Jean-Michel Coron
email: coron@ann.jussieu.fr

Lars Grüne
email: lars.gruene@uni-bayreuth.de

Karl Worthmann
email: karl.worthmann@tu-ilmenau.de

Model Predictive Control, Cost Controllability, and Homogeneity *

We are concerned with the design of Model Predictive Control (MPC) schemes such that asymptotic stability of the resulting closed loop is guaranteed -even if the linearization at the desired set point fails to be stabilizable. Therefore, we propose to construct the stage cost based on the homogeneous approximation and rigorously show that applying MPC yields an asymptotically stable closed-loop behavior if the homogeneous approximation is asymptotically null controllable. To this end, we verify cost controllability -a condition relating the current state, the stage cost, and the growth of the value function w.r.t. time -for this class of systems in order to provide stability and performance guarantees for the proposed MPC scheme without stabilizing terminal costs or constraints.

Introduction

Model Predictive Control (MPC) is nowadays a well-established and widely applied control technique for linear and nonlinear systems, see, e.g. the textbooks [START_REF] Grüne | Nonlinear model predictive control[END_REF][START_REF] Kouvaritakis | Model Predictive Control[END_REF][START_REF] Rawlings | Model Predictive Control: Theory, Computation, and Design[END_REF] and the survey papers [START_REF] Forbes | Model predictive control in industry: Challenges and opportunities[END_REF][START_REF] Lee | Model predictive control: Review of the three decades of development[END_REF]. One of the main reasons for its success is the simplicity of the underlying idea: measure the current state of the system in consideration, optimize its behaviour on a finite prediction window, and implement the first piece of the computed control function before repeating the procedure ad infinitum. However, the stability analysis (or the construction of stabilizing terminal costs and constraints) is often based on the (explicit or implicit) assumption that the linearization at the desired set point is stabilizable [START_REF] Chen | A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability[END_REF]. Otherwise, the origin may not be asymptotically stable w.r.t. the MPC closed loop if, e.g. a quadratic stage cost is used -independently of the length of the optimization horizon as rigorously shown in [START_REF] Müller | Quadratic costs do not always work in MPC[END_REF] for the mobile robot example, i.e. a finite-time controllable system.

We propose to design the stage cost based on the homogeneous approximation [START_REF] Kawski | Stability and nilpotent approximations[END_REF][START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF] and [START_REF] Coron | Control and nonlinearity[END_REF]Section 12.3]. Homogeneity -which is a generalization of linearity to nonlinear systems -is a property, which was extensively used to construct stabilizing (homogeneous) control laws [START_REF] Kawski | Homogeneous stabilizing feedback laws[END_REF][START_REF] Coron | Adding an integrator for the stabilization problem[END_REF][START_REF] Hermes | Homogeneous feedback controls for homogeneous systems[END_REF][START_REF] Morin | Time-varying exponential stabilization of chained form systems based on a backstepping technique[END_REF][START_REF] Morin | Design of homogeneous time-varying stabilizing control laws for driftless controllable systems via oscillatory approximation of lie brackets in closed loop[END_REF]. We show that, if the homogeneous approximation is stabilizable, the desired set point is asymptotically stable w.r.t. the MPC closed-loop without terminal costs or constraints. Here, asymptotic stability is understood in a global (homogeneous system with degree zero), semi-global (negative degree of homogeneity), or local (homogeneous approximation) sense. All results are motivated and illustrated by examples.

To this end, we verify cost controllability -a condition relating the growth of the value function w.r.t. time with the stage cost. Then, stability guarantees or a desired performance bound for the proposed MPC scheme can be deduced if the prediction horizon is sufficiently large using techniques originally developed in a discrete-time setting, see [START_REF] Grüne | Analysis and design of unconstrained nonlinear MPC schemes for finite and infinite dimensional systems[END_REF][START_REF] Grüne | Analysis of unconstrained nonlinear mpc schemes with time varying control horizon[END_REF], adopted to the cost-controllability condition [START_REF] Tuna | Shorter horizons for model predictive control[END_REF][START_REF] Worthmann | Stability Analysis of Unconstrained Receding Horizon Control Schemes[END_REF], and transferred to the continuous-time setting [START_REF] Reble | Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems[END_REF][START_REF] Worthmann | The role of sampling for stability and performance in unconstrained nonlinear model predictive control[END_REF]. While we first show our results for homogeneous systems, we make use of homogeneous Lyapunov functions (see, in particular, [START_REF] Rosier | Homogeneous lyapunov function for homogeneous continuous vector field[END_REF][START_REF] Coron | Adding an integrator for the stabilization problem[END_REF][START_REF] Grüne | Homogeneous state feedback stabilization of homogenous systems[END_REF][START_REF] Bacciotti | Liapunov functions and stability in control theory[END_REF] and [START_REF] Coron | Control and nonlinearity[END_REF]Section 12.3]) in order to show that our results are locally preserved for (general) nonlinear systems, i.e. that cost controllability and local asymptotic stability of the origin for the MPC closed loop hold if the homogeneous approximation at the origin is stabilizable.

The paper is organized as follows: In Section 2, cost controllability and its relationship with MPC are recalled and adapted to our setting. Then, we introduce homogeneous control systems in accordance to [START_REF] Grüne | Homogeneous state feedback stabilization of homogenous systems[END_REF] and [START_REF] Bhat | Geometric homogeneity with applications to finitetime stability[END_REF] and give an example that MPC with quadratic costs may fail for finite-time stabilizable systems with zero degree of homogeneity. In Section 3, we establish cost controllability and asymptotic stability of the origin for the MPC closed loop for control systems with zero degree of homogeneity and the local counterparts of these results if the degree of homogeneity is negative. Then, in Section 4, we extend our results to systems with globally asymptotically null controllable homogeneous approximation before conclusions are drawn in Section 5.

Notation: We call a continuous function ρ : [0, ∞) → [0, ∞) of class K ∞ if it maps zero to zero, is strictly monotonically increasing, and unbounded. For further details on comparison functions, we refer to the compendium [START_REF] Kellett | A compendium of comparison function results[END_REF] and the references therein. The set co K denotes the convex hull of the set K.

Problem Formulation

We consider the control system governed by the autonomous ordinary differential equation ẋ(t) = f (x(t), u(t))

with state x(t) ∈ R n and control u(t) ∈ R m at time t ≥ 0. We assume continuity of the map f : R n ×R m → R n and local Lipschitz continuity w.r.t. its first argument on R n \{0} such that, for each initial value x 0 and each control function u ∈ L ∞ loc ([0, ∞), R m ), the solution trajectory x(•; x 0 , u) uniquely exists (on its maximal interval I x 0 of existence).

Model Predictive Control

Let f (0, 0) = 0 hold, i.e. the origin 0 ∈ R n is a (controlled) equilibrium. Our goal is the stabilization of system (1) at the origin. As a preliminary condition, we require a stage cost ℓ :

R n × R m → R ≥0 satisfying ρ( x ) ≤ ℓ ⋆ (x) := inf u∈R m ℓ(x, u) ≤ ̺( x ) ∀ x ∈ R n , (2)
for K ∞ -functions ρ and ̺.

We briefly introduce model predictive control. The basic idea is simple: For the current state x ∈ R n , optimize a cost functional on a finite prediction horizon [0, T ] w.r.t. the system dynamics (1), the initial condition x(0) = x, and, if present, state and control constraints. Then, the first portion u ⋆ (t), t ∈ [0, δ), of the computed optimal control function u

⋆ = u ⋆ (•; x) ∈ L ∞ ([0, T ], R m
) is implemented before the prediction window is shifted forward in time. The procedure is repeated ad infinitum. Assuming existence of a minimizer for simplicity of exposition, the MPC scheme reads as displayed in Algorithm 1.

Algorithm 1 Model Predictive Control

Set t = 0 and let the prediction horizon T and the time shift δ ∈ (0, T ) be given.

1. Measure the current state x := x(t)

2. Solve the optimal control problem min

u∈L ∞ ([0,T ],R m ) T 0 ℓ(x(s; x, u), u(s)) ds subject to x(0; x, u) = x and (1)
and denote its solution by u ⋆ = u ⋆ (•; x).

3. Implement u ⋆ (s)| s∈[0,δ) at the plant, increment t by δ, and goto Step 1.

We define the value function

V t : R n → R ≥0 ∪ {∞} by V t (x) := inf u∈L ∞ ([0,t],R m ) t 0 ℓ(x(s; x, u), u(s)) ds subject to x(0; x, u) = x and (1), with the convention that V t (x) = ∞ if there exists no u ∈ L ∞ ([0, t], R m ) such that x(t; x, u), t ∈ [0, T ],
is defined. This definition allows us to state the following condition relating the stage cost ℓ at the current state x and the value function.

Assumption 1 (Cost controllability on a set K). Consider a set K ⊆ R n containing 0 in its interior. We assume that there exists a monotonically increasing, bounded function

B K : [0, ∞) → R ≥0 , which satisfies the inequality V t (x) ≤ B K (t) ℓ ⋆ (x) ∀ (t, x) ∈ R ≥0 × K. (3)
For the formulation of the following theorem, which is an extension of a result from [START_REF] Reble | Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems[END_REF], we moreover need the sublevel set

V -1 T ([0, L]) := {x ∈ R n | V T (x) ≤ L}.
Theorem 1. Consider the MPC scheme given in Algorithm 1 and let condition (2) and Assumption 1 for a set K ⊆ R n be satisfied for the stage cost ℓ and the system dynamics (1). Then, for given δ > 0, there exists a sufficiently large prediction horizon T > δ such that the origin is locally asymptotically stable for the MPC closed loop. Moreover, the domain of attraction contains all level sets

V -1 T ([0, L]) ⊆ K. If K = R n ,
the origin is globally asymptotically stable for the MPC closed loop. Moreover, if the assumptions hold for each compact set K ⊂ R n containing 0 in its interior (with B K possibly depending on K), then the origin is semiglobally asymptotically stable for the MPC closed loop, i.e., for each compact set K ⊂ R n and each δ > 0 there is T > 0, such that the origin is locally asymptotically stable for the MPC closed loop and K is contained in the domain of attraction.

Proof. The assertion for K = R n was already shown in [START_REF] Reble | Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems[END_REF]. To prove the non-global version, i.e. K R n , we proceed similarly to the discrete time counterpart given in [START_REF] Grüne | Nonlinear model predictive control[END_REF]Theorem 6.35] (see also [START_REF] Boccia | Stability and feasibility of state constrained mpc without stabilizing terminal constraints[END_REF] for related results). Hence we only give a sketch. First one shows using the same techniques as in [START_REF] Reble | Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems[END_REF] that for sufficiently large T the optimal value function V T decreases along solutions in K. This implies that each level set V -1 ([0, L]) ⊂ K is forward invariant. On this level set asymptotic stability with domain of attraction V -1 ([0, L]) follows with usual Lyapunov function arguments. The second statement follows by estimating the size of the level sets V -1

T ([0, L]) exactly as in the proof of [START_REF] Grüne | Nonlinear model predictive control[END_REF]Theorem 6.35].

Assumption 1 is a sufficient condition for stability that is particularly suitable for the class of systems and cost functions that we will investigate in this paper. Of course, for other systems and cost functions other sufficient conditions may be better suited. Two examples will be discussed in Remark 2 at the end of Section 3.

Monotonicity of the (growth) function B K in Assumption 1 can be assumed w.l.o.g. since the left hand side of inequality (3) is monotonically increasing due to the absence of a terminal cost. Furthermore, the condition lim sup t→∞ B K (t)/t < 1 suffices to ensure the existence of a bounded (growth) function B K , see [START_REF] Müller | Quadratic costs do not always work in MPC[END_REF]Proposition 6]. Using the abbreviation γ := lim sup t→∞ B K (t), Assumption 1 ensures the inequality

sup x∈K\{0} V t (x) ℓ ⋆ (x) ≤ γ ∀ t ≥ 0.
In particular, this implies that the value function V t cannot contain terms of lower order than the minimized stage cost ℓ ⋆ (x) = inf u∈R m ℓ(x, u). Otherwise Assumption 1 cannot hold in a neighbourhood of the origin and, thus, on the set K.

Homogeneous Systems and quadratic stage costs

The following definition of homogeneity is taken from [START_REF] Grüne | Homogeneous state feedback stabilization of homogenous systems[END_REF] with a slightly adapted notation.

Definition 1 (Homogeneous System). For given r = (r 1 , . . . , r n ) ∈ (0, +∞) n , s = (s 1 , . . . , s m ) ∈ (0, +∞) m , and τ ∈ (-min i r i , ∞), System (1) is said to be (r, s, τ )homogeneous if

f (Λ α x, ∆ α u) = α τ Λ α f (x, u) ∀ (x, u) ∈ R n × R m and α ≥ 0 (4)
is satisfied, where the matrices Λ α ∈ R n×n and ∆ α ∈ R m×m are defined as

Λ α =       α r 1 0 • • • 0 0 . . . . . . . . . . . . . . . . . . 0 0 • • • 0 α rn       and ∆ α =       α s 1 0 • • • 0 0 . . . . . . . . . . . . . . . . . . 0 0 • • • 0 α sm       .
Homogeneity can be considered as a generalization of linearity. For linear systems, i.e. ẋ(t) = Ax(t) + Bu(t), homogeneity is trivially satisfied with τ = 0 and r i = 1 = s j for (i, j) ∈ {1, . . . , n} × {1, . . . , m}. Using Definition 1, one easily obtains the identity

x(t; Λ α x 0 , ∆ α u(α τ •)) = Λ α x(α τ t; x 0 , u) (5)
for all t ∈ I x 0 ∩ [0, ∞). For τ = 0, the identity (5) simplifies to x(t; Λ α x 0 , ∆ α u) = Λ α x(t; x 0 , u). We consider the following example in order to show that the design of stage cost in MPC matters.

Example 1. Let the system dynamics (1) be given be

ẋ(t) = f (x(t), u(t)) :=   u 1 (t) x 3 (t)u 1 (t) u 2 (t)   =   1 x 3 (t) 0   u 1 (t) +   0 0 1   u 2 (t). ( 6
)
Example 1 is governed by homogeneous system dynamics since Condition (4) holds with τ = 0, r 2 = 2, and r 1 = r 3 = s 1 = s 2 = 1. Moreover, the system is null controllable in finite time, which means that starting from any given state one can steer the control system to 0 ∈ R 3 in finite time (even in arbitrary small positive time). This property follows from the Rashevski-Chow Theorem [START_REF] Coron | Control and nonlinearity[END_REF]Theorem 3.18]. It can also be checked directly as follows. Ensure that the x 3 -component is not equal to 0 by using u 2 . Then take u 2 = 0 (so x 3 does not change) and use u 1 to steer the x 2 -component to 0. Then take u 1 = 0 (so x 2 remains equal to 0) and set the x 3 -component equal to 0 using u 2 . Finally, take u 2 = 0 (so both x 2 and x 3 remain equal to 0) and use u 1 to steer the x 1 -component to 0. It follows from this method to steer the control system (6) to 0 ∈ R 3 that the value function

V t is uniformly bounded w.r.t. t ∈ R ≥0 on compact sets of R 3 .
Nevertheless, one can show that there does not exist a (finite) prediction horizon such that the origin is asymptotically stable w.r.t. the MPC closed loop.

Proposition 1. We consider the MPC scheme presented in Algorithm 1 for the combination of system (6) and the quadratic stage cost ℓ(x, u)

= x ⊤ Qx + u ⊤ Ru with positive definite matrices Q ∈ R 3×3 and R ∈ R 2×2 .
Then, for arbitrary but fixed prediction horizon T , the origin is not (locally) asymptotically for the MPC closed loop.

Proof. Firstly, it can be shown that, for every ε > 0, there exists an initial value x 0 ∈ {x ∈ R 3 | x ≤ ε} such that u ⋆ ≡ 0 is the unique optimal control. The proof is almost literally the same as its analogon for the nonholonomic robot presented in [START_REF] Müller | Quadratic costs do not always work in MPC[END_REF]Subsection 4.1] if Condition ( 13) is replaced by (Qx 0 ) 3 = 0 = (Qx 0 ) 1 + (Qx 0 ) 2 θ 0 in view of the slightly different adjoint equation. Hence, the closed-loop trajectory emanating at x 0 stays at x 0 forever and does, thus, not converge to the origin, which shows the assertion.

We note that this implies that Assumption 1 must be violated. In the following, our goal is to construct a stage cost such that asymptotic stability of the origin w.r.t. the MPC closed loop is ensured for a sufficiently long prediction horizon T for a class of systems including both Example 1 and the mentioned nonholonomic robot example provided that the homogeneous (approximation of the) system is globally asymptotically null controllable. Moreover, we want to keep the construction simple by solely replacing the quadratic performance measure by one based on the dilated norm N :

R n → [0, ∞), i.e. N (x) := n i=1 x d r i i 1 d with d := 2 n i=1 r i . (7)
Note that the dilated norm satisfies the equality

N (Λ α x) = d n i=1 (α r i x i ) d r i = d α d n i=1 x d r i i = αN (x) ∀ (α, x) ∈ R ≥0 × R n (8)
and that it is positive on R n \ {0}. However it is not a norm unless Λ α = αI.

Cost Controllability for Homogeneous Systems

We consider homogeneous systems and propose to define the stage cost ℓ :

R n × R m → R ≥0 by ℓ(x, u) := N (x) d + m j=1 u d s j j = n i=1 x d r i i + m j=1 u d s j j . (9)
The stage cost is based on the dilated norm and the coefficients r i , i ∈ {1, . . . , n}, and s j , j ∈ {1, . . . , m}, from Definition 1. Note that the definition of ℓ in (9) leads to quadratic stage cost for linear systems ẋ(t) = Ax(t)+Bu(t). Moreover, the definition is compatible with homogeneity analogously to (8).

Proposition 2. The stage costs ℓ are homogeneous, i.e.

ℓ(x(s; Λ α x 0 , ∆ α u(α τ •)), ∆ α u(α τ s)) = α d ℓ(x(α τ s; x 0 , u), u(α τ s)) (10)
holds for α ≥ 0. For τ = 0, equality (10) simplifies to

ℓ(x(s; Λ α x 0 , ∆ α u), ∆ α u(s)) = α d ℓ(x(s; x 0 , u), u(s)). (11)
Proof. A direct calculation shows

ℓ(x(s; Λ α x 0 , ∆ α u(α τ •)), ∆ α u(t)) = N (x(s; Λ α x 0 , ∆ α u(α τ •)) (5)
= Λαx(α τ s;x 0 ,u)

) d + m j=1 (α s j u j (t)) d s j (8) = α d N (x(α τ s; x 0 , u)) d + α d m j=1 u j (t) d s j = α d ℓ(x(α τ s; x 0 , u), u(t)),
i.e. the assertion if t is replaced by α τ s.

Our goal in this section is to verify cost controllability for homogeneous, globally asymptotically null controllable systems.

Definition 2. The system (1) is said to be globally asymptotically null controllable if, for each x 0 ∈ R n , there exists a control function

u 0 ∈ L ∞ loc ([0, ∞), R m ) such that lim t→∞ x(t; x 0 , u 0 ) = 0 holds.
Note that a locally asymptotically null controllable homogeneous system is always globally asymptotically null controllable.

Inequality (3) is equivalent to

V (t) ≤ B K (t)N (x 0 ) d ∀ (t, x 0 ) ∈ R ≥0 × K, (12)
using the stage cost [START_REF] Coron | A necessary condition for feedback stabilization[END_REF]. Under this condition, asymptotic stability of the origin w.r.t. the MPC closed loop is ensured for a sufficiently large prediction horizon T using Theorem 1 since Condition (2) is satisfied. Before we proceed, we consider a simple example in order to illustrate and motivate our results.

Example 2. Let the system dynamics (1) be given by

ẋ(t) = |x(t)| k sign(x) + u with k ∈ (0, +∞). ( 13
)
Since we have

f (Λ α x, ∆ α u) = |α r x| k sign(x) + α s u, α τ Λ α f (x, u) = α τ +r (|x| k sign(x) + u),
we get the (r, s, τ )-homogeneity of (13) if and only if τ + r = rk and τ + r = s. This leads to the following three cases:

• k = 1: Degree of homogeneity zero, i.e. τ = 0 with coefficients r = s

• k ∈ (0, 1): Negative degree of homogeneity, i.e. τ < 0 with coefficients r = s/k

• k > 1: Positive degree of homogeneity, i.e. τ > 0 with r = s/k

Next, we compute the infinite horizon optimal value function using the Hamilton-Jacobi-Equation min

u∈R m ∇V (x), f (x, u) + ℓ(x, u) = 0 ( 14
)
with m = 1. To this end, we quadratically penalize the control effort to simplify the following calculations by using the stage cost ℓ :

R n ×R m , n = m = 1, defined by ℓ(x, u) = (x 2 ) d + u 2 , which yields u ⋆ = -1 2 V ′ (x).
Then, plugging this expression into (14

), i.e. V ′ (x) |x| k sign(x) + u ⋆ + ℓ(x, u ⋆ ) = 0, leads to V ′ (x)|x| k sign(x) -1 4 V ′ (x) 2 + (x 2 ) d = 0
, or (only the root having the sign of x has to be taken into account since V is increasing on [0, +∞) and decreasing on (-∞, 0])

V ′ (x) = 2|x| k sign(x) + sign(x) 4|x| 2k + 4(x 2 ) d d=k = 2(1 + √ 2)|x| k sign(x).
Overall, we get the value function

V (x) = 2(1 + √ 2) k + 1 |x| k+1 and, thus, V (x) ℓ ⋆ (x) = 2(1 + √ 2) k + 1 |x| 1-k .
In conclusion, we get for

• k = 1:
The growth bound is uniformly upper bounded by 1 + √ 2, i.e. cost controllability in the sense of Assumption 1 with

K = R n = R. • k ∈ (0, 1): The growth bound is uniformly upper bounded on each compact set K by 2(1+ √ 2)
k+1 max x∈K |x| 1-k , i.e. cost controllability in the sense of Assumption 1.

• k > 1: the quotient V (x)/ℓ ⋆ (x) is uniformly upper bounded on R \ (-ε, ε) for arbitrary but fixed ε > 0 by 2(1+ √ 2) k+1 max x∈R\(-ε,ε) |x| 1-k .
Moreover, note that the choice d = 1, i.e. purely quadratic costs, implies for k ∈ (0, 1) that the value function still exhibits a term of order 1 + k but ℓ ⋆ (x) = x 2 holds. Hence, there does not exist a uniform bound for the quotient V (x)/ℓ ⋆ (x) for x → 0, x = 0.

Remark 1. In general, the degree of homogeneity (τ < 0, τ = 0, and τ > 0) is not unique, which can be seen from the (driftless) system Example 1. Here, Condition (4) is also satisfied for the combinations τ = 0.5, r 1 = r 3 = 1, r 2 = 2, and s 1 = s 2 = 1.5 and τ = -0.5, r 1 = r 3 = 1, r 2 = 2, and s 1 = s 2 = 0.5.

We emphasize that the relative weighting of the control in comparison to the penalization of deviations w.r.t. the state depends on the choice, which explains the different behavior of the MPC closed loop. For control affine systems, we set s j = 2 whenever possible in view of the Hamilton-Jacobi-Equation.

Degree of homogeneity zero

In this subsection, we derive cost controllability and, then, conclude asymptotic stability for globally asymptotically null controllable systems with degree of homogeneity τ = 0.

Theorem 2 (Cost Controllability). Let the system (1) be globally asymptotically null controllable and (r, s, τ = 0)-homogeneous. In addition, let the stage cost be defined by [START_REF] Coron | A necessary condition for feedback stabilization[END_REF]. Then, cost controllability, i.e. inequality [START_REF] Forbes | Model predictive control in industry: Challenges and opportunities[END_REF], holds. In particular, given δ > 0, for any sufficiently large T > 0 the origin is globally asymptotically stable for the MPC closed loop.

Proof. First, we show inequality [START_REF] Forbes | Model predictive control in industry: Challenges and opportunities[END_REF] only for initial values x 0 contained in the compact set

N c 1 ,c 2 := {x ∈ R n | c 1 ≤ N (x) d ≤ c 2 } with 0 < c 1 < c 2 < ∞.
Later on, we show that the same growth function B can be used for arbitrary initial values x 0 ∈ R n . Global asymptotical null controllability implies that, for each x 0 ∈ N c 1 ,c 2 , there exists a control function u 0 ∈ L ∞ loc ([0, ∞), R m ) such that lim t→∞ x(t; x 0 , u 0 ) = 0 holds. Hence, continuity of the solution trajectory x(•; x 0 , u 0 ) implies the existence of a time instant T 0 ∈ (0, ∞) (depending on x 0 ) such that

N (x(T 0 ; x 0 , u 0 )) d = α c 1 + c 2 2
where α ∈ (0, 1) is chosen such that α(c

1 +c 2 )/2 < c 1 holds, i.e. x(T 0 ; x 0 , u 0 ) is contained in the sublevel set {x ∈ R n | N (x) d < c 1 }.
Due to continuity and compactness of the time interval [0, T 0 ], there exists δ 0 > 0 sufficiently small (depending on x 0 ) such that the chain of inequalities

αc 1 < N (x(T 0 ; x, u 0 )) d < αc 2 (15)
holds for all x ∈ B δ 0 (x 0 ). Since

x 0 ∈Nc 1 ,c 2 B δ 0 (x 0
) is an open cover of the compact set N c 1 ,c 2 , there exists a finite subcover and, thus, there exist finitely many initial values x1 , . . . , x k , k ∈ N, with corresponding u i , T i and δ i , satisfying

N c 1 ,c 2 ⊂ k i=1 B δ i (x i ).
Let us set T ⋆ := min i=1,...,k T i ∈ (0, ∞). Then, the following function is obviously a growth function B on the finite interval [0, T ⋆ ] for all x 0 ∈ N c 1 ,c 2 :

B(t) := max i=1,...,k sup x∈B δ i (x i )∩ Nc 1 ,c 2 N (x) -d t 0 ℓ(x(s; x, u i ), u i (s)) ds ∀ t ∈ [0, T ⋆ ), B(T ⋆ ) := max i=1,...,k sup x∈B δ i (x i )∩ Nc 1 ,c 2 N (x) -d T i 0 ℓ(x(s; x, u i ), u i (s)) ds. ( 16
)
Note that the preceding line of argumentation implicitly implies well-posedness and boundedness of B on [0, T ⋆ ] in view of the fact that N (x) is uniformly bounded from below by c 1 . 1 This construction is quite conservative and does not rely on homogeneity. But it is essential to have a bounded B on a compact time interval of positive length for the following line of argumentation. Before we proceed, let us present some preliminary considerations. Let x0 / ∈ N c 1 ,c 2 be arbitrary but fixed. Then, if x0 = 0, there exists a scaling factor c ∈ (0, ∞) such that Λ c x0 ∈ N c 1 ,c 2 . This implies the existence of an index j ∈ {1, 2, . . . , k}, such that Λ c x0 ∈ B δ j (x j ). Hence, using x0 = Λ c -1 (Λ c x0 ), we get

ℓ(x(s; x0 , ∆ c -1 u j ), ∆ c -1 u j (s)) (11)
= c -d ℓ(x(s; Λ c x0 , u j ), u j (s)) [START_REF] Hermes | Homogeneous feedback controls for homogeneous systems[END_REF] and, for ℓ ⋆ (x 0 ) := inf u∈R m ℓ(x 0 , u),

ℓ ⋆ (x 0 ) = N (x 0 ) d = N (Λ c -1 Λ c x0 ) d (8) = c -d N (Λ c x0 ) d = c -d ℓ ⋆ (Λ c x0 ). (18)
Hence, showing the desired inequality (see Assumption 1) for x 0 ∈ N c 1 ,c 2 suffices. The identity ( 17) is also the key ingredient to show the assertion for t > T ⋆ based on the following contraction argument. For each x 0 ∈ N c 1 ,c 2 , there exists an index q ∈ {1, . . . , k}, and a time instant t 0 ∈ [T ⋆ , max {i=1,...,k} T i ] such that x(t 0 ; x 0 , u q ) ∈ αN c 1 ,c 2 and

t 0 ℓ(x(s; x 0 , u q ), u q (s)) ds ≤ B(min{t, T ⋆ }) N (x 0 ) d ∀ t ∈ [0, t 0 ]
hold. Next, using the identity (17) for x0 = x(t 0 ; x 0 , u q ) with c = α -1 > 1 allows us to repeat the presented line of reasoning to show that the stage cost are scaled with α d . In addition,

x(T j ; x0 , ∆ α u j ) = Λ α x(T j ; Λ α -1 x0 , u j ) ∈αNc 1 ,c 2 ∈ α 2 N c 1 ,c 2 .
Hence, we get the desired growth condition if we periodically extend the growth function B on (T ⋆ , ∞) by setting

B(t) := ⌊t/T ⋆ ⌋-1 i=0 α id B(T ⋆ ) + α d⌊t/T ⋆ ⌋ B(t -T ⋆ ⌊t/T ⋆ ⌋) ∀ t > T ⋆ , which is bounded by ∞ i=0 α id B(T ⋆ ) = (1 -α d ) -1 B(T ⋆ ) ∈ (0, ∞).
This completes the proof of the first statement. Asymptotic stability then follows from Theorem 1.

Degree of homogeneity not equal to zero

We begin with globally asymptotically null controllable systems with negative degree of homogeneity, i.e. τ < 0. Here, we show semiglobal asymptotic stability of the origin w.r.t. the MPC closed loop, i.e. local asymptotic stability of the origin with an arbitrary but fixed compact set being the domain of attraction, cf. Theorem 1. We refer to Example 2, which shows the necessity of this restriction for the derivation of cost controllability.

In Proposition 3 we extend Theorem 2 to homogeneous systems with negative degree.

Proposition 3 (Cost Controllability for τ < 0). Let the system (1) be globally asymptotically null controllable and (r, s, τ )-homogeneous with τ < 0. In addition, let the stage cost be defined by [START_REF] Coron | A necessary condition for feedback stabilization[END_REF]. Then, (3), i.e. cost controllability, holds on any compact set K ⊂ R n with 0 ∈ int K. In particular, given δ > 0, for any sufficiently large T > 0 the origin is semiglobally asymptotically stable for the MPC closed loop.

Proof. Since the set K is compact there exists c 2 ∈ (0, ∞) such that

K ⊆ {x 0 ∈ R n | N (x 0 ) d ≤ c 2 }
holds. We show that inequality [START_REF] Forbes | Model predictive control in industry: Challenges and opportunities[END_REF] holds for all x 0 contained in this sublevel set. To this end, we start analogously to the verification of Theorem 2 with c 1 ∈ (0, c 2 ) but define the growth function B : R ≥0 → R ≥0 differently. Let B denote the growth function defined analogously as in the proof of Theorem 2 and α, α ∈ (0, 1), be an arbitrarily chosen but fixed contraction parameter. Then, the adapted growth function B is given by

B(t) := B(α τ t) for all t ≥ 0. ( 19
)
Note that α τ > 1 holds since α ∈ (0, 1) and τ < 0. Since B is monotonically increasing, B(t) ≥ B(t) holds for all t ≥ 0.2 Next, we show that inequality [START_REF] Forbes | Model predictive control in industry: Challenges and opportunities[END_REF] holds for all t ≥ 0 for x 0 ∈ N c 1 ,c 2 . We choose x 1 , . . . , x k with corresponding u i , T i and δ i , i = 1, . . . , k, as in the proof of Theorem 2.

For given initial state x 0 ∈ N c 1 ,c 2 there exists an index q ∈ {1, 2, . . . , k} such that x0 := x(T q ; x 0 , u q ) ∈ αN c 1 ,c 2 holds. In addition, there exits an index j ∈ {1, 2, . . . , k} such that Λ α -1 x0 ∈ B δ j (x j ) holds and the analogon of identity [START_REF] Hermes | Homogeneous feedback controls for homogeneous systems[END_REF] for τ < 0 is

ℓ(x(s; x0 , ∆ c -1 u j (c -τ •)), ∆ c -1 u j (c -τ s)) (10) = c -d ℓ(x(c -τ s; Λ c x0 , u j ), u j (c -τ s)).
Next, we use the scaling factor c := α -1 > 1 for x0 . Note that equation ( 18) yields ℓ ⋆ (x 0 ) = α d ℓ ⋆ (Λ α -1 x0 ). Overall, for t ≤ α -τ T j , we get

V t (x 0 ) ℓ ⋆ (x 0 ) = α -d V t (x 0 ) ℓ ⋆ (Λ α -1 x0 ) ≤ α -d ℓ ⋆ (Λ α -1 x0 ) t 0 ℓ(x(s; x0 , ∆ α u j (α τ •)), ∆ α u j (α τ s)) =ℓ(x(α τ s;Λ α -1 x0 ,u j ),u j (α τ s)) ds = α -d α -τ α τ t 0 ℓ(x(s; Λ α -1 x0 , u j ), u j (s)) ds ℓ ⋆ (Λ α -1 x0 ) ≤ α -d α -τ B(min{t, α -τ T ⋆ }) ≤ α -d B(min{t, α -τ T ⋆ })
where we have used that α τ > 1 and α -τ < 1 hold. Furthermore, for a scaling factor c > 1 (e.g. c = α -1 ), we get

x(c τ T j ; x0 , ∆ c -1 u j (c -τ •)) = x(c τ T j ; Λ c -1 Λ c x0 , ∆ c -1 u j (c -τ •)) (5) = Λ c -1 x(T j ; Λ c x0 , u j ) ∈αNc 1 ,c 2
where c τ < 1 and c -1 < 1. This shows that a contraction is attained already at time α -τ T j (c = α -1 ). For general initial value, the index j has to be suitably chosen. Hence, an iterative application of this line of reasoning shows the claimed inequality not only for x 0 ∈ α i N c 1 ,c 2 , i ∈ N ∪ {0}, but indeed for all x 0 ∈ K.

Semiglobal asymptotic stability then follows from Theorem 1.

Remark 2. As already mentioned after the proof of Theorem 1, Assumption 1 is a sufficient condition for stability that is particularly suited for the class of homogeneous problems we investigated in this section. However, there are other types of conditions that may be advantageous for other settings. In this remark we discuss two of them that apply when Assumption 1 is violated. (i) In discrete time, [START_REF] Grüne | Nonlinear model predictive control[END_REF]Theorem 6.37] states that if there is a function ρ ∈ K ∞ such that the inequality

V t (x) ≤ ρ(ℓ ⋆ (x)) ( 20
)
holds for all (t, x) ∈ R ≥0 × R n , then the MPC closed loop is semiglobally practically asymptotically stable. If this result were also available in continuous time, then we expect that it provides a way to handle homogeneous systems and running costs with positive degree. In fact, for such systems we would even expect ρ to be bounded from above by an affinely linear function, due to which we conjecture that global practical asymptotic stability can be concluded. Unfortunately, the derivation of these results in continuous time is a nontrivial task and beyond the scope of this paper. It will thus be addressed in future research.

(ii) Both Assumption 1 and inequality [START_REF] Kellett | A compendium of comparison function results[END_REF] require that sup t≥0 V t (x) < ∞ for all x ∈ R n . However, stability of the closed loop may also hold if V t (x) grows unboundedly for t tending to infinity. A sufficient condition for asymptotic stability that applies in this case is the following: we assume the existence of a continuous function C : R ≥0 × R n → [0, 1] that is nonincreasing in t with C(t, x) < 1 for all x ∈ R n and t > 0, such that

V t (x) ≤ tC(t, x)ℓ ⋆ (x) ( 21
)
holds for all (t, x) ∈ R ≥0 × R n . One easily checks that this condition is satisfied, e.g., for the 1d example

ẋ(t) = -|x(t)|(x(t) + u(t)), ℓ(x, u) = |x| + |u|.
Abbreviating the optimal trajectory by x ⋆ (t) := x(t; x 0 , u ⋆ ), condition (21) implies

ℓ ⋆ (x ⋆ (T )) = inf t∈[0,T ] ℓ ⋆ (x ⋆ (t)). ( 22
)
To see this, assume that (22) does not hold. Then there is ε > 0 and a τ ∈ [0, T )

with ℓ ⋆ (x ⋆ (τ )) ≤ ℓ ⋆ (x ⋆ (T )) -ε. Chosing τ maximal with this property, it follows that ℓ ⋆ (x ⋆ (t)) ≥ ℓ ⋆ (x ⋆ (T )) -ε ≥ ℓ ⋆ (x ⋆ (τ )) for all t ∈ [τ, T ]. By the optimality principle we obtain V T -τ (x ⋆ (τ )) = T τ ℓ(x ⋆ (t), u ⋆ (t))dt ≥ (T -τ )ℓ ⋆ (x ⋆ (τ )
), which contradicts (21) and thus proves [START_REF] Lee | Model predictive control: Review of the three decades of development[END_REF]. Now consider the value J δ (x) := δ 0 ℓ(x ⋆ (t), u ⋆ (t)) dt. Since ℓ ≥ 0 holds, there exists τ ⋆ ∈ [0, δ] with ℓ ⋆ (x ⋆ (τ ⋆ )) ≤ J δ (x)/δ. Using (21) and [START_REF] Lee | Model predictive control: Review of the three decades of development[END_REF], this implies V δ (x ⋆ (T )) ≤ C(δ, x ⋆ (T ))δℓ ⋆ (x ⋆ (T )) ≤ C(δ, x ⋆ (T ))J δ (x). From this we can conclude

V T (x ⋆ (δ)) ≤ V T -δ (x ⋆ (δ)) + V δ (x ⋆ (T )) = V T (x) -J δ (x) + V δ (x ⋆ (T )) ≤ V T (x) -(1 -C(δ, x ⋆ (T )))J δ (x) < V T (x)
for x = 0. Thus, V T is a Lyapunov function for the closed loop and asymptotic stability of the closed loop follows.

Systems with globally asymptotically null controllable homogeneous approximation

In this section, we consider systems with homogeneous approximation at the origin in the following sense [START_REF] Andrieu | Homogeneous approximation, recursive observer design, and output feedback[END_REF].

Definition 3. Consider two controlled vector fields f, h : R n × R m → R n , where h is homogeneous with parameters r i > 0, s j > 0 and τ ∈ (-min i r i , ∞). Then h is called a homogeneous approximation of f near the origin, if the relation

f (x, u) = h(x, u) + R(x, u) holds with |R i (Λ α x, ∆ α u)| ≤ M α r i +τ +η (23)
for all x, u with x ≤ ρ, u ≤ ρ and all α ∈ (0, 1], where ρ, M, η are positive constants.

In case the approximation h is globally asymptotically null controllable, we will establish cost controllability for the system governed by f if the stage cost [START_REF] Coron | A necessary condition for feedback stabilization[END_REF] are employed in the sense of Assumption 1, where K is a neighborhood of the origin. As a consequence, by means of Theorem 1 we obtain local asymptotic stability of the origin for the MPC closed loop.

Our main tool for deriving these results are homogeneous control Lyapunov functions, whose existence was established in [START_REF] Grüne | Homogeneous state feedback stabilization of homogenous systems[END_REF]. We first show preliminary results in Section 4.1, i.e. in particular robustness of homogeneous Lyapunov functions. Then, we generalize Theorem 2 (τ = 0) and its extension Proposition 3 (τ < 0) before our findings are illustrated by means of the nonholonomic robot example, a variant of Brockett's famous nonholonomic integrator.

Properties of Homogeneous Lyapunov Functions

In this section we assume that approximation h is globally asymptotically null controllable. Then [13, Theorem 3.5] establishes the existence of a homogeneous control Lyapunov function V : R n → R ≥0 , which 1) is continuous on R n and Lipschitz on R n \ {0}, 2) obeys the equality V (Λ α x) = α 2k V (x), and 3) satisfies the inequality min v∈co h(x,Wx) DV (x; v) ≤ -µN (x) τ V (x),

where k := min i=1,...,n r i , µ > 0, W x = ∆ N (x) U for a suitable compact set U ⊂ R m , and DV (x; v) denotes the lower Dini derivative

DV (x; v) := lim inf tց0,v ′ →v 1 t V (x + tv ′ ) -V (x) .
Next, we show an auxiliary result. Namely, that a suitably scaled version of the homogeneous Lyapunov function V is locally Lipschitz.

Lemma 1. Let p be defined by max i=1,...,n r i and let the homogeneous Lyapunov function V : R n → R ≥0 satisfy Properties 1) and 2). Then, for each compact set K ⊂ R n , there is L > 0 such that V := V p/(2k) is Lipschitz continuous on K with Lipschitz constant L.

Proof. A straightforward computation yields that V satisfies the (in)equalities

V (Λ α x) = α p V (x) (24)
and min

v∈co h(x,Wx) D V (x; v) ≤ -σN (x) τ V (x) (25)
with σ := pµ/(2k) > 0. Moreover, Property 1) is preserved. Based on this observation, we can prove the improved Lipschitz property of V .

Throughout the proof we use the 1-norm • = • 1 . Because of the homogeneity, it suffices to show that V is Lipschitz on the compact set K := {x ∈ R n | N (x) ≤ 1}. To this end, first observe that the Lipschitz continuity on R n \ {0} implies the existence of a Lipschitz constant L 1 > 0 on the set

K 1 := {x ∈ R n | 1/2 ≤ N (x) ≤ 1}. Moreover, by continuity of V there exists a constant C 1 > 0 such that for all x ∈ R n with N (x) = 1 and all ŷ ∈ R n with N (ŷ) ≤ 1/2 the inequality | V (x) -V (ŷ)| ≤ C 1 holds.
Together with the fact that there is a constant C 2 > 0 with xŷ ≥ C 2 for all such x and ŷ, for

L 2 = C 1 /C 2 we obtain | V (x) -V (ŷ)| ≤ C 1 ≤ C 1 /C 2 x -ŷ = L 2 x -ŷ .
This yields that, for all x ∈ R n with N (x) = 1 and all ŷ ∈ K, the inequality

| V (x) -V (ŷ)| ≤ L x -ŷ (26)
holds with L := max{L 1 , L 2 }. Now consider two arbitrary points x, y ∈ K. Without loss of generality assume N (x) ≥ N (y). Let α := N (x) ≤ 1 and define x := Λ -1 α x and ŷ := Λ -1 α y. Then we obtain

x -ŷ = n i=1 |x i -ŷi | = n i=1 α -r i |x i -y i | ≤ α -p n i=1 |x i -y i | = α -p x -y .
Observing that N (x) = 1 and N (ŷ) ≤ 1, we can apply [START_REF] Müller | Quadratic costs do not always work in MPC[END_REF] in order to conclude

| V (x) -V (y)| = | V (Λ α x) -V (Λ α ŷ)| = α p | V (x) -V (ŷ)| ≤ α p L x -ŷ ≤ α p Lα -p x -y = L x -y ,
which shows the claim.

Based on the Lipschitz continuity of the suitably scaled homogeneous Lyapunov function, we can now derive a robustness statement for the Dini derivative of the Lyapunov function along the vector field f that is homogeneously approximated by h. Proposition 4. Consider a vector field f with homogeneous approximation h according to Definition 3 and a homogeneous control Lyapunov function V : R n → R ≥0 satisfying Properties 2) and 3). Then, for each δ ∈ (0, 1), there exists ε > 0 such that the inequality

min v∈co f (x,Wx) DV (x; v) ≤ -δσN (x) τ V (x). ( 27
)
holds for all x ∈ R n with N (x) ≤ ε.

Proof. At the beginning of the proof of Lemma 1 we made the observation that scaling of the homogeneous Lyapunov function does not change Properties 2) and 3). Analogously, it can be seen that once inequality ( 27) is established for V := V p/(2k) , a straightforward computation shows that it also holds (with δσ replaced by δσq) for V q , q > 0, and, thus, in particular for V . Hence, it is sufficient to prove inequality [START_REF] Rawlings | Model Predictive Control: Theory, Computation, and Design[END_REF] for V . In what follows, we again write V instead of V in order to simplify the notation within the remainder of the proof.

From the form of f it follows that for all α ∈ (0, 1] and each

v h ∈ co h(Λ α x, W Λαx ) there exists v ∈ co f (Λ α x, W Λαx ) with v = v h + v R (v h , x, α), where |[v R (v h , x, α)] i | ≤ M α r i +τ +η , ( 28
)
provided x is sufficiently close to 0. We choose ν > 0 small enough such that (28) holds whenever N (x) ≤ ν. inequality [START_REF] Reble | Unconstrained model predictive control and suboptimality estimates for nonlinear continuous-time systems[END_REF] implies that v R (v h , x, α) can be written as

v R (v h , x, α) = α τ +η Λ α vR for a vector vR = vR (v h , x, α) ∈ R n with |[v R ] i | ≤ M . Any sequence v ′ → v can thus be written as v ′ = v ′ h + α τ +η Λ α vR with v ′ h → v h .
From the Lipschitz continuity of V we obtain

|V (Λ α x + tv ′ ) -V (Λ α x + tv ′ h )| ≤ α p |V (x + tΛ -1 α v ′ ) -V (x + tΛ -1 α v ′ h )| ≤ α p tL Λ -1 α α τ +η Λ α vR = α p+τ +η tL vR .
For the Dini-derivative this implies min

v∈co f (Λαx,W Λαx ) DV (Λ α x; v) ≤ min v h ∈co h(Λαx,W Λαx ) DV (Λ α x; v) + α p+τ +η M R .
with M R being a bound on vR . Now let m V := min{V (x) | N (x) = ν}. Then [START_REF] Morin | Design of homogeneous time-varying stabilizing control laws for driftless controllable systems via oscillatory approximation of lie brackets in closed loop[END_REF] implies

α p+τ +η M R ≤ (1 -δ)σα p+τ V (x)ν τ for all x ∈ R n with N (x) = ν, whenever α ≤ α δ := ((1 -δ)σm V ν τ /M R ) 1/η . Given
x ∈ R n with N (x) ≤ ν, setting α := N (x)/ν ∈ (0, 1] and x := Λ -1 α (x) we obtain N (x) = ν. Hence, for ε = α δ ν and N (x) ≤ ε (implying α ≤ α δ ) we obtain min

v∈co f (x,Wx) DV (x; v) = min v∈co f (Λα x,W Λα x) DV (Λ α x; v) ≤ min v h ∈co h(Λα x,W Λα x) DV (Λ α x; v) + α p+τ +η M R ≤ -σ N (Λ α x) τ =α τ ν τ V (Λ α x) =α p V (x) + α p+τ +η M R ≤(1-δ)σα p+τ V (x)ν τ ≤ -δσα p+τ ν τ V (x) = -δσN (Λ α x) τ V (Λ α x) = -δσN (x) τ V (x).
This shows the claim.

Cost controllability via homogeneous control Lyapunov functions

We now show that cost controllability can be concluded from the existence of a homogeneous control Lyapunov function for the homogeneous approximation. To this end, we use the observation made in the proof of [30, Theorem 1] (based on a similar statement for differential inclusions from [START_REF] Aubin | Differential Inclusions[END_REF]), that inequality [START_REF] Rawlings | Model Predictive Control: Theory, Computation, and Design[END_REF] implies the existence of a measurable control function u : [0, t] → R m for which the integral inequality

V (x(t, x 0 , u)) -V (x 0 ) ≤ t 0 -qδσN (x(s; x 0 , u)) τ V (x(s, x 0 , u)) ds (29)
and the inclusion u(s) ∈ W x(s,x 0 ,u) hold for all almost all s ≥ 0. While the construction in [START_REF] Sontag | Nonsmooth control-Lyapunov functions[END_REF] is for global control Lyapunov functions, standard Lyapunov function arguments show that it remains valid if the control Lyapunov function is only defined in a neighbourhood of the origin. Of course, in this case (29) will only hold for x 0 from a (possibly different) neighbourhood of the origin.

Theorem 3. Consider a vector field f with homogeneous approximation h of degree τ ≤ 0 according to Definition 3. Assume that the homogeneous system with vector field h is globally asymptotically null controllable to the origin. Then there exists a neighborhood N of the origin such that (3) is satisfied with K = N , i.e. the cost controllability is satisfied for the vector field f and B K (t) is uniformly bounded. In particular, given δ > 0, for any sufficiently large T > 0 the origin is locally asymptotically stable for the MPC closed loop.

Proof. By [13, Theorem 3.5] and Proposition 4 there exists a control Lyapunov function V satisfying inequality [START_REF] Rawlings | Model Predictive Control: Theory, Computation, and Design[END_REF] and, by suitably scaling, we have

V (Λ α x) = α d V (x) (30)
and the inequality min

v∈co f (x,Wx) DV (x; v) ≤ -γN (x) τ V (x) (31)
for γ > 0 in a neighbourhood N of the origin. From equation [START_REF] Sontag | Nonsmooth control-Lyapunov functions[END_REF] it follows that there exists

C 1 > 0 with V (x) ≤ C 1 inf u∈R m ℓ(x, u) (32)
for all x ∈ R n . Now pick an initial condition x 0 from the neighbourhood N of the origin on which (29) with q = d/p holds. Then V decreases along the solution which implies that V (x(t, x 0 , u)) and thus N (x(t, x 0 , u)) are bounded uniformly in t and x 0 . As a consequence, there exists C 2 > 0, independent of x 0 , such that N (x(t, x 0 , u)) τ ≥ C 2 for all t ≥ 0 using τ ≤ 0. Then, along the solution in [START_REF] Rosier | Homogeneous lyapunov function for homogeneous continuous vector field[END_REF], again [START_REF] Sontag | Nonsmooth control-Lyapunov functions[END_REF] and the inclusion u(s) ∈ W x(s,x 0 ,u) (which implies N (u(s)) ≤ DN (x(s, x 0 , u)) for an appropriate constant D > 0) yield the existence of C 3 > 0 with ℓ(x(s, x 0 , u), u(s)) ≤ C 3 V (x(s, x 0 , u), u(s)) for almost all s ∈ (0, t). Thus, using (29) we obtain

V t (x 0 ) ≤ t 0 ℓ(x(s, x 0 , u), u(s)) ds ≤ t 0 C 3 V (x(s, x 0 , u), u(s)) ds ≤ C 3 C 2 γ V (x 0 ) -V (x(t, x 0 , u)) ≤ C 3 C 2 γ V (x 0 ) (32) ≤ C 1 C 3 C 2 γ inf u∈R m ℓ(x 0 , u).
This shows the first claim for C = C 1 C 3 /(C 2 γ). Asymptotic stability of the origin for the MPC closed loop then follows from Theorem 1.

Example: Nonholonomic Robot

We illustrate our findings by means of the following example.

Example 3. We consider the nonholonomic mobile robot given by

ẋ(t) = f (x(t), u(t)) :=   cos(x 3 (t)) sin(x 3 (t)) 0   u 1 (t) +   0 0 1   u 2 (t), (33)
which is a control affine system, i.e. f (x, u) = f 0 (x) + m i=1 u i f i (x) with mappings f 0 , f 1 , . . . , f m : R n → R n . The system dynamics (33) describe a driftless system since f 0 (x) ≡ 0.

Let us recall that, as shown by Brockett in [START_REF] Brockett | Asymptotic stability and feedback stabilization[END_REF] (see also [START_REF] Coron | A necessary condition for feedback stabilization[END_REF] and [START_REF] Coron | Control and nonlinearity[END_REF]Theorem 11.1]), the control system (33) cannot be locally asymptotically stabilizable by means of continuous feedback laws. We show that the proposed MPC method allows to locally asymptotically stabilize the control system [START_REF] Worthmann | Regulation of differential drive robots using continuous time mpc without stabilizing constraints or costs[END_REF]. To this end, we firstly state the homogeneous approximation and show that the approximation property, i.e. Inequality [START_REF] Liberzon | Switching in systems and control, ser. systems & control: Foundations & applications[END_REF], is satisfied. Note that the parameters r i and s j are not unique because the homogeneity condition (4) remains satisfied if they are multiplied by an arbitrary positive scalar. For the nonholonomic mobile robot, i.e. Example 3, MPC does not work if (purely) quadratic stage cost are used, i.e. ℓ(x, u) := x ⊤ Qx + u ⊤ Ru with matrices Q ∈ R n×n and R ∈ R m×m as rigorously shown in [START_REF] Müller | Quadratic costs do not always work in MPC[END_REF]. Using the proposed stage cost (9) reads for the nonholonomic robot (and its homogeneous approximation at the origin) as

ℓ(x, u) = x 4 1 + x 2 2 + x 4 3 + u 4 1 + u 4 2 , (35)
i.e. precisely the stage costs used in [START_REF] Worthmann | Regulation of differential drive robots using continuous time mpc without stabilizing constraints or costs[END_REF], for which asymptotic stability of the origin w.r.t. the MPC closed loop holds if a sufficiently large prediction horizon T is employed, cf. [START_REF] Worthmann | Model predictive control of nonholonomic mobile robots without stabilizing constraints and costs[END_REF]. Using the results derived in this paper, local asymptotic stability of the origin is a direct corollary of the simple calculations presented in Proposition 5. Moreover, we conjecture that the observations w.r.t. essentially using the homogeneous approximation may significantly facilitate (and improve) the quantitative estimates on the length of the prediction horizon.

Remark 3. Note that the particular definition of d has not been used so far. This explains, e.g., why the stage cost [START_REF] Worthmann | The role of sampling for stability and performance in unconstrained nonlinear model predictive control[END_REF] for the nonholonomic robot may be replaced by

ℓ(x, u) = n i=1 q x i |x i | d/r i + m j=1
q u j |u j | d/s j with arbitrary d ∈ (0, ∞) where q x i , i ∈ {1, 2, . . . , n}, and q u j , j ∈ {1, 2, . . . , m} are positive weighting factors, see [35, Subsection 2.2] for an example. 3Remark 4. Other homogeneous approximations can also been used to locally stabilize in small-time the control system (33) by means of periodic time-varying feedback laws as shown in [START_REF] Andréa-Novel | Small-time stabilization of nonholonomic or underactuated mechanical systems: the unicycle and the slider examples[END_REF].

Conclusions

For nonlinear systems with stabilizable linearization at the origin, MPC works (with quadratic stage cost) for sufficiently large prediction horizon (without adding stabilizing terminal costs and/or constraints). We have generalized this methodology to nonlinear systems with globally asymptotically null controllable homogeneous approximation at the desired set point. To this end, we have set up a general framework for rigorously showing cost controllability and, thus, local asymptotic stability of the origin w.r.t. the MPC closed loop for systems by designing the stage cost based on the homogeneous approximation.

A major advantage of the presented approach is its simplicity from the user's perspective: calculating the homogeneous approximation, checking (local) asymptotic stabilizability using the already existing results from the literature, and finally running MPC with the stage costs based on the coefficients from the homogeneity definition.

Proposition 5 .r 1 = r 3 = s 1 = s 2 = 1 , and r 2 = 2 .≤ α 2 x 2 3 / 2 .

 513121232 Inequality (23) holds for the homogeneous approximation of Example 3 at the origin, which is given by ẋ(t) = h(x(t), u(t)) := Proof. For i = 3, the approximation error is zero. For i = 1, we have to show the inequality |(cos(αr 3 x 3 )-1)α r 1 u 1 | ≤ M 1 α r 1 +τ +η 1 . Using r 1 = r 3 = 1, we have | cos(αx 3 )-1| Hence, |(cos(α r 3 x 3 ) -1)α r 1 u 1 | is smaller than α 3 ρ 3 /2,which shows the assertion for M 1 := ρ 3 /2 and η 1 ∈ [0, 2]. For i = 2, we analogously derive the estimate| sin(α r 3 x 3 )α r 3 x 3 |α r 1 |u 1 | ≤ α 4 |x 3 | 3 |u 1 |/6 ≤ α 4 ρ 4 /6and, thus, get the estimate for M 2 := ρ 4 /6 and η 2 ∈ [0, 2]. Overall, the desired property holds with M := max{ρ 3 /2, ρ 4 /6} and η ∈ (0, 2].

Every trajectory x(•; x i , u i ) is bounded on [0, T ⋆ ]. In addition, u i is essentially bounded. Hence, the stage costs are also bounded. Then, continuity of the solution w.r.t. the initial value and continuity of the stage cost imply the assertion.

This preliminary step is important to uniformly (independent of the initial value) reparametrize the time argument in the former growth bound by the factor α τ

In this reference, Brockett's nonholonomic integrator example was considered. However, this example is (locally) equivalent to the nonholonomic robot (unicycle), see[23, pp. 83-89].

* Jean-Michel Coron acknowledges funding from the Miller Institute and from the Agence Nationale de La Recherche (ANR), grant ANR Finite4SoS. He also thanks the Miller Institute and UC Berkeley for their hospitality. Karl Worthmann gratefully acknowledges funding from the German Research Foundation (Deutsche Forschungsgemeinschaft, grant WO 2056/6-1).


Dilys Lam

BSc (Hons Marie-Laure Ancelin

PhD Karen A Ritchie

PhD; Richard Saffery

PhD Joanne Ryan
email: joanne.ryan@monash.edu

DNA methylation and genetic variation of the angiotensin converting enzyme (ACE) in depression

Keywords: Angiotensin converting enzyme (ACE), cortisol, depression, DNA methylation, epigenetics, genetic variation

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Depression is one of the most prevalent psychiatric disorders, yet despite considerable advances in clinical practice, case detection remains relatively poor at all ages. In older persons, untreated depression has been furthermore associated with both high rates of functional impairment and comorbidity, notably cardiovascular disease (CVD) [START_REF] Alexopoulos | Depression in the elderly[END_REF]. One of the principal factors contributing to poor case identification is insufficient knowledge of the mechanisms underlying disease onset, and in particular the causes of variations in clinical presentation and treatment response.

Development of specific biomarkers would significantly contribute to a better understanding of disease etiology and heterogeneity, as well as improving case detection and monitoring.

Growing evidence suggests that depression results from complex interactions between genetic and environmental factors [START_REF] Caspi | Influence of life stress on depression: moderation by a polymorphism in the 5-HTT gene[END_REF][START_REF] Booij | Looking beyond the DNA sequence: the relevance of DNA methylation processes for the stress-diathesis model of depression[END_REF]. Epigenetic mechanisms such as DNA methylation may play an important role due to their ability to respond to both genetic and environmental influences to potentiate changes in gene activity [START_REF] Tsankova | Epigenetic regulation in psychiatric disorders[END_REF]. Whilst depression is a brainbased disorder, the effects on an individual are more wide-spread, and physiological changes in peripheral tissues have been observed [START_REF] O'donovan | Clinical anxiety, cortisol and interleukin-6: evidence for specificity in emotion-biology relationships[END_REF][START_REF] Thomson | Peripheral inflammation is associated with remote global gene expression changes in the brain[END_REF]. Furthermore, brain tissue cannot be accessed in living humans, so human epigenetic marks have been predominantly studied in peripheral tissues including blood and buccal (mouth epithelial cells) [START_REF] Lockwood | The role of epigenetics in depression and suicide: A platform for gene-environment interactions[END_REF], which offer an advantage of being easily accessible, thus contributing to their potential as biomarkers. Biomarkers do not always lie along causal pathways, but must robustly associate with the phenotype to ensure diagnostic utility.

The hypothalamic-pituitary-adrenal (HPA) axis is a neuroendocrine system that is central to regulating responses to stress [START_REF] Palazidou | The neurobiology of depression[END_REF]. In older persons, a high dysfunction of the HPA axis is observed [START_REF] Otte | A meta-analysis of cortisol response to challenge in human aging: importance of gender[END_REF] which may be enhanced by stimulatory effects of genes involved in the axis, such as that coding for the angiotensin converting enzyme (ACE). ACE converts Angiotensin I into Angiotensin II to regulate cardiovascular homeostasis [START_REF] Sayed-Tabatabaei | ACE polymorphisms[END_REF]. ACE also has a wellcharacterised role in regulating the HPA axis following stress-activation through stimulatory effects on two hormones, adrenocorticotropin hormone (ACTH) and corticotropin-releasing-hormone (CRH) [START_REF] Armando | Angiotensin II AT1 receptor blockade prevents the hypothalamic corticotropin-releasing factor response to isolation stress[END_REF][START_REF] Pavlatou | Chronic administration of an angiotensin II receptor antagonist resets the hypothalamic-pituitary-adrenal (HPA) axis and improves the affect of patients with diabetes mellitus type 2: preliminary results[END_REF]. Hyperactivity of the axis is observed consistently in depressed patients, as attested by elevated levels of cortisol, CRH and ACTH [START_REF] Lloyd | The role of corticotropin-releasing hormone in the pathophysiology of depression: therapeutic implications[END_REF]. ACE is notably considered a candidate susceptibility factor for the bi-directional relationship between depression and CVD [START_REF] Bondy | Common genetic factors for depression and cardiovascular disease[END_REF]. Genetic studies have linked several ACE genetic variants to an increased risk of depression [START_REF] Baghai | Polymorphisms in the angiotensin-converting enzyme gene are associated with unipolar depression, ACE activity and hypercortisolism[END_REF][START_REF] Lopez-Leon | Meta-analyses of genetic studies on major depressive disorder[END_REF][START_REF] Ancelin | Angiotensin-converting enzyme gene variants are associated with both cortisol secretion and late-life depression[END_REF] and to antidepressant treatment response [START_REF] Baghai | Possible influence of the insertion/deletion polymorphism in the angiotensin Iconverting enzyme gene on therapeutic outcome in affective disorders[END_REF][START_REF] Baghai | The angiotensin I converting enzyme insertion/deletion polymorphism influences therapeutic outcome in major depressed women, but not in men[END_REF]. ACE polymorphisms have also been shown to regulate cortisol secretion, a measure of HPA axis activity [START_REF] Baghai | Hypothalamic-pituitary-adrenocortical axis dysregulation in patients with major depression is influenced by the insertion/deletion polymorphism in the angiotensin Iconverting enzyme gene[END_REF][START_REF] Baghai | Polymorphisms in the angiotensin-converting enzyme gene are associated with unipolar depression, ACE activity and hypercortisolism[END_REF][START_REF] Ancelin | Angiotensin-converting enzyme gene variants are associated with both cortisol secretion and late-life depression[END_REF].

To date, few studies have investigated DNA methylation of ACE and its association with depression.

This study used data gathered from a large population based cohort of older individuals to investigate the associations between depression status and methylation at the promoter region of the ACE gene in peripheral blood. Analyses were adjusted for participants' lifestyle, health and medical history.

Potential modifying effects of genetic variation across the ACE gene and cortisol levels were also considered.

Methods

Study participants

The ESPRIT study is a French longitudinal general population study assessing neuropsychiatric disorders in older individuals [START_REF] Ritchie | Prevalence of DSM-IV psychiatric disorder in the French elderly population[END_REF]. Participants were recruited by random selection from electoral rolls within the Montpellier district, between 1999 and 2001. Non-institutionalised individuals aged 65 years and over were eligible and were contacted via mail with an invitation to participate, along with information detailing study procedures. The ESPRIT study protocol was approved by the regional ethics committee (Ethical Committee of University Hospital of Kremlin-Bicêtre, France). Following recruitment, participants underwent standardised clinical assessments, as well as health and psychiatric interviews. Information was collected on the participants' lifestyle, present state of health, medical history, medication use and exposure to adverse environmental factors.

All participants provided written informed consent.

Depression assessment

Current major depressive disorder (MDD) was diagnosed using a standardised psychiatric interview; the Mini International Neuropsychiatric Interview (MINI, French version 5.00), following the Diagnostic and Statistical Manual of Mental Disorders-IV (DSM-IV) (American Psychiatric Association, 1994). The MINI has been widely used in clinical and research settings, and validated within the general population setting [START_REF] Sheehan | The Mini-International Neuropsychiatric Interview (M.I.N.I.): the development and validation of a structured diagnostic psychiatric interview for DSM-IV and ICD-10[END_REF]. Participants that met the DSM-IV criteria for MDD were further reviewed by a panel of three psychiatrists and a psychologist with access to information from other health assessments. The Centre for Epidemiologic Studies-Depression (CES-D) scale was also administered to assess the severity of depressive symptoms [START_REF] Radloff | The CES-D Scale: A Self-Report Depression Scale for Research in the General Population[END_REF]. This questionnaire has been previously validated in the older population [START_REF] Radloff | The CES-D Scale: A Self-Report Depression Scale for Research in the General Population[END_REF][START_REF] Beekman | Criterion validity of the Center for Epidemiologic Studies Depression scale (CES-D): results from a community-based sample of older subjects in The Netherlands[END_REF]. In contrast to the MINI, the CES-D scale is a self-reporting questionnaire, with a score of 16 or above considered the threshold for significant depressive symptoms warranting further clinical investigation [START_REF] Radloff | The CES-D Scale: A Self-Report Depression Scale for Research in the General Population[END_REF]. To take into account that late-life depression covers a range of mild to severe depressive symptoms [START_REF] Fiske | Depression in older adults[END_REF], late-life depression assessment was based on two criteria; having a current diagnosis of MDD, or severe depressive symptoms (CES-D≥16), in keeping with prior work [START_REF] Ancelin | Angiotensin-converting enzyme gene variants are associated with both cortisol secretion and late-life depression[END_REF][START_REF] Januar | BDNF promoter methylation and genetic variation in late-life depression[END_REF].

Genotyping

Of the 1863 recruited participants, 785 agreed to provide blood samples at baseline for research.

Genomic DNA was extracted from white blood cells following a standard protocol [START_REF] Ritchie | Association of adverse childhood environment and 5-HTTLPR Genotype with late-life depression[END_REF], with samples stored at -80°C until use. Genotyping was performed by LGC Genomics, Hoddesdon, UK using their KASP SNP genotyping system [START_REF] Ancelin | Angiotensin-converting enzyme gene variants are associated with both cortisol secretion and late-life depression[END_REF]. This included data for seven single nucleotide polymorphisms (SNPs) across the ACE gene: rs1800764, rs4291, rs4295, rs4311, rs4333, rs4343, rs4351. These variants were chosen based on prior associations with depression and to ensure adequate coverage across the gene [START_REF] Ancelin | Angiotensin-converting enzyme gene variants are associated with both cortisol secretion and late-life depression[END_REF]. The distribution of genotypes was examined using a chi-squared test for Hardy-Weinberg equilibrium. Linkage disequilibrium between the SNPs were also calculated, with relatively high linkage disequilibrium across 3 SNPs (rs1800764, rs4291, rs4295) as estimated by r2 values (>0.75) [START_REF] Ancelin | Angiotensin-converting enzyme gene variants are associated with both cortisol secretion and late-life depression[END_REF].

ACE promoter methylation assay

A 356 base pair (bp) assay (hg_38: chr17: 63476511-63476866) was designed using the Epidesigner package (epidesigner.com) to cover a region of the ACE promoter CpG island, spanning 32 CpG sites, where differential methylation of several CpG sites was previously reported (Figure S1) [START_REF] Zill | DNA methylation analysis of the angiotensin converting enzyme (ACE) gene in major depression[END_REF]. Sodium bisulphite conversion of 500ng DNA from each participant blood sample was performed using the EZ-96 DNA Methylation-Lightning MagPrep Kit (Zymo Research; Irvine, USA) following the manufacturer's protocol. PCR amplification of the region of interest was performed in technical triplicates for all participant samples (Table S1), to account for possible variability [START_REF] Coolen | Genomic profiling of CpG methylation and allelic specificity using quantitative high-throughput mass spectrometry: critical evaluation and improvements[END_REF].

The primers used were F: 5'-TAGAGGAAGTTGGAGAAAGGGTT-3' and R: 5'-CCTACAAAACACCTAAAATCCCC-3', with an adapter (5'-aggaagagag) and T7 tag (5'cagtaatacgactcactatagggagaaggct) attached respectively. DNA methylation was quantified for each triplicate sample using the SEQUENOM MassARRAY EpiTYPER platform [START_REF] Coolen | Genomic profiling of CpG methylation and allelic specificity using quantitative high-throughput mass spectrometry: critical evaluation and improvements[END_REF].

Mean methylation values were calculated from replicates within 10% of the median [START_REF] Martino | Longitudinal, genome-scale analysis of DNA methylation in twins from birth to 18 months of age reveals rapid epigenetic change in early life and pair-specific effects of discordance[END_REF]. Further quality control included the exclusion of CpG units or participants where less than 50% of methylation data was available, to help ensure that spurious data was not analysed. Clearly outlying data points, i.e. those more than 3 times the interquartile range, were also excluded (n=21). Following this quality control, data was obtained for 16 CpG units encompassing 25 CpG sites (Table S2, Figure S1).

Cortisol measurement

Salivary cortisol was collected from a subsample of participants not taking medications likely to affect cortisol levels [START_REF] Ancelin | Angiotensin-converting enzyme gene variants are associated with both cortisol secretion and late-life depression[END_REF] and measured by direct radioimmunoassay (Diagnostic systems Laboratories, Webster, TX, USA) [START_REF] Hellhammer | Measurement of salivary cortisol under psychological stimulation[END_REF]. Cortisol samples were obtained at four time points throughout the day. Participants were told to withhold from eating, drinking or smoking at least 30 minutes before saliva collection. Salivary samples were collected by participants carrying out normal daily activities at home at least 1 hour after awakening, and subsequently 3, 7 and 14 hours after the first sampling, with the exact time recorded as published previously [START_REF] Beluche | Persistence of abnormal cortisol levels in elderly persons after recovery from major depression[END_REF].

Statistical analysis

After quality control, ACE methylation data was obtained for 552 participants, ranging from 414 to 550 at individual CpG units (Table S2). Participants excluded from our analysis (n=233) were not significantly different in key variables (Table 1) from those who were included. However, when compared with all recruited participants (n=1863), those included in our analyses were more likely to be male, have hypertension and less likely to be taking anxiolytic medication (p<0.05). The raw mean methylation values were first log-transformed to normalise the data and these values were used for all subsequent parametric statistical tests. To avoid undefined log-transformed values, such as when raw mean methylation equals 0, a value of 0.001 was added prior to log transformation. Univariate analysis (t-tests, chi-squared tests) were used to investigate potential associations between depression and methylation, as well as between population characteristics with depression and/or methylation levels independently. Multivariate linear regression analysis adjusted for potential confounding factors; population characteristics associated with both depression (shown in Table 1) and methylation at a conservative level of p<0.15, namely sex, age, smoking, functional impairment, ischemic disease, diabetes, thyroid disease, comorbidities, impaired cognition, antidepressant use, and batch effects (from different Sequenom chips). Genetic variants were also considered as potential effect modifiers of the association between depression and DNA methylation, by including a multiplicative interaction term in the regression models. When evidence of potential effect modification was found (p<0.15), stratified analysis was then performed to determine the association between depression status and DNA methylation across different genotype groups. Sensitivity analysis excluding antidepressant use was also performed (n=31), given that antidepressants may potentially mask depression status and influence methylation independently of depression [START_REF] Menke | Epigenetic alterations in depression and antidepressant treatment[END_REF]. For the 79 participants with both cortisol and methylation data, associations between salivary cortisol and methylation levels were examined using Pearson's correlation tests. Cortisol area under the curve (AUC) was standardised and calculated between the absolute morning (1hr after awakening) and evening levels (14hrs after first sampling) for each subject by extrapolating values from the equation of the regression line as published previously [START_REF] Beluche | Persistence of abnormal cortisol levels in elderly persons after recovery from major depression[END_REF]. Based on the number of non-depressed and depressed individuals, our study will have >90% power to detect a methylation change of 0.3% with 1% standard deviation (2-sided α=0.05). All statistical analyses were performed using the statistical software package Stata 14.1 (StataCorp, College Station, USA).

Results

Study Population

Characteristics of participants of this current study according to depression status are shown in Table 1. Depressed individuals (29.7%) were significantly more likely to be female, older, living alone, functionally impaired, have poorer health and take psychotropic or anticholinergic medication, as compared to non-depressed persons. Habitual smokers were less likely to be depressed.

Association between depression and ACE methylation

We first investigated the association between depression and methylation at the ACE promoter. In unadjusted analysis, mean methylation across the assayed region was not significantly different (∆= -0.10%, 95%CI: -0.23; 0.03%, p=0.941) between non-depressed (3.53%, 95%CI: 3.45;3.60%) and depressed individuals (3.43%, 95%CI: 3.32;3.53%). No significant differences in methylation were found at any individual CpG units (Figure 1, Table S3) and further adjustment for potential confounders including age, sex, and antidepressant use (see Methods) did not modify this pattern.

Sensitivity analysis with the exclusion of antidepressant users (n=31) did not alter the significance of the associations.

Modification of the association between depression and methylation by ACE genotype

As genetic variation is known to influence DNA methylation, seven polymorphisms across the ACE gene (rs1800764, rs4291, rs4295, rs4311, rs4333, rs4343, rs4351) were investigated. The genotype frequencies in non-depressed and depressed groups were in Hardy-Weinberg equilibrium (p>0.10 for all SNPs) and two SNPs, rs4333 and rs4351, were significantly associated with depression (Table S4).

In the whole sample, there was some evidence that specific genetic variation influenced methylation at several CpG sites (Table 2): rs4291 was associated with methylation at CpG 10.11.12 while rs4333 was associated with methylation at CpG 28.29.30. Homozygous TT individuals had significantly lower methylation in both cases.

In investigating the potential modifying effects of genetic variation, all seven SNPs were found to modify the association between depression and methylation at multiple CpGs. In particular, genotypes for four ACE variants were found to significantly modify the association between depression and methylation at CpG 28.29.30 (the p-values for interaction terms were for each SNP; rs1800764, p=0.061; rs4351, p=0.036; rs4295, p=0.016; rs4311, p=0.016). Following stratification analysis according to genotype, depression associated with significantly lower methylation at CpG 28.29.30 (Figure 2), for these specific homozygote genotypes: rs1800764 (TT, ∆= -1.57%, 95%CI: -2.85; -0.29%, p=0.037, Figure 2a); rs4351 (AA, ∆= -2.05%, 95%CI: -3.65; -0.46%, p=0.039, Figure 2b); rs4295 (CC, ∆= -1.52%, 95%CI: -2.70; -0.34% p=0.023, Figure 2c) and rs4311 (CC, ∆= -2.02%, 95%CI: -3.50; -0.54%, p=0.031, Figure 2d). Genotype also modified the associations between depression and methylation at several other CpGs; CpG 4.5.6.7 (rs4291, p-values for interaction terms = 0.013), CpG 14 (rs4333, p=0.147), and CpG 17 (rs4343, p=0.146). However, in these cases a positive association was observed, with significant higher methylation levels in the depressed individuals with these homozygote genotypes of rs4291 (TT, ∆= 0.59%, 95%CI: -0.02; 1.20%, p=0.019, Figure 2e) at CpG 4.5.6.7; rs4333 (CC, ∆= 1.74%, 95%CI: 0.14; 3.35%, p=0.033, Figure 2f) at CpG 14; and rs4343 (AA, ∆0.74%, 95%CI: 0.073; 1.40%, p=0.020, Figure 2g) at CpG 17. All associations, with the exception of CpG 14, remained significant following adjustment for potential confounders.

Association between salivary cortisol levels and ACE methylation

Finally, as the ACE gene is involved in the regulation of cortisol secretion, the association between DNA methylation and AUC cortisol levels was examined, for a sub-sample of 79 participants (as detailed in the Methods). Baseline cortisol significantly negatively correlated with average methylation levels (r= -0.336, p=0.042) (Figure 3) and methylation at CpG 1 (r= -0.247, p=0.039). Given the small numbers in this sub-sample (n=79), we were unable to investigate whether these associations differed by depression status. However, when excluding the 14 depressed individuals in sensitivity analysis, only the association between average methylation (but not CpG 1) and basal cortisol levels remained significant (r=-0.417, p=0.030, n=65).

Discussion

No significant association between depression status and ACE promoter methylation in isolation was found in this older population of 552 individuals after adjustment for a wide range of potential confounding factors. However, there was evidence that ACE genetic variants were associated with methylation, and modified the association between depression and methylation. In particular, depression was associated with significantly decreased ACE methylation at CpG 28.29.30 for four variants, and significantly increased ACE methylation at CpGs 1, 14 and 17 for three variants. These novel findings were not confounded by sex, age, antidepressant use or other factors examined, including vascular disorder. Furthermore, in a small sub-sample, we provide some preliminary support that lower ACE methylation was correlated with higher cortisol levels, independent of depression status.

Previous studies have suggested the ACE gene to be implicated in depression onset [START_REF] Baghai | Polymorphisms in the angiotensin-converting enzyme gene are associated with unipolar depression, ACE activity and hypercortisolism[END_REF][START_REF] Lopez-Leon | Meta-analyses of genetic studies on major depressive disorder[END_REF][START_REF] Ancelin | Angiotensin-converting enzyme gene variants are associated with both cortisol secretion and late-life depression[END_REF], antidepressant response [START_REF] Baghai | Possible influence of the insertion/deletion polymorphism in the angiotensin Iconverting enzyme gene on therapeutic outcome in affective disorders[END_REF][START_REF] Baghai | The angiotensin I converting enzyme insertion/deletion polymorphism influences therapeutic outcome in major depressed women, but not in men[END_REF]) and cortisol signaling [START_REF] Baghai | Hypothalamic-pituitary-adrenocortical axis dysregulation in patients with major depression is influenced by the insertion/deletion polymorphism in the angiotensin Iconverting enzyme gene[END_REF][START_REF] Baghai | Polymorphisms in the angiotensin-converting enzyme gene are associated with unipolar depression, ACE activity and hypercortisolism[END_REF][START_REF] Ancelin | Angiotensin-converting enzyme gene variants are associated with both cortisol secretion and late-life depression[END_REF]. However, only one prior study has investigated ACE promoter methylation in the context of depression [START_REF] Zill | DNA methylation analysis of the angiotensin converting enzyme (ACE) gene in major depression[END_REF]. In a case-control study, [START_REF] Zill | DNA methylation analysis of the angiotensin converting enzyme (ACE) gene in major depression[END_REF] analysed methylation in peripheral white blood cells, targeting a similar region of the ACE promoter to that examined in our study. In contrast to our findings however, they reported an overall pattern of hypermethylation of the ACE promoter in 81 MDD-diagnosed individuals compared to 81 non-depressed adults. They observed significant associations at 3 of 24 CpG sites examined, corresponding to CpG 14, 19-23 and 24.25 in our study.

We also observed hypermethylation at CpG 14 in depressed participants, however with specific genotypes (see below). It should be noted that CpG 19-23 and 24.25 were not included in any of our analyses (failing initial quality control-see Methods) and CpG 28.29.30 was not examined in their study.

Additionally, the average methylation levels at each CpG in our study appeared to be consistently lower (ranging from 0-9%) than that observed by [START_REF] Zill | DNA methylation analysis of the angiotensin converting enzyme (ACE) gene in major depression[END_REF] (up to 16%).

Our study, like that of [START_REF] Zill | DNA methylation analysis of the angiotensin converting enzyme (ACE) gene in major depression[END_REF] included European Caucasians, although their population was much smaller (n=162) and more heterogeneous (19-72 years). Further, potential confounding or effect modification by other health, lifestyle or genetic factors was not considered. These differences may account for the divergent findings. Methylation levels change with age (the "epigenetic clock"), with reports of global hypomethylation [START_REF] Johnson | The role of DNA methylation in aging, rejuvenation, and age-related disease[END_REF] and gene-specific hypermethylation with increasing age [START_REF] Jung | Aging and DNA methylation[END_REF], hence decreases in ACE methylation may be specifically associated with late-life depression. Furthermore, increased dysfunction of the HPA axis has been observed in older individuals [START_REF] Otte | A meta-analysis of cortisol response to challenge in human aging: importance of gender[END_REF], which may be differentially influenced by ACE. [START_REF] Zill | DNA methylation analysis of the angiotensin converting enzyme (ACE) gene in major depression[END_REF] also focused on patients diagnosed with MDD, which represents only 4.3% (n=16) of our study population. The majority of our study participants had severe depressive symptoms (as assessed with the CES-D), which may differ to clinically diagnosed MDD in its causation and associated physiological processes [START_REF] Fiske | Depression in older adults[END_REF][START_REF] Fournier | Antidepressant drug effects and depression severity: a patient-level meta-analysis[END_REF].

To our knowledge, this is the first study to show that ACE promoter methylation was significantly negatively correlated with basal cortisol levels. Although in only a small sub-sample of participants, it is interesting to note that the direction of these associations supports the main findings of our study: decreased methylation was observed in depressed individuals (with specific genotypes) and was associated with heightened basal cortisol levels. An over-activation of cortisol signalling is frequently reported as a feature of depression [START_REF] Lloyd | The role of corticotropin-releasing hormone in the pathophysiology of depression: therapeutic implications[END_REF]. Cortisol secretion has also previously been shown to be modulated by ACE polymorphisms [START_REF] Baghai | Polymorphisms in the angiotensin-converting enzyme gene are associated with unipolar depression, ACE activity and hypercortisolism[END_REF][START_REF] Ancelin | Angiotensin-converting enzyme gene variants are associated with both cortisol secretion and late-life depression[END_REF][START_REF] Baghai | Hypothalamic-pituitary-adrenocortical axis dysregulation in patients with major depression is influenced by the insertion/deletion polymorphism in the angiotensin Iconverting enzyme gene[END_REF]. Our findings are in line with previous research demonstrating potential functional consequences of methylation at this region, with promoter methylation inversely correlated with ACE expression and serum concentration in a dose-dependent manner [START_REF] Riviere | Epigenetic regulation of somatic angiotensin-converting enzyme by DNA methylation and histone acetylation[END_REF][START_REF] Zill | DNA methylation analysis of the angiotensin converting enzyme (ACE) gene in major depression[END_REF].

Peripheral methylation levels have also been observed to be comparable to levels in post-mortem cortex and hippocampal tissues, which are part of a network that is regulated by the HPA axis and dysregulated in depressive disorders [START_REF] Zill | DNA methylation analysis of the angiotensin converting enzyme (ACE) gene in major depression[END_REF]. Therefore, peripheral ACE methylation could be correlated with HPA axis activity, as measured by cortisol secretion. This suggests that ACE methylation may be more suited as a biomarker for cortisol levels and/or HPA axis activity than for depression. However, no definitive conclusions can be drawn from this result, with further research needed, particularly considering the potential role of cortisol in depression [START_REF] Herbert | Cortisol and depression: three questions for psychiatry[END_REF]. Also unique to our study is the inclusion of six ACE polymorphisms shown to be independently associated with methylation levels at several CpGs within the ACE promoter. Such genetic variants that influence DNA methylation at specific sites are referred to as methylation quantitative trait loci or mQTLs. This fits with emerging literature highlighting the importance of the genome in regulating the epigenome [START_REF] Gaunt | Systematic identification of genetic influences on methylation across the human life course[END_REF]. The effects of mQTLs tend to be more pronounced at CpG sites in close proximity (cis effect) with a distance of less than 1 Megabase (MB), although some act more distally (trans effect) [START_REF] Zhang | Genetic control of individual differences in gene-specific methylation in human brain[END_REF]. The studied polymorphisms may be acting in cis on specific CpGs, as our promoter (rs1800764, rs4291) and intronic SNPs (rs4295,rs4311,rs4333,rs4343,rs4351) are within a distance of 13kb from the promoter. Further studies are now needed to investigate the functionality of this putative layer of regulation.

An interesting finding from our study was that all seven polymorphisms investigated significantly modified the association between methylation and depression at several CpG units. This suggests that an effective biomarker would need to be a combination of ACE methylation and genetic variants. For example, lower methylation levels at CpG 28.29.30 were associated with depression for four variants, but only in individuals homozygous for the T allele of rs1800764, the A allele of rs4351, or the C allele of rs4295 or rs4311. On the other hand, ACE methylation levels at CpG 4.5.6.7, 14, and 17 were higher in depressed individuals, but only for those homozygous for T allele of rs4291, the C allele of rs4333, or the A allele of rs4343 genotypes. In modulating susceptibility to diseases, genetic variants may modify the probability of DNA methylation and in turn regulate gene expression [START_REF] Rakyan | Epigenome-wide association studies for common human diseases[END_REF] and/or combine to modify depression risk [START_REF] Zhi | SNPs located at CpG sites modulate genome-epigenome interaction[END_REF]. While these findings add to the growing body of research in this field, further investigation is needed to clarify these complex genotype and sitespecific methylation interactions of studied polymorphisms and their functional consequences.

The major strength of this study is the large sample size which increases the statistical power to detect the likely subtle differences in peripheral methylation observed in psychiatric disorders [START_REF] Olsson | Prospects for epigenetic research within cohort studies of psychological disorder: a pilot investigation of a peripheral cell marker of epigenetic risk for depression[END_REF]. Larger studies help provide more robust estimates of true associations, as small underpowered studies are prone to false positives, or type 1 errors [START_REF] Ioannidis | Why most published research findings are false[END_REF]. Another strength of our study is the simultaneous investigation of both ACE promoter methylation and genetic variation, and their association with depression. Other strengths of our study include the diagnosis of depression which was based on a clinical assessment (MINI, DSM-IV) of MDD, as well as a validated questionnaire of severe depressive symptoms (CES-D≥16).

This study was conducted on an older French Caucasian population and while applying validated assessments, did not take into account age at onset of first depressive disorder. The extent to which these results can be applied to earlier onset depression with possible different etiologies in younger age groups remains to be determined [START_REF] Blazer | Depression in late life: review and commentary[END_REF]. DNA methylation patterns vary with age and ethnicity [START_REF] Zhang | Significant differences in global genomic DNA methylation by gender and race/ethnicity in peripheral blood[END_REF][START_REF] Johnson | The role of DNA methylation in aging, rejuvenation, and age-related disease[END_REF]. The association between methylation and depression can be confounded by many lifestyle and health factors that affect methylation and depression status and we controlled for these in our analysis, however further analysis is still needed. Of particular importance are functional impairment, a characteristic of depressive disorders as listed in the DSM criteria (American Psychiatric Association, 1994); antidepressant use, which has been shown to modify methylation levels [START_REF] Menke | Epigenetic alterations in depression and antidepressant treatment[END_REF]; and ACE inhibitors, which may have antidepressant-life effects [START_REF] Vuckovic | The use of captopril in treatment-resistant depression: an open trial[END_REF]. Given our cross-sectional analysis, our findings may result from an association with one or more of these confounding factors. While the present study was able to consider a much wider range of potential confounders than the previous study, the potential for residual confounding from factors where information was not available or collected, remains possible. For example, factors known to influence DNA methylation include stress (beyond the cortisol marker) [START_REF] Booij | Looking beyond the DNA sequence: the relevance of DNA methylation processes for the stress-diathesis model of depression[END_REF] , methyl donors and nutrition [START_REF] Anderson | Nutrition and epigenetics: an interplay of dietary methyl donors, one-carbon metabolism and DNA methylation[END_REF]. There may also be unknown factors that influence DNA methylation and/or depression that have yet to be determined.

Finally, no adjustment for multiple testing was made, and none of the associations would remain significant at the stringent Bonferroni corrected p value of 0.00045 (accounting for 16 CpG units and 7 SNPs, thus 112). However, such an adjustment would be overly conservative, as Bonferroni correction assumes independence of tests, and the tests we performed were not independent. Three of the SNPs for example, are in high linkage disequilibrium as stated in the text (page 7, section 2.3) and methylation levels at all but one of the CpG units was significantly correlated with other CpG units (p<0.05).

Our study focuses on examining potential peripheral biomarkers of depression. Depression is a brainbased disorder, however it is becoming increasingly recognised as a systemic disease, with physiological changes observed in peripheral tissue [START_REF] O'donovan | Clinical anxiety, cortisol and interleukin-6: evidence for specificity in emotion-biology relationships[END_REF][START_REF] Thomson | Peripheral inflammation is associated with remote global gene expression changes in the brain[END_REF].

DNA methylation levels between blood and post-mortem brain tissues may be correlated, at least for some genes [START_REF] Horvath | Aging effects on DNA methylation modules in human brain and blood tissue[END_REF]. Furthermore, to be a useful peripheral biomarker, blood methylation levels do not need to reflect brain methylation levels or underlying mechanisms of depression, as long as they can be shown to be able to consistently distinguish depressed from non-depressed individuals across multiple independent studies.

Another limitation of our study is the candidate gene approach although there is evidence for the involvement of ACE in depression. This candidate approach requires a strong a priori hypothesis, and being a complex disorder, may however, only have captured a small part of depression pathogenesis.

For example, ACE levels or methylation may only be relevant to manifestations of depression that involve dysregulation of the HPA axis. Epigenome-wide association studies (EWAS) could be an alternative approach, enabling interrogation of large proportions of genomic locations without an a priori hypothesis. EWAS studies have the added advantage of being able to adjust for cellular heterogeneity through computational estimation of cell composition, which may have distinct methylation profiles [START_REF] Jaffe | Accounting for cellular heterogeneity is critical in epigenome-wide association studies[END_REF]. This was not achievable in our study, potentially confounding our methylation analyses and may account for the lack of associations observed between depression and ACE methylation. Unfortunately costs remain very high for large cohort EWAS studies such as our study cohort [START_REF] Rakyan | Epigenome-wide association studies for common human diseases[END_REF].

Conclusion

Our study of late-life depression provides evidence of genotype-specific associations between depression and ACE methylation, as well as inverse correlations between ACE methylation and cortisol secretion. These findings highlight the potential need to consider methylation levels in combination with genetic variation in the development of effective depression biomarkers. Given this novel finding, the small effect sizes observed and the potential relationship between age and DNA methylation patterns, replication in large independent populations are needed to confirm these findings. Extending methylation analysis to cover the entire gene and associated regulatory regions, and to other potential HPA axis genes should be considered. Importantly, future investigation of the mechanisms behind the genomic control of the epigenome in ACE need to be undertaken.

Figure 1 .

 1 Figure 1. Methylation differences between depressed and non-depressed individuals at each CpG

Figure 2 .

 2 Figure 2. Differences in methylation levels at CpG units according to depression status, stratified by angiotensin converting enzyme (ACE) genotypes: (a) CpG 28.29.30 rs1800764 (n=219); (b) CpG 28.29.30 rs4351 (n=219); (c) CpG 28.29.30 rs4295 (n=225); (d) CpG 28.29.30 rs4311 (n=224); (e) CpG 4.5.6.7 (rs4291, n=542); (f) CpG 14 (rs4333, n=212); (g) CpG 17 (rs4343, n=532). Data presented as the geometric mean methylation ± 95% CI. T-tests were used to calculate p-values, with significant associations at p<0.05 indicated in the figure.

Figure 3 .

 3 Figure 3. Correlation between baseline cortisol AUC levels and average methylation (r= -0.336,

Table 1 .

 1 Characteristics of study participants according to depression status at baseline

	a Not all participants with methylation data (n=552) had information for each population characteristic
	listed. Missing data did not exceed 2.4%
	b Underwent post-secondary education
	c Independently unable to complete two items on both or either of the IADL and ADL scales
	d Hypertension: resting blood pressure≥160/95mmHg
	e Ischemic disease: History of angina pectoris, myocardial infarction, stroke, cardiovascular surgery
	and/or arteritis
	f Comorbidities: Ischemic or chronic (high blood pressure, high cholesterol, diabetes, thyroid disease,
	asthma) disease, or recent cancer (diagnosed within the last 2 years)
	g MMSE: Mini-Mental State Examination
	h Chi-squared tests were used to assess p-value for all variables except age, where a t-test was used

Table 2 .

 2 Potential associations between angiotensin converting enzyme (ACE) genotype and DNA methylation a Mean methylation presented as geometric mean b one-way ANOVAs were used to calculate p-values and only those with p-value<0.15 are shown

	Polymorphism	CpG unit	Methylation a (%)		p-value b
	rs4291	10.11.12	AA (7.72)	AT (7.72)	TT (7.04)	0.041
	rs4295	2	CC (0.59)	CG (0.57)	GG (0.81)	0.137
		14	CC (7.06)	CG (7.82)	GG (8.04)	0.114
		15.16	CC (6.11)	CG (5.45)	GG (5.25)	0.086
		28.29.30	CC (5.20)	CG (4.86)	GG (4.27)	0.146
	rs4311	14	TT (8.11)	TC (7.65)	CC (6.78)	0.081
		15.16	TT (5.45)	TC (5.48)	CC (6.24)	0.102
	rs4333	28.29.30	TT (4.45)	TC (5.25)	CC (4.84)	0.020
	rs4343	14	GG (8.27)	GA (7.75)	AA (7.59)	0.090
		26	GG (0.28)	GA (0.21)	AA (0.31)	0.079
	rs4351	27	GG (1.12)	GA (1.05)	AA (1.43)	0.136

Table S2 .

 S2 Individual CpG units assayed and analysed in this study

	CpG unit	Position (chr17) Included	Reason for exclusion	n =	Outliers excluded (n)	Mean methylation (%)	SD
	1	63476818	Yes		526	18	0.93	1.09
	2	63476807	Yes		550	17	0.56	0.67
	3	63476798	Yes		473	9	0.87	1.20
	4.5.6.7	63476785
		63476781 63476777	Yes		542	5	2.75	1.19
		63476775
	8.9	63476767 63476763	Yes		449	5	7.51	3.57
	10.11.12	63476745
		63476738	Yes		525	5	7.61	2.18
		63476733
	13	63476719	Yes		414	12	8.00	6.79
	14	63476711	Yes		505	16	7.89	2.80
	15.16	63476701 63476695	Yes		533	15	5.53	1.94
	17	63476681	Yes		532	17	3.16	1.66
	18	63476662	Yes		545	19	2.44	1.71
	19.20.21.22.23	63476656
		63476653
		63476649	No	<50% data NA	NA	NA	NA
		63476645
		63476642
	24.25	63476626 63476616	No	<50% data NA	NA	NA	NA
	26	63476593	Yes		532	33	0.25	0.42
	27	63476586	Yes		547	19	1.17	1.13
	28.29.30	63476580
		63476577	Yes		514	20	4.92	2.34
		63476570
	31	63476553	Yes		543	20	1.33	1.12
	32	63476533	Yes		537	22	0.97	0.99

Table S4 .

 S4 Prevalence of ACE genotypes in the study population according to depression status a Chi-squared tests were used to calculate p-value

	SNP and genotype	No Depression (%)	Depression(%)	p-value a
	rs1800764 (n=235)
	TT	25.3	39.3
	TC	47.1	44.3	0.065
	CC	27.6	16.4
	rs4291 (n=552)
	AA	32.2	39.0
	AT	50.8	48.2	0.222
	TT	17.0	12.8
	rs4295 (n=241)
	CC	29.8	41.3
	CG	50.5	47.6	0.142
	GG	19.7	11.1
	rs4311 (n=240)
	TT	33.9	20.6
	TC	49.2	52.4	0.075
	CC	16.9	27.0
	rs4333 (n=235)
	TT	37.8	22.2
	TC	46.5	49.2	0.025
	CC	15.7	28.6
	rs4343 (n=552)
	GG	35.3	26.2
	GA	46.7	53.1	0.114
	AA	18.0	20.7
	rs4351 (n=234)
	GG	36.8	20.6
	GA	48.6	52.4	0.020
	AA	14.6	27.0

Acknowledgments

We thank all of the participants, and Dr Benjamin Ong for assistance with the Sequenom MassARRAY platform. The ESPRIT project is financed by the regional government of Languedoc-Roussillon, the Agence Nationale de la Recherche Project 07 LVIE 004, and an unconditional grant from Norvartis.

This work was also supported by the National Health and Medical Research Council, through a Senior Research Fellowship (APP1045161 to RS); the Murdoch Children's Research Institute (studentship to DL); the Victorian Government's Operational Infrastructure Support Program. The funders had no role in the design and conduct of the study, in data collection, management, analysis or interpretation of the data and were not involved with the writing, preparation, review or approval of the manuscript.

Conflict of interest

All authors declare no conflicts of interest.

Supplementary Material

DNA methylation and genetic variation of the Angiotensin converting enzyme (ACE) in depression

Dilys Lam, BSc (Hons); Marie-Laure Ancelin, PhD; Karen Ritchie, PhD; Richard Saffery, PhD; Joanne Ryan, PhD