Discrete Approximation of Impulsive Differential Inclusions

The article deals with the approximation of the solution set and the reachable sets of an impulsive differential inclusion with variable times of impulses. It is strongly connected to [11] and is its continuation. We achieve order of convergence 1 for the Euler approximation under Lipschitz assumptions on the set-valued right-hand side and on the functions describing the jump surfaces and jumps themselves. Another criterion prevents the beating phenomena and generalizes available conditions. Several test examples illustrate the conditions and the practical evaluation of the jump conditions.


PRELIMINARIES
We study discrete approximations of finite dimensional autonomous impulsive systems, having the form: Here, F : n ⇒ n is a multifunction with nonempty compact and convex values, S i : n → n yields the magnitude of the impulses (jumps) and i : n → are the impulsive surfaces. If a solution hits the impulsive surface, it has to jump (time of impulse). Sometimes we will simply write i or i (x) 654 R. Baier and T. Donchev (not i (x(t ))). We assume that t = 0 is not a time of impulse, otherwise we simply replace the initial condition x 0 by x 0 + S i (x 0 ).
Recall (compare [23, p. 34]) that the piecewise absolutely continuous function x(·) is said to be a solution of (1)- (2), if: a) x(·) is left continuous, i.e., x(t ) = x(t − 0), and satisfies (1) for almost all t ∈ I , t = i (x(t )) and (2). b) It has jumps on t = i (x(t )) (discontinuities of the first kind), defined by Given a solution x(·) of (1)-(2) we define a function, which zeros coincide with the jump times: The multiple hitting of one switching surface is called beating phenomena.
In this article, we study the approximation of the solution set and the reachable set of (1)- (2) with the help of the following approximation scheme: Let k = t j = j k | j =0,1, ,k be a uniform grid of I (if we have to cover I := [0, 1 − ] with > 0, we drop some grid points). To (1)-(2) we juxtapose the following discrete system below.
We let x k 0 = x 0 and for j = 0, 1, , k − 1 we define x k j +1 = x(t j +1 + 0) inductively as follows: x k (t ) t = i (x k (t )), ∀t ∈ (t j , t j +1 ], x k (t j +1 ) + S i (x k (t j +1 )) otherwise (5) The formula above means that the jump times are permitted to be only on the grid points, that is, if some discrete solution x k (·) must have a jump time t j < i (x(t )) = t < t j +1 , then the jump is "postponed" to the time t j +1 . For simplicity we did not explicitly express the dependence of x and the superscript k for such solutions and jump times.
On every interval (t j , t j +1 ], we check the signs of f x i (·). If f x i (t j ) = 0, then there will be no jump on (t j , t j +1 ] (at least for step-sizes small enough). If f x i (t j ) > 0 (f x i (t j ) < 0) and f x i (t j +1 ) ≤ 0 (f x i (t j +1 ) ≥ 0), then x(·) has a jump in (t j , t j +1 ] and we set x k j +1 = x k (t j +1 ) + S i (x k (t j +1 )). Here, i is the minimal index for which the change of the inequality from ">" to "≤" (or vice versa) occurs for f x i (·). We prove in case of Lipschitzian F (·) under some natural assumption that the order of convergence is (h) (with respect to a kind of distance (·, ·) from Definition 1).
Our other aim is to test this scheme in some concrete examples, where we compare (whenever it is possible) the graph of the approximate and the graph of the exact systems.
We refer to [18,24] for the theory of impulsive differential equations. Among others we note the interesting books [21] and [23] (the second one studies impulsive differential inclusions with averaging methods in the first chapter). Notice that the impulsive systems have connection with so-called hybrid systems (see [1] and [3,15]) which are out of the scope of this article. We do not study measure-driven systems, where intensive research is done and mention in this direction [5,14].
There are a lot of articles devoted to approximation of solutions and of the reachable set of systems without impulses. We mention only [12,16], survey papers [13,19] and references therein. Numerical realizations of set-valued Euler's method and careful estimates of the space discretization of reachable sets are contained in [6,7].
This article studies a similar subject to [10,11]. In the first paper, some qualitative properties play a major role, in the second one discrete approximations of impulsive differential inclusions are considered, however, there are some difficulties to approximate numerically such systems. The main goal here is the numerical justification of the approximation of those nonlinear systems. For an Euler-type approximation of impulsive systems with linear dependence on the measure (stateindependent jump times), the graph convergence is shown for nonlinear systems with measure decomposition [26] as well as convergence order 1 for linear systems with polyhedral approximations in [17].
The multifunction G : n ⇒ n with nonempty compact values is said to be upper semi continuous (USC) at x 0 , if for every > 0 there exists > 0 such that G (x 0 + ) ⊂ G (x 0 ) + , where is the closed unit ball. We call a map G : I × n ⇒ n almost USC, if for any > 0 there exists a compact I ⊂ I with meas(I \I ) < such that G (·, ·) is USC on I × n . The multifunction G : n ⇒ n is said to be Lipschitzian with a constant L, if for every x, y ∈ n one has sup b∈B inf a∈A |b − a| be the Hausdorff distance between the bounded sets A and B. For compact sets A we let (l , A) = max a∈A l , a -the support function.
We refer to [2] or [9] for all the concepts used but not explicitly given here.

Standing Hypotheses
F. F (·) is Lipschitz (with a constant L) or it is locally Lipschitz, satisfying the linear growth condition |F (x)| ≤ (1 + |x|) for some positive constant .

R. Baier and T. Donchev
for every x and every j = i. Each i (·) is Lipschitz with a constant N and Clarke's subdifferential i (·). A2. The functions S i : n → n are Lipschitz with a constant . A3. Just one of the following assumptions holds for every x ∈ n : Assumptions A3 a), resp. b) will guarantee strong monotonicity of f x i (·) such that even in A3 b) with the condition i+1 (x) < i (x) we have that the jump times i < * i+1 for a given function x(·) are ordered in the right way.

Lemma 1.
Under the standing hypotheses there exist < 1 and > 1 such that it follows for every i = 1, , p: Proof. Let G i : n ⇒ n be the minimal USC map with convex compact values, which graph contains the graph of x → i (x). From [8, Section 1.2, (4)]) we know that G i (x) = i (·). Since the support function (l , F (·)) is Lipschitz and (·, F (x)) is convex, the proof is complete.
Let (s, t ) be the interval of continuity of x(·). Applying the chain rule for Clarke's subdifferential yields Discrete Approximation of Impulsive Differential Inclusions

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The inequalities f x i (·) < f x i+1 (·) and i < i+1 follow immediately from i (x) < i+1 (x) and 1 Hence, the functions f x i (·) are strongly monotone except in the jump times for every i and hence f x i ( ) = 0 has at most one solution, which is a jump time.
Remark 1. If i (·) is differentiable, then A3 a) is a nonsmooth and setvalued generalization of the corresponding pointwise condition x i (x) · f (x) < 1 as for example, in [18,Theorem 1.3.5]. Condition A3 a) replaces the condition A3 in [10,11], that is, is a new condition and helps to deal with situations in which A3 a) is not fulfilled, cf. Example 1, case 2).
Let us provide a simple illustrative example. Example 1. Consider the following one dimensional differential inclusion:ẋ Here, the impulsive surface is the (black) dashed line x = t + 1 in Figure 1 and the right-hand side is It is easy to see that the system (7) satisfies A1 and since the impulsive surface is just one, also F and A2. But neither a) nor b) of A3 is true.
There are "good" solutions as x(t ) = 1 3 + t with no jumps. However, if we study the solution x * (·) withẋ * (t ) = 3x * (t ), we see after some standard calculations that this solution hits several times the impulsive surface (beating phenomena), see Figure 1.
, if the zeros of the function are calculated numerically.
Here, no multiple hitting of the jumping surface is possible, because if a solution intersects the impulsive surface t = (x(t )), then x(t + 0) > t + 1. Since the derivativeẋ(t ) ≥ 1, x(t ) > t + 1 is valid for all times after the only jump time. In this case b) of A3 holds.
In case 2) of Example 1 the functions f x i (·) are strongly increasing except in its jump times.
Given > 0 and motivated by USC case (see [11]) we study inner perturbations: First we prevent the beating phenomena: Lemma 3. Under A1-A3 for every solution of (1)-(2) (if it exists) and i = 1, , p there exists at most one t ∈ I with t = i (x(t )). Furthermore, the solution set is nonempty and bounded.
Proof. Let x(·) be a solution of (1)- (2). Due to Lemma 2, the functions f x i (·) are strongly monotone for every i except in the jump times and hence f x i ( ) = 0 has at most one solution, which is a jump time. The solution exists on each interval (s, t ] of continuity due to A1) and the reachable set is bounded. If the solution has a jump at due to the ith jump surface, then the new value x( + 0) = x( ) + S i (x( )) will form a new starting value for the same differential inclusion (8). Since A2) holds, we can estimate Since all values x( ) of each solution before the jump are uniformly bounded, the values after the jump are also uniformly bounded which results in the precompactness of the reachable set due to the maximal number p of the possible jumps.
The existence is already proved in [10] in a more general form.
For Lemma 3, we may allow ≤ 1 and ≥ 1 in A3 a) resp. A3 b) as long as we have a strict inequality between i (x) and i (x + S i (x)).
Using the following standard arguments one can also show the same existence result for problem (8), that is, for small > 0 there exist ). In this case we will replace by¯ and by¯ .

Remark 2.
Calculating the solution set under the scheme (4)-(5) (or some other one), the problem arises to verify, if the approximate solution say (·) = x k (·) has a jump time between two grid points t k j < t k j +1 . The criterion is very easy: the jump occurs if and only if f i changes its sign. Corollary 1. Assume that the conditions A1, A2, A3 hold, then there exists a constant > 0 such that for every solution y(·) of (8) all of the absolute values The proof for A3 a) and the inequality for the first term is given in [11], however, we include here the full proof for the reader's convenience.
Proof. Since i (·) are Lipschitz for i = 1, , p, one has that on every compact set K ⊂ n the sets i := x∈K (x, i (x)) and j ,i := x∈K (x, j (x + S i (x))) are compacts. Due to A1, A3 i ∩ j = ∅ and i ∩ j ,i = ∅ for any i = j . Consequently the first estimation follows from min i =j D H ( i , j ) = > 0, similar arguments help for the estimation of the other terms.
The above corollary shows that the jump surfaces have a minimum distance of > 0, if both surfaces are evaluated at the values of a solution y(·). The next corollary shows that the jump times for a solution also have a minimal distance to each other.

Corollary 2.
Under the conditions of Lemma 3 there exists a constant > 0 such that for every solution y(·) of (8) and every jump time Hereby One can take the number k of subintervals so big that the step-size is less than . Due to Corollary 2 it is impossible for an approximate solution (say x k (·)) to have more that one jump in [t j , t j +1 ].

APPROXIMATION OF THE SOLUTION SET
The numerical approximation of the Lipschitzian case has been studied among others in [12,13,16,19]. Euler's method in the presence of state constraints is realized in [7]. Definition 1. Let x(·) and y(·) be solutions of (8) we say that (x(·), y(·)) ≤ if they intersect successively the impulsive surfaces, i.e., i (x) ≥ j (y) is impossible for i < j and vice versa. Moreover To study the approximation of the solution set of (1)- (2) in C (I , n ) with discrete trajectories (the solution set of (4)-(5)), we will use the Filippov-Plis lemma. To prove this lemma, we first mention the following known lemma (Lemma 2 in [22]), a discrete version of Gronwall inequality.
We need a variant of Filippov-Plis lemma, which is proved in [11] for F being OSL and USC under slightly stronger assumptions on the jumps, but not for the assumption A3 b). Here, we study the Lipschitzian case.
Denote a 2 = e L . We let b := (e L − 1) and Modifying the proof of Theorem 2 of [22] and applying the Gronwall inequality for |y(t ) − x(t )| of (10) we will derive the following estimates: where a 1 will be defined later in (16), Inequality (12) holds, since there is no jump in follows due to (10). Since − i = i (x), + i = i (y) holds and we can use the C -Lipschitz continuity of y(·) on [ − i , + i ] as well as Lemma 1 a) and A3 a) to show that (13) in case of A3 a).

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In case of A3 b) one uses Lemma 1 b) to estimate Hence, we have in case of A3 a) by (15) and A2 which is just (11). In this case In case of A3 b) one has to replace 1 − by − 1.
If i (x(t )) > i (y(t )), the proof is very similar. Now Lemma 4 applies. Due to A1, A2 there exists a constant > 0 such that − i+1 − − i ≥ , thanks to Corollary 2. It follows from (13) and (14) that Therefore, there exists a (generic) constant K > 0 such that: (x(·), y(·)) ≤ K Now we study the discrete approximations. Notice that it is difficult to estimate the distance between the solution set of (1)-(2) and the piecewise linear approximations (4)-(5). So we successively approximate the solution set of (1)-(2) with other solution sets for which such an estimate is simpler.
First we prove a lemma for the approximation of solutions of (1)-(2) with the solutions oḟ where f j (·) is an integrable selection of the constant set-valued map F (x k j ). If ∃t ∈ (t j , t j +1 ) with t = i (x k (t )), then we redefine the solution x k (·) as x k (t ) = x k (t ) + S i (x k (t )) + t t f j (s)ds on (t , t j +1 ], wheref j (t ) ∈ F (x k (t ) + S i (x k (t ))).
Recall that we use the same partition k as in (4)-(5).
Proof. (Vice Versa) Let x k (·) be a solution of (17). Then, Hence, x k (·) is a solution of (8) with = C · h. Now, the result holds due to Theorem 1.
(Other Assertion) Let y(·) be a solution of (1)-(2). We define . Define the multifunction: It is easy to see that we define x k (·) on the whole subinterval. If there exists a jump l (z(t * )) = t * ∈ (t k i , t k i+1 ), then we redefine x k (t ) for t = t * + 0 by setting x k (t * + 0) = x k (t * ) + S l (x k (t * )). Now we define G i (t ) := u ∈ F (x k (t * + 0)) : |ẏ(t ) − u| ≤ L|y(t ) − x k (t * + 0)| and extend x k (·) on (t * , t k i+1 ] as x k (t ) = x k (t * + 0) + t t * gi (s)ds, whereg i (s) ∈ G i (s) is a measurable selection. As in the proof of Theorem 1 we may assume only one jump in the subinterval and that x k (·) and y(·) intersect the impulse surfaces successively. In this way, one can define x k (·) on the whole interval I .
follows as well as Hence, (12) holds true.

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Let A3 a) be valid. As in the proof of Theorem 1 we have and, hence, (13) is also true. In the case of A3 b), one can also proceed in a similar way to show that (14) of Theorem 1 is fulfilled. Due to Lemma 4 for sufficiently large k there exists a constant M > 0 (not depending on k) such that for every solution y(·) of (1)-(2) there exists an approximate solution x k (·) (obtained by (18)) with (x k (·), y(·)) ≤ M k . In case of A3 b) one obtains the same estimate with − 1 replacing 1 − .
Consider now the new discrete system having the similar form as (17): If ∃t ∈ (t j , t j +1 ) with t = i (x(t )), then we postpone the jump and proceed as in the scheme (4)-(5). As a next step, we want to estimate the distance between the solution sets of (1)- (2) and (20). Similarly as in the proof of Lemma 5, one can prove as a corollary of Theorem 1:
Proof. We only have to estimate the distance between the solution sets of (20) and (17) and afterwards to use Lemma 5. Let (·) be a solution of (17). We look for a solution x k (·) of (20), which is sufficiently close to (·). Let 1 ∈ (t j , t j +1 ) its (possible) first jump time and set x k (t ) = (t ) on [0, 1 ]. The first jump of x k (·) will be at t j +1 . Using the linear growth condition F, the Lipschitz continuity of S 1 and Lemma 3 we derive ]. Now we consider the problems (20) and (17) On every impulsive point i we will add (1 + C ) to the estimation of | (t ) − x k (t )|. Since the solutions (·) and x k (·) differ by (h), this estimation also holds for the jump times i and x k i by A1. The rest of the proof is standard and will be omitted.
The following theorem is a corollary of Theorem 2:
Proof. For the system without impulses the proof is trivial. It is known that (t − s)F (x(t i )) = t s F (x(t i ))d by the convexity of Aumann's integral. Since the strategy of postponing jumps in (20) and (5) coincides too, the result is just a reformulation of Theorem 2.
Due to Definition 1 of (·, ·), the jump times of solutions in Theorem 3 also differ as O 1 k . Let us consider the following nonlinear example: Example 2 (Kenderov's Example). Consider the following two dimensional systeṁ where we add the impulse hyperplane (x) = 1 16 |x| and the jump function S (x) = − x 2 . Let the matrixes A = be given, where k ∈ (0, 1) is a fixed number (we will take k = 1 3 ). Notice that without impulses this example is due to Kenderov and can be found in [7, Example 5.2.1]. It is easy to see that A1, A2 are true with N = 1 16 . We will show that NC < 1 which implies that A3 a) holds due to Remark 1.
Hence, consider C (u) = A + uB. It is easy to see that |C (u)| ≤ √ 34 3 . Of course F (·) is not uniformly bounded, but it is sufficient to check the assumptions on a compactum containing all solution values as we noticed earlier.
The reachable set t (t ∈ I ) for the system without impulses has the form (in polar coordinates (r , )): In our case (r 0 , 0 ) = √ 2, 4 .

FIGURE 2
Reachable set for Example 2 (full picture resp. details).
All trajectories have the same jump time * which is approximated by * N , the common jump time of all Euler iterates (cf. Table 1). The logarithmic least squares approximation by log(Ch This example does not seem to fit in our problem class, since jumps x 2 ( + 0) = − · x 2 ( ) appear at a time t , if x 2 (t ) = 0 and x 1 (t ) ≤ 0. Introducing an artificial variable x 3 (·) with x 3 (t ) = 1, x 3 (0) = 0 which measures the time, we set The following model also incorporates uncertainty (modeled by · U = 0 00335 · [−1, 1]) on the value g = 9 80665: Here, A := The jump is given by The number s N of detected jump times of the Euler iteration depends on the number of subintervals used (cf.   Single trajectory for Example 4 (coordinates resp. phase portrait) and N = 500.

FIGURE 5
Reachable set for Example 4 with N = 500 (full picture resp. details).
Example 5. Consider the following system: where  Figure 6).
It is easy to see that all our conditions hold. For the special trajectory x(·) with the control function u(t ) = 1 we get the following approximation of the switching time * by the Euler iterates (cf. Table 3). The logarithmic least squares approximation by log(Ch Different jump times occur in this example for different solutions (cf. Table 4).
674 We consider just one impulsive surface: (x) = 1 3 · x + 23 10 and S (x) = − 1 2 · x. In Figures 7 and 8, the impulsive surface are plotted as dashed dotted line, the solution points for non-jump times with solid lines and the end points y(t ) of the solutions before the jump time and after the jump, that is, y(t + 0), are depicted in gray color.

R. Baier and T. Donchev
In the details of Figures 9 and 10, the linear convergence of the Euler iteration could be clearly observed as well as the delay of the jump times to the right endpoint of the subinterval as in (5). In Figure 9, only 3 of 20 trajectories jump at * 1 = 0 9, the others jump at * 1 = 1 0. In Figure 10, all trajectories jump before t = 1 0. The violation of the jump surface by Euler iterates tends linearly to zero in Figure 10.

CONCLUSIONS
The results in Section 2 show that one could apply at least three strategies to deal with jumps in trajectories when implementing a discretization. For numerical approximations, it is crucial to justify how the algorithm should react, if jump times of the original solution of the impulsive differential inclusion would not lie on grid points used for the discretization.
The first strategy is a kind of nonlinear interpolation strategy on grid points in (17), that is, it uses an integrable selection f j (·) with f j (t ) ∈ F (x k j ) andẋ yielding x k (t ) = x k j + t t j f j (s)ds 676 R. Baier and T. Donchev This (nonlinear) selection is used to test the jump condition i (x k ( i )) = i (24) and allow jump times even within the subinterval [t j , t j +1 ]. Lemma 5 shows that one could reach order of convergence 1 w.r.t. step-size. The second strategy is similar and uses the (nonlinear) selection f j (·), but the jump within a subinterval [t j , t j +1 ] is postponed to the subsequent grid point t j +1 as in (20). Theorem 2 tells that this simple strategy still gives us the same order of convergence than the first strategy.
In contrary to the first two approaches, the third strategy uses a piecewise linear strategy on (4)-(5) with This interpolation is used to evaluate the jump condition (24) on the subinterval. Nevertheless, the jump is again postponed to the following grid point. Theorem 3 justifies the resulting order of convergence 1 for the simplest strategy among the three ones.
The proof of Theorem 3 was done step by step by increasing the simplicity for testing the jump condition. Lemma 5 justifies the first strategy with nonlinear selections and jumps within subintervals. It relies heavily on the Filippov-Plis lemma in Theorem 1. Within the convergence proof for the second strategy, solutions of the first strategy are approximated by the second and vice versa, also applying the result on inner perturbations of the original differential inclusion. Finally, the third strategy is justified by the convexity of Aumann's integral.
Evidently, not all the conditions of A3 hold for Example 4 which demonstrates that this condition is important to prevent the multiple hitting of jump surfaces. However, we have only one impulsive surface which causes this beating phenomena for solutions. In this example, f x (·) is still strongly monotone on the intervals of the continuity of a given solution x(·) and our method gives a good approximation. It is interesting to investigate the approximation of solutions with beating phenomena and we leave this interesting task for a further article.
In a forthcoming article, we want to apply our results to the Lagrange optimal control problem: min J [x] := 1 0 g (t , x(t )) dt (25) subject to (1)- (2). Assuming that f (t , ·) is (locally) Lipschitz, Mordukhovich studies more general kind of problem, if the system is without impulses.
Our future aim is to extend Mordukhovich's results in [20] to the case of an impulsive system.

ACKNOWLEDGMENTS
This paper is dedicated to Frank Lempio on the occassion of his 65th birthday.
This work is supported by the Hausdorff Research Institute of Mathematics in Bonn within the framework of the Junior Trimester Program "Computational Mathematics" (February-April 2008).