ABSTRACT
By using the notion of Carathéodory solution to differential equations, the present work studies the boundedness of solutions of discontinuous differential equations. For these discontinuous systems determined by discontinuous differential equations, results are obtained that guarantee sufficient conditions to boundedness of solutions in terms of nonsmooth Lyapunov functions.
Keywords:
boundedness of solutions; discontinuous differential equations; Carathéodory solutions
1 INTRODUCTION
Ordinary differential equations with the discontinuous right side, also called discontinuous differential equations, determine discontinuous systems. The boundedness of solutions to discontinuous systems determined by autonomous differential equations was studied by22 A.F. Armaki, N. Pariz & R. Asgharian. Construction of nonpathological Lyapunov functions for discontinuous systems with Carathéodory solutions. Asian J Control, 14(1) (2012), 284-287..
Motivated by the study of the boundedness of solutions for ordinary differential equations in terms of smooth Lyapunov functions (see for instance99 R.K. Miller & A.N. Michel. “Ordinary differential equations”. Academic Press, New York (1982). and1515 T. Yoshizawa. “Stability theory by Liapunov’s second method”. The Mathematical Society of Japan, Tokyo (1966).), this report studies the boundedness of solutions of discontinuous systems determined by nonautonomous differential equations.
The study carried out here makes use of the notion of Carathéodory solution for discontinuous differential equations. In addition, nonpathological functions are used as Lyapunov functions.
Qualitative results via nonsmooth Lyapunov functions for solutions to discontinuous differential equations can be found, for example, in33 A. Bacciotti & F. Ceragioli. Nonpathological Lyapunov functions and discontinuous Carathéodory systems. Automatica, 42(3) (2006), 453-458.), (77 M. Grzanek, A. Michalak & A. Rogowski. A nonsmooth Lyapunov function and stability for ODE’s of Carathéodory type. Nonlinear Anal, 69(1) (2008), 337-342.), (1212 I.L.D. Santos. Lyapunov stability for discontinuous systems. Ciência e Natura, 42 (2020), 1-08.), (1313 D. Shevitz & B. Paden. Lyapunov stability theory of nonsmooth systems. IEEE Trans Automat Control, 39(9) (1994), 1910-1914.. By using the notion of Filippov solution,1313 D. Shevitz & B. Paden. Lyapunov stability theory of nonsmooth systems. IEEE Trans Automat Control, 39(9) (1994), 1910-1914. studies Lyapunov stability of equilibria of discontinuous systems. On the other hand,33 A. Bacciotti & F. Ceragioli. Nonpathological Lyapunov functions and discontinuous Carathéodory systems. Automatica, 42(3) (2006), 453-458.), (1212 I.L.D. Santos. Lyapunov stability for discontinuous systems. Ciência e Natura, 42 (2020), 1-08. and77 M. Grzanek, A. Michalak & A. Rogowski. A nonsmooth Lyapunov function and stability for ODE’s of Carathéodory type. Nonlinear Anal, 69(1) (2008), 337-342. address Lyapunov stability of equilibria of discontinuous systems by using the notion of Carathéodory solution.
The discontinuous systems studied in this work are determined by
and by
where f: ℝ × ℝn → ℝn and g: ℝn → ℝn . For the study of the boundedness of solutions of the system (1.1) and (1.2), it is assumed throughout the work that for every x 0 ∈ ℝn , each of the differential equations given in (1.1) and (1.2) admit at least one Carathéodory solution with the initial condition x(t 0) = x 0. It is also assumed that all Carathéodory solutions to (1.1) and (1.2) satisfying x(t 0) = x 0, are defined on the interval [t 0, + ∞). The existence of Carathéodory solutions for (1.1) has been treated in55 E.A. Coddington & N. Levinson. “Theory of ordinary differential equations”. McGraw-Hill Book Company, Inc., New York-Toronto-London (1955). and88 J.K. Hale. “Ordinary differential equations”. Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., second ed. (1980).. On the other hand, a study on the continuation of solutions can be found in1414 P.G. Stechlinski & P.I. Barton. Dependence of solutions of nonsmooth differential-algebraic equations on parameters. J Differential Equations, 262(3) (2017), 2254-2285..
Thus, in the present study results are established (Theorems 3.1, 3.2, 3.3 and 3.4) that provide sufficient conditions for the boundedness of solutions of (1.1) and (1.2) in terms of nonsmooth Lyapunov functions. Theorems 3.1 and 3.3 establish results for the boundedness of solutions of (1.1), while Theorems 3.2 and 3.4 establish results for the boundedness of solutions of (1.2).
2 PRELIMINARIES
In this section are considered basic concepts and results that will be used throughout the work.
2.1 Carathéodory solution
Solutions to the differential equation (1.1) will be understood as absolutely continuous functions. A approach to absolutely continuous functions can be found in1010 I.P. Natanson. “Theory of functions of a real variable”. Frederick Ungar Publishing Co., New York (1955).. Below, the concept of absolutely continuous function on an interval [a, b] is defined.
Definition 2.1. A function x : [a,b] → ℝ n is said to be absolutely continuous if for any ε > 0, there exists δ > 0 such that, for any countable collection of disjoint subintervals [a k , b k ] of [a, b] obeying
implies that
Let I ⊂ ℝ be an interval. It is said that a statement P holds almost everywhere (a.e.) on I, if the set N given by
has Lebesgue measure zero. A more complete approach to Lebesgue measure can be found in1111 H.L. Royden. “Real analysis”. The Macmillan Co., New York (1963)..
Definition 2.2.Let I ⊂ ℝ be an interval. A function x : I → ℝnis a Carathéodory solution of(1.1)on I if x(t) is absolutely continuous andfor a.e. t ∈ I.
Consider, for example, the following discontinuous differential equation
where sgn(·) is the sign function, that is,
As discussed in11 K.A. Alshammari. “Filippov’s operator and discontinuous differential equations”. Ph.D. thesis, Louisiana State University and Agricultural and Mechanical College, Baton Rouge, LA (2006)., the Carathéodory solution of (2.1) that satisfies the condition x(0) = c is given by x(t) = |t| + c.
2.2 Nonpathological functions
The following is an approach on nonpathological functions. The considerations made here on nonpathological functions are based on33 A. Bacciotti & F. Ceragioli. Nonpathological Lyapunov functions and discontinuous Carathéodory systems. Automatica, 42(3) (2006), 453-458..
In the following, ∂V (x) denotes Clarke’s generalized gradient for the function V (x) at x. See44 F.H. Clarke. “Optimization and nonsmooth analysis”. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, Inc., New York (1983). for a better approach to generalized gradients.
Definition 2.3. Let V : ℝ n → ℝ be a locally Lipschitz function. The generalized gradient of V at x is defined by
wheredenotes the closure of the convex hull, and ΩVis the set of measure zero (in the sense of Lebesgue measure) such that the gradient of V is not defined.
From the previous definition, if V: ℝ × ℝn → ℝ is given by V(t, x) = V 1(x), where V 1: ℝn → ℝ, then ∂V(t, x) = (0, ∂V 1(x)).
Definition 2.4.It is said that a function V : ℝn→ ℝ is nonpathological if it is locally Lipschitz continuous and for every absolutely continuous function φ : I ⊂ ℝ → ℝnand for a.e. t ∈ I, the set ∂V(φ(t)) is a subset of an affine subspace orthogonal to.
The proposition stated below can be founded in33 A. Bacciotti & F. Ceragioli. Nonpathological Lyapunov functions and discontinuous Carathéodory systems. Automatica, 42(3) (2006), 453-458..
Proposition 2.1.If V : ℝn→ ℝ is nonpathological, and φ : ℝ × ℝn→ ℝ is absolutely continuous, then the set {p ‧ φ(t) : p ∈ ∂V(φ(t))} is reduced to the singletonfor a.e. t.
Nonpathological derivative is defined below.
Definition 2.5. Let V : ℝ × ℝ n → ℝ be a nonpathological function and let A V be the set given by
If (t, x) ∈ A V , the nonpathological derivative of the map V with respect to (1.1) at (t, x) is the number
where p is any vector in ∂V (t, x).
From Proposition 2.1 follows the following corollary.
Corollary 2.0.Consider the nonpathological function V : ℝ × ℝn→ ℝ, and let φ be any solution of(1.1). Then (t, φ(t)) ∈ AVandfor a.e. t.
In33 A. Bacciotti & F. Ceragioli. Nonpathological Lyapunov functions and discontinuous Carathéodory systems. Automatica, 42(3) (2006), 453-458., nonpathological functions V: ℝn → ℝ are used to study the autonomous system (1.2). In this case, the set A V is defined as
If x ∈ A V , the nonpathological derivative of the map V with respect to (1.2) at x is defined by
where p is any vector in ∂V (x).
The next corollary is stated in33 A. Bacciotti & F. Ceragioli. Nonpathological Lyapunov functions and discontinuous Carathéodory systems. Automatica, 42(3) (2006), 453-458..
Corollary 2.0.Let the function V : ℝn→ ℝ be nonpathological, and let φ be any solution of system(1.2). Then φ(t) ∈ AVandfor a.e. t.
3 BOUNDEDNESS OF SOLUTIONS
In what follows are obtained the main results of the work, the Theorems 3.1, 3.2, 3.3 and 3.4. Below are defined the concepts of boundedness that are used in the present work.
Suppose that the set of Carathéodory solutions of (1.1) with initial condition x(t 0) = x 0 is denoted by S(x 0, t 0). Also suppose that for any initial conditions (x 0, t 0) the set S(x 0, t 0) is nonempty and all solutions are defined on the interval [t 0, ∞). If x(t) ∈ S(x 0, t 0), the notation x(t, t 0, x 0) will also be used to denote x(t).
Definition 3.6.A solution x(t) ∈ S(x0, t0) is bounded if there exists a β > 0 such that |x(t)| < β for t ≥ t0, where β may depend on each solution.
Definition 3.7.The solutions of(1.1)are uniformly bounded if for any α > 0 and t0≥ 0, there is a β = β(α) > 0 (independent of t0) such that |x0| < α implies |x(t)| < β for t ≥ t0and x(t) ∈ S(x0, t0).
Definition 3.8.The solutions of(1.1)are uniformly ultimately bounded (with bound L) if there exists a L > 0 and if for any α > 0 and any t0≥ 0, there is a T = T(α) > 0 (independent of t0) such that |x0| < α implies that |x(t)| < L for t ≥ t0+ T and x(t) ∈ S(x0, t0).
Similarly, the boundedness of solutions for (1.2) is defined.
It can be seen that the solutions of (2.1) are not bounded, or even uniformly bounded.
Theorems 3.1 and 3.3 and their proofs are similar to their smooth counterparts. The smooth version of Theorem 3.1 can be founded in [99 R.K. Miller & A.N. Michel. “Ordinary differential equations”. Academic Press, New York (1982)., Theorem 9.13] and [1515 T. Yoshizawa. “Stability theory by Liapunov’s second method”. The Mathematical Society of Japan, Tokyo (1966)., Theorem 10.2]. On the other hand, the smooth version of Theorem 3.3 can be founded in [99 R.K. Miller & A.N. Michel. “Ordinary differential equations”. Academic Press, New York (1982)., Theorem 9.14] and [1515 T. Yoshizawa. “Stability theory by Liapunov’s second method”. The Mathematical Society of Japan, Tokyo (1966)., Theorem 10.4]. For completeness and ease of reading, the proofs of Theorems 3.1 and 3.3 will be played.
Before stating Theorems 3.1, 3.2, 3.3 and 3.4, the definitions of the functions of class and class are considered.
Definition 3.9.It is said that a continuous function ψ : [0, r1] → [0, ∞) (respectively, ψ : [0, ∞) → [0, ∞) belongs to class, that is, , if ψ(0) = 0 and if ψ is strictly increasing on [0, r1] (respectively, on [0, ∞)). It is said that a functiondefinided on [0, ∞) belongs to classif limr → ∞ψ(r) = +∞.
Theorem 3.1. Let V : ℝ × ℝ n → ℝ be a nonpathological function and let R > 0. Suppose that there exist functions satisfying
for all |x| ≥ R and all t ≥ 0. Iffor all (t, x) ∈ AVwith |x| ≥ R and t ≥ 0, then the solutions of(1.1)are uniformly bounded.
Proof. Take k > R and consider with |x 0| > R, where is the open ball of radius k centered at the origin. Now, let Φ0(t) ∈ S(x 0, t 0) be denoted by Φ (t, t 0, x 0) and define v(t) by v(t) = V (t, Φ0(t)) for as long as |Φ0(t)| > R. Since v(t) is an absolutely continuous function,
From Corollary 2.0, for a.e. t, and using the hypothesis , it can be concluded that v(t) ≤ v(t 0). Hence,
Since , its inverse exists and for as long as |Φ0(t)| > R.
If |Φ0(t)| starts at a value smaller than R or if it reaches a value less than R for some t > t 0, then Φ0(t) can remain in for all subsequent t or else it may leave on a interval t 1 < t < t 2 ≤ +∞. On the interval I = (t 1, t 2), the aforementioned argument yields |Φ0(t)| ≤ β on I. Therefore |Φ0(t)| ≤ max{R, β} for all t ≥ t 0.
It is also possible to state a result analogous to the Theorem 3.1 for systems determined by autonomous differential equations.
Theorem 3.2. Let V : ℝ n → ℝ be a nonpathological function and let R > 0. Suppose that there exist functions obeying
for all |x| ≥ R. Iffor all x ∈ AVwith |x| ≥ R, then the solutions of(1.2)are uniformly bounded.
Theorem 3.3. Let V : ℝ n → ℝ be a nonpathological function and let R > 0. Suppose that there exist functions such that
for all |x| ≥ R and all t ≥ 0. If in addition, there exists a functionsatisfyingfor all (t, x) ∈ AVwith |x| ≥ R and t ≥ 0, then the solutions of(1.1)are uniformly ultimately bounded.
Proof. Consider k 1 > R and let B > k 1 be such that ψ2(k 1) < ψ1(B). This is possible because . Take k 2 > B and let T = [ψ2(k 2)/ψ3(k 1)] + 1. With B < |x 0| ≤ k 2 and t 0 ≥ 0, let Φ0(t) = Φ (t, t 0, x 0) ∈ S(x 0, t 0) and v(t) = V (t, Φ0(t)). Since v(t) is a absolutely continuous function,
It follows from Corollary 2.0 that for a.e. t, and then
The function |Φ0(t)| must satisfy |Φ0(t ∗)| ≤ k 1 for some t ∗ ∈ (t 0, t 0 + T), otherwise, suppose that |Φ0(t)| > k 1 for all t ∈ (t 0, t 0 + T). It follows from hypothesis that
Hence, if t = T + t 0,
So, t ∗ must exist.
Now, suppose that |Φ0(t ∗)| = k 1 and |Φ0(t)| > k 1 for t ∈ (t ∗, t 1), where t 1 ≤ +∞. Since v(t) is nonincreasing in t,
for all t ≥ t ∗. Thus, |Φ0(t)| < B for all t ≥ t ∗.
Below is a result similar to the Theorem 3.3 for systems determined by autonomous differential equations.
Theorem 3.4. Let V : ℝ n → ℝ be a nonpathological function and let R > 0. Suppose that there exist functions such that
for all |x| ≥ R. If in addition, there exists a functionobeyingfor all x ∈ AVwith |x| ≥ R, then the solutions of(1.2)are uniformly ultimately bounded.
As an example for the use of Theorem 3.2, consider the Artstein’s circles example
where u ∈ ℝ. In the same way that [33 A. Bacciotti & F. Ceragioli. Nonpathological Lyapunov functions and discontinuous Carathéodory systems. Automatica, 42(3) (2006), 453-458., Example 1], let
Let V: ℝn → ℝ be the nonpathological function . Take (x, y) ∈ A V arbitrary and consider where g(x, y) = ((x 2 - y 2)u, 2xyu). If x ≠ 0, then ∂V (x, y) = ∇V (x, y). Thence , since
whenever x ≠ 0. Otherwise, suppose that x = 0. If y > 0,
and if y < 0,
In its turn, if y = 0 then
Anyways, if x = 0 it follows that g(x, y) = (−y 2 u, 0) and so
Hence for all (x, y) ∈ A V . From Theorem 3.2 the solutions of (3.1) are uniformly bounded.
Now, for the use of Theorem 3.1, it will be considered a system of differential equations treated by77 M. Grzanek, A. Michalak & A. Rogowski. A nonsmooth Lyapunov function and stability for ODE’s of Carathéodory type. Nonlinear Anal, 69(1) (2008), 337-342.. In this way, consider the two dimensional system
where f 1, f 2: [0, ∞) × ℝ2 → ℝ, f 2(t, x, y) ≥ sgn(y) and f 1(t, x, y) ≥ sgn(x). For the application of Theorem 3.1, consider the function V: ℝ × ℝ2 → ℝ given by V (t, x, y) = V 1(x, y), where V 1(x, y) = |x| + |y| is nonpathological. Take (t, x, y) ∈ AV arbitrary. Hence,
where f (t, x, y) = (−xf 2(t, x, y) + |y|, −yf 1(t, x, y) + |x|). If x → 0 and y ≠ 0, it follows that ∂V (t, x, y) = (0, ∇V 1(x, y)) = (0, sgn(x), sgn(y)). Consequently , since
Now, suppose that y = 0. If x > 0,
and thus
seeing that f 2(t, x, 0) ≥ sgn(0) = 0. If x < 0,
and thus
On the other hand, suppose that x = 0. If y > 0,
and then
because f 1(t, 0, y) ≥ sgn(0) = 0. If y < 0,
and then
Lastly, if (x, y) = (0, 0),
and then . Whence for all (t, x, y) ∈ A V , and from Theorem 3.1 the solutions of (3.2) are uniformly bounded.
Next, a fashion equation arising from the Physics of Fluids in the past 30 years is considered. More specifically, consider Eq. (6.2) of paper66 A. Degasperis, D.D. Holm & A.N.W. Hone. A new integrable equation with peakon solutions. Theor Math Phys, 133(2) (2002), 1463-1474. with N = 2,
where j ∈ {1, 2}. The Eq. (6.2) of66 A. Degasperis, D.D. Holm & A.N.W. Hone. A new integrable equation with peakon solutions. Theor Math Phys, 133(2) (2002), 1463-1474. is a discontinuous differential equation that is widely studied. Note that Theorems 3.2 and 3.4 do not apply to (3.3), since p 1(t) ≡ 0, p 2(t) ≡ 1, q 1(t) = t and q 2(t) = t provide a solution not bounded to (3.3).
4 CONCLUSIONS
The work contributes to the qualitative theory of discontinuous systems. More specifically, the work studies the boundedness of solutions to discontinuous systems determined by discontinuous differential equations. The boundedness results established here refer to the concepts of solutions uniformly bounded and solutions uniformly ultimately bounded. The main results of the work are stated in the Theorems 3.1, 3.2, 3.3 and 3.4. Theorems 3.1 and 3.3 establish boundedness results to systems determined by nonautonomous differential equations, while Theorems 3.2 and establish boundedness results to systems determined by autonomous differential equations.
REFERENCES
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1K.A. Alshammari. “Filippov’s operator and discontinuous differential equations”. Ph.D. thesis, Louisiana State University and Agricultural and Mechanical College, Baton Rouge, LA (2006).
-
2A.F. Armaki, N. Pariz & R. Asgharian. Construction of nonpathological Lyapunov functions for discontinuous systems with Carathéodory solutions. Asian J Control, 14(1) (2012), 284-287.
-
3A. Bacciotti & F. Ceragioli. Nonpathological Lyapunov functions and discontinuous Carathéodory systems. Automatica, 42(3) (2006), 453-458.
-
4F.H. Clarke. “Optimization and nonsmooth analysis”. Canadian Mathematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, Inc., New York (1983).
-
5E.A. Coddington & N. Levinson. “Theory of ordinary differential equations”. McGraw-Hill Book Company, Inc., New York-Toronto-London (1955).
-
6A. Degasperis, D.D. Holm & A.N.W. Hone. A new integrable equation with peakon solutions. Theor Math Phys, 133(2) (2002), 1463-1474.
-
7M. Grzanek, A. Michalak & A. Rogowski. A nonsmooth Lyapunov function and stability for ODE’s of Carathéodory type. Nonlinear Anal, 69(1) (2008), 337-342.
-
8J.K. Hale. “Ordinary differential equations”. Robert E. Krieger Publishing Co., Inc., Huntington, N.Y., second ed. (1980).
-
9R.K. Miller & A.N. Michel. “Ordinary differential equations”. Academic Press, New York (1982).
-
10I.P. Natanson. “Theory of functions of a real variable”. Frederick Ungar Publishing Co., New York (1955).
-
11H.L. Royden. “Real analysis”. The Macmillan Co., New York (1963).
-
12I.L.D. Santos. Lyapunov stability for discontinuous systems. Ciência e Natura, 42 (2020), 1-08.
-
13D. Shevitz & B. Paden. Lyapunov stability theory of nonsmooth systems. IEEE Trans Automat Control, 39(9) (1994), 1910-1914.
-
14P.G. Stechlinski & P.I. Barton. Dependence of solutions of nonsmooth differential-algebraic equations on parameters. J Differential Equations, 262(3) (2017), 2254-2285.
-
15T. Yoshizawa. “Stability theory by Liapunov’s second method”. The Mathematical Society of Japan, Tokyo (1966).
Publication Dates
-
Publication in this collection
06 Sept 2021 -
Date of issue
2021
History
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Received
09 July 2020 -
Accepted
15 Feb 2021