Stability for Impulsive Control Systems
Fernando L. Pereira
Institute for Systems and Robotics-Porto Faculdade de Engenharia da Universidade do Porto
Rua Dr. Roberto Frias, 4200-465 Porto, Portugal flp@fe.up.pt
Tel/fax: +351 22 508 1857/1443 and
Geraldo N. Silva
Dept. Computer Science and Statistics, Universidade Estadual Paulista 15054-000 - S. J. Rio Preto-SP, Brasil
gsilva@dcce.ibilce.unesp.br Tel/fax: +55 17 221 2209/ 221 2203
September 5, 2002
Abstract
In this paper we extend the notion of control Lyapounov pair of functions and derive an stability theory for impulsive control systems. The control system is a measure driven differential inclusion that is partly absolutely continuous and partly singular. Some examples illustrating the features of the Lyapounov stability are provided.
Keywords: Control Lyapounov pair of functions, Stability, Impulsive control, Mea- sure driven differential inclusion.
1 Introduction
In this article, we address stability conditions for a class of impulsive control problems, in which the dynamics are defined by a differential inclusion driven by a vector valued control measure. A concept of proper solution is presented which provides a meaning to the stabilization problem in the context defined by the set of assumptions specified for the data of the problem. An important feature of the adopted solution concept
concerns the fact that it encompasses systems whose singular dynamics do not satisfy the so-called Frobenius condition.
The addressed class of problems arise in a wide variety of application areas such as finance, impact vibro-mechanics, management of renewable resources, and aerospace navigation, (consider (Clark, Clarke & Munro 1979, Baumeister 2001, Gurman 1981, Brogliato 1996, Marec 1979), to name just a few references), for which the solution should be found within the set of control processes with trajectories of bounded variation.
Naturally, this fueled the relatively recent rapid development of a, by now, considerable body of theory for this class of systems (see, for example, (Arutyunov, Jacimovic &
Pereira 2001, Bressan & Rampazzo 1991, Bressan & Rampazzo 1994, Dykhta 1990, Dykhta & Samsonyuk 2000, Gurman 1972, Kolokolnikova 1996, Miller 1996, Motta &
Rampazzo 1996, Pereira & Silva 2000, Rishel 1965, Rockafellar 1976, Rockafellar 1981, Silva & Vinter 1997, Vinter & Pereira 1988, Warga 1972, Zavalischin & Sesekin 1991), and references therein) and supporting control strategies computation schemes, (Dykhta
& Derenko 1994, Baturin & Goncharova 1999, Baturin & Ourbanovich 1997, Krotov &
Gurman 1987).
Furthermore, the pervasive availability of computational capabilities associated to technological evolution led to the emergence of control systems paradigms, generically designated by Hybrid Systems, whose description involves a combination of (continuum) time driven and event dynamics, (Witsenhausen 1966, Branicky, Borkar & Mitter 1998, de Sousa & Pereira 2001, R. Grossman & Rischel 1993). Albeit in a modeling context that differs from the one adopted in this article, (Aubin 2000) draws the attention for the fact that the impulsive control framework is suitable to capture important features of hybrid systems. Therefore, besides the motivation naturally associated with the above mentioned range of applications addressed by impulsive control, its relevance in dealing with systems of hybrid nature (in the sense defined above) depends on the type of results and algorithms that can be developed in this framework. This is an additional incentive to pursue the research issues addressed in this article.
There is a vast amount of literature addressing stability problems for conventional control systems, i.e., when dynamic control systems such as ˙x = f(x, u), x(0) = x0
or, its differential inclusion counterpart, ˙x ∈ F(x) are considered. A lot of effort have been spent in providing conditions which are necessary and or sufficient for the system to satisfy various types of stability. An important role has been played by Control Lyapounov Function (CLF) (see (Clarke, Ledyaev, Stern & Wolenski 1998) for a defi- nition), V : <n → <, which in the control or game paradigms underlies the synthesis of stabilizing feedback control laws, i.e., u = k(x) by the so-called Lyapounov design method1.
Clearly, a number of questions of utmost importance which have been the subject of very active research, (see (Coron & Rosier 1994, Sontag & Sussmann 1995, Sontag 1999, Sontag & Ledyaev 1999, Clarke, Ledyaev, Rifford & Stern 2000), to name just a few references): a) Existence of regular Lyapounov functions, b) existence of contin-
1It involves two stages: a) Find a Control Lyapounov Function pair (V, W) and b) Select a function u=k(x) satisfying∇V(x)·f(x, k(x))≤ −W(x),∀x∈ <n.
uous feedback functions, c) solution concepts for ordinary differential equations with discontinuous r.h.s. and d) robustness to errors, being measurement errors of particular importance for systems with discontinuous feedback control laws.
Brockett condition, (Ryan 1994), states that a dynamic system ˙x=f(x, u) is stabi- lizable by continuous feedback only if, for every neighborhood N of 0, f(N,Ω) contains the origin. A famous counterexample is the nonholonomic integrator which even is glob- ally asymptotically controllable (GAC). In (Clarke, Ledyaev, Sontag & Subbotin 1997), it is shown that a GAC system possesses a, possibly discontinuous, stabilizable feedback controller, being the adopted solution concept defined by x := limN→∞xN, where xN satisfies
˙
xN(t) =f(xN(t), k(xNi )) ∀t∈[ti, ti+1),
with xNi =xN(ti), andπN ={t0, t1, ..., tN} is a partition of the considered time interval satisfying maxi|ti−ti−1| →0 as N → ∞, (Clarke et al. 1998).
It is shown in (Sontag 1983) that a GAC system always has a continuous Lyapounov function satisfying
infu∈ΩDV(x;f(x, u))≤ −W(x)<0 ∀x6= 0
where DV(x;f) is the Dini derivative of V at x in the direction f (for definitions, see (Clarke et al. 1998)). In (Clarke et al. 2000), it is shown that, under very mild assump- tions on the data of the control system, there exists always a locally Lipschitz Lyapounov function that “practically stabilizes the system”, i.e., that provides a feedback driving the system to an arbitrarily small neighborhood of the equilibrium point in finite time and a converse Lyapounov theorem is shown. In (Rifford 2000), it is shown that a global locally Lipschitz Lyapounov function always exists for a GAC system.
Robustness to measurement errors is ensured for sufficiently smooth Lyapounov func- tions. In the nonsmooth case, robustness can only be guaranteed if dynamic feedback is considered, (Sontag 1999).
The construction of a Lyapounov function remains an important issue in control de- sign which has been addressed when either there is a clear physical interpretation, or the specific structure of the problem can be exploited. In this later category, backstepping and sliding modes are two of the most popular classes of methods.
In spite of the extensive stability literature for conventional control systems and the vast effort in the control of measure driven dynamic systems, little is known in what concerns stability results for impulsive dynamic systems. In (Miller 1978), the stability problem for impulsive systems is addressed in the context of robustness of the solution with respect to perturbations in the control measure. Characterizations of necessary and sufficient stability conditions are derived after the introduction of an appropriate notion of closure of the set of trajectories. In the next section, a result on robustness of the solution concept to the measure differential inclusion adopted in this article, proved in (Silva & Vinter 1996), is presented.
In a modeling context substantially distinct from the one considered in this article, (Aubin 2000) provides stability conditions for dynamic systems with discontinuous tra- jectories in terms of inequalities involving a certain function, designated by Lyapounov
function, which is shown to be a contingent solution to Hamilton-Jacobi variational inequalities.
This paper is organized as follows. In Section 2, the problem addressed in this article is formulated in detail. This includes the presentation of the solution concept, and its robustness properties. Then, in section 3, the extension to the measure driven differential inclusion context of the concept of Control Lyapounov function is presented and used to state and prove the stability conditions. In the ensuing section, we present a few examples addressing the main points of the developed stability conditions for impulsive systems and, then, some conclusions are drawn in the last section.
2 The impulsive system and solution concept
We consider the solution concept defined in (Pereira & Silva 2000) to the general im- pulsive dynamic control system whose dynamics are given by the following differential inclusion:
dx(t) ∈ F(t, x(t))dt+G(t, x(t))µ(dt),
x(0) = x0. (1)
Here,F : [0,∞)×Rn ,→ P(Rn), andG: [0,∞)×Rn,→ P(Rn×q) are given set-valued functions andµ∈C∗([0,∞);K), the set in the dual space of continuous functions from [0,∞) into Rq with values in K, a positive, convex, and pointed cone in Rq.
Definition 1. The trajectory x, withx(0) =x0, is admissible for (1) ifx(t) = xac(t) + xs(t)∀t ∈[0,∞), where
x˙ac(t) ∈ F(t, x(t)) +G(t, x(t))·wac(t) a.e.
xs(t) = R
[0,t]gc(τ)wc(τ)d¯µsc(τ) +R
[0,t]ga(τ)d¯µsa(τ).
Here, ¯µ is the total variation measure associated with µ, µsc, µsa and µac are, re- spectively, the singular continuous, the singular atomic, and the absolutely continuous components ofµ,wac is the time derivative ofµac,wsc is the Radon-Nicodym derivative of µsc with respect to its total variation, gc(·) is a ¯µsc measurable selection of G(·, x(·)) and ga(·) is a ¯µsa measurable selection of the multifunction
G(t, x(t˜ −);µ({t})) : [0,∞)× <n×K ,→ P(<n)
specifying the reachable set of the singular dynamics at (t, x(t−)) when the control measure has an atom of “weight” µ({t}). In order to define this multifunction, the following concept of graph completion of a time reparametrized function is required (see (Pereira & Silva 2000)).
Let, ¯η(t) = [η(t−), η(t)] if ¯µ({t})>0, and ¯η(t) = {η(t)} otherwise, where
• η(t) := t+Pq
i=1Mi(t), and
• M(·) = col(M1(·), ..., Mq(·)), with Mi(t) =R
[0,t]µi(ds), ∀t >0, andMi(0) = 0.
Definition 2. A family ofgraph completionsassociated with a measureµ∈C∗(0,1;K) is the set of pairs (θ, γ) : [0,∞),→ <+× P(K) where θ : [0,∞)→ <is the “inverse” of
¯
η, and the function γ : ¯η(t)→ <q takes values γ(s) :=
M(θ(s)) if ¯µ({t}) = 0 M(t−) +Rs
η(t−)v(σ)dσ if ¯µ({t})>0, where ¯µ(dt) = Pq
i=1µi(dt) andv(·) is inVt, the set of functions from ¯η(t) to<qsatisfying v(s)∈K, and R
¯
η(t)v(s)ds=µ({t}).
Now, we are in position of defining the set-valued function ˜G(t, z;α) which is given by{G(t, z)w(t)} when |α|= 0 and by
{[ξ(η(t))−ξ(η(t−))]/|α|: ˙ξ(s)∈G(t, ξ(s)) ˙γ(s) ¯η(t)-a.e. ξ(0) =z and γ(η(t))−γ(η(t−))) =α} when |α|>0. Here |α|=Pq
i=1αi, w(·) is the Radon-Nicodym derivative ofµ w.r.t. ¯µ, (ξ, γ) are in AC([0,1];<n× <q), and the pair (θ, γ) is agraph completion with ˙θ(s)≡0 on ¯η(t).
Notice that, in spite of the nonuniqueness of the graph completions of the control measure, for a given pair (x(t−), µ({t})) with µ({t})) > 0 and a given measurable selection of G, x(t+) is only unique in the case of commutativity of the singular vector fields.
Before pursuing with considerations concerning the robustness property, we would like to remark that a motivation for this solution concept concerns important classes of engineering problems involving the coordinated control of multiple dynamic systems. In this context, it is of interest to control a dynamical system with several viable config- urations. Although transitions between different configurations correspond to non pro- ductive phases, the way they evolve might affect the overall performance of the system and, therefore, it is relevant to incorporate their management in the global optimization problem.
The result concerning the robustness of this solution was established in (Silva &
Vinter 1996) for the particular case of scalar valued measures. The extension to systems with vector-valued control measures with commutative singular vector fields is straight- forward as this problem can be easily reduced to the former. Furthermore, the necessary modifications in order to encompass the noncommutative case, the one considered here, are minor and are omitted.
Proposition 1. Consider multifunctionsF andGwith domain [0,+∞)×<nsatisfying:
(i) F(t,·) and G(·,·) have closed graphs and take compact sets in, respectively, <n and <n×q as values,
(ii) F is Lebesgue × Borel measurable andG is Borel measurable, and (iii) F(t, x) andG(t, x) are convex valued for all (t, x).
ConsiderT > 0 and take a sequence{xi0}in<nand a sequence{µi}inC∗([0, T];K), and elementsx0 ∈ <nand µ∈C∗([0, T];K) such that, asi→ ∞,xi0 →x0 andµi →∗ µ.
Take also a sequence {xi} inBV+([0, T];<n) such that xi(·) is a solution to (1) in the sense of Definition 1 with µi in place of µ. Consider the following inclusion
˙
y(s)∈F(θ(s), y(s)) ˙θ(s) +G(θ(s), y(s)) ˙γ(s) (2) almost everywhere in [0, T].
For eachi, assume the existence ofβ(t)∈L1 andc >0 such thatF(t, xi(t))⊂β(t)B a.e. and G(t, xi(t))⊂cB for all t.
Then, there exist:
• a sequence of processes (yi, θi, γi), solution to (2) withyi(0) =xi0,
• (y, θ, γ), solution to (2), with y(0) =x0, and
• a solution x to (1), such that,∀t ∈(0, T],
xi(t) =yi(ηi(t)) andx(t) = y(η(t)).
Along a subsequence, we have dxi →∗ dxandxi(t)→x(t) for allt ∈([0, T]\Mµ)∪{0, T} (whereMµ denotes the atoms of µ) and yi →y strongly in C([0, T];<n).
Here, the notation dxi →∗ dx means the weak* convergence of measures dxi to dx.
We also need the following auxiliary technical result for the proof of the main result of this article.
Lemma 1. Let (θ,Γ) be a family ofgraph completionsofµ∈C∗([0,+∞);K) as defined above and take γ(·) a Borel measurable selection of Γ(·). Then
(i) θ and γ are Lipschitz continuous, non-negative functions satisfying θ(s) +˙
q
X
i=1
˙
γi(s) = 1 La.e..
(ii) For all Borel measurableµ∈C∗([0,∞);K), integrable functionG: [0,∞)7→ <n×q and Borel set T ⊂[0,∞), we have
Z
θ−1(T)
G(θ(s)) ˙γ(s)ds = Z
T
G(τ)µ(dτ).
(iii) For all measurable functionf : [0,∞)7→ <n and Borel setS⊂[0,∞),θ(S) is also
Borel set Z
S
f(θ(s)) ˙θ(s)ds= Z
θ(S)
f(τ)dτ.
Proof. The arguments of the proof of this result can be found, for example, in (Silva
& Vinter 1996, Silva & Vinter 1997, Pereira & Silva 2000).
3 Lyapounov Stability
3.1 Main result
An equilibrium point, ¯x, is globally asymptotically stable if, for any x0 ∈ <n, there exists a feasible trajectory x, solution to (1) with x(0) = x0, satisfying x(t) → x¯ as t → ∞. A characterization of asymptotic stability for impulsive control systems in terms of Lyapounov functions analogous to the one for conventional control systems (see, for example (Clarke et al. 1998)) is addressed here. To simplify matters, we will assume, from now on, that F and Gdepend on the state variable only.
Before proceeding we should set up in what sense a point ¯x is an equilibrium point of the impulsive dynamical system (1). For this purpose, define the set-valued map F˜ :={F v0+Gv : (v0, v) ∈ V¯} where ¯V :={(v0, v) : v0 ≥ 0, v ∈ K,Pq
i=0vi = 1}. We say that ¯x is an equilibrium point of (1) if 0∈F˜(¯x).
Definition 3. Functions (V, W) form a Lyapounov pair for (1) at ¯x if they satisfy the following properties:
1) Positive definiteness. V ≥0,W ≥0 with W(x) = 0 iff x= ¯x;
2) Growth. The set{x∈ <n|V(x)≤r} is compact for all r ≥0;
3) Infinitesimal decrease. ∀z∈<n, ∀ξ∈∂pV(z),
min{< ξ, ν >:ν ∈F(z)v0+G(z)v,(v0, v)∈V¯} ≤ −W(z). (3) Here, ¯V is the set previously defined and ∂PV(z) denotes the set of proximal gradients of V atz (see, e.g. (Clarke et al. 1998)).
The third item deserves additional considerations. If the control measure is abso- lutely continuous w.r.t. the Lebesgue measure, then v0 > 0, z = x(t), and condition 3) coincides with the conventional one. However, if there is a component of the control measure which is an atom, then, on its support, we have v0 = 0, and inequality in con- dition 3) above has to considered along the graph completion of the trajectory between the jump endpoints, that is, z = ˜x(s) fors∈η(t).¯
Now, consider, at some time t, an incremental jump ∆δt(t). Since we have V(x(t+t))−V(x(t−t )) =
Z
¯ η(t)
<ξ(s), ζ(s)w(s)>ds for some selections
• w(·) ofK, withR
¯
η(t)w(s)ds = ∆,
• ξ(·) of ∂PV(˜x(·)), and
• ζ(·) ofG(t,x(˜ ·)), with ˜x satisfying ˜x(η(t−)) = x(t−) and ˙˜x(s) = ζ(s)w(s),
an alternative convenient expression of the decreasing condition in terms of the original parametrization is given by
−W(x(t−)) ≥ min{min{<ξ, ν >:ν ∈F(t, x(t))v0+G(t, x(t))v, (v0, v)∈V¯ s.t.
Z
¯ η(t)
v(s)ds = 0},
min{V(x(t))−V(x(t−)) : x(t)∈RG(t;x(t−),∆), |∆|<ε,∀ε>0} }. where W is a suitably defined function satisfying 1) of Definition 3, ¯V is as above, and RG(t+, z;α) denotes the reachable set of the singular dynamics when the control measure exhibits an atomα and the state variable takes the valuez att−.
Let us now state the assumptions on the set-valued mapsF andG, which will remain in force throughout this work.
(h1) For every x∈ <n, F(x) and G(x) are nonempty compact convex sets;
(h2) F and Gare upper semicontinuous; and
(h3) there exist constants a and b such that for all x∈ <n,
v ∈ F(x) =⇒ kvk ≤akxk+b; (4) v ∈ G(x) =⇒ kvk ≤akxk+b. (5) We recall that F is upper semicontinuous on x if, given any > 0, there exists δ > 0, such that
ky−xk< δ =⇒F(y)⊂F(x) +B
where B stands for the unit ball of <n. The condition (h3) of the hypotheses is known as the linear growth condition and is a natural generalization of the case where F is a function (single valued).
Now, we are prepared to state the main result of this paper.
Theorem 1. Let ¯x∈ <n be an equilibrium point of (1), i.e. 0∈F˜(¯x), and assume that the following regularity condition is satisfied at ¯x:
0∈ {/ G(¯x)v : (0, v)∈V¯}. (6) Suppose further that, for ¯x, there exists a Lyapounov pair of functions,V andW, which are lower semi-continuous.
Then, for any x0 ∈ <n, there exists a process (x, µ) of (F,G) in [0,∞) such that x(0) =x0 and x(t)→x, when¯ t → ∞.
3.2 Proof of Theorem 1
Let ¯x ∈ <n be an equilibrium point and (V, W) be a Control Lyapounov pair for ¯x.
First, observe that ˜F inherits the hypothesis (h1)-(h3) from F and G. Then, from Theorem 4.5.5 of (Clarke et al. 1998),∀x0 ∈ <n, there exists a trajectoryy(·) of ˜F with y(0) =x0, such that y(s)→x¯as s goes to infinity.
Now, it follows, by construction of the trajectory y as the minimal selection of ˜F and from the definition of ˜F itself that, for some selectors f(s) ∈ F(y(s)) a.e. and G(s)¯ ∈G(y(s)) a.e.
˙
y(s)∈ {f(s)v0+ ¯G(s)v : (v0, v)∈V¯}a.e. s ∈[0,∞).
At this point we apply standard measurable selection theorem (see, e.g. (Clarke et al. 1998): 151) to conclude that there exists a measurable function (v0, v)(s) ∈ V¯ a.e.
in [0,∞), such that
˙
y(s) =f(s)v0(s) +
q
X
i=1
gi(s)vi(s). (7)
We claim that v0 : [0,∞) → [0,∞) is nonvanishing in unbounded subintervals of [0,∞). Indeed. Suppose, by contradiction, that v0 ≡ 0 in [t0,∞) for some t0 ≥ 0. It follows from (7) that
y(t)−x0 = Z t
t0
q
X
i=1
gi(s)vi(s)ds.
Taking the limit whent → ∞in the equation above we obtain
¯
x−x0 = Z ∞
t0
q
X
i=1
gi(s)vi(s)ds.
This implies that the integral on right side is finite. So, we can arrange a sequence sn → ∞ such that the integrand
q
P
i=1
gi(sk)vi(sk) tend to zero. This together with the upper semicontinuity of the set-valued map x ,→ {G(x)v : (0, v) ∈ V¯}, imply that 0 ∈ {G(¯x)v : (0, v) ∈ V¯}, contradicting the regularity assumption (6) at ¯x. The proof of the claim is complete.
The next step consists in the construction of a control process (x, µ) from (y,(v0, v)), that is a solution to (1), with x of bounded variation converging asymptotically to ¯x.
Let us first construct a control measure µ by defining µ(A) := R
θ−1(A)v(s)ds with θ(s) :=Rs
0 v0(σ)dσ for any A∈ B, the Borel σ-algebra of sets contained in the interval [0,∞). Since v0(s) is nonnegative and different from zero in unbounded intervals, then θ is nondecreasing and has [0,∞) as image. These properties of θ imply that µ is a Borel measure with domain in the Borel σ−algebra B. Moreover, µ takes finite values in bounded Borel sets, i.e. kµ(A)k < ∞ for all bounded set A in B, where the k · k means the euclidian norm of <q.
Define η(t) := sup{s : θ(s) ≤ t}. Notice that η is a strictly increasing function.
Let {ti : i = 1,2, . . .} be a set of points in [0,∞) such that, for each i, θ−1(ti) is a
non-degenerated interval. By construction, {ti}∞i=1 is an ordering of the atoms of the measure µ.
Set ˜f(t) := f(η(t)) and ˜g(t) to
G(η(t))¯ ·v(η(t)) ∀t ∈[0,∞)\ ∪∞i=1{ti} 1/(η(t)−η(t−))R
¯
η(t)G(s)¯ ·v(s)ds ∀t∈ ∪∞i=1{ti}.
Note that x(θ(s)) = y(s) in [0,∞)\ ∪i{θ−1(ti)} and ˙θ(s) = 0 in ∪i{θ−1(ti)}. Observe that
0=
Z
[0,∞)
χF(y(s))( ˜f(θ(s))) ˙θ(s)ds=
Z
[0,∞)
χF(x(t))( ˜f(t))dt
(the second equality follows from Lemma 1). This implies that ˜f(t)∈F(x(t)) a.e., The fact that ˜g(t)∈G(t, x(t˜ −);µ({t})), ∀t ∈ ∪i{ti} follows from the definition of ˜g.
To see that ¯G(η(t))∈G(x(t))µ-a.e. in [0,∞), notice that 0 =
Z
[0,∞)\∪i{θ−1(ti)}
χG(x(θ(s)))( ¯G(θ(s)))·v(s)ds
= Z
[0,∞)\∪i{ti}
χG(x(t))( ¯G(η(t)))µ(dt).
In the above another application of Lemma 1 involved.
Because of the definition of ˜g on [0,∞)\ ∪i{ti}, it follows from the construction of µ that µ(dt) = v(η(t))¯µ(dt), i.e., w(t) = v(η(t)) is the Radon-Nicodym derivative of µ with respect to the total variation measure ¯µ. Thus, we have ˜g(t)∈G(t, x(t˜ −);µ({t})),
¯
µ-a.e. in [0,∞)\ ∪i{ti}. Therefore, ˜g(t)∈G(t, x(t˜ −);µ({t})), ¯µ-a.e. in [0,∞).
Fix some t >0. Then, by applying Lemma 1 again, we obtain x(t)−x0 = y(η(t))−x0
= Z η(t)
0
f(s)v0(s)ds+ Z η(t)
0
G(s)v¯ (s)ds
= Z η(t)
0
f(θ(s))v˜ 0(s)ds+ Z
[0,η(t)]\∪iθ−1(ti)
G(s)v(s)ds¯ + Z
[0,η(t)]∩(∪iθ−1(ti))
G(s)v(s)ds¯
= Z t
0
f˜(τ)dτ + Z
[0,t]
˜
g(τ)¯µ(dτ).
Sincet is arbitrary, this shows thatxa solution of (1) corresponding to measure control µand initial valued x0, i.e. (x, µ) is a an admissible process of (1) such that x(0) =x0. It remains to show that x(t)↑ x¯ as t ↑ ∞. Observe that if t → ∞ then η(t) → ∞. Since x(t) =y(η(t)) andy(s)→x¯ as s↑ ∞, the desired conclusion follows.
4 Examples
In this section we provide a few examples to show how our result is applied and to reveal some issues raised by adding singular dynamics to the conventional control problem.
Before pursuing, we would like to notice that, although the stated results apply to the more general context of differential inclusions, our examples are given only for single- valued dynamics in order to not obscure their main purpose.
4.1 Linear examples
Now, let us address three simple and related examples. Consider x := col(x1, x2), the dynamics in (1) given by
dx(t) = Ax(t)dt+G(x(t))dµ(t) x(0) = x0
(F and Gare single valued multi functions), and the candidate Lyapunouv function V(x) = 1
2(x21+x22).
Let A :=
0 1 1 0
and both µ1 and µ2 take values in <+. Consider the following examples of singular dynamics:
a) G(x) =
0 −1
−1 0
x and dµ= dµ1,
b) G(x)dµ=B1xdµ1+B2xdµ2 with B1 as in a) and B2 =
0 0 0 −1
,
c) G(x) =
0 1
−1 0
x and dµ= dµ1.
In the analysis of the above example, a critical role is played by dV(x(t)) :=
∇V(x(t))·dx(t).
In casea)we have dV(x(t)) = 2x1(t)x2(t)(dt−dµ1(t)). Notice that the path joining the endpoints of feasible jumps coincide with segments of absolutely continuous trajec- tories. Therefore, unless x0 is on the eigendirection associated with the eigenvalue −1 for either the matrices in the absolutely continuous and singular dynamics, the only possibility of the trajectory to converge to a point arbitrarily close to the origin is to start in a point arbitrarily close to the above mention eigendirection. In fact, conditions of Theorem 1 are not met since, in the set C := {x ∈ <2 : x 6= 0, x1 = 0 or x2 = 0}, Lyapounov inequality would hold only withW = 0 for some x6= 0.
The lack of controllability in example a) is overcome in case b) for which we have dV(x(t)) = 2x1(t)x2(t)(dt−dµ1(t))−x2(t)2dµ2. Takex0 to be in the first quadrant with
x0,2 > x0,1. Then ∇V(x0)Ax0 > 0 and we necessarily have ∇V(x0)Bix0 < 0, for both i = 1,2. Notice that, if we jump with µ2, then x0,2 will approach asymptotically to 0 withx0,1constant and the other vector fields will not allow any further approximation to the origin. Therefore, we should jump with µ1. Let dµ1 =αδ0(t) withα = ln(xx00,2−x0,1
,1+x0,2).
Since 2˜x1(α) =−eα[x0,2−x0,1] +e−α[x0,2+x0,1] and 2˜x2(α) =eα[x0,2−x0,1] +e−α[x0,2+ x0,1], we have x(0+) = ˜x(α) =col(−x0,1, x0,2).
Another value of α could have been chosen. In particular, it could have been such that x1(0+) = 0. Then, we could choose to jump with dµ2 in order to approach the origin. However, the problem with this strategy is thatx2 evolves exponentially and an arbitrarily large impulse would be required to approximate the state arbitrarily close to the origin.
Now, we have ∇V(x(0+))Ax(0+)<0. If we proceed with the absolutely continuous dynamics, the state variable swings back to the first quadrant. Therefore, it remains to jump with dµ2. In order to drive the state to the pointx(0+) in the eigendirection of the absolutely continuous dynamics associated with the eigenvalue −1, dµ2 = βδ0(t) with β = ln(−˜x˜x21(α)(α)) = ln(xx21(0)(0)). As a consequence x(0+) = ˜x(α+β) =col(−x0,1, x0,1) and, from here, the trajectory approaches the origin without any additional control effort.
Finally, let us consider the last situation. Here, dV(x(t)) = 2x1(t)x2(t)dt and, clearly, Theorem 1 is not satisfied for points of the phase plane such that x1 ·x2 > 0.
Let us consider such an initial point, x0 and let m0 = kx0k and θ0 = arctan(xx00,2
,1).
Then, if we pick dµ1(t) = αδ0(t), we obtain ˜x(α) = 2m0col(cos(α−θ0),sin(θ0 −α)).
By choosing α = π4 +θ0, we get x(0+) = √
2m0col(1,−1). As in the previous case, the trajectory approaches asymptotically the origin with no control effort. Clearly, the derived conditions are merely sufficient.
4.2 A nonlinear example
Let us now consider the following system with nonlinear dynamics.
Consider x:= col(x1, x2). Letµtake values in <+ and let the (in fact single-valued) dynamics in (1) be given by
dx = A(x)xdt+B(x)xdµ x(0) = x0
with A(x) :=
α(x) −1 1 α(x)
, B(x) :=
β(x) −1 1 β(x)
, being α(x) = min{a21−x21−x22, x21+x22−a22} β(x) = max{x21+x22−s21, s22−x21−x22}
with s1 > a1 > a2 > s2. An observation relevant to the discussion that follows consists in the fact that the real component of the eigenvalues of matrices A(x) and B(x) are, respectively, α(x) and β(x). We consider also the candidate Lyapounov function
V(x) = 1
2(x21+x22).
It is straightforward to conclude that
dV(x(t)) = 2α(x(t))V(x(t))dt+ 2β(x(t))V(x(t))µ(dt).
Let us consider the following regions
• R1 :={x∈ <2 :x21+x22 ≥s21},
• R2 :={x∈ <2 :s21 > x21+x22 > a21},
• R3 :={x∈ <2 :a22 > x21+x22 > s22}, and
• R4 :={x∈ <2 :s22 > x21+x22}.
and analyze the behaviour of the system in the various regions.
Let us assume that x ∈ R1, then α(x) = a21 −x21 −x22 < 0 and β(x) = x21+x22 − s21 > 0. Clearly, if the singular dynamics were used the state would progress towards points further away from the origin. But, by using the absolutely continuous dynamics, the trajectory will evolve towards an orbit of radius a1 since the (complex conjugate) eigenvalues of A(x) are given byα(x)±j. Once it has entered the region R2 and it is kept outside the region R4, the real part of the eigenvalues of B(x) are negative, i.e., β(x)±j, allowing this the trajectory to jump towards a point in the region R3. Notice that, if, on the one hand, the jump can not pursue intoR4 becauseβ(x) becomes positive there, on the other hand, driving the system with the absolutely continuous dynamics from an intermediate point in the region {x∈ <2 : a21 > x21 +x22 > a22} onwards would imply that the state variable would move further away from the origin to the orbit of radius a1 since α(x)>0 in this region.
Finally, in R3∪R4, α(x) remains strictly negative implying this that the trajectory approaches the origin without any additional control effort.
5 Conclusions
In this article, the notions of an equilibrium point and the global asymptotic stability of such a point were extended from standard differential inclusions to measure differential inclusions. A framework of control Lyapounov pair of functions was defined for impulsive systems. It was shown that the existence of such functions implies the asymptotic stability of the equilibrium point. The derived stability conditions are provided in the conventional differential form. The main difficulty in the proof arises in the construction of a regular measure on the unbounded interval [0,∞), which gives rise to the trajectory of the impulsive differential inclusion. A few examples were provided to illustrate the applications of the derived conditions. These reveal new challenges in bridging the gap between the necessity and the sufficiency of the stability conditions that emerge when impulsive controls are considered in the dynamics.
Acknowledgements
The research of the first author is supported by INVOTAN and Funda¸c˜ao da Ciˆencia e Tecnologia project CorDyAL and the research of the second author is jointly supported by Conselho Nacional de Desenvolvimento Cient´ıfico e Tecnol´ogico (CNPq) of Brazil and Funda¸c˜ao de Amparo a Pesquisa do Estado de S˜ao Paulo (FAPESP).
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