output values.
6. Example
We show the effectiveness of the results presented in this work by constructing a bisimilar symbolic model
for the model of a road network, which is divided in 5 cells of 250 meters with 2 entries and 2 ways out, as
depicted schematically in Figure 2. The model is borrowed from [CGG13], however it is now affected by noise
and newly described in continuous time.

u1

1

2

3

4

5

u2

Figure 2. Model of a road divided in 5 cells with 2 entries and 2 ways out.
The two entries are controlled by traffic lights, denoted by u1 and u2 , that enable (green light) or not (red
light) the vehicles to pass. In this model the length of a cell is in kilometres (0.25 km), and the flow speed of

16

M. ZAMANI, I. TKACHEV, AND A. ABATE

the vehicles is 70 kilometres per hour (km/h). Moreover, during the sampling time interval τ , it is assumed
that 6 vehicles pass the entry controlled by the light u1 , 8 vehicles pass the entry controlled by the light u2 ,
and one quarter of vehicles that leave cell 1 goes out on the first exit. We assume that both lights cannot be
red at the same time. The model of Σ is described by:
Σ : {d ξ = (Aξ + Bυ) d t + ξ d Wt ,

(6.1)

where


−0.0541
0.3224
−0.7636
2.1122
−6.2132

0
−0.1370
0.3224
−0.7636
2.1122

0
0
−0.0541
0.1260
−0.2205

0.0696
 −0.2743

B = 104 × 
 0.7075
 −2.0081
5.9802

0
0.1402
−0.2743
0.7075
−2.0081

0
0
0.0696
−0.0924
0.1911



A = 10 × 


4




0

0

,
0


0
−0.0541

0
0

0
0

,
0
0


0.0696
0
−0.0924 0.0696

0
0
0
−0.0541
0.1260

U = {u0 , u1 , u2 } = {[6 0 8 0 0]T , [6 0 0 0 0]T , [0 0 8 0 0]T }, and ξi is the number of vehicles in cell i of the road.
Note that Uτ contains curves taking values in U. Since U is finite, as explained in Remark 5.14, µ = 0 is to be
used in (5.9), (5.11), (5.13), and (5.14). One can readily verify that the function V (x, x0 ) = (x − x0 )T P (x − x0 ),
for any x, x0 ∈ R5 , where


76763.4393 −2101.1583 3790.9182 −155.6576 −125.9871
 −2101.1583 10676.9437 1237.3552
−86.6855
100.5718 


4

1237.3552 1823.02431 171.1549
−71.1162 
P = 10 ×  3790.9182
,
 −155.6576
−86.6855
171.1549
229.2134
−5.5649 
−125.9871
100.5718
−71.1162
−5.5649
33.3977
satisfies conditions (i)-(iii) in Definition 3.2 with q = 2, κ = 300, α(r) = 1/2λmin (P )r, α(r) = 1/2λmax (P )r,
ρ(r) = 5kBk2 kP k/(2κ)r2 , ∀r ∈ R+
0 . Hence, Σ is δ-ISS-M2 , equipped with the δ-ISS-M2 Lyapunov function V .
Using the results ofTheorem 3.3, provided in [ZEM+ 14], one gets that functions β(r, s) = α−1 (α(r)e−κs ) and
1
γ(r) = α−1 eκ
ρ(r) satisfy property (3.1) for Σ. We choose the source state as the one proposed in [CGG13],
i.e. xs = [3.8570 3.3750 3.3750 8.5177 8.5177]T .
For a given precision ε = 0.5 and fixed sampling time τ = 0.00277 h (10 sec), the parameter N for S q (Σ),
based on inequality (5.9) in Theorem 5.7, is obtained as 14. Therefore, the resulting cardinality of the set of
14
states for S q (Σ) is |U| = 314 = 4782969. Using the aforementioned parameters, one gets η ≤ 6.0776 × 10−6 ,
where η is given in (5.3). Note that the results in Theorems 5.9 and 5.12 cannot be applied here because
−κτ n
1
(β(εq , τ )) q > ε. Using criterion (5.16), one has |U| e q = 0.37, implying that the approach proposed in this
paper is more appropriate in terms of the size of the abstraction than the one proposed in [ZEM+ 14]. We
elaborate more on this at the end of the section.
Remark 6.1. By considering the non-probabilistic control system Σ and using the results in Corollary 5.8 and
the same parameters q as the ones in S q (Σ), one obtains ε = 0.01 in (5.10). Therefore, as expected, S q (Σ)
(i.e. symbolic model for the non-probabilistic control system Σ) provides much smaller precision than S q (Σ)
(i.e. symbolic model for the stochastic control system Σ) while having the same size as S q (Σ).
Now the objective, as inspired by the one suggested in [CGG13], is to design a schedule for the coordination of
traffic lights enforcing Σ to satisfy a safety and a fairness property. The safety part is to keep the density of
traffic lower than 16 vehicles per cell which can be encoded via the LTL specification4 2ϕW , where W = [0 16]5 .
The fairness part requires to alternate the accesses between the two traffic lights and to allow only 3 identical
4Note that the semantics of LTL are defined over the output behaviors of S (Σ).
q

TOWARDS SCALABLE SYNTHESIS OF STOCHASTIC CONTROL SYSTEMS

17

consecutive modes of red light ensuring fairness between two traffic lights. Starting from the initial condition
x0 = [1.417 4.993 10.962 9.791 14.734]T , we obtain a periodic schedule υ = (u0 u0 u0 u2 u1 u0 u0 u2 u1 u0 u0 u2 u1 u2 )ω
keeping u0 as much as possible in each period in order to maximize number of vehicles accessing the road.
Figure 3 displays a few realizations of the closed-loop solution process ξx0 υ . In Figure 3 bottom right, we
show the average value (over 100000 experiments) of the distance (in the 2nd moment metric) in time of
the solution process ξx0 υ to the set W , namely kξx0 υ (t)kW , where the point-to-set distance is defined as
kxkW = inf w∈W kx − wk. Notice that the empirical average distance is as expected lower than the precision
ε = 0.5.

8
6

15

4

4

0.15

0

16

14

14

0.1
Time

0.1
Time

0.15

ε
0.4

0

(E[||ξ

10

8

0.05

x υ

12

10

5
0

0.15

ξ5

ξ4

12

0.05

W

0.1
Time

(t)||2 ])1/2

0.05

16

6
0

10

2

2
0

ξ3

2

ξ

ξ1

6

0.2

8
0.05

0.1
Time

0.15

0

0.05

0.1
Time

0.15

0
0

0.05

0.1
Time

0.15

Figure 3. A few realizations of the closed-loop solution process ξx0 υ (top panel and the
first two figures from the left in the bottom panel) and the average values (over 100000
experiments) of the distance of the solution process ξx0 υ to the set W (bottom right panel).
To compute exactly the size of the symbolic model, proposed in Theorem 5.1 in [ZEM+ 14], we consider the
dynamics of Σ over the subset D = [0 16]5 of R5 . Note that Theorem 5.3 in [ZEM+ 14] cannot be applied
1
here because (β(εq , τ )) q > ε. Using the same precision ε = 0.5 and sampling time τ = 0.00277 as the ones
here, and the inequalities (5.3) and (5.4) in [ZEM+ 14], we obtain the state space quantization parameter as
ν ≤ 6.0776 × 10−6 . Therefore, if one uses ν = 6.0776 × 10−6 , the cardinality of the state set of the symbolic
5
model, provided by the results in Theorem 5.1 in [ZEM+ 14], is equal to 16
= 1.2645 × 1032 which is much
ν
higher than the one proposed here, amounting instead to 4782969 states.
7. Conclusions
In this paper we have proposed a symbolic abstraction technique for incrementally stable stochastic control
systems (and corresponding non-probabilistic model), which features only the discretization of the input set.
The proposed approach is potentially more scalable than the one proposed in [ZEM+ 14] for higher dimensional
stochastic control systems.
Future work will concentrate on efficient implementations of the symbolic models proposed in this work using
Binary Decision Diagrams, on top of the recently developed synthesis toolbox SCOTS [RZar], as well as on
more efficient controller synthesis techniques.

18

M. ZAMANI, I. TKACHEV, AND A. ABATE

References
[Ang02]
[BK08]
[CGG13]

[dAHM01]
[GP07]
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[LAB09]
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[MNA03]
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[Oks02]
[PAV+ 13]
[PGT08]
[PT09]
[Rei11]
[RWR15]
[RZar]

[Tab09]
[TI09]

[ZA14]
[ZAG15]
[ZEAL13]
[ZEM+ 14]

[ZPJT12]

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[ZTA14]

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pages 41–50. ACM New York, NY, USA, April 2014.

Appendix
Proof. of Lemma 5.2: Let xq ∈ Xq , where xq = (u1 , u2 , . . . , uN ), and uq ∈ Uq . Using the definition of S q (Σ),
one obtains x0q = (u2 , . . . , uN , uq ) ∈ Postuq (xq ). Since V is a δ-ISS-Mq Lyapunov function for Σ, we have:

 q

≤ V (ξ H q (xq )uq (τ ), H q x0q )
α ξ H q (xq )uq (τ ) − H q x0q
= V (ξ ξx

s xq

(N τ )uq (τ ), ξ xs x0q (N τ ))

= V (ξ ξx

s u1

(τ )(u2 ,...,uN ,uq ) (N τ ), ξ xs (u2 ,...,uN ,uq ) (N τ ))

≤ e−κN τ V (ξ xs u1 (τ ), xs ).

(7.1)

We refer the interested readers to the proof of Theorem 3.3 in [ZEM+ 14] to see how we derived the inequality
(7.1). Hence, one gets

(7.2)
kξ H q (xq )uq (τ ) − H q x0q k ≤ (α−1 (e−κN τ V (ξ xs u1 (τ ), xs )))1/q ,
because of α ∈ K∞ . Since the inequality (7.2) holds for all xq ∈ Xq and uq ∈ Uq , and α ∈ K∞ , inequality
(5.2) holds.

Proof. of Lemma 5.3: Let xq ∈ Xq , where xq = (u1 , u2 , . . . , uN ), and uq ∈ Uq . Using the definition of S q (Σ),
one obtains x0q = (u2 , . . . , uN , uq ) ∈ Postuq (xq ). Since Σ is δ-ISS-Mq and using inequality (3.1), we have:

kξ H q (xq )uq (τ ) − H q x0q kq = kξ ξx x (N τ )uq (τ ) − ξ xs x0q (N τ )kq
s q

= kξ ξx

s u1

(τ )(u2 ,...,uN ,uq ) (N τ )

− ξ xs (u2 ,...,uN ,uq ) (N τ )kq ≤ β(kξ xs u1 (τ ) − xs kq , N τ ).

Hence, one gets
kξ H q (xq )uq (τ ) − H q (x0q )k ≤ (β(kξ xs u1 (τ ) − xs kq , N τ ))1/q .

(7.3)

Since the inequality (7.3) holds for all xq ∈ Xq and all uq ∈ Uq , and β is a K∞ function with respect to its
first argument when the second one is fixed, inequality (5.4) holds.

Proof. of Theorem 5.7: We start by proving that R is an ε-approximate simulation relation from Sτ (Σ) to
S q (Σ). Consider any (xτ , xq ) ∈ R. Condition (i) in Definition 4.2 is satisfied because
1

1

(E[kxτ − H q (xq )kq ]) q ≤ (α−1 (E[V (xτ , H q (xq ))])) q ≤ ε.

(7.4)

We used the convexity assumption of α and the Jensen inequality [Oks02] to show the inequalities in (7.4).
Let us now show that condition (ii) in Definition 4.2 holds. Consider any υτ ∈ Uτ . Choose an input uq ∈ Uq
satisfying
kυτ − uq k∞ = kυτ (0) − uq (0)k ≤ µ.
(7.5)
Note that the existence of such uq is guaranteed by U being a finite union of boxes and by the inequality
S
υτ
- x0τ = ξx υ (τ ) P-a.s.
µ ≤ span(U) which guarantees that U ⊆ p∈[U]µ Bµ (p). Consider the transition xτ
τ τ
τ

in Sτ (Σ). Since V is a δ-ISS-Mq Lyapunov function for Σ and using inequality (7.5), we have (cf. equation
(3.3) in [ZEM+ 14])
E[V (x0τ , ξH q (xq )uq (τ ))] ≤ E[V (xτ , H q (xq ))]e−κτ +

1
1
ρ(kυτ − uq k∞ ) ≤ α (εq ) e−κτ + ρ(µ).
eκ
eκ

(7.6)

20

M. ZAMANI, I. TKACHEV, AND A. ABATE

Note that existence of uq , by the definition of S q (Σ), implies the existence of xq

uq

- x0q in S q (Σ). Using

q

Lemma 3.4, the concavity of γ
b, the Jensen inequality [Oks02], equation (5.3), the inequalities (5.1), (5.9),
(7.6), and triangle inequality, we obtain
E[V (x0τ , H q (x0q ))] = E[V (x0τ , ξH q (xq )uq (τ )) + V (x0τ , H q (x0q )) − V (x0τ , ξH q (xq )uq (τ ))]
= E[V (x0τ , ξH q (xq )uq (τ ))] + E[V (x0τ , H q (x0q )) − V (x0τ , ξH q (xq )uq (τ ))]
1
ρ(µ) + E[b
γ (kξH q (xq )uq (τ ) − H q (x0q )k)]
eκ
1
≤ α(εq )e−κτ + ρ(µ) + γ
b(E[kξH q (xq )uq (τ ) − ξ H q (xq )uq (τ ) + ξ H q (xq )uq (τ ) − H q (x0q )k])
eκ
1
≤ α(εq )e−κτ + ρ(µ) + γ
b(E[kξH q (xq )uq (τ ) − ξ H q (xq )uq (τ )k] + kξ H q (xq )uq (τ ) − H q (x0q )k)
eκ
1
1
≤ α(εq )e−κτ + ρ(µ) + γ
b((hxs ((N + 1)τ )) q + η) ≤ α(εq ).
eκ

Therefore, we conclude that x0τ , x0q ∈ R and that condition (ii) in Definition 4.2 holds.
≤ α(εq )e−κτ +

Now we prove that R−1 is an ε-approximate simulation relation from S q (Σ) to Sτ (Σ). Consider any (xτ , xq ) ∈ R
(or equivalently (xq , xτ ) ∈ R−1 ). As showed in the first part of the proof, condition (i) in Definition 4.2 is
satisfied. Let us now show that condition (ii) in Definition 4.2 holds. Consider any uq ∈ Uq . Choose the
input υτ = uq and consider x0τ = ξxτ υτ (τ ) P-a.s. in Sτ (Σ). Since V is a δ-ISS-Mq Lyapunov function for Σ,
one obtains (cf. equation 3.3 in [ZEM+ 14]):
E[V (x0τ , ξH q (xq )uq (τ ))] ≤ e−κτ E[V (xτ , H q (xq ))] ≤ e−κτ α (εq ) .

(7.7)

Using Lemma 3.4, the definition of S q (Σ), the concavity of γ
b, the Jensen inequality [Oks02], equation (5.3),
the inequalities (5.1), (5.9), (7.7), and triangle inequality, we obtain
E[V (x0τ , H q (x0q ))] = E[V (x0τ , ξH q (xq )uq (τ )) + V (x0τ , H q (x0q )) − V (x0τ , ξH q (xq )uq (τ ))]
= E[V (x0τ , ξH q (xq )uq (τ ))] + E[V (x0τ , H q (x0q )) − V (x0τ , ξH q (xq )uq (τ ))]
≤ e−κτ α(εq ) + E[b
γ (kξH q (xq )uq (τ ) − H q (x0q )k)]
b(E[kξH q (xq )uq (τ ) − ξ H q (xq )uq (τ ) + ξ H q (xq )uq (τ ) − H q (x0q )k])
≤ e−κτ α(εq ) + γ
≤ e−κτ α(εq ) + γ
b(E[kξH q (xq )uq (τ ) − ξ H q (xq )uq (τ )k] + kξ H q (xq )uq (τ ) − H q (x0q )k)
1

≤ e−κτ α(εq ) + γ
b((hxs ((N + 1)τ )) q + η) ≤ α(εq ).

Therefore, we conclude that (x0τ , x0q ) ∈ R (or equivalently x0q , x0τ ∈ R−1 ) and condition (ii) in Definition 4.2
holds.

Proof. of Theorem 5.9: We start by proving that R is an ε-approximate simulation relation from Sτ (Σ) to
S q (Σ). Consider any (xτ , xq ) ∈ R. Condition (i) in Definition 4.2 is satisfied by the definition of R. Let us now
show that condition (ii) in Definition 4.2 holds. Consider any υτ ∈ Uτ . Choose an input uq ∈ Uq satisfying
kυτ − uq k∞ = kυτ (0) − uq (0)k ≤ µ.

(7.8)

Note that the existence of such uq is guaranteed by U being a finite union of boxes and by the inequality
S
υτ
- x0τ = ξxτ υτ (τ ) P-a.s.
µ ≤ span(U) which guarantees that U ⊆ p∈[U]µ Bµ (p). Consider the transition xτ
τ

in Sτ (Σ). It follows from the δ-ISS-Mq assumption on Σ and (7.8) that:
E[kx0τ − ξH q (xq )uq (τ )kq ] ≤ β(E[kxτ − H q (xq )kq ], τ ) + γ(kυτ − uq k∞ ) ≤ β(εq , τ ) + γ(µ).

(7.9)

TOWARDS SCALABLE SYNTHESIS OF STOCHASTIC CONTROL SYSTEMS

21

uq

- x0q in S q (Σ). Using equation (5.3),

Existence of uq , by the definition of S q (Σ), implies the existence of xq

q

the inequalities (3.3), (5.11), (7.9), and triangle inequality, we obtain
1

1

(E[kx0τ − H q (x0q )kq ]) q = (E[kx0τ − ξH q (xq )uq (τ ) + ξH q (xq )uq (τ ) − ξ H q (xq )uq (τ ) + ξ H q (xq )uq (τ ) − H q (x0q )kq ]) q
1

1

≤ (E[kx0τ − ξH q (xq )uq (τ )kq ]) q + (E[kξH q (xq )uq (τ ) − ξ H q (xq )uq (τ )kq ]) q
1

+ (E[kξ H q (xq )uq (τ ) − H q (x0q )kq ]) q
1

1

≤ (β(εq , τ ) + γ(µ)) q + (hxs ((N + 1)τ )) q + η ≤ ε.

Therefore, we conclude that x0τ , x0q ∈ R and that condition (ii) in Definition 4.2 holds.
Now we prove that R−1 is an ε-approximate simulation relation from S q (Σ) to Sτ (Σ). Consider any (xτ , xq ) ∈ R
(or equivalently (xq , xτ ) ∈ R−1 ). Condition (i) in Definition 4.2 is satisfied by the definition of R. Let us now
show that condition (ii) in Definition 4.2 holds. Consider any uq ∈ Uq . Choose the input υτ = uq and consider
x0τ = ξxτ υτ (τ ) P-a.s. in Sτ (Σ). Since Σ is δ-ISS-Mq , one obtains:
E[kx0τ − ξH q (xq )uq (τ )kq ] ≤ β(E[kxτ − H q (xq )kq ], τ ) ≤ β(εq , τ ).

(7.10)

Using definition of S q (Σ), equation (5.3), the inequalities (3.3), (5.11), (7.10), and the triangle inequality, we
obtain
1

1

(E[kx0τ − H q (x0q )kq ]) q = (E[kx0τ − ξH q (xq )uq (τ ) + ξH q (xq )uq (τ ) − ξ H q (xq )uq (τ ) + ξ H q (xq )uq (τ ) − H q (x0q )kq ]) q
1

1

≤ (E[kx0τ − ξH q (xq )uq (τ )kq ]) q + (E[kξH q (xq )uq (τ ) − ξ H q (xq )uq (τ )kq ]) q
1

+ (E[kξ H q (xq )uq (τ ) − H q (x0q )kq ]) q
1

1

≤ (β(εq , τ )) q + (hxs ((N + 1)τ )) q + η ≤ ε.

Therefore, we conclude that (x0τ , x0q ) ∈ R (or equivalently x0q , x0τ ∈ R−1 ) and condition (ii) in Definition 4.2
holds.


Proof. of Theorem 5.21: Denote θ̂ := θ − r/2 > 0, and dM (a) :=

1
M

follows from [KP92, Theorem 4.5.4] that for all p ≥ 1 and a ∈ Rn


E kξxs xq (N τ ) − akp ≤ b(a, p).

M
P
i=1

kξxi s xq − akq

 q1

for all a ∈ Rn . It

Hence, by Chernoff’s inequality for any a0 ∈ Ar we obtain:

 b(a0 , 2q)
q
.
P (d(Hq (xq ), a0 )) − (dM (a0 ))q ≥ θ̂ ≤
M θ̂2
Furthermore, since x 7→ xq is Hölder continuous with power q,

 b(a0 , 2q)
P |d(Hq (xq ), a0 ) − dM (a0 )| ≥ θ̂ ≤
.
M θ̂2q
Thus, for the union of such events over a0 ∈ Ar , we have

 |Ar |b(a∗ , 2q)
P ∃a0 ∈ Ar s.t. |d(Hq (xq ), a0 ) − dM (a0 )| ≥ θ̂ ≤
,
(7.11)
M θ̂2q
due to the fact that the probability of a union is dominated by the sum of probabilities. Let [·] : A → Ar be
any surjective map such that ka − [a]k ≤ r/2 for all a ∈ A, i.e. [·] chooses an r/2-close point in the grid Ar .
Using this map, we can extrapolate the inequality (7.11) to the whole set A since
|d(Hq (xq ), a) − dM ([a])| ≤ |d(Hq (xq ), a) − d(Hq (xq ), [a])| + |d(Hq (xq ), [a]) − dM ([a])|
≤ r/2 + |d(Hq (xq ), [a]) − dM ([a])| ,

22

M. ZAMANI, I. TKACHEV, AND A. ABATE

where we used the fact that |d(Hq (xq ), a) − d(Hq (xq ), [a])| ≤ ka − [a]k by the triangle inequality. As a result,
the following inequality holds:


P (∃a ∈ A s.t. |d(Hq (xq ), a) − dM ([a])| ≥ θ) ≤ P ∃a0 ∈ Ar s.t. |d (Hq (xq ), a0 ) − dM (a0 )| ≥ θ̂ . (7.12)
On the other hand, since for any two functions f, g : A → R it holds that
inf f (a) − inf g(a) ≤ sup |f (a) − g(a)|,

a∈A

a∈A

a∈A

we obtain that
P (|d(Hq (xq ), A) − drM | ≥ θ) ≤ P (∃a ∈ A s.t. |d(Hq (xq ), a) − dM ([a])| ≥ θ) .
Combining the latter inequality with (7.11) and (7.12) yields:
P (|d(Hq (xq ), A) − drM | ≥ θ) ≤

|Ar |b(a∗ , 2q)

,
M θ̂2q
and in case M satisfies the assumption of the theorem, the right-hand side is bounded above by π as desired. 

1 Department

of Electrical and Computer Engineering, Technische Universität München, D-80290 Munich, Germany.

E-mail address: zamani@tum.de
URL: http://www.hcs.ei.tum.de
2 Delft

Center for Systems and Control, Delft University of Technology, Mekelweg 2, 2628 CD, Delft, The
Netherlands.
E-mail address: tkachev.ilya@gmail.com
URL: http://www.dcsc.tudelft.nl/∼itkachev
3 Department

of Computer Science, University of Oxford, Wolfson Building, Parks Road, Oxford, OX1 3QD, UK.

E-mail address: alessandro.abate@cs.ox.ac.uk
URL: https://www.cs.ox.ac.uk/people/alessandro.abate

