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Nonpathological Lyapunov functions and discontinuous Carathéodory systems
Automatica 42 (2006) 453 – 458 www.elsevier.com/locate/automatica
Brief paper
Nonpathological Lyapunov functions and discontinuous Carathéodory systems夡 Andrea Bacciotti, Francesca Ceragioli ∗ Dipartimento di Matematica del Politecnico di Torino, C.so Duca degli Abruzzi, 24-10129 Torino, Italy Received 27 October 2004; received in revised form 11 October 2005; accepted 20 October 2005
Abstract Differential equations with discontinuous right-hand side and solutions intended in Carathéodory sense are considered. For these equations, sufficient conditions which guarantee both Lyapunov stability and asymptotic stability in terms of nonsmooth Lyapunov functions are given. An invariance principle is also proven. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Lyapunov functions; Stability; Stabilizability; Discontinuous control; Invariance principle; Nonpathological functions; Carathéodory solutions
1. Introduction The interest in the study of Lyapunov-like theorems for discontinuous systems is essentially motivated by the connection between stability and stabilizability problems. Since the papers by Sussmann (1979), Artstein (1983) and Brockett (1983) it is clear that, in order to deal with general stabilization problems, discontinuous feedback laws are needed. With the introduction of discontinuous feedback laws, the theoretical problem of giving an appropriate definition of solution for a system with discontinuous right-hand side, comes out. Different approaches have been followed in the literature (Filippov, 1988; Clarke et al., 1997; Ancona & Bressan, 1999). Filippov solutions in particular have been used in order to deal with discontinuous stabilizability problems. For these solutions, Lyapunov methods have been widely developed (see, e.g., Aubin & Cellina, 1994; Clarke et al., 1998; Bacciotti & Rosier, 2001). Some recent papers show that the stabilization problem can be very
夡 This paper is not presented in IFAC meeting. This paper was presented at NOLCOS 2004. This paper is recommended for publication in revised form by Associate Editor Andrew R. Teel under the direction of Editor Hassan Khalil. ∗ Corresponding author. Tel.: +39 01 15 64 75 18; fax: +39 01 15 64 75 99. E-mail addresses: [email protected] (A. Bacciotti), [email protected] (F. Ceragioli).
0005-1098/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2005.10.014
well approached by means of Carathéodory solutions (Ancona & Bressan, 1999, 2002, 2004; Rifford, 2002, 2003; Kim & Ha, 2004). If the notion of Carathéodory solution is accepted as a good notion of solution for discontinuous systems, in particular for systems coming from discontinuous stabilization problems, it is then interesting to develop an appropriate Lyapunov theory for them. Indeed, as far as the authors know, such a generalization of Lyapunov methods has not been treated in the literature before. We consider nonsmooth Lyapunov functions. Resorting to nonsmooth Lyapunov functions gives more flexibility in the stability analysis, in particular of piecewise linear, switched and discontinuous systems. They are also convenient in order to achieve general converse theorems (see Sontag, 1999; Rifford, 2002). In the paper we take advantage of a notion of derivative for Lipschitz continuous functions which are, in addition, nonpathological (Valadier, 1989). This notion was introduced in Shevitz and Paden (1994), improved in Bacciotti and Ceragioli (1999) and then applied in Bacciotti and Ceragioli (1999, 2003). The paper consists of two main sections. In Section 2, the fundamental tools needed in order to obtain the main results are collected. In particular, the definition and basic properties of nonpathological functions are recalled. In Section 3, a Lyapunov like theorem and an invariance principle are stated and proved. In Section 4, these results are illustrated by means of some examples and counterexamples. Finally, some conclusions are stated in Section 5.
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t ∈ I , the set jC V ((t)) is a subset of an affine subspace orthogonal to (t). ˙
2. Tools First of all, we introduce Carathéodory solutions. Consider the autonomous differential equation x˙ = f (x),
(1)
where x ∈ Rn , f : Rn → Rn . Definition 1. Let I be an interval of R. A function : I → Rn is said to be a Carathéodory solution of (1) on I if (t) is absolutely continuous and (d/dt)(t) = f ((t)) for almost every t ∈ I . In the following, only Carathéodory solutions are considered and they are simply addressed as solutions. Moreover, Sx0 denotes the set of maximal solutions of (1) with initial condition x(0) = x0 and I = [0, T ) denotes the maximal right interval where ∈ Sx0 is defined. The vector field f in (1) is in general not continuous, and consequently the existence of solutions of (1) is not guaranteed by classical theorems. General sufficient conditions on the vector field f in order to have existence of solutions of (1) have been studied in Bressan (1988) and Ancona and Bressan (1999). We will make the following basic assumptions: (H1) for any initial condition x0 ∈ Rn at least one solution of (1) exists, (H2) f is Lebesgue measurable and locally bounded.
Remark 1. It is not difficult to prove that under assumptions (H1) and (H2), if ∈ Sx0 is bounded, then T = +∞. Definitions of stability in the case of discontinuous systems with solutions intended in Carathéodory sense do not differ from the usual ones. We report them for reader’s convenience. Definition 2. System (1) is said to be • Lyapunov stable at the origin if for any > 0 there exists > 0 such that for any x0 with x0 < and for any ∈ Sx0 , (t) < for all t ∈ [0, T ); • globally asymptotically stable at the origin if it is Lyapunov stable and moreover there exists > 0 such that for any x0 with x0 < and for any ∈ Sx0 , limt→T (t) = 0. Note that if the origin is locally stable then f (0) = 0. From now on, the origin is then assumed to be an equilibrium position for the system, i.e. f (0) = 0. We now recall the definition of nonpathological function given in Valadier (1989). In the following, jC V (x) denotes the Clarke gradient of the real function V at the point x (see Clarke, 1983). Definition 3. A function V : Rn → R is said to be nonpathological if it is locally Lipschitz continuous and for every absolutely continuous function : I → Rn and for almost every
Nonpathological functions form a wide class which includes Clarke regular functions, semiconcave and semiconvex functions. Precise definitions and a collection of relationships and properties (with appropriate references) for these classes of functions can be found in Bacciotti and Ceragioli (2003). Roughly speaking, a semiconvex (respectively, semiconcave) is the sum of a convex (respectively, concave) function and a C 1 function. A typical semiconvex function is V1 (x, y) = |x| + |y|, which will be used later as a Lyapunov function in Example 2, while a typical semiconcave function is V2 (x, y) = (4x 2 + 3y 2 )1/2 − |x|, that we need in Example 1. A nonpathological function which is neither √ semiconvex nor 2 − 3 3/2|x|y. Note semiconcave is V3 (x, y) = 47 x 2 + 13 y 4 that, very roughly speaking, nonpathological functions may have level sets with both concave and convex angles. Nonpathological functions can be easily handled thanks to the following proposition (Valadier, 1989). Proposition 1. If V : Rn → R is nonpathological, and : R → Rn is absolutely continuous, then the set {p · (t), ˙ p∈ jC V ((t))} is reduced to the singleton {(d/dt)V ((t))} for almost every t. The notion of nonpathological derivative of a map with respect to a differential equation is now introduced. This notion is analogous to the notion of set-valued derivative of a map with respect to a differential inclusion introduced in Shevitz and Paden (1994) and improved in Bacciotti and Ceragioli (1999). Let V : Rn → R be a nonpathological function and let (1) be given. Let AV = {x ∈ Rn : p1 · f (x) = p2 · f (x) ∀p1 , p2 ∈ jC V (x)}. The definition of nonpathological derivative is now given. Definition 4. If x ∈ AV , the nonpathological derivative of the map V with respect to (1) at x is the number V˙ f (x) = p · f (x), where p is any vector in jC V (x). The use of the nonpathological derivative is explained by the following corollary of Proposition 1. Corollary 1. Let the function V : Rn → R be nonpathological, and let be any solution of system (1). Then (t) ∈ AV and (d/dt)V ((t)) = V˙ f ((t)) for almost every t. Remark 2. The previous corollary shows that Carathéodory solutions, nonpathological functions and the nonpathological derivative fit very well together. In fact it turns out that in order to understand the behaviour of the function V along solutions of system (1), the object to be studied is the nonpathological derivative of V with respect to (1) at the points of AV . We remark that, depending on the set of nondifferentiability points
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of V and on the vector field f, AV may be a proper subset of Rn or all Rn (this is the case, in particular, when V is everywhere differentiable). The examples of Section 4 will clarify this point. We emphasize that, generally, the set AV is “big”. More precisely, it can be proven that if V is nonpathological and assumption (H2) is satisfied then AV is dense in Rn and, if V is moreover Clarke regular or semiconcave or semiconvex, then Rn \AV has null Lebesgue measure.
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(d/dt)V ((t)) = V˙ f ((t)) is nonpositive thanks to (2). The statement is then a consequence of Lemma 1(b). (ii) The proof is an immediate consequence of Corollary 1 and Lemma 1. Next a LaSalle-like invariance principle is proved (see Shevitz & Paden, 1994; Ryan, 1998; Bacciotti & Ceragioli, 1999 for other versions of the invariance principle). In order to get it, some regularity for the vector field is needed.
3. Results In this section two propositions which generalize Lyapunov method to systems with discontinuous right-hand side and solutions intended in Carathéodory sense are proved. Proposition 2. Let V : Rn → R be positive definite, nonpathological and radially unbounded. Let AV be defined as in Definition 4. Assume that ∀x ∈ AV ,
V˙ f (x)0.
(2)
Then (i) system (1) is Lyapunov stable at the origin; (ii) if moreover there exists a function W : Rn → R continuous and positive definite such that V˙ f (x) − W (x) for all x ∈ AV then system (1) is globally asymptotically stable. Lemma 1. Let V : Rn → R be positive definite, locally Lipschitz continuous and radially unbounded and let : [0, T ) → Rn be any solution of (1). Then (a) V ◦ : [0, T ] → [0, +∞] is differentiable almost everywhere, (b) if (d/dt)V ((t))0 for almost every t, then T = +∞ and the origin is Lyapunov stable, (c) if there exists a function W : Rn → R continuous and positive definite such that (d/dt)V ((t)) − W ((t)) for almost every t, then the origin is globally asymptotically stable. Proof. (a) V ◦ is absolutely continuous and then almost everywhere differentiable (see Wheeden & Zygmund, 1977). (b) Since V ◦ is absolutely continuous, then t2 d V ((t2 )) − V ((t1 )) = V ((s)) ds (3) t1 ds (see Wheeden & Zygmund, 1977). Then (d/dt)V ((t)) 0 implies that V ◦ is nonincreasing and from the fact that V is radially unbounded it follows that is bounded. Remark 1 implies that T =+∞. The remaining part of the proof is standard (see Khalil, 1992). (c) The proof is standard (see Khalil, 1992), after recalling (3). Proof of Proposition 2. (i) According to Corollary 1, for any solution of (1), (t) ∈ AV for almost every t, and
Definition 5. A vector field f : Rn → Rn is said to have the solutions closure property if for any sequence {n } of solutions of (1) such that n → uniformly on compact subsets of R, one has that also is a solution of (1). Of course any continuous vector field has the solutions closure property. An important class of discontinuous vector fields with the solutions closure property is the class of patchy vector fields (see Ancona & Bressan, 1999). In the statement of Proposition 3 the notion of weakly invariant set is needed. Definition 6. A set M is said to be weakly invariant for (1) if for any x0 ∈ M there exists ∈ Sx0 such that (t) ∈ M for all t 0. Proposition 3. Assume that the vector field f has the solutions closure property. Let V : Rn → R be positive definite, nonpathological and radially unbounded. Let AV be defined as in Definition 4 and assume (2). Let ZfV = {x ∈ AV : V˙ f (x) = 0} and let M be the largest weakly invariant subset of ZfV . Then for any x0 and any ∈ Sx0 lim dist ((t), M) = 0.
t→+∞
(4)
Lemma 2. If f has the solutions closure property, then for any x0 and any ∈ Sx0 the fact that the positive limit set () is nonempty implies that it is weakly invariant. The proof of this lemma is very similar to the proof of the analogous lemma for Filippov solutions (see Filippov, 1988, p. 130). Proof. Let ∈ Sx0 be given. First of all note that, as in the proof of Lemma 1, is bounded and then T = +∞. Since is bounded then () = ∅ and limt→+∞ dist ((t), ()) = 0. Next it is proved that () ⊂ ZfV . From the fact that () is weakly invariant it will follow (4). Let z ∈ (). Consider the composite function V ◦ , which is nonincreasing and bounded from below. From this fact it follows that there exists limt→+∞ V ((t)) = c and then V (z) = c for every z ∈ (). In fact, if z ∈ (), there exists a sequence {tn }, tn → +∞ such that limn→+∞ (tn ) = z and, thanks to the continuity of V, V (z) = limn→+∞ V ((tn )) = limt→+∞ V ((t)) = c. Due to Lemma 2, () is weakly invariant, then there exists ˜ ∈ Sz such that (t) ˜ ∈ () for all t 0. It follows that V ((t)) ˜ =c
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for all t 0 and (d/dt)V ((t)) ˜ = 0 for all t 0. Since (t) ˜ ∈ ˙ ˜ = (d/dt)V ((t)) ˜ = 0 for almost all t 0, AV and V f ((t)) (t) ˜ ∈ ZfV for almost all t 0. Let now {ti } be a sequence such that ti → 0 and (t ˜ i ) ∈ ZfV for all i. Since (t) ˜ is continuous it finally follows that z = limi→+∞ (t ˜ i ) ∈ ZfV .
f + (x, y) = −
f (x, y) =
Corollary 2. Assume that f : Rn → Rn has the solutions closure property. Let V : Rn → R be positive definite, nonpathological and radially unbounded, and let AV be defined as in Definition 4. If (2) holds and V˙ f (x) = 0 if and only if x = 0, then system (1) is globally asymptotically stable. 4. Examples Propositions 2 and 3 are now illustrated by means of some examples. Example 1, besides being an application of Proposition 2, aims to explain the use of the nonpathological derivative. Examples 2 and 3 are counterexamples showing that Proposition 2 is not true if either assumption (H1) or assumption (H2) do not hold. In Example 3, Proposition 3 is applied to prove the asymptotic stability of a planar bilinear “switched system”. This example aims also to remark that discontinuous feedback laws also appear when control systems with finite control set are studied. Finally, Example 4 is a counterexample showing that the assumption about the solutions closure property of the vector field in Proposition 3 is actually needed. Example 1. Consider the two-dimensional, single input, driftless system x˙ = (x 2 − y 2 )u, y˙ = 2xyu
Consider now the slightly modified vector fields
(5)
(the so-called Artstein’s circles example, see Artstein, 1983). This system cannot be stabilized by means of a continuous feedback. A discontinuous stabilizing feedback is 1 if x < 0, (6) u(x, y) = −1 if x 0. Denote by f (x, y) the right-hand side of the implemented system. It is shown in Artstein (1983), that a C 1 Lyapunov function for this system does not exist. On the other hand, V (x, y)=V2 (x, y)= 4x 2 + 3y 2 −|x| satisfies the assumptions of Proposition 2. In fact V is nonpathological, AV =R2 \{(x, y) : x = 0} and if (x, y) ∈ AV then V˙ f (x, y) = ∇V (x, y)f (x, y) = −W (x, y), where 4|x|3 + 2|x|y 2 − (x 2 − y 2 ) 4x 2 + 3y 2 W (x, y) = , 4x 2 + 3y 2 is continuous and positive definite. We can conclude that the implemented system is asymptotically stable. Note that in this example the set where V is not differentiable is a curve, the y-axis, which is not included in AV , because the vector field f is transversal to it. In this case, in order to apply Proposition 2, the condition involving the nonpathological derivative need not to be checked at all points of R2 .
f (x, y) (0, y)
if x = 0, if x = 0,
(7)
f (x, y) if x = 0, (0, −y) if x = 0.
(8)
Note that for x = 0 both these vector fields are parallel to the yaxis, i.e. to the curve where V is not differentiable, and AV =R2 . The condition involving the nonpathological derivative needs now to be checked at all points of R2 . Roughly speaking, Rn \AV either coincides or does not coincide with the set NV where V is not differentiable depending on the fact that either there are or there are not trajectories of the system lying on NV . √ √ V˙ f + (x, y) = 3|y| and V˙ f − (x, y) = − 3|y|, then the system defined by f − (x, y) satisfies the assumptions of Proposition 2 and hence it is asymptotically stable, while the system defined by f + (x, y) does not satisfy the assumptions of Proposition 2. The system defined by f + (x, y) is clearly unstable. Example 2. Consider the two dimensional system (1) where f is given by ⎧ ⎪ 0 if y 0, ⎨ −1 f (x, y) = (9) ⎪ ⎩ 0 if y < 0, 1 and the function V (x, y) = V1 (x, y) = |x| + |y|. It can be computed that AV = R2 \{(x, y) : y = 0}. Note that in this case the set NV where V is not differentiable is the union of the x-axis and the y-axis, but Rn \AV is only the x-axis due the fact that there are trajectories of the system lying on the y-axis. For all (x, y) ∈ AV , V˙ f (x, y) = −1, but the system is not asymptotically stable. Note that Proposition 2 cannot be applied because assumption (H1) is not satisfied. In fact there are no solutions issuing from the points of the x-axis. Example 3. Consider the two-dimensional system whose right-hand side is ⎧0 ⎪ if = 0, ⎪ ⎪ 0 ⎪ ⎪
⎪ y 2 ⎨ −x − sin −1/n (−1/n)2 1 if ∈ ( n1 , n−1 ], y x 2 ⎪ − + sin 2 ⎪ −1/n −1/n) ( ⎪ ⎪ ⎪ ⎪ ⎩ −x if > 1, −y where = x 2 + y 2 , n ∈ N, n 2. Consider the function V (x, y)=(x 2 +y 2 )/2. Here V is everywhere differentiable then AV = R2 . ⎧ ⎨0 ˙ V f (x, y) = − x 2 + y 2 ⎩ −(x 2 + y 2 )
if = 0, if 0 < 1, if > 1,
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then V˙ f (x, y) − min{ x 2 + y 2 , (x 2 + y 2 )/2}. Proposition 2(ii) cannot be applied due to the fact that (H2) does not hold since the vector field is not locally bounded. The trajectories of the system in polar coordinates are: (t) = 0 − t, 1 1 (t) = sin2 −t−1/n + 0 − sin2 −1/n . 0
0
If 0 ∈ (1/n, 1/(n − 1)], they are defined on the intervals [0, 0 − 1/n). Note that the system is Lyapunov stable, but not asymptotically stable. Example 4. Consider the two-dimensional control system x˙ x y =u + (1 − u) , (10) y˙ −y −x and the feedback law 1 if (x, y) ∈ D, u(x, y) = 0 if (x, y) ∈ R2 \D,
(11)
where D = {(x, y) : x > 0 and 2x < y < 3x or x < 0 and 3x < y < 2x}. The function V (x, y) = (x 2 + y 2 )/2 is C ∞ , nevertheless we cannot use it in order to apply classical Lyapunov theorems to the implemented system due to the fact the feedback law is discontinuous. Denote by f (x, y) the right-hand side of the implemented system. AV = R2 , V˙ f (x, y) = ∇V (x, y)f (x, y)0 V
for all (x, y) ∈ R2 and Z f = R2 \D. The largest weakly invariV
ant subset of Z f is the origin, then the implemented system is asymptotically stable. Example 5. Consider the equation x˙ = f (x) where x ∈ R, f (0)=0, f (x)=−f (−x) and f is defined in the following way: 1 −x + n1 if x ∈ ( n1 , n−1 ], f (x) = (12) −x + 1 if x > 1, where n ∈ N, n2. The unique solutions of this equation with initial condition (0) ∈ (1/n, 1/(n − 1)] is (t) = e−t (0) + (1/n)(1 − e−t ), which is defined on [0, +∞). Remark that the origin is stable, but not asymptotically stable. Consider V (x) = x 2 /2. It holds 1 x(−x + n1 ) if x ∈ ( n1 , n−1 ], ˙ V f (x) = x(−x + 1) if x > 1, then ZfV = {0}. In this case Proposition 3 cannot be applied due to the fact that the vector field f does not have the solutions closure property. In fact note that for any solution with initial condition in (1/n, 1/(n − 1)], the positive limit set is the point 1/n, which is not invariant. Finally, remark that there does not exist a positive definite and continuous function W such that V˙ f (x) − W (x). 5. Conclusion Differential equations with discontinuous right-hand side and solutions intended in Carathéodory sense have been
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considered. The interest of such equations is motivated by the fact that they naturally arise in stabilizability problems in connection with important classes of discontinuous feedback laws. For these equations sufficient conditions which guarantee both Lyapunov stability and asymptotic stability in terms of nonsmooth Lyapunov functions are proved. Moreover an appropriate version of LaSalle invariance principle is given. References Ancona, F., & Bressan, A. (1999). Patchy vector fields and asymptotic stabilization. Esaim-Cocv, 4, 445–472. Ancona, F., & Bressan, A. (2002). Flow stability of patchy vector fields and robust feedback stabilization. SIAM Journal on Control and Optimization, 5, 1455–1476. Ancona, F., & Bressan, A. (2004). Stability rates for patchy vector fields. Esaim-Cocv, 10, 168–200. Artstein, Z. (1983). Stabilization with relaxed controls. Nonlinear Analysis, 7, 1163–1173. Aubin, J. P., & Cellina, A. (1994). Differential inclusions. Berlin: Springer. Bacciotti, A., & Ceragioli, F. (1999). Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions. Esaim-Cocv, 4, 361–376. Bacciotti, A., & Ceragioli, F. (2003). Nonsmooth optimal regulation and discontinuous stabilization. Abstract and Applied Analysis, 20, 1159–1195. Bressan, A. (1988). Unique solutions for a class of discontinuous differential equations. Proceedings of the American Mathematical, 104, pp. 772–778. Bacciotti, A., & Rosier, L. (2001). Lyapunov functions and stability in control theory. London: Springer. Brockett, R. (1983). Asymptotic stability and feedback stabilization. In: R. Millman, R. Brockett, & H. Sussmann (Eds.), Differential geometric control theory (pp. 181–191). Boston: Birkhäuser. Clarke, F. H. (1983). Optimization and nonsmooth analysis. New York: Wiley. Clarke, F. H., Ledyaev, Yu. S., Sontag, E. D., & Subbotin, A. I. (1997). Asymptotic controllability implies feedback stabilization. IEEE Transactions on Automatic Control, 42, 1394–1407. Clarke, F. H., Ledyaev, Yu. S., Stern, R. J., & Wolenski, P. R. (1998). Nonsmooth analysis and control theory. New York: Springer. Filippov, A. F. (1988). Differential equations with discontinuous righthand sides. Dordrecht: Kluwer. Khalil, H. K. (1992). Nonlinear systems. New York: Macmillan. Kim, S. J., & Ha, I. J. (2004). On the existence of Carathéodory solutions in nonlinear systems with discontinuous switching control laws. IEEE Transactions on Automatic Control, 49, 1167–1171. Rifford, L. (2002). Semiconcave control–Lyapunov functions and stabilizing feedbacks. SIAM Journal on Control and Optimization, 41, 659–681. Rifford, L. (2003). Singularities of viscosity solutions and the stabilization problem in the plane. Indiana University Mathematics Journal, 52, 1373–1396. Ryan, E. P. (1998). An integral invariance principle for differential inclusions with applications in adaptive control. SIAM Journal on Control and Optimization, 36, 960–980. Shevitz, D., & Paden, B. (1994). Lyapunov stability theory of nonsmooth systems. IEEE Transactions on Automatic Control, 39, 1910–1914. Sontag, E. D. (1999). Stability and stabilization: Discontinuities and the effect of disturbances. In: F.H. Clarke, & R.J. Stern (Eds.), Nonlinear analysis, differential equations and control (pp. 551–598). Dordrecht: Kluwer. Sussmann, H. (1979). Subanalytic sets and feedback control. Journal of Differential Equations, 31, 371–393. Valadier, M. (1989). Entraıˆnement unilatéral, lignes de descente, fonctions lipschitziennes non pathologiques. Comptes Rendu de l’Academie des Sciences. Series I, Mathematique. Paris, 8, 241–244. Wheeden, R., & Zygmund, A. (1977). Measure and integration. New York: Marcel Dekker.
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A. Bacciotti, F. Ceragioli / Automatica 42 (2006) 453 – 458 Andrea Bacciotti received a degree in Mathematics at the University of Florence. He is Full Professor of Mathematical Analysis since 1981. He has been with the Mathematics Department of the University of Florence, Siena, Torino (Technical University). His research interests include ordinary differential equations, differential inclusions, mathematical control theory.
Francesca Ceragioli received the laurea degree from the University of Torino in 1995 and the Ph.D. degree from the University of Florence in 2000, both in Mathematics. She is currently Assistant Professor of Mathematical Analysis at the Technical University of Torino. Her research interests include ordinary differential equations, nonsmooth analysis and mathematical control theory.
Brief paper
Nonpathological Lyapunov functions and discontinuous Carathéodory systems夡 Andrea Bacciotti, Francesca Ceragioli ∗ Dipartimento di Matematica del Politecnico di Torino, C.so Duca degli Abruzzi, 24-10129 Torino, Italy Received 27 October 2004; received in revised form 11 October 2005; accepted 20 October 2005
Abstract Differential equations with discontinuous right-hand side and solutions intended in Carathéodory sense are considered. For these equations, sufficient conditions which guarantee both Lyapunov stability and asymptotic stability in terms of nonsmooth Lyapunov functions are given. An invariance principle is also proven. 䉷 2005 Elsevier Ltd. All rights reserved. Keywords: Lyapunov functions; Stability; Stabilizability; Discontinuous control; Invariance principle; Nonpathological functions; Carathéodory solutions
1. Introduction The interest in the study of Lyapunov-like theorems for discontinuous systems is essentially motivated by the connection between stability and stabilizability problems. Since the papers by Sussmann (1979), Artstein (1983) and Brockett (1983) it is clear that, in order to deal with general stabilization problems, discontinuous feedback laws are needed. With the introduction of discontinuous feedback laws, the theoretical problem of giving an appropriate definition of solution for a system with discontinuous right-hand side, comes out. Different approaches have been followed in the literature (Filippov, 1988; Clarke et al., 1997; Ancona & Bressan, 1999). Filippov solutions in particular have been used in order to deal with discontinuous stabilizability problems. For these solutions, Lyapunov methods have been widely developed (see, e.g., Aubin & Cellina, 1994; Clarke et al., 1998; Bacciotti & Rosier, 2001). Some recent papers show that the stabilization problem can be very
夡 This paper is not presented in IFAC meeting. This paper was presented at NOLCOS 2004. This paper is recommended for publication in revised form by Associate Editor Andrew R. Teel under the direction of Editor Hassan Khalil. ∗ Corresponding author. Tel.: +39 01 15 64 75 18; fax: +39 01 15 64 75 99. E-mail addresses: [email protected] (A. Bacciotti), [email protected] (F. Ceragioli).
0005-1098/$ - see front matter 䉷 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2005.10.014
well approached by means of Carathéodory solutions (Ancona & Bressan, 1999, 2002, 2004; Rifford, 2002, 2003; Kim & Ha, 2004). If the notion of Carathéodory solution is accepted as a good notion of solution for discontinuous systems, in particular for systems coming from discontinuous stabilization problems, it is then interesting to develop an appropriate Lyapunov theory for them. Indeed, as far as the authors know, such a generalization of Lyapunov methods has not been treated in the literature before. We consider nonsmooth Lyapunov functions. Resorting to nonsmooth Lyapunov functions gives more flexibility in the stability analysis, in particular of piecewise linear, switched and discontinuous systems. They are also convenient in order to achieve general converse theorems (see Sontag, 1999; Rifford, 2002). In the paper we take advantage of a notion of derivative for Lipschitz continuous functions which are, in addition, nonpathological (Valadier, 1989). This notion was introduced in Shevitz and Paden (1994), improved in Bacciotti and Ceragioli (1999) and then applied in Bacciotti and Ceragioli (1999, 2003). The paper consists of two main sections. In Section 2, the fundamental tools needed in order to obtain the main results are collected. In particular, the definition and basic properties of nonpathological functions are recalled. In Section 3, a Lyapunov like theorem and an invariance principle are stated and proved. In Section 4, these results are illustrated by means of some examples and counterexamples. Finally, some conclusions are stated in Section 5.
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t ∈ I , the set jC V ((t)) is a subset of an affine subspace orthogonal to (t). ˙
2. Tools First of all, we introduce Carathéodory solutions. Consider the autonomous differential equation x˙ = f (x),
(1)
where x ∈ Rn , f : Rn → Rn . Definition 1. Let I be an interval of R. A function : I → Rn is said to be a Carathéodory solution of (1) on I if (t) is absolutely continuous and (d/dt)(t) = f ((t)) for almost every t ∈ I . In the following, only Carathéodory solutions are considered and they are simply addressed as solutions. Moreover, Sx0 denotes the set of maximal solutions of (1) with initial condition x(0) = x0 and I = [0, T ) denotes the maximal right interval where ∈ Sx0 is defined. The vector field f in (1) is in general not continuous, and consequently the existence of solutions of (1) is not guaranteed by classical theorems. General sufficient conditions on the vector field f in order to have existence of solutions of (1) have been studied in Bressan (1988) and Ancona and Bressan (1999). We will make the following basic assumptions: (H1) for any initial condition x0 ∈ Rn at least one solution of (1) exists, (H2) f is Lebesgue measurable and locally bounded.
Remark 1. It is not difficult to prove that under assumptions (H1) and (H2), if ∈ Sx0 is bounded, then T = +∞. Definitions of stability in the case of discontinuous systems with solutions intended in Carathéodory sense do not differ from the usual ones. We report them for reader’s convenience. Definition 2. System (1) is said to be • Lyapunov stable at the origin if for any > 0 there exists > 0 such that for any x0 with x0 < and for any ∈ Sx0 , (t) < for all t ∈ [0, T ); • globally asymptotically stable at the origin if it is Lyapunov stable and moreover there exists > 0 such that for any x0 with x0 < and for any ∈ Sx0 , limt→T (t) = 0. Note that if the origin is locally stable then f (0) = 0. From now on, the origin is then assumed to be an equilibrium position for the system, i.e. f (0) = 0. We now recall the definition of nonpathological function given in Valadier (1989). In the following, jC V (x) denotes the Clarke gradient of the real function V at the point x (see Clarke, 1983). Definition 3. A function V : Rn → R is said to be nonpathological if it is locally Lipschitz continuous and for every absolutely continuous function : I → Rn and for almost every
Nonpathological functions form a wide class which includes Clarke regular functions, semiconcave and semiconvex functions. Precise definitions and a collection of relationships and properties (with appropriate references) for these classes of functions can be found in Bacciotti and Ceragioli (2003). Roughly speaking, a semiconvex (respectively, semiconcave) is the sum of a convex (respectively, concave) function and a C 1 function. A typical semiconvex function is V1 (x, y) = |x| + |y|, which will be used later as a Lyapunov function in Example 2, while a typical semiconcave function is V2 (x, y) = (4x 2 + 3y 2 )1/2 − |x|, that we need in Example 1. A nonpathological function which is neither √ semiconvex nor 2 − 3 3/2|x|y. Note semiconcave is V3 (x, y) = 47 x 2 + 13 y 4 that, very roughly speaking, nonpathological functions may have level sets with both concave and convex angles. Nonpathological functions can be easily handled thanks to the following proposition (Valadier, 1989). Proposition 1. If V : Rn → R is nonpathological, and : R → Rn is absolutely continuous, then the set {p · (t), ˙ p∈ jC V ((t))} is reduced to the singleton {(d/dt)V ((t))} for almost every t. The notion of nonpathological derivative of a map with respect to a differential equation is now introduced. This notion is analogous to the notion of set-valued derivative of a map with respect to a differential inclusion introduced in Shevitz and Paden (1994) and improved in Bacciotti and Ceragioli (1999). Let V : Rn → R be a nonpathological function and let (1) be given. Let AV = {x ∈ Rn : p1 · f (x) = p2 · f (x) ∀p1 , p2 ∈ jC V (x)}. The definition of nonpathological derivative is now given. Definition 4. If x ∈ AV , the nonpathological derivative of the map V with respect to (1) at x is the number V˙ f (x) = p · f (x), where p is any vector in jC V (x). The use of the nonpathological derivative is explained by the following corollary of Proposition 1. Corollary 1. Let the function V : Rn → R be nonpathological, and let be any solution of system (1). Then (t) ∈ AV and (d/dt)V ((t)) = V˙ f ((t)) for almost every t. Remark 2. The previous corollary shows that Carathéodory solutions, nonpathological functions and the nonpathological derivative fit very well together. In fact it turns out that in order to understand the behaviour of the function V along solutions of system (1), the object to be studied is the nonpathological derivative of V with respect to (1) at the points of AV . We remark that, depending on the set of nondifferentiability points
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of V and on the vector field f, AV may be a proper subset of Rn or all Rn (this is the case, in particular, when V is everywhere differentiable). The examples of Section 4 will clarify this point. We emphasize that, generally, the set AV is “big”. More precisely, it can be proven that if V is nonpathological and assumption (H2) is satisfied then AV is dense in Rn and, if V is moreover Clarke regular or semiconcave or semiconvex, then Rn \AV has null Lebesgue measure.
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(d/dt)V ((t)) = V˙ f ((t)) is nonpositive thanks to (2). The statement is then a consequence of Lemma 1(b). (ii) The proof is an immediate consequence of Corollary 1 and Lemma 1. Next a LaSalle-like invariance principle is proved (see Shevitz & Paden, 1994; Ryan, 1998; Bacciotti & Ceragioli, 1999 for other versions of the invariance principle). In order to get it, some regularity for the vector field is needed.
3. Results In this section two propositions which generalize Lyapunov method to systems with discontinuous right-hand side and solutions intended in Carathéodory sense are proved. Proposition 2. Let V : Rn → R be positive definite, nonpathological and radially unbounded. Let AV be defined as in Definition 4. Assume that ∀x ∈ AV ,
V˙ f (x)0.
(2)
Then (i) system (1) is Lyapunov stable at the origin; (ii) if moreover there exists a function W : Rn → R continuous and positive definite such that V˙ f (x) − W (x) for all x ∈ AV then system (1) is globally asymptotically stable. Lemma 1. Let V : Rn → R be positive definite, locally Lipschitz continuous and radially unbounded and let : [0, T ) → Rn be any solution of (1). Then (a) V ◦ : [0, T ] → [0, +∞] is differentiable almost everywhere, (b) if (d/dt)V ((t))0 for almost every t, then T = +∞ and the origin is Lyapunov stable, (c) if there exists a function W : Rn → R continuous and positive definite such that (d/dt)V ((t)) − W ((t)) for almost every t, then the origin is globally asymptotically stable. Proof. (a) V ◦ is absolutely continuous and then almost everywhere differentiable (see Wheeden & Zygmund, 1977). (b) Since V ◦ is absolutely continuous, then t2 d V ((t2 )) − V ((t1 )) = V ((s)) ds (3) t1 ds (see Wheeden & Zygmund, 1977). Then (d/dt)V ((t)) 0 implies that V ◦ is nonincreasing and from the fact that V is radially unbounded it follows that is bounded. Remark 1 implies that T =+∞. The remaining part of the proof is standard (see Khalil, 1992). (c) The proof is standard (see Khalil, 1992), after recalling (3). Proof of Proposition 2. (i) According to Corollary 1, for any solution of (1), (t) ∈ AV for almost every t, and
Definition 5. A vector field f : Rn → Rn is said to have the solutions closure property if for any sequence {n } of solutions of (1) such that n → uniformly on compact subsets of R, one has that also is a solution of (1). Of course any continuous vector field has the solutions closure property. An important class of discontinuous vector fields with the solutions closure property is the class of patchy vector fields (see Ancona & Bressan, 1999). In the statement of Proposition 3 the notion of weakly invariant set is needed. Definition 6. A set M is said to be weakly invariant for (1) if for any x0 ∈ M there exists ∈ Sx0 such that (t) ∈ M for all t 0. Proposition 3. Assume that the vector field f has the solutions closure property. Let V : Rn → R be positive definite, nonpathological and radially unbounded. Let AV be defined as in Definition 4 and assume (2). Let ZfV = {x ∈ AV : V˙ f (x) = 0} and let M be the largest weakly invariant subset of ZfV . Then for any x0 and any ∈ Sx0 lim dist ((t), M) = 0.
t→+∞
(4)
Lemma 2. If f has the solutions closure property, then for any x0 and any ∈ Sx0 the fact that the positive limit set () is nonempty implies that it is weakly invariant. The proof of this lemma is very similar to the proof of the analogous lemma for Filippov solutions (see Filippov, 1988, p. 130). Proof. Let ∈ Sx0 be given. First of all note that, as in the proof of Lemma 1, is bounded and then T = +∞. Since is bounded then () = ∅ and limt→+∞ dist ((t), ()) = 0. Next it is proved that () ⊂ ZfV . From the fact that () is weakly invariant it will follow (4). Let z ∈ (). Consider the composite function V ◦ , which is nonincreasing and bounded from below. From this fact it follows that there exists limt→+∞ V ((t)) = c and then V (z) = c for every z ∈ (). In fact, if z ∈ (), there exists a sequence {tn }, tn → +∞ such that limn→+∞ (tn ) = z and, thanks to the continuity of V, V (z) = limn→+∞ V ((tn )) = limt→+∞ V ((t)) = c. Due to Lemma 2, () is weakly invariant, then there exists ˜ ∈ Sz such that (t) ˜ ∈ () for all t 0. It follows that V ((t)) ˜ =c
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for all t 0 and (d/dt)V ((t)) ˜ = 0 for all t 0. Since (t) ˜ ∈ ˙ ˜ = (d/dt)V ((t)) ˜ = 0 for almost all t 0, AV and V f ((t)) (t) ˜ ∈ ZfV for almost all t 0. Let now {ti } be a sequence such that ti → 0 and (t ˜ i ) ∈ ZfV for all i. Since (t) ˜ is continuous it finally follows that z = limi→+∞ (t ˜ i ) ∈ ZfV .
f + (x, y) = −
f (x, y) =
Corollary 2. Assume that f : Rn → Rn has the solutions closure property. Let V : Rn → R be positive definite, nonpathological and radially unbounded, and let AV be defined as in Definition 4. If (2) holds and V˙ f (x) = 0 if and only if x = 0, then system (1) is globally asymptotically stable. 4. Examples Propositions 2 and 3 are now illustrated by means of some examples. Example 1, besides being an application of Proposition 2, aims to explain the use of the nonpathological derivative. Examples 2 and 3 are counterexamples showing that Proposition 2 is not true if either assumption (H1) or assumption (H2) do not hold. In Example 3, Proposition 3 is applied to prove the asymptotic stability of a planar bilinear “switched system”. This example aims also to remark that discontinuous feedback laws also appear when control systems with finite control set are studied. Finally, Example 4 is a counterexample showing that the assumption about the solutions closure property of the vector field in Proposition 3 is actually needed. Example 1. Consider the two-dimensional, single input, driftless system x˙ = (x 2 − y 2 )u, y˙ = 2xyu
Consider now the slightly modified vector fields
(5)
(the so-called Artstein’s circles example, see Artstein, 1983). This system cannot be stabilized by means of a continuous feedback. A discontinuous stabilizing feedback is 1 if x < 0, (6) u(x, y) = −1 if x 0. Denote by f (x, y) the right-hand side of the implemented system. It is shown in Artstein (1983), that a C 1 Lyapunov function for this system does not exist. On the other hand, V (x, y)=V2 (x, y)= 4x 2 + 3y 2 −|x| satisfies the assumptions of Proposition 2. In fact V is nonpathological, AV =R2 \{(x, y) : x = 0} and if (x, y) ∈ AV then V˙ f (x, y) = ∇V (x, y)f (x, y) = −W (x, y), where 4|x|3 + 2|x|y 2 − (x 2 − y 2 ) 4x 2 + 3y 2 W (x, y) = , 4x 2 + 3y 2 is continuous and positive definite. We can conclude that the implemented system is asymptotically stable. Note that in this example the set where V is not differentiable is a curve, the y-axis, which is not included in AV , because the vector field f is transversal to it. In this case, in order to apply Proposition 2, the condition involving the nonpathological derivative need not to be checked at all points of R2 .
f (x, y) (0, y)
if x = 0, if x = 0,
(7)
f (x, y) if x = 0, (0, −y) if x = 0.
(8)
Note that for x = 0 both these vector fields are parallel to the yaxis, i.e. to the curve where V is not differentiable, and AV =R2 . The condition involving the nonpathological derivative needs now to be checked at all points of R2 . Roughly speaking, Rn \AV either coincides or does not coincide with the set NV where V is not differentiable depending on the fact that either there are or there are not trajectories of the system lying on NV . √ √ V˙ f + (x, y) = 3|y| and V˙ f − (x, y) = − 3|y|, then the system defined by f − (x, y) satisfies the assumptions of Proposition 2 and hence it is asymptotically stable, while the system defined by f + (x, y) does not satisfy the assumptions of Proposition 2. The system defined by f + (x, y) is clearly unstable. Example 2. Consider the two dimensional system (1) where f is given by ⎧ ⎪ 0 if y 0, ⎨ −1 f (x, y) = (9) ⎪ ⎩ 0 if y < 0, 1 and the function V (x, y) = V1 (x, y) = |x| + |y|. It can be computed that AV = R2 \{(x, y) : y = 0}. Note that in this case the set NV where V is not differentiable is the union of the x-axis and the y-axis, but Rn \AV is only the x-axis due the fact that there are trajectories of the system lying on the y-axis. For all (x, y) ∈ AV , V˙ f (x, y) = −1, but the system is not asymptotically stable. Note that Proposition 2 cannot be applied because assumption (H1) is not satisfied. In fact there are no solutions issuing from the points of the x-axis. Example 3. Consider the two-dimensional system whose right-hand side is ⎧0 ⎪ if = 0, ⎪ ⎪ 0 ⎪ ⎪
⎪ y 2 ⎨ −x − sin −1/n (−1/n)2 1 if ∈ ( n1 , n−1 ], y x 2 ⎪ − + sin 2 ⎪ −1/n −1/n) ( ⎪ ⎪ ⎪ ⎪ ⎩ −x if > 1, −y where = x 2 + y 2 , n ∈ N, n 2. Consider the function V (x, y)=(x 2 +y 2 )/2. Here V is everywhere differentiable then AV = R2 . ⎧ ⎨0 ˙ V f (x, y) = − x 2 + y 2 ⎩ −(x 2 + y 2 )
if = 0, if 0 < 1, if > 1,
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then V˙ f (x, y) − min{ x 2 + y 2 , (x 2 + y 2 )/2}. Proposition 2(ii) cannot be applied due to the fact that (H2) does not hold since the vector field is not locally bounded. The trajectories of the system in polar coordinates are: (t) = 0 − t, 1 1 (t) = sin2 −t−1/n + 0 − sin2 −1/n . 0
0
If 0 ∈ (1/n, 1/(n − 1)], they are defined on the intervals [0, 0 − 1/n). Note that the system is Lyapunov stable, but not asymptotically stable. Example 4. Consider the two-dimensional control system x˙ x y =u + (1 − u) , (10) y˙ −y −x and the feedback law 1 if (x, y) ∈ D, u(x, y) = 0 if (x, y) ∈ R2 \D,
(11)
where D = {(x, y) : x > 0 and 2x < y < 3x or x < 0 and 3x < y < 2x}. The function V (x, y) = (x 2 + y 2 )/2 is C ∞ , nevertheless we cannot use it in order to apply classical Lyapunov theorems to the implemented system due to the fact the feedback law is discontinuous. Denote by f (x, y) the right-hand side of the implemented system. AV = R2 , V˙ f (x, y) = ∇V (x, y)f (x, y)0 V
for all (x, y) ∈ R2 and Z f = R2 \D. The largest weakly invariV
ant subset of Z f is the origin, then the implemented system is asymptotically stable. Example 5. Consider the equation x˙ = f (x) where x ∈ R, f (0)=0, f (x)=−f (−x) and f is defined in the following way: 1 −x + n1 if x ∈ ( n1 , n−1 ], f (x) = (12) −x + 1 if x > 1, where n ∈ N, n2. The unique solutions of this equation with initial condition (0) ∈ (1/n, 1/(n − 1)] is (t) = e−t (0) + (1/n)(1 − e−t ), which is defined on [0, +∞). Remark that the origin is stable, but not asymptotically stable. Consider V (x) = x 2 /2. It holds 1 x(−x + n1 ) if x ∈ ( n1 , n−1 ], ˙ V f (x) = x(−x + 1) if x > 1, then ZfV = {0}. In this case Proposition 3 cannot be applied due to the fact that the vector field f does not have the solutions closure property. In fact note that for any solution with initial condition in (1/n, 1/(n − 1)], the positive limit set is the point 1/n, which is not invariant. Finally, remark that there does not exist a positive definite and continuous function W such that V˙ f (x) − W (x). 5. Conclusion Differential equations with discontinuous right-hand side and solutions intended in Carathéodory sense have been
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considered. The interest of such equations is motivated by the fact that they naturally arise in stabilizability problems in connection with important classes of discontinuous feedback laws. For these equations sufficient conditions which guarantee both Lyapunov stability and asymptotic stability in terms of nonsmooth Lyapunov functions are proved. Moreover an appropriate version of LaSalle invariance principle is given. References Ancona, F., & Bressan, A. (1999). Patchy vector fields and asymptotic stabilization. Esaim-Cocv, 4, 445–472. Ancona, F., & Bressan, A. (2002). Flow stability of patchy vector fields and robust feedback stabilization. SIAM Journal on Control and Optimization, 5, 1455–1476. Ancona, F., & Bressan, A. (2004). Stability rates for patchy vector fields. Esaim-Cocv, 10, 168–200. Artstein, Z. (1983). Stabilization with relaxed controls. Nonlinear Analysis, 7, 1163–1173. Aubin, J. P., & Cellina, A. (1994). Differential inclusions. Berlin: Springer. Bacciotti, A., & Ceragioli, F. (1999). Stability and stabilization of discontinuous systems and nonsmooth Lyapunov functions. Esaim-Cocv, 4, 361–376. Bacciotti, A., & Ceragioli, F. (2003). Nonsmooth optimal regulation and discontinuous stabilization. Abstract and Applied Analysis, 20, 1159–1195. Bressan, A. (1988). Unique solutions for a class of discontinuous differential equations. Proceedings of the American Mathematical, 104, pp. 772–778. Bacciotti, A., & Rosier, L. (2001). Lyapunov functions and stability in control theory. London: Springer. Brockett, R. (1983). Asymptotic stability and feedback stabilization. In: R. Millman, R. Brockett, & H. Sussmann (Eds.), Differential geometric control theory (pp. 181–191). Boston: Birkhäuser. Clarke, F. H. (1983). Optimization and nonsmooth analysis. New York: Wiley. Clarke, F. H., Ledyaev, Yu. S., Sontag, E. D., & Subbotin, A. I. (1997). Asymptotic controllability implies feedback stabilization. IEEE Transactions on Automatic Control, 42, 1394–1407. Clarke, F. H., Ledyaev, Yu. S., Stern, R. J., & Wolenski, P. R. (1998). Nonsmooth analysis and control theory. New York: Springer. Filippov, A. F. (1988). Differential equations with discontinuous righthand sides. Dordrecht: Kluwer. Khalil, H. K. (1992). Nonlinear systems. New York: Macmillan. Kim, S. J., & Ha, I. J. (2004). On the existence of Carathéodory solutions in nonlinear systems with discontinuous switching control laws. IEEE Transactions on Automatic Control, 49, 1167–1171. Rifford, L. (2002). Semiconcave control–Lyapunov functions and stabilizing feedbacks. SIAM Journal on Control and Optimization, 41, 659–681. Rifford, L. (2003). Singularities of viscosity solutions and the stabilization problem in the plane. Indiana University Mathematics Journal, 52, 1373–1396. Ryan, E. P. (1998). An integral invariance principle for differential inclusions with applications in adaptive control. SIAM Journal on Control and Optimization, 36, 960–980. Shevitz, D., & Paden, B. (1994). Lyapunov stability theory of nonsmooth systems. IEEE Transactions on Automatic Control, 39, 1910–1914. Sontag, E. D. (1999). Stability and stabilization: Discontinuities and the effect of disturbances. In: F.H. Clarke, & R.J. Stern (Eds.), Nonlinear analysis, differential equations and control (pp. 551–598). Dordrecht: Kluwer. Sussmann, H. (1979). Subanalytic sets and feedback control. Journal of Differential Equations, 31, 371–393. Valadier, M. (1989). Entraıˆnement unilatéral, lignes de descente, fonctions lipschitziennes non pathologiques. Comptes Rendu de l’Academie des Sciences. Series I, Mathematique. Paris, 8, 241–244. Wheeden, R., & Zygmund, A. (1977). Measure and integration. New York: Marcel Dekker.
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A. Bacciotti, F. Ceragioli / Automatica 42 (2006) 453 – 458 Andrea Bacciotti received a degree in Mathematics at the University of Florence. He is Full Professor of Mathematical Analysis since 1981. He has been with the Mathematics Department of the University of Florence, Siena, Torino (Technical University). His research interests include ordinary differential equations, differential inclusions, mathematical control theory.
Francesca Ceragioli received the laurea degree from the University of Torino in 1995 and the Ph.D. degree from the University of Florence in 2000, both in Mathematics. She is currently Assistant Professor of Mathematical Analysis at the Technical University of Torino. Her research interests include ordinary differential equations, nonsmooth analysis and mathematical control theory.