Global finite-time stability of differential equation with discontinuous right-hand side

In the paper new sufficient conditions for global finite-time stability of a stationary solution to differential equation with discontinuous right-hand side are given. Time-dependent Lyapunov function which is only continuous is used. Properties of Lyapunov function are described by presubdifferential and contingent derivative.

The aim of this paper is to present sufficient conditions for global finite-time stability of the origin for the differential equation where f : [0, ∞) × R n → R n is a Carathéodory function.
Global finite-time stability was considered in e.g.[2,7,8].In our paper we use only continuous time-varying Lyapunov function.Therefore, in the opposite to the articles cited above, instead of differentiating (even in Dini sense) this function we use presubdifferential and contingent derivative, first considered for this purpose in our previous article [13].
Moreover, we weaken the condition which the Lyapunov function must satisfy in relation to conditions given in [2], [7] and [8].More precisely, we admit in the crucial inequality (3.1) the presence only of a measurable function which can take zero value even on sets of positive measure.In [2] the authors use only a positive constant.In [7] the function must be greater than a positive constant and in [8] this function is a.e.positive.
In this paper we prove two global finite-time stability theorems.In the first one we do not need to assume uniqueness of solutions to the differential equation.In this theorem some crucial condition which should be satisfied by Lyapunov function must occur in the whole space.In the second one we have to assume the uniqueness of solutions to the differential equation but in that case the condition which should satisfy Lyapunov function can occur only in an arbitrary small neighborhood of the origin.

Preliminaries
Definitions, assumptions, lemmas, propositions and theorems presented in this section come from [13], where complete necessary proofs are given.
The following assumption holds throughout this section.Assumption 2.1.
1. G ⊆ R n is an open set containing zero.
2. The function t → f (t, x) is measurable in [0, ∞) for all x ∈ G.
Let us see that V(t, x) is upper right contingent derivative of V in (t, x) towards (1, f (t, x)).Now we give definition of the presubdifferential from [14].
Definition 2.2.Let W : R j → R, j ∈ N. The presubdifferential of function W in point v we name the following set Definition 2.3.By K ∞ 0 we name the class of continuous and increasing functions Lyapunov function is an important tool which allows investigating stability as well as global finite-time stability of the solution to the differential equation.In the literature there are commonly known conditions for smooth Lyapunov function (see e.g.[3,8,9]).In this paper Lyapunov function is only continuous, therefore like in [13] we give other conditions which can be easily checked.
and Γ ⊆ [0, ∞) be a set of measure zero such that In addition, there exists at most countable set C ⊆ [0, ∞) such that for all t ∈ (0, ∞) \ C and x ∈ G \ {0} there exists ε tx ∈ (0, t) and P tx > 0 such that for s ∈ (t Let V be a function defined in Assumption 2.4.By ν ϕ we denote a function The following Lemma was proved in [13, Lemma 2.5].
An obvious consequence of Lemma 2.5 is the following lemma.
The above propositions allow to prove the Lyapunov stability theorem.Theorem 2.12.If differential equation (1.1) has continuous function V satisfying (2.1)-(2.4),then the origin for the differential equation is stable.
Definition 2.15.We tell that the origin is finite-time stable for the differential equation (1.1) if it is stable and for any t 0 ≥ 0 there exists δ = δ(t 0 ) > 0 such that for x 0 ∈ G satisfying x 0 < δ, the values of T(t 0 , x 0 ) are finite.
Definition 2.16.We denote by P a class of nonnegative functions c : [0, ∞) → [0, ∞), which are measurable and upperbounded on each compact subinterval [0, ∞) such that there exists t 0 ≥ 0 for which ∞ t 0 c(τ)dτ = ∞.Let us consider a simple example of a differential equation for which the origin is a finitetime stable equilibrium.In the proof of the global finite-time stability theorem properties of solutions to this differential equation are used.
Let us take any function c ∈ P, t ∈ [0, ∞), z ∈ R, α ∈ (0, 1) and consider Cauchy problem ) Remark 2.17.It is easy to see that for any c ∈ P and t ≥ 0 the function Hence for any z ∈ R and α ∈ (0, 1) there exists t ≥ t such that It is easy to check that the solutions to the Cauchy problem (2.7)-(2.8)are functions (2.10) An important tool being used in the proof of finite-time stability theorem of the solution to the differential equation (1.1) is the Comparison Lemma from [15].The essence of this lemma is assuming only measurability with respect to time and absence of any assumption about monotonicity of the right-hand side of the differential equation and using only Dini derivatives.Therefore it is enough that the inequality (2.3) holds only almost everywhere.
The proof of this lemma will be given in [15] but we include it here for the benefit of readers.
Lemma 2.18 (Comparison Lemma, [15]).Let E ⊆ R be an open interval, σ : [0, ∞) × E → R a function measurable with respect to t for each x ∈ E and continuous with respect to x for all t ∈ [0, ∞).
with boundary condition u(t 0 ) = u 0 (2.12) as well as: 1. for each t 1 ∈ (0, T) and x ∈ E there exists a neighbourhood V t 1 ,x ⊆ E of the point x and a constant L t 1 ,x such that, for all (t, y) ∈ [0, and there exists at most countable set C ⊆ [t 0 , T) such that then ν(t) ≤ u(t) for all t ∈ [t 0 , T).
Proof.Choose any t 1 ∈ (t 0 , T).Since the interval [t 0 , the following properties: • for all x ∈ E k 1 and µ ∈ (−1, 1) the function f is Lebesgue measurable with respect to t; • from the assumption 1, the function f is continuous with respect to x for all (t, µ) ∈ [0, ∞) × (−1, 1) and u is the unique solution to the equation (2.11); • for all t ∈ [0, ∞), the function f is continuous with respect to (x, µ); • from the assumption 2, the constant Therefore, using [4, Thm.4.2, p. 59], for any ε > 0 there exists δ > 0 such that, for |µ| < δ each right-maximal solution u µ of the equation x = σ(t, x) + µ in the set E 0 with boundary condition (2.12) can be defined at least in the interval [t 0 , t 1 ].Moreover, for t ∈ [t 0 , t 1 ] the following inequality takes place To prove the thesis of the theorem we first prove that ν(t) ≤ u µ (t) for all µ ∈ (0, δ) and t ∈ [t 0 , t 1 ].Indeed, in the opposite case, it would exist a point β ∈ (t 0 , By assumption 1 of the theorem, there exists L β > 0 such that, for all t ∈ [t 0 , β] and x, y Continuity of ν and u µ follows, that the set W is open, and hence, the set Y = [t 0 , β] \ W is nonempty (because at least t 0 ∈ Y) and compact.Therefore, there exists some α ∈ [t 0 , β] being the maximum of Y. Since ν(β) > u µ (β), therefore β / ∈ Y and consequently, α < β.Continuity of ν i u µ and the Darboux condition follows that α satisfies Since ν also satisfies (2.14), we easily conclude from [11,Thm. 7.4.14],that the function which is in the contradiction with (2.18).This proves, that Therefore, the inequality ν(t) ≤ u(t) holds for all t ∈ [t 0 , t 1 ].From arbitrariness of t 1 < T it follows, that ν(t) ≤ u(t) for all t ∈ [t 0 , T), which ends the proof of the lemma.

Main results
We shall prove the global finite-time stability theorems basing on the following definition.The following assumption, similar to Assumption 2.1 holds throughout this section.Assumption 3.2.
In this paper, condition (3.1) which is satisfied by Lyapunov function, is weaken then the conditions given in [2], [7] and [8].In (3.1) we use only measurable functions c(t) which can take zero on sets of positive measure.In [2] instead of function c(t) only a positive constant can be used.In [7] the function h(t) which plays the role of function c(t) in this paper must be greater than a positive constant and in [8] this function must be a.e.positive.
With the above assumptions the origin for the differential equation (3.4) is globally finite-time stable.More precisely, for any initial conditions (t, x) ∈ [0, ∞) × (R n \ {0}) the settling-time function can be estimated by the formula Proof.It is easy to see that the function f : [0, ∞) × R n → R n defined in (3.5) satisfies Assumption 3.2.Indeed, according to the assumptions on functions η and Ψ, the function and z ∈ B(0, l), l ∈ N, where m l (t) = L 1t l + L 2t l .We will show that the origin for the differential equation (3.4) is globally finite-time stable.For this purpose let us consider Lyapunov function V(t, x) = g(t) x δ , where t ∈ [0, ∞), x ∈ R n , g(t) = t 0 γ(τ)dτ + 1 and δ satisfy (3.6).By the definition of function g and assumptions of function γ we get immediately that and for all t ∈ (0, ∞) \ C there exists g (t) = γ(t).The function V(t, x) satisfies of course conditions (2.1) and (2.2).We will show that V satisfies (2.4) and (3.1) which are also required in Theorem 3.3.Indeed: • Take any t ∈ (0, ∞) \ C and x = 0, where C is at most countable set described above.
Therefore the condition (2.4) is satisfied for C and ε tx given above and Of course this set is measure zero.For any t ∈ [0, ∞) \ Γ and x = 0 we receive From above, for t ∈ [0, ∞) \ Γ and x ∈ R n \ {0} we receive that and and therefore c ∈ P.
So, from Theorem 3.3 it follows that the origin for the differential equation (3.4) is globally finite-time stable.
We can estimate the settling-time function.For any t ≥ 0 and x = 0 and from inequality (3.9), the settling-time function T satisfies In the example given below the formula estimating precisely enough of the settling-time function is given.
Example 3.5.Let us consider the following differential equation where . ., m = 0, 1, . .., where for k = 1, 2, . .., m = 0, 1, . . .and Let us see that γ ∈ P and that this function is continuous in every point of the set (0, ∞) \ C, where C = k − 1 2 m : k = 1, 2, . . ., m = 0, 1, . . . is countable and closed set.Denote and for k = 1, 2, . .., m = 0, 1, . . .Let t 1 > 0.Then, for any t ∈ [0, t 1 ] we receive |γ(t)| ≤ 1 and |g(t)| ≤ 1 3 t 1 + 1 + 1.As a consequence, for any t ∈ [0, t 1 ], z ∈ B(0, x + l) and l ∈ N we receive and (because g(t) ≥ 1) (3.13) and x ∈ R n we have γ(t) = 0 and as a consequence the following inequalities hold for t ≥ 0, where γ and g are given by the formula (3.11) and (3.12) respectively.It is easy to see that c ∈ P. As a consequence, from Theorem 3.4 the origin for the differential equation (3.10) is globally finite-time stable.The settling-time function satisfies Below we prove the second global finite-time stability theorem.In the proof of this theorem global asymptotic stability is used.Therefore we must strengthen the assumptions which Lyapunov function should satisfy.We do this in the following assumption.Proof.We know from Theorem 2.12 that the origin for the differential equation (1.1) is stable.We will show that the origin for this differential equation is globally asymptotically stable.Let (t 0 , x 0 ) be any element from [0, ∞) × (R n \ {0}) and let ϕ ∈ S t 0 ,x 0 , ϕ : [t 0 , b) → R n , b ∈ (t 0 , ∞) ∪ {∞}, be any right-maximally defined solution to the differential equation (1.1).From (2.3) we know that the function V(t, x) is upperbounded by function κ for t ∈ [0, ∞) \ Γ and x ∈ R n \ {0}.As in Theorem 3.3 we receive that b = ∞.Moreover, there exists the limit β = lim t→∞ V(t, ϕ(t)) ≥ 0. We will show that β = 0. Let us assume contrary that β > 0. In this case there exists γ > 0 such that ϕ(t) ≥ γ for t ≥ t 0 .Indeed, otherwise it would exist two possibilities.One of them is existence t and a sequence (t n k ) such that t n k → t and ϕ( t) = 0 and hence for t ≥ t the condition ϕ(t) = 0 it would be satisfy.Therefore V(t, ϕ(t)) = 0 for t ≥ t, so β = lim t→∞ V(t, ϕ(t)) = 0.The second case is existence of a sequence (t n ), t n ≥ t 0 , t n → ∞ such that ϕ(t n ) → 0 and β = lim n→∞ V(t n , ϕ(t n )) ≤ lim x→0 sup t≥t 0 V(t, x) = 0, what leads to contradiction in both cases with assumption that β > 0. Therefore, if β > 0, then there exists γ > 0 such that to (3.2) M ≥ ϕ(t) ≥ γ for all t ≥ t 0 .From the definition of the function κ there exists a constant L = max{κ(s) : s ∈ [γ, M]} < 0. Let (t j ), t j > t 0 , be any sequence such that t j → ∞.Of course ϕ(t j ) ≥ γ.Because Assumption 3.6 implies Assumption 2.4, then from Proposition 2.9, for j ∈ N, we receive 0 <  Let us see that in Theorem 3.3 the condition (3.1) is satisfied for x from the whole space R n (except the origin).Below we prove Theorem 3.9, in which the condition (3.1) can occur only in an arbitrarily small neighborhood of the origin Ω \ {0}, Ω ⊆ R n .In this case it is necessary to assume the right-uniqueness of solutions to the differential equation (1.1) and strengthening Assumption 2.4 -see condition (3.17) in Assumption 3.6.Theorem 3.9.Let us assume that for the differential equation (1.1) there exists a continuous function In addition we assume that for any initial conditions from [0, ∞) × (R n \ {0}), differential equation (1.1) has the right-unique solutions in [0, ∞) × (R n \ {0}).
Then the origin for the differential equation (1.1) is globally finite-time stable.

Definition 3 . 1 .
We call the origin global finite-time stable for the differential equation (1.1) if it is stable and the settling-time function T : [0, ∞) × R n → R + ∪ {∞} has only finite values.
for a.a.t ≥ 0 and for all x ∈ Q k .Now we prove the first global finite-time stability theorem of the solution to the differential equation (1.1).

Assumption 3 . 6 .Definition 3 . 7 .Theorem 3 . 8 .
Let us assume that all conditions from Assumption 2.4 are satisfied for G = R n and for the continuous and negative function κ.Additionally assume that lim We say that the origin for the differential equation (1.1) is globally asymptotically stable if it is stable and every solution ϕ to the differential equation (1.1) can be extended to infinity and lim t→∞ ϕ(t) = 0. Now we prove the global asymptotic stability theorem.If for the differential equation (1.1) there exists function V satisfying Assumption 3.6, then the origin for this differential equation is globally asymptotically stable.