Stabilizability in optimal control

We extend the classical concepts of sampling and Euler solutions for control systems associated to discontinuous feedbacks by considering also the corresponding costs. In particular, we introduce the notions of Sample and Euler stabilizability to a closed target set (p0,W)-regulated cost, for some continuous, state-dependent function W and some positive constant p0: it roughly means that we require the existence of a stabilizing feedback K such that all the corresponding sampling and Euler solutions starting from a point z have suitably defined finite costs, bounded above by W(z)/p0. Then, we show how the existence of a special, semiconcave Control Lyapunov Function W, called p0-Minimum Restraint Function, allows us to construct explicitly such a feedback K. When dynamics and Lagrangian are Lipschitz continuous in the state variable, we prove that K as above can be still obtained if there exists a p0-Minimum Restraint Function which is merely Lipschitz continuous. An example on the stabilizability with (p0,W)-regulated cost of the nonholonomic integrator control system associated to any cost with bounded Lagrangian illustrates the results.


Introduction
In this paper we investigate in an optimal control perspective the stabilizability to a set C ⊂ R n of the nonlinear control systeṁ to which we associate a cost of the form Tx 0 l(x(τ ), u(τ )) dτ, (1.2) where l ≥ 0, and T x ≤ +∞ -the exit-time of x-verifies (for any y ∈ R n , d(y) denotes the distance of y from C). In particular, for any r > 0, we set B r (C) := {y ∈ R n : d(y) ≤ r} and assume that: (H0) the target set C ⊂ R n is closed, with compact boundary; the control set U ⊆ R m is closed, not necessarily bounded; the functions f : (R n \ C) × U → R n , l : (R n \ C) × U → [0, +∞) are uniformly continuous on K × U for any compact set K ⊂ R n \ C; for any R > 0 there is some M (R) > 0 such that Hence, for any admissible control, given by a function u ∈ L ∞ loc ([0, T x ), U ) 1 , every Cauchy problem associated to (1.1) has in general multiple solutions and the cost may be finite even if T x = +∞.
In order to obtain sufficient conditions for the stabilizability of the system with regulated cost, we consider the Hamiltonian H(x, p 0 , p) := inf u∈U p , f (x, u) + p 0 l(x, u) , (1.5) and the following notion, firstly introduced in [23] (in a slightly weaker form).
A p 0 -Minimum Restraint Function is at once a Control Lyapunov Function for (1.1) and, by [23,Prop. 5.1], also a strict viscosity supersolution to the Hamilton-Jacobi-Bellman equation Hence, when a p 0 -Minimum Restraint Function exists, on the one hand, the (open-loop) global asymptotic controllability of the control system (1.1) to C -namely, that for any initial condition z ∈ R n \ C there is an admissible trajectory-control pair (x, u) to (1.1) with x(0) = z, such that lim t→T − x d(x(t)) = 0 in a certain uniform and stable manner that we will not dwell upon here-is expected (see e.g. [30,32,3]). On the other hand, if p 0 > 0, the existence of an admissible trajectory-control pair (x, u) with x(0) = z satisfying the cost estimate Tx 0 l(x(t), u(t)) dt ≤ W (z) p 0 (1.7) follows by known optimality principles (see e.g. [33,22,25]). The main contribution of [23,19] was to prove that the existence of a p 0 -Minimum Restraint Function W for some p 0 > 0 allows to produce a pair (x, u) that meets both of these properties. A related and important goal is, given a p 0 -Minimum Restraint Function W for some p 0 > 0, to provide a state feedback K : R n \ C → U such that the systemẋ(t) = f (x(t), K(x(t)) is globally asymptotically stable to C and has (p 0 , W )-regulated cost, that is, such that for any stable trajectory x with x(0) = z, the cost Tx 0 l(x(t), K(x(t))) dt is not greater than W (z)/p 0 . In this paper we address the question, left by [23,19] as an open problem, of how to define such a feedback law through the use of W . In the ideal case in which W is differentiable and there exists a continuous feedback K(x) such that DW (x) , f (x, K(x)) + p 0 l(x, K(x)) ≤ −γ(W (x)) for all x ∈ R n \ C, one easily derives global asymptotic stabilizability with (p 0 , W )-regulated cost. However, it is a classical matter in nonlinear control systems that a differentiable Control Lyapunov Function W may not exist and, even if a smooth W exists, a continuous feedback K does not generally exist (see e.g. [1,7,31,30,32,15]). From here, the need of considering the nonsmooth version (1.6) of the decrease condition and of defining a discontinuous feedback K : R n \ C → U , which, because of the unboundedness of U , we can only assume locally bounded (see Prop. 3.2 below). In particular, we might have lim sup x→x∈∂C |K(x)| = +∞.
Identified a feedback K as above, the main issue is how to interpret the discontinuous differential equation and the associated exit-time cost, so that the control system (1.1) can be stabilized to the target C with (p 0 , W )regulated cost by K. For the trajectories ofẋ(t) = f (x(t), K(x(t)), we simply adapt to our setting the nowadays classical notions of sampling and Euler solutions in [13,14], inspired by differential games theory [17]. However, our primary objective is to introduce a suitable concept of cost associated to a stable sampling or Euler solution starting from z, so that such cost is bounded above by W (z)/p 0 . Postponing the precise definitions to Section 2, given some constants r, R such that 0 < r < R, we call a sampling pair (x, u) (r, R)stable when, starting from some z with r < d(z) ≤ R, x reaches in a uniform manner the r-neighborhood B r (C) of the target and, after a timeT r x , remains definitely in B r (C). In this case, for all t ≥ 0 we define the corresponding sampling cost as l(x(s), u(s)) ds and show that x 0 (t) ≤ W (z)/p 0 . The difficulty in proving the latter inequality lies in the fact thatT r x may not be the first instant in which x enters B r (C). Consequently, we must estimate the cost in a time interval where we basically have no information on x, except that it is in B r (C) (see Subsection 3.1 below). Nevertheless, this is the correct notion of sampling cost. Indeed, let us now define an Euler cost-solution pair (X 0 , X) as the locally uniform limit on [0, +∞) of a sequence of sampling cost-trajectory pairs as above, when the sampling times tend to zero. We obtain that X approaches uniformly asymptotically C, while lim t→T − Subsection 4.6).
Furthermore, inspired by [28], we prove that, when f and l are locally Lipschitz continuous in x uniformly w.r.t. u ∈ U up to the boundary of C, the existence of a p 0 -Minimum Restraint Function W with p 0 > 0, possibly not locally semiconcave but merely locally Lipschitz continuous on R n \ C, still guarantees stabilizability of (1.1) to C with (p 0 , W )-regulated cost, in the sample and Euler sense (see Theorem 4.3 and Corollary 4.4). This result can be useful for the effective implementation of our feedback construction, as shown by the example in Section 5.
We are motivated to consider U unbounded and the minimization in the decrease condition on the whole set U (and not on bounded subsets, as usual) mainly because these assumptions are met, for instance, in the stabilization of mechanical systems with vibrating controls. These are nonlinear control systems, affine or quadratic in the derivative of the control, which is considered as an impulsive control (see [6]). In particular, the reparameterized systems usually introduced in the study of control-polynomial systems satisfy (H0). As a consequence, as shown in [18], the results of the present paper are the starting point for the stabilizability (with and without regulated cost) of impulsive control systems. In fact, our assumptions include also the case where U is bounded and f and l are continuous on R n × U . For U unbounded, they are satisfied, for instance, by the class of control problems in which the input appears inside a saturation nonlinearity, such as . . , f m ∈ C(R n ) and σ 0 , . . . , σ m are bounded, uniformly continuous maps on U . The stabilizability of such control systems plays a relevant role both in the literature and in the applications (see e.g. [4,10,11,34]).
Finally, the value function is clearly bounded above by any p 0 -Minimum Restraint Function divided by p 0 . Hence our approach could be useful to design approximated optimal closed-loop strategies, when there exists a sequence of p 0 -Minimum Restraint Functions approaching V , as in [25], or at least to obtain "safe" performances, keeping the cost under the value W . Moreover, when V ≤ W/p 0 , then V is continuous on the targets boundary and this is crucial to establish comparison, uniqueness, and robustness properties for the associated HamiltonJa-cobiBellman equation [22,24,25] and to study associated asymptotic and ergodic problems [26]. From this PDE point of view, problem (1.1)-(1.2) has been widely investigated; a likely incomplete bibliography, also containing applications (for instance, the Füller and shape-from-shading problems), includes [16,20,8,33] and the references therein. The paper is organized as follows. In the remaining part of the Introduction we provide some preliminary definitions and notation. In Section 2 we define precisely the sample and Euler stabilizability of (1.1) to C with (p 0 , W )-regulated cost, which is shown to be guaranteed by the existence of a p 0 -Minimum Restraint Function in Theorem 3.1, our main result. In Section 4 we consider the case of Lipschitz continuous data, postponing the proofs in the Appendix. An example on the stabilizability with regulated cost of the non-holonomic integrator control system concludes the paper (see Section 5).

Notation and preliminaries
For every r ≥ 0 and Ω ⊂ R N , we set B r (Ω) : where d is the usual Euclidean distance. When Ω = {z} for some z ∈ R N , we also use of the notation B(z, r) := B r ({z}). For any map F : Ω → R M we call modulus (of continuity) of F any increasing, continuous function ω : [0, +∞) → [0, +∞) such that ω(0) = 0, ω(r) > 0 for every r > 0 • Ω to denote the closure and the interior of the set Ω, respectively.
for all x,x ∈ Ω such that [x,x] ⊂ Ω. The constant ρ above is called a semiconcavity constant for F in Ω. F is said to be locally semiconcave on Ω if it semiconcave on every compact subset of Ω.
We remind that locally semiconcave functions are locally Lipschitz continuous. Actually, they are twice differentiable almost everywhere.
where ∇ denotes the classical gradient operator and DIF F (F ) is the set of differentiability points of F .
The set-valued map x D * F (x) is upper semicontinuous on Ω, with nonempty, compact, and not necessarily convex values. When F is a locally semiconcave function, D * F coincides with the limiting subdifferential ∂ L F , namely, As usual, for every x ∈ Ω, the proximal subdifferential of F at x is given by For locally Lipschitz continuous functions, the Clarke subdifferential ∂ C F (x) of F at x, can be defined as ∂ C F (x) :=co∂ L F (x). Finally, locally semiconcave functions enjoy the following properties (see [9, Propositions 3.3.1, 3.6.2]).

Lemma 1.5.
Let Ω ⊆ R N be an open set and let F : Ω → R be a locally semiconcave function. Then for any compact set K ⊂ Ω there exist some positive constants L and ρ such that, for any x ∈ K 3 , for any pointx ∈ K such that [x,x] ⊂ K.

Sample and Euler stabilizability with regulated cost
Let us introduce the notions of sampling and Euler solutions with regulated cost. Hypothesis (H0) is assumed throughout the whole section.
Definition 2.1 (Admissible trajectory-control pairs and costs). For every point z ∈ R n \ C, we will say that (x, u) is an admissible trajectory-control pair from z for the control systeṁ x is a Carathéodory solution of (2.1) in [0, T x ) corresponding to u, verifying x(0) = z and We shall use A f (z) to denote the family of admissible trajectory-control pairs (x, u) from z for the control system (2.1). Moreover, we will call cost associated to (x, u) ∈ A f (z) the function If T x < +∞, we extend continuously (x 0 , x) to [0, +∞), by setting From now on, we will always consider admissible trajectories and associated costs defined on [0, +∞).
Observe that for any admissible trajectory-control pair defined on [0, T x ), when T x < +∞ the above limit exists by (H0). In particular, this follows by the compactness of ∂C and the boundedness of f and l in any bounded neighborhood of the target.
A partition of [0, +∞) is a sequence π = (t j ) such that t 0 = 0, t j−1 < t j ∀j ≥ 1, and lim j→+∞ t j = +∞. The value diam(π) := sup j≥1 (t j − t j−1 ) is called the diameter or the sampling time of the sequence π. A feedback for (2.1) is defined to be any locally bounded function K : R n \ C → U . Definition 2.2 (Sampling trajectory and sampling cost). Given a feedback K : R n \ C → U , a partition π = (t j ) of [0, +∞), and a point z ∈ R n \ C, a π-sampling trajectory for (2.1) from z associated to K is a continuous function x defined by recursively solvinġ from the initial time t j−1 up to time where x(t 0 ) = x(0) = z. In this case, the trajectory x is defined on the rightopen interval from time zero up to time t − := inf{τ j : τ j < t j }. Accordingly, for every j ≥ 1, we set The pair (x, u) will be called a π-sampling trajectory-control pair of (2.1) from z (corresponding to the feedback K). The sampling cost associated to (x, u) is given by Definition 2.3 (Sample stabilizability with (p 0 , W )-regulated cost). A feedback K : R n \ C → U is said to sample-stabilize (2.1) to C if there is a function β ∈ KL satisfying the following: for each pair 0 < r < R there exists δ = δ(r, R) > 0, such that, for every partition π of [0, +∞) with diam(π) ≤ δ and for any initial state z ∈ R n \ C such that d(z) ≤ R, any π-sampling trajectory-control pair (x, u) of (2.1) from z associated to K belongs to A f (z) and verifies: Such (x, u) are called (r, R)-stable (to C) sampling trajectory-control pairs. If the system (2.1) admits a sample-stabilizing feedback to C, then it is called sample stabilizable (to C).
When there exist p 0 > 0 and a continuous map W : R n \ C → [0, +∞) which is positive definite and proper on R n \ C, such that the sampling cost x 0 associated to any (r, R)-stable sampling pair (x, u) verifies Observe that, when d(z) ≤ r, the timeT r x may be zero. In this case (2.5) imposes no conditions on the cost.
Let us now introduce Euler solutions and the associated costs and a notion of Euler stabilizability to C with (p 0 , W )-regulated cost.
Definition 2.4 (Euler trajectory and Euler cost). Let (π i ) be a sequence of partitions of [0, +∞) such that δ i := diam(π i ) → 0 as i → ∞. Given a feedback K : R n \ C → U and z ∈ R n \ C, for every i, let (x i , u i ) ∈ A f (z) be a π i -sampling trajectory-control pair of (2.1) from z associated to K and let x 0 i be the corresponding cost. If there exists a map X : [0, +∞) → R n , verifying we call X an Euler trajectory of (2.1) from z (corresponding to the feedback K).
we call X 0 the Euler cost associated to X.
Remark 2.5. As Euler trajectories are not, in general, classical solutions to the control system (2.1), an Euler cost may not coincide with the integral of the Lagrangian along the corresponding Euler trajectory, for some control. Nevertheless, this is true in special situations, as, for instance, when the function l is continuous, bounded and does not depend on the control, that is l(x, u) =l(x) for all (x, u). Indeed, in this case if there exists a sequence of sampling trajectories x i → X locally uniformly in [0, +∞), the dominated convergence theorem implies that the associated costs x 0 i converge locally uniformly to the function X 0 verifying when ω denotes a modulus ofl on a suitable compact neighborhood of Definition 2.6 (Euler stabilizability with (p 0 , W )-regulated cost). The system (2.1) is Euler stabilizable to C with Euler stabilizing feedback K, if there exists a function β ∈ KL such that for each z ∈ R n \ C, every Euler solution X of (2.1) from z associated to K verifies When there exist some p 0 > 0 and a continuous map W : which is positive definite and proper on R n \ C, such that every Euler cost X 0 associated to X verifies where

Main result
Theorem 3.1. Assume hypothesis (H0) and let W be a p 0 -MRF with p 0 > 0. Then there exists a locally bounded feedback K : R n \ C → U that sample and Euler stabilizes system We split the proof of Theorem 3.1 in two subsections, concerning with the sample stabilizability and the Euler stabilizability, respectively.
Preliminarily, let us observe that for any (x, p) ∈ (R n \ C) × R n the infimum in the definition of the Hamiltonian H can be taken over a compact subset of U , in view of the following result.
As an immediate consequence of Proposition 3.2, the existence of a p 0 -MRF W guarantees the existence of a feedback K with the following properties. Proposition 3.3. Let W be a p 0 -MRF with p 0 ≥ 0 and fix a selection p(x) ∈ D * W (x) for any x ∈ R n \ C. Given a function N as in Proposition 3.2, then any map K : R n \ C → U such that We call any map K as above, a W -feedback for the control system (2.1). When the dependence of K on W is clear, we simply call K a feedback.  . Let W be a p 0 -MRF with p 0 ≥ 0, define N accordingly to Proposition 3.2, and let K be a W -feedback. Moreover, let ε,μ, σ verify ε > 0 and 0 <μ < σ. Then there exists somê δ =δ(μ, σ) > 0 such that, for every partition π = (t j ) of [0, +∞) with diam(π) ≤δ and for each z ∈ R n \ C satisfying W (z) ∈ (μ, σ], any πsampling trajectory-control pair (x, u) oḟ

5)
associated to the feedback K is defined on [0,t) and enjoys the following properties: where Proposition 3.4 describes the behavior of any sampling trajectory-control pair (x, u) with sampling time not greater thanδ just until its first exit-timê t from the set {x ∈ R n \ C : W (x) >μ}. In [19] this was enough to derive global asymptotic controllability. Global asymptotic stabilizability, instead, requires also that, loosely speaking, any x is defined in [0, +∞) and stays in the sublevel set {x ∈ R n \ C : W (x) ≤μ} for every t ≥t, for somē t =t(μ, σ). This is the content of the next proposition, which can be seen as an extension of [13, Lemma IV.2] to the setting considered here.
Fix j ≥n and assume by induction that an arbitrary π-sampling trajectory x, eventually extended accordingly to Definition 2.1, is defined up to time t j−1 and verifies and verifies (3.12) is obviously satisfied. When 0 < W (x(t j−1 )) ≤μ, we distinguish the following situations: Case 1. W (x(t j−1 )) ≥μ/2. Then by Proposition 3.4 (choosing in particular z = x(t j−1 ) and the partition π j : 4,μ]) and this implies (3.12).
Case 2. W (x(t j−1 )) <μ/2. Any π-sampling solution x of (3.5) with this property can be defined on a maximal interval [t j−1 ,t). Assume first that t > t j , so that x is defined for all t ∈ [t j−1 , t j ] and suppose by contradiction Then there exist t i−1 < t j <t j ≤ t j such that This yields the required contradiction, since we havê µ/2 = W (x(t j )) − W (x(t j )) ≤ Lmδ ≤μ/4. Therefore x verifies (3.12). Let us now assumet ≤ t j . By standard properties of the ODEs, the blow-up timet verifies either lim t→t − |x(t)| = +∞ ort = T x . Notice that if we had we could find t i−1 < t j <t j <t and obtain a contradiction arguing as above. Hence x([t j−1 ,t)) ⊆ W −1 ([0,μ]) andt = T x necessarily, since the set W −1 ([0,μ]) is compact. By the boundedness of f on W −1 ((0,μ]) × U this implies that ∃ lim t→t x(t) =z ∈ ∂C and the extension of x to [t j−1 , t j ] given by x(t) =z for all t ∈ [t, t j ] verifies (3.12). The proof is thus concluded.
Finally, let us relate the level sets of a p 0 -MRF W with the ones of the distance function d using the following general result. Moreover, one has Then for any r > 0 there existᾱ,ᾱ 1 > 0 such that S α ⊂ S 1< r for all α ≤ᾱ and S α ⊃ S 1 r for all α ≥ᾱ 1 , so thatḡ(r) and g(r) turn out to be well-defined. Moreover, r →ḡ(r), g(r) are clearly increasing and verify the limits (3.14).
The timet might be zero when d(z) ≤ r. Of course, condition (ii) is significant only ift > 0.
Observing that (3.20) implies the construction of a KL function β such that can be obtained arguing as in [19, p.600], hence we omit it. Together with (iii), this yields that d(x(t)) ≤ max{β(d(z), t), r} ∀t ≥ 0.
Moreover, whent > 0 by summing up j from 0 to the last indexñ such that tñ <t, from (ii) it follows that p 0 and this concludes the proof sincē Remark 3.7. When d(z) > r, the timeT r x , after which any (r, R) stable π-sampling trajectory x starting from z remains definitively in B r (C), is uniformly bounded by a positive constant. Precisely, using the above notations, by the previous proof one can easily deduce the following upper bound 3.2. Proof of the Euler stabilizability with (p 0 , W )-regulated cost Let us start with some preliminary results. In the sequel we make use of all the notations introduced in the previous subsection.
Next result allows us to determine, given a p 0 -MRF W , a positive constant R and a sampling time δ > 0 small enough, a radius r < R such that any π-sampling trajectory-control pair for (3.5) with initial point z verifying d(z) ≤ R and with diam(π) = δ is (r, R)-stable. Lemma 3.9. Assume (H0). Let W be a p 0 -MRF with p 0 ≥ 0 and for any pair r, R > 0 with r < R, let δ = δ(r, R) be defined accordingly to (3.19). Then, for every fixed R > 0, δ(·, R) is positive and increasing and We note thatμ → L(μ, σ) is decreasing in (0, σ): this implies at once conditions (b) and (c) and the fact that, for every σ > 0, the mapμ →μ/4L(μ, σ)m is increasing. To conclude it is left to show that, for every σ > 0, the map µ →δ(μ, σ) is increasing in (0, σ). This monotonicity that can be easily derived arguing as above, by the definitions of the constants in the proof of [19, Proposition 3.5], hence we omit to prove it.
Owing to Lemma 3.9, given a p 0 -MRF W and a positive constant R, we can assume without loss of generality that δ(·, R) defined as above is strictly increasing and continuous. Therefore, for any R > 0 we can define the inverse of the map r → δ(r) := δ(r, R), given by which is continuous, strictly increasing and such that r(0) = 0 and r(δ(R)) = R. As an immediate consequence, by the sample stabilizability of (2.1) with (p 0 , W )-regulated cost we get the following result.
Remark 3.11. When f and l verify hypothesis (H0) and system (2.1) is sample stabilizable to C with (p 0 , W )-regulated cost, there always exist continuous Euler solutions to (2.1). Indeed, for any z with 0 < d(z) ≤ R and any sequence (x i , u i ) of π i -sampling trajectory-control pairs of (2.1) with x(0) = z, δ i :=diam(π i ) → 0 as i → +∞, and associated costs x 0 i , it turns out that (x 0 i , x i ) is equi-Lipschitz continuous on [0, +∞) with Lipschitz constant m > 0. Hence the existence of continuous, actually m-Lipschitz continuous Euler solutions to (2.1) from z with m-Lipschitz continuous Euler costs follows straightforwardly by Ascoli-Arzelá's Theorem.
We are now in position to prove that, if we assume (H0) and W is a p 0 -MRF with p 0 > 0, the feedback K Euler-stabilizes the system (2.1) to C with (p 0 , W )-regulated cost. Given z ∈ R n \ C, let (X 0 , X) be an Euler solution of (2.1) with initial condition X(0) = z. By definition there exist a sequence of partitions (π i ) of [0, +∞) such that δ i := diam(π i ) → 0 as i → ∞, a sequence of π i -sampling trajectory-control pairs (x i , u i ) for (2.1) with x i (0) = z for each i, and associated costs To conclude the proof it remains only to show that where the limit is well defined, since X 0 , pointwise limit of monotone nondecreasing functions, is monotone nondecreasing. Passing eventually to a subsequence, we setT := lim iT r(δi) xi . In view of Lemma 3.8,T satisfies Then for any t ∈ [0,T ) one hasT r(δi) xi > t for all i sufficiently large and, taking the limit as i → ∞ in (3.32), by (3.30) it follows that Moreover, by the locally uniform convergence of x i to X and the m-Lipschitz continuity of X, we get the following estimate where the r.h.s. tends to zero as i → +∞. Thus we have in any case where the first inequality is again a consequence of the m-Lipschitz continuity of X. This concludes the proof, since (3.36) implies now the thesis (3.34).

On the notion of p 0 -Minimum Restraint Function
In For instance, one can choose W 1 = d, distance function from C.
To prove that (ii) ⇒ (i), it is enough to invert the roles of W and W 1 . Precisely, if (4.2) is verified for some W , W 1 andγ as in the statement of the proposition, arguing as above one obtains that W verifies (4.1) choosing as γ : (0, +∞) → (0, +∞) any continuous, strictly increasing approximation from below of the increasing map r →γ • g W1,W (r).
Proof of Proposition 4.1. The only non trivial fact in order to derive Proposition 4.1 from Proposition 4.2, is that (4.1) involves the proximal subdifferential ∂ P W (x) at x instead of the set of limiting gradients D * W (x) at x, considered in the decrease condition for a p 0 -MRF. However, when W is locally Lipschitz continuous, condition (4.1) with Ω = R n \ C implies readily the following inequality: where ∂ L W (x) denotes the limiting subdifferential at x. This concludes the proof, since a p 0 -MRF W is locally semiconcave and therefore ∂ L W (x) = D * W (x) at any x.

Lipschitz continuous p 0 -MRF
Under the following hypothesis: (H1) The sets U ⊂ R m , C ⊂ R n are closed and the boundary ∂C is compact. f : (R n \ C) × U → R n , l : (R n \ C) × U → [0, +∞) are continuous functions such that for every compact subset K ⊂ R n \ C there exist M f , M l , L f , we obtain the main result of this section: Theorem 4.3. Assume (H1) and let p 0 ≥ 0. Let W : R n \ C → [0, +∞) be a locally Lipschitz continuous map on R n \ C, such that W is positive definite, and proper on R n \ C, and verifies the decrease condition for some continuous, strictly increasing functionγ : (0, +∞) → (0, +∞) and some continuous function W 1 : R n \ C → [0, +∞), positive definite, and proper on R n \ C. Then there exists a p 0 2 -MRFW , which also satisfies Theorem 4.3, whose proof is postponed to Appendix A.1, generalizes the result on the existence of a semiconcave Control Lyapunov Function obtained in [28, sect. 5] to the present case, where the decrease condition involves also the Lagrangian l and the target is not the origin, but an arbitrary closed set C with compact boundary.
Let us call a map W as in Theorem 4.3 a Lipschitz continuous p 0 -MRF. As an immediate consequence of Theorems 4.3 and 3.1, we have the following: Corollary 4.4. Assume (H1) and let p 0 > 0. Let W : R n \ C → [0, +∞) be a Lipschitz continuous p 0 -MRF. Then there exists a locally bounded feedback K : R n \ C → U that sample and Euler stabilizes system (1.1) to C with (p 0 /2, W )-regulated cost.
Note that, using the notation of Theorem 4.3, the feedback K in Corollary 4.4 is actually aW -feedback and the claim above relies on the inequalitȳ W ≤ W .  4.5 (p 0 -OMRF). Let W : R n \ C → [0, +∞) be a continuous function, and let us assume that W is locally semiconcave, positive definite, and proper on R n \ C. We say that W is a p 0 -OMRF for some p 0 ≥ 0 if it verifies H(x, p 0 , D * W (x)) < 0 ∀x ∈ R n \ C. (4.4) A p 0 -MRF is obviously a p 0 -OMRF, but the converse might be false. By [23] we have the following result.
Proposition 4.6 clarifies the difference between the two notions: the existence of a p 0 -OMRF implies that there exists a rate function γ σ , which is in general, not global. In particular, γ σ can become smaller and smaller as σ tends to +∞. Consequently, also the feedback K can be defined only given a σ > 0, on W −1 ((0, σ]).
it is not difficult to prove that, under the assumptions of Proposition 4.6, there exists a continuous, strictly increasing function γ : (0, +∞) → (0, +∞) independent of σ, such that (4.5) holds for all x ∈ R n \ C (see [23,Remark 3.1]). In other words, The proof of Theorem 3.1 can be easily adapted to derive the following result.
As in the case of p 0 -MRF, when f , l are locally Lipschitz continuous in x, we can replace the semiconcavity assumption in the definition of a p 0 -OMRF with local Lipschitz continuity. Precisely, we establish what follows. Theorem 4.9. Assume (H1) and let p 0 ≥ 0. Let W : R n \ C → [0, +∞) be a locally Lipschitz continuous map on (R n \ C), such that W is positive definite, and proper on R n \ C, and verifies the decrease condition Then there exists a p 0 2 -OMRFW which also satisfiesW (x) ≤ W (x) for all x ∈ R n \ C.
The proof of this theorem is sketched in Appendix A.2.
Let us call a map W as in Theorem 4.9 a Lipschitz continuous p 0 -OMRF.

An example: stabilization of the non-holonomic integrator control system with regulated cost
Let us illustrate the preceding theory through a classical example. Precisely, in the first part of this section we provide a p 0 -MRF W 1 for the non-holonomic integrator control system associated to a Lagrangian l, that verifies a suitable growth condition (see (5.3) below). In view of Theorem 3.1, this implies the existence of a possibly discontinuous feedback K that sample and Euler stabilizes the non-holonomic integrator to the origin with the cost bounded above by W 1 /p 0 . Furthermore, we show how weakening the requirements on the p 0 -MRF (by replacing semiconcavity with Lipschitz continuity) may be crucial for the effective construction of a p 0 -MRF. In particular, for any bounded Lagrangian l which cannot satisfy assumption (5.3) below for any C > 0 (as, for instance, in the case of the minimum time problem, where l ≡ 1), we provide a less regular, Lipschitz continuous but not semiconcave p 0 -MRF W 2 . In this case, the sample and Euler stabilizability of the control system with (p 0 /2, W 2 )-regulated cost is guaranteed by Corollary 4.4.
Set U := {u = (u 1 , u 2 ) ∈ R 2 : u 2 1 + u 2 2 ≤ 1}, C := {0} ⊂ R 3 and consider the non-holonomic integrator control system: The following map W 1 , introduced in [21], given by is proper, positive definite, locally semiconcave in R 3 \ {0}, and verifies Therefore, W 1 is a Control Lyapunov Function for the control system (5.1) and, consequently, any W 1 -feedback sample and Euler stabilizes (5.1) to the origin [29]. When the Lagrangian l satisfies, for some positive constant C, then W 1 is also a p 0 -MRF for every p 0 ∈ (0, 1/C). Indeed, for all x ∈ R 3 \ {0} and for all p ∈ D * W 1 (x), one has However, the Control Lyapunov Function W 1 cannot be a p 0 -MRF when lim x→0 inf u∈U l(x, u) Since V (x) tends to 0 + as x → 0, condition (5.4) does not hold, for instance, for the minimum time problem, where l ≡ 1.
For any bounded Lagrangian l, let M l > 0 verify l(x, u) ≤ M l for all (x, u) ∈ (R n \ C) × U . Then a discontinuous feedback that sample and Euler stabilizes (5.1) and at the meantime provides strategies for which the target is reached with regulated cost, can be obtained if we consider the following Control Lyapunov Function W 2 , introduced in [27]: The map W 2 is locally semiconcave only outside the set S := {(x 1 , x 2 , x 3 ) ∈ R 3 | x 2 3 = 4(x 2 1 + x 2 2 )}, therefore, it is not a p 0 -MRF. However, W 2 matches the weaker definition of Lipschitz continuous p 0 -MRF for p 0 < 1/M l : it is indeed a locally Lipschitz continuous map in R 3 , which is positive definite and proper in R 3 \ {0}, and a direct computation shows that (see also [29] and [21]). Since the data f and l verify assumption (H1), it follows by Corollary 4.4 that (5.1)-(5.2) is sample and Euler stabilizable with (l/2, W 2 )-regulated cost as soon as the Lagrangian l is bounded.

Conclusions
In this paper we addressed sample and Euler stabilizability of nonlinear control system in an optimal control theoretic framework. We introduced the notion of sample and Euler trajectories with regulated cost, which conjugate stabilizability with an upper bound on the payoff, depending on the initial state. Under mild regularity hypotheses on the vector field f and on the Lagrangian l and for a closed, possibly unbounded control set, we proved that the existence of a special Control Lyapunov Function W , called a p 0 -Minimum Restraint Function, p 0 -MRF, implies that all sample and Euler stabilizing trajectories have (p 0 , W )-regulated costs. The proof is constructive: it is based indeed on the synthesis of appropriate feedbacks derived from W . As in the case of classical Control Lyapunov Functions, this construction requires that W is locally semiconcave. However, by generalizing an earlier result by Rifford [28] we established that it is possible to trade regularity assumptions on f and l with milder regularity assumptions on W . In particular, we showed that if the vector field f and the Lagrangian l are locally Lipschitz up to the boundary of the target, then the existence of a mere locally Lipschitz p 0 -MRF W provides sample and Euler stabilizability with (p 0 /2, W )-regulated cost.
The present work is part of an ongoing, wider investigation of global asymptotic controllability and stabilizability in an optimal control perspective. A slightly weaker notion of p 0 -MRF -called here p 0 -OMRF-was introduced in [23] and further extended in [19] to more general optimization problems, in order to yield global asymptotic controllability with regulated cost. This paper represents the stability-oriented counterpart of [23]. In a forthcoming paper we will address the question of stabilizability with regulated cost for possibly non-coercive optimization problems with unbounded controls. Other interesting research directions include the relation between p 0 -MRFs and input-to-state stability (in the fashion of [21]) and the study of a possible inverse Lyapunov theorem for p 0 -MRFs -i.e., whether the results in [30] can be extended to p 0 -MRFs by showing that their existence is also a necessary condition for the global asymptotic controllability of the control system with regulated cost.
Step 1. As it is not restrictive in view of Proposition 4.2, let us assume that W 1 ≡ d. We extendγ continuously to R, by settingγ(t) =γ(0) := lim s→0 +γ(s) for every t < 0. Without loss of generality, we can supposẽ γ 1-Lipschitz continuous in R. Otherwise, we can replaceγ in the decrease condition (A.1) withγ(t) := inf s∈R {γ(s) + |t − s|} for every t ∈ R. Indeed, it is not difficult to see thatγ ≤γ, andγ is strictly increasing and 1-Lipschitz continuous. Therefore, the map W :=γ • W 1 =γ • d is 1-Lipschitz continuous and positive definite on R n \ C. As a consequence of these results, (A.1) implies that W verifies Step 2. For any integer n ≥ 1, let us set By the Lipschitz properties of f , l and W , let us denote L n f , L n l , L n W ≥ 1 the Lipschitz constants of f (·, u), l(·, u) and W , respectively, on the sublevel set W −1 ([0, M n ]). Finally, let us set α n := max 8n(L n W ) 2 + 1 , Let us extend W to R n by setting W (x) = 0 for all x in the interior of C.
Precisely, one has what follows.

A.2. Proof of Theorem 4.9
Let us sketch how to adapt the arguments of the proof of Theorem 4.3 to the case of a Lipschitz continuous p 0 -OMRF, for which only the existence of a local rate function γ σ is ensured. Let W : R n \ C → [0, +∞) be a locally Lipschitz continuous map, positive definite and proper on R n \ C and verifying H(x, p 0 , ∂ L W (x)) < 0 ∀x ∈ R n \ C.
Our aim is to prove the existence of a p 0 2 -OMRFW ≤ W .
Step 1. By the Lipschitz continuity of W , the set-valued map x ∂ L W (x) has closed graph with compact values, so that it is upper semicontinuous (see [35,Props. 4 we can suppose without loss of generality thatγ σ is 1-Lipschitz continuous and consider the map W σ :=γ σ • d, which is 1-Lipschitz continuous and positive definite in R n \ C. Therefore, recalling that ∂ P W (x) ⊆ ∂ L W (x) for every x, W verifies H(x, p 0 , ∂ P W (x)) ≤ −W σ (x) ∀x ∈ W −1 ((0, σ]). (A.10) Step 2. For any integer n ≥ 1, let us set σ n := 11n. Hence W verifies H(x, p 0 , ∂ P W (x)) ≤ −W σn (x) ∀x ∈ W −1 ((0, σ n ]), (A.11) where it is easy to see that (W σn ) n is a decreasing sequence. From now on, the proof proceeds similarly to Appendix A.1, with the crucial differences that the decrease rate W σn in (A.11) depends on σ n and that the condition (A.11) is satisfied only in W −1 ((0, σ n ]). In particular, these facts imply that, for any n ≥ 1, the inf-convolution W αn of W depends on W σn , since α n is given by α n := max 8n(L n W ) 2 + 1 , Lemma A.6. Let z ∈ W −1 1 2n , σ n and p ∈ ∂ P W αn (z). Then H(z, p 0 , p) ≤ − W σn (z) 2 . (A.12) Proof. The only delicate point in order to adapt the proof of Lemma A.3 to the present setting, is that, given z ∈ W −1 ((0, σ n ]), one has to apply the decrease condition in (A.11) not at z, but at the pointȳ where the minimum in definition (A.4) of W αn (x) is obtained. This can be done sinceȳ belongs to the sublevel set W −1 ((0, σ n ]) too; indeed, Lemma A.7. For all integer n ≥ 1 and for all z ∈ W −1 1 n , σ n − n , one hasW (z) =W n0 (z) for some n 0 ∈ {1, . . . , n}. Furthermore, if p ∈ ∂ PW (z), then Proof. Going through the proof of Lemma A.5, the crucial remark is that, whenever the minimum W (z) =W n0 (z) := min 1≤k≤nW n (z) is obtained for some n 0 < n, then W (z) ≤ σ n0 . The last inequality implies that, when W (z) ≥ 1 2n0 , the point z belongs to the strip W −1 1 2n0 , σ n0 . Therefore, arguing as in the proof of Lemma A.5, one can apply Lemma A.6 to derive that H z, p 0 2 , p ≤ − W σn 0 (z) 4 .
Recalling that the sequence (W σn ) is decreasing, this yields the decrease condition (A.13). The proof in the case W (z) < 1 2n0 , where one can assume n 0 = n, can be obtained again by Lemma A.6 and the arguments of the proof of Lemma A.5.
The decrease condition (4.4) follows now by the arbitrariness of n and, consequently, we have thatW is a p 0 /2-OMRF.