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Abstract and Applied Analysis Ordinary differential systems describing hysteresis effects and numerical simulations
Ordinary differential systems describing hysteresis effects and numerical simulations
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巻:
7
年:
2002
言語:
english
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Abstract and Applied Analysis
DOI:
10.1155/s108533750220603x
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ORDINARY DIFFERENTIAL SYSTEMS DESCRIBING HYSTERESIS EFFECTS AND NUMERICAL SIMULATIONS EMIL MINCHEV, TAKANOBU OKAZAKI, AND NOBUYUKI KENMOCHI Received 1 May 2002 We consider a general class of ordinary diﬀerential systems which describes inputoutput relations of hysteresis types, for instance, play or stop operators. The system consists of two firstorder nonlinear ODEs and one of them includes a subdiﬀerential operator depending on the unknowns. Our main objective of this paper is to give an existenceuniqueness result for the system as well as to give various numerical simulations of inputoutput relations which the system describes as typical cases. 1. Introduction Consider a nonlinear system of ODEs of the following form: a1 u(t),w(t) u (t) + a2 u(t),w(t) w (t) = g u(t),w(t) , 0 < t < T, h u(t),w(t) , 0 < t < T, (1.1) b1 u(t),w(t) u (t) + b2 u(t),w(t) w (t) + ∂Iu(t) w(t) (1.2) subject to the initial conditions u(0) = u0 , w(0) = w0 , (1.3) where 0 < T < +∞ and ai (·, ·), bi (·, ·), i = 1,2, g(·, ·), h(·, ·) are functions on R2 satisfying some conditions (see Section 2), and for each u ∈ R, ∂Iu (·) is the subdiﬀerential of the indicator function Iu (·) of the interval [ f∗ (u), f ∗ (u)] in R; Copyright © 2002 Hindawi Publishing Corporation Abstract and Applied Analysis 7:11 (2002) 563–583 2000 Mathematics Subject Classification: 34C55, 34A34 URL: http://dx.doi.org/10.1155/S108533750220603X 564 Systems with hysteresis eﬀect namely ∅ [0,+∞) ∂Iu (w) = {0} (−∞,0] (−∞,+∞) for w > f ∗ (u) or w < f∗ (u), for w = f ∗ (u) > f∗ (u), for f∗ (u) < w < f ∗ (u), for w = f∗ (u) < f ∗ (u), for w = f∗ (u) = f ∗ (u), (1.4) f∗ (·) and f ∗ (·) being nondecreasing functions such that f∗ ≤ f ∗ on R (see Section 2 for precise conditions). Equation (1.2) describes a lot of inputoutput relations u → w which are physically relevant. For instance, when b1 ≡ 0 (resp., −1), b2 ≡ 1, and h ≡ 0, the relati; on assigning to a function u(t) the solution w(t) of (1.2) is called a play (resp., stop) operator (cf. [5, 6, 7]). These operators are typical examples of hysteresis inputoutput relations, and are used to describe irreversible phenomena such as solidliquid phase transition with supercooling eﬀect and martensiteaustenite phase transition in shape memory alloys (cf. [2, 8]). In a particular case when a1 ≡ 1, a2 ≡ 1, b1 ≡ 0, b2 ≡ 1, g ≡ 0, and h ≡ 0, the system (1.1), (1.2) was studied in detail by Visintin [7] in a very general framework; the idea for uniqueness proof is based on the socalled L1 theory of nonlinear semigroups (cf. [1, 4]), and the same idea was applied to the uniqueness proof of the Cauchy problem for (1.1), (1.2) with nonzero righthand sides and diﬀusion eﬀects in [3]. Our main objective of this paper is to give an existenceuniqueness result of the Cauchy problem for system (1.1), (1.2) under some restrictions on coeﬃcients ai , bi , i = 1,2, general enough. One of the main points in our proof is to eliminate the term u from (1.2) to get w (t) + ∂Iu(t) w(t) h̃ u(t),w(t) , 0 < t < T, (1.5) with a function h̃ satisfying the same property as h, and then consider the coupling of (1.1) and (1.5). Another objective of this paper is to show by some numerical simulations that our nonlinear system covers many of physically relevant relations u → w arising in the mathematical descriptions of phase transition phenomena. In fact, a suitable choice of the set of coeﬃcients ai , bi and forcing terms g, h, can create the clockwise or anticlockwise trend of the orbit (u(t),w(t)) as t increases. 2. Theoretical results In this section, we mention the precise assumptions on the functions ai , bi , i = 1,2, g, h, and theoretical results on the existence and uniqueness of a solution of (1.1), (1.2), and (1.3). Emil Minchev et al. 565 Let f∗ and f ∗ be functions on R such that f∗ , f ∗ are nondecreasing and Lipschitz continuous on R, the derivatives f∗ := f∗ = f ∗ df∗ ∗ df ∗ , f := are Lipschitz continuous on R, du du (2.1) f∗ ≤ f ∗ on R, on − ∞, −k0 ∪ k0 ,+∞ for a positive number k0 , and put Ᏺ := (u,w) ∈ R2 ; f∗ (u) ≤ w ≤ f ∗ (u) . (2.2) For these functions f∗ , f ∗ , we consider the indicator function 0 for f∗ (u) ≤ w ≤ f ∗ (u), Iu (w) := +∞ otherwise, (2.3) associated with the interval [ f∗ (u), f ∗ (u)] for every u ∈ R, and denote its subdiﬀerential by ∂Iu (·) given by (1.4). Next, let ai , bi , i = 1,2, g, h be functions on Ᏺ such that ai , bi , i = 1,2, are Lipschitz continuous on Ᏺ, a1 ≥ c0 a1 b2 − a2 b1 ≥ c0 on Ᏺ, b2 ≥ c0 on Ᏺ, (2.4) on Ᏺ for a positive constant c0 , g, h are Lipschitz continuous on Ᏺ. (2.5) Now we give the definition of a solution of (1.1), (1.2). Definition 2.1. A pair of (scalar) functions {u,w} is called a solution of system (1.1), (1.2), if u,w ∈ W 1,2 (0,T) and they satisfy (1.1) and (1.2) a.e. on [0,T]; hence u(t),w(t) ∈ Ᏺ ∀t ∈ [0,T]. (2.6) Definition 2.2. A pair of functions {u,w} is called a solution of (1.1), (1.2), and (1.3), if it is a solution of (1.1), (1.2) and the initial conditions u(0) = u0 , w(0) = w0 are satisfied. The existence of a solution is proved under an additional condition. 566 Systems with hysteresis eﬀect Theorem 2.3. Suppose that (2.1), (2.2), (2.3), (2.4), and (2.5) hold, and moreover suppose that there is a positive constant δ0 such that a1 u, f∗ (u) + a2 u, f∗ (u) f∗ (u) ≥ δ0 , a1 u, f ∗ (u) + a2 u, f ∗ (u) f ∗ (u) ≥ δ0 for any u ∈ R. (2.7) Then (1) for any initial data u0 , w0 satisfying (u0 ,w0 ) ∈ Ᏺ, there exists at least one solution {u,w} of system (1.1), (1.2), and (1.3); (2) there is a positive constant K0 , depending only on the quantities in assumptions (2.1), (2.2), (2.3), (2.4), (2.5), (2.6), and (2.7), such that u(t) + w(t) ≤ eK0 (1+T) 1 + u0 u (t) + w (t) ≤ eK0 (1+T) 1 + u0 3 ∀t ∈ [0,T], for a.e. t ∈ [0,T], (2.8) for any solution {u,w} of (1.1), (1.2), and (1.3) with initial data u0 , w0 . Our second result is concerned with the uniqueness of a solution of (1.1), (1.2), and (1.3). Theorem 2.4. Suppose that (2.1), (2.2), (2.3), (2.4), and (2.5) hold as well as a2 ≥ c0 on Ᏺ, (2.9) where c0 is the same positive constant as in (2.4). Then system (1.1), (1.2), and (1.3) admits at most one solution. More precisely, there is a positive constant M0 , depending only on the quantities in assumptions (2.1), (2.2), (2.3), (2.4), (2.5), and (2.9), and the length T of the time interval, such that u1 (t) − u2 (t) + w1 (t) − w2 (t) ≤ eM0 (1+u01  7 +u 7 ) 02 u01 − u02 + w01 − w02 ∀t ∈ [0,T], (2.10) for any two solutions {ui ,wi } of (1.1), (1.2) with initial values u0i , w0i , i = 1,2. Remark 2.5. We see from the proof of Theorem 2.4 given in Section 4 that the constant M0 is of the form eK1 (1+T) for a positive constant depending only on the quantities in assumptions (2.1), (2.2), (2.3), (2.4), (2.5), and (2.9). Remark 2.6. As is easily understood, for the construction of a local in time solution of (1.1), (1.2), and (1.3) it is enough to assume that the functions ai , bi , g, h, f∗ , f ∗ are locally Lipschitz continuous and the inequalities in (2.4) and (2.7) are satisfied as well in a neighborhood of the initial point (u0 ,w0 ). As is easily checked, without loss of generality, we may assume that the functions ai , bi , i = 1,2, g, and h are globally Lipschitz continuous on R2 and all the inequalities in (2.4) hold on R2 . In fact, it is enough to extend them outside Ᏺ, Emil Minchev et al. 567 for instance, in the following manner: ai (u,w) = ai u, f ∗ (u) , bi (u,w) = bi u, f ∗ (u) , for w > f ∗ (u), ai (u,w) = ai u, f∗ (u) , ∗ bi (u,w) = bi u, f∗ (u) , for w < f∗ (u), g(u,w) = g u, f (u) , h(u,w) = h u, f (u) , for w > f ∗ (u), g(u,w) = g u, f∗ (u) , h(u,w) = h u, f∗ (u) , for w < f∗ (u). ∗ (2.11) In the rest of this paper, we always assume such extended conditions for functions ai , bi , i = 1,2, g, and h. The key for the proofs of our theorems is found in the following lemma. Lemma 2.7. Under conditions (2.1), (2.2), (2.3), (2.4), and (2.5), system (1.1), (1.2) is equivalent to the coupling of (1.1) and the following inclusion: w (t) + ∂Iu(t) w(t) h̃ u(t),w(t) for a.e. t ∈ [0,T], (2.12) where h̃(u,w) := a1 (u,w)h(u,w) − b1 (u,w)g(u,w) . a1 (u,w)b2 (u,w) − a2 (u,w)b1 (u,w) (2.13) Proof. In order to eliminate the term u from (1.2), compute ((1.2) × a1 − (1.1) × b1 )/(a1 b2 − a2 b1 ). Then we have w + a1 ∂Iu (w) h̃ a.e. on [0,T], a1 b2 − a2 b1 (2.14) where h̃ is the same as given by (2.13). Here we note the invariance of ∂Iu (w) under multiplication by positive numbers, namely ∂Iu (w) = k∂Iu (w) for every positive k. In fact, this property is immediately seen from (1.4). Therefore, by (2.4), we have (a1 /(a1 b2 − a2 b1 ))∂Iu (w) = ∂Iu (w). Hence (2.12) is obtained. This lemma shows that it is enough to prove Theorems 2.3 and 2.4 to the system {(1.1),(2.12)} instead of {(1.1),(1.2)}. 3. A priori bounds of solutions In this section, we give a priori bounds of the form (2.8) for solutions under the same assumptions as Theorem 2.3. For the sake of simplicity of notation we denote by L0 , chosen so that L0 > 1, a common Lipschitz constant of functions ai , bi , i = 1,2, g, h on R2 ; note the extended condition (2.11). Let {u,w} be any solution of (1.1), (1.2) with given initial data u0 ,w0 ; of course, the relation f∗ (u0 ) ≤ w0 ≤ f ∗ (u0 ) is satisfied. Then we prove the following lemma. 568 Systems with hysteresis eﬀect Lemma 3.1. Put N0 := k0 + u0 , where k0 is the same number as in (2.1). Then, g − N0 , f ∗ − N0 − N0 − 2L20 ≤ u(t) ≤ N0 + exp 2L20 T δ0 g N0 , f ∗ N0 2L20 exp 2L20 T δ0 (3.1) for all t ∈ [0,T]. Proof. We multiply (1.1) by (u − N0 )+ to get a1 u u − N0 + + a2 w u − N0 + + = g u − N0 . (3.2) Since w = f ∗ (u) for u ≥ N0 by (2.1), it follows from the above equality that d + 1 a1 + a2 f ∗ (u) u − N0 2 dt 2 + = g u − N0 . (3.3) Also, we note that g u − N0 + + = g u, f ∗ (u) − g N0 , f ∗ N0 u − N0 + + g N0 , f ∗ N0 u − N0 + 2 + ≤ 2L20 u − N0 + g N0 , f ∗ N0 u − N0 . (3.4) By (3.3) and (3.4) with assumption (2.7) we have + 2L2 + 1 d u − N0 ≤ 0 u − N0 + g N0 , f ∗ N0 , dt δ0 δ0 (3.5) and the second inequality of (3.1) is obtained. The first inequality is similarly obtained, too. Corollary 3.2. There is a positive constant K0(1) , depending only on the quantities in (2.1), (2.2), (2.3), (2.4), (2.5), (2.6), and (2.7), such that (1) u(t) + w(t) ≤ eK0 (1+T) 1 + u0 , 0 ≤ ∀t ≤ T. (3.6) Proof. We note that w is bounded by a linear function of u and hence so are g(N0 , f ∗ (N0 )) and g(−N0 , f∗ (−N0 )). Therefore, from (3.1) together with this fact we immediately derive (3.6) for a certain positive constant K0(1) . Emil Minchev et al. 569 Lemma 3.3. There is a positive constant K0(2) , depending only on the quantities in (2.1), (2.2), (2.3), (2.4), (2.5), (2.6), and (2.7), such that (2) u (t) + w (t) ≤ eK0 (1+T) 1 + u0 3 for a.e. t ∈ [0,T]. (3.7) Proof. We have a1 u 2 + a2 w u = gu by multiplying (1.1) by u . Here, since u, w satisfy (2.12), we observe that w = h̃ a.e. on t; f∗ (u) < w < f ∗ (u) , a.e. on t;w = f∗ (u) , a.e. on t;w = f ∗ (u) , f∗ (u)u ∗ f (u)u (3.8) so that g u  ≥ a1 u 2 + a2 w u 2 1 u  + a2 h̃u a 2 = a1 + a2 f∗ (u) u  a1 + a2 f ∗ (u) u 2 a.e. on t; f∗ (u) < w < f ∗ (u) , a.e. on t;w = f∗ (u) , a.e. on t;w = f ∗ (u) . (3.9) Accordingly, using our assumptions (2.4) and (2.7), we see from the above equality that u 2 ≤ 2 2 a2 h̃ c0 g 2 2 δ0 2 + g 2 a.e. on t; f∗ (u) < w < f ∗ (u) , (3.10) a.e. on t;w = f∗ (u) or f ∗ (u) . Moreover, note that h̃ ≤ const(1 + u2 ) and hence a2 h̃ ≤ const(1 + u3 ). Therefore, it follows from (3.10) with (3.6) and (3.8) that an estimate of the form (2.8) holds for a certain positive constant K0(2) . Now, putting K0 := max{K0(1) ,K0(2) }, we see the estimate (2.8) for all solutions {u,w } with initial data u0 , w0 . 4. Proof of uniqueness In this section, we prove (2.10) for two solutions {uk ,wk }, k = 1,2, of system (1.1), (1.2), and (1.3) for given initial data u0k , w0k , assuming always that conditions (2.1), (2.2), (2.3), (2.4), (2.5), and (2.9) are satisfied. 570 Systems with hysteresis eﬀect For simplicity we put a(k) i := ai uk ,wk , bi(k) := bi uk ,wk , h(k) := h uk ,wk , h̃(k) := h̃ uk ,wk , ū := u1 − u2 , ū0 := u01 − u02 , g (k) := g uk ,wk , i,k = 1,2, w̄ := w1 − w2 , (4.1) w̄0 := w01 − w02 . With these notations, by taking the diﬀerence of two solutions, we have (1) (1) (2) (1) (2) (1) a(1) − g (2) − a1 − a1 u2 − a2 − a2 w2 . 1 ū + a2 w̄ = g (4.2) Now, take any measurable function sū in time so that 1 sū ∈ sign ū = [−1,1] −1 if ū > 0, if ū = 0, if ū < 0, (4.3) and multiply (4.2) by sū and use the Lipschitz continuity of functions ai , g. Then, since ū sū = (d/dt)ū, we have a(1) 1 d (1) ū + w̄  . ū + a2 w̄ sū ≤ L0 1 + u2 + w2 dt (4.4) We arrange this inequality in the following form: d (1) ū + w̄ ). (4.5) a ū + a(1) 2 w̄ sū ≤ L0 1 + u1  + u2 + w1 + w2 dt 1 Next, we show that w̄ sū ≥ d w̄  − 2L1 ū + w̄  dt a.e. on [0,T], (4.6) where L1 is a Lipschitz constant of h̃ on R2 ; note that L1 is dominated by a positive number of the form const(1 + u01 3 + u02 3 ). We show (4.6) on the following four subsets of time t ∈ [0,T]: (a) E1 := {t; ū ≥ 0, w̄ < 0}, (b) E2 := {t; ū ≤ 0, w̄ > 0}, (c) E3 := {t; ū ≥ 0, w̄ ≥ 0}, (d) E4 := {t; ū ≤ 0, w̄ ≤ 0}. On the set E3 ∪ E4 , by the definition of the function sign(·), we can choose sw̄ so that sw̄ = sū . Therefore, (4.6) trivially holds. Next, we consider the case (a). In this case observe that w1 < f ∗ u1 , w2 > f∗ u2 on E1 . (4.7) Emil Minchev et al. 571 In fact, if not, then w1 = f ∗ (u1 ) ≥ f ∗ (u2 ) ≥ w2 or w2 = f∗ (u2 ) ≤ f∗ (u1 ) ≤ w1 , which contradicts our assumption. By the definition of ∂Iu1 (w1 ) we see that h̃(1) − w1 z − w1 ≤ 0, f ∗ u1 ≤ ∀z ≤ f ∗ u1 . (4.8) Taking f ∗ (u1 ) as z in (4.8) yields that h̃(1) ≤ w1 on E1 and similarly h̃(2) ≥ w2 on E1 . Consequently, it follows that w̄ − h̃(1) − h̃(2) ≥ 0 on E1 . (4.9) Since sw̄ = −1 ≤ sū on E1 , we derive by multiplying the above inequality by sū − sw̄ (≥ 0) that w̄ sū ≥ d w̄  + h̃(1) − h̃(2) sū − sw̄ dt a.e. on E1 . (4.10) This is easily arranged to the form (4.6). Just as (a), we see that inequality (4.6) holds a.e. on E2 , too. Thus (4.6) holds a.e. on [0,T]. Moreover, multiply (4.6) by a(1) 2 (≥ c0 ) to get (1) a(1) 2 w̄ sū ≥ a2 ≥ d (1) w̄  − 2a2 L1 ū + w̄  dt d (1) a w̄ dt 2 (1) − L0 + 2a2 L1 1 + u1 + w1 ū + w̄  a.e. on [0,T]. (4.11) We then obtain from (4.5) with the above inequality that d (1) a ū + a(1) 2 w̄  dt 1 (1) ≤ 2 L0 + a2 L1 1 + u1 + u2 + w1 + w2 ū + w̄  (4.12) 2 L0 + a(1) 2 L1 ≤ 1 + u1 + u2 c0 + w1 + w2 (1) a(1) 1 ū +a2 w̄  a.e. on [0,T]. Here, on account of estimates (2.8) with the fact that a(1) 2 L1 is dominated by a positive number of the form const(1 + u01 4 + u02 4 ), the last inequality implies that d (1) a ū + a(1) 2 w̄  dt 1 ≤ eK1 (1+T) 1 + u01 7 + u02 7 (1) a(1) 1 ū + a2 w̄  (4.13) a.e. on [0,T], 572 Systems with hysteresis eﬀect where K1 is a positive constant depending only on the quantities in assumptions (2.1), (2.2), (2.3), (2.4), (2.5), and (2.9). Therefore, (1) a(1) 1 (t) ū(t) + a2 (t) w̄(t) 7 7 ≤ exp eK1 (1+T) 1 + u01 + u02 T (1) (1) · a1 (0) ū0 + a2 (0) w̄0 ∀t ∈ [0,T]. (4.14) (1) Now, noting that a(1) 1 (0) and a2 (0) are dominated by a number of the form const(1 + u01 ), from (4.14) we infer the required inequality (2.10) for a positive number M0 of the form eK1 (1+T) , where K1 is a positive constant depending only on the quantities in (2.1), (2.2), (2.3), (2.4), (2.5), and (2.9). 5. Proof of existence Throughout this section assume that the extended assumptions (2.1), (2.2), (2.3), (2.4), (2.5), (2.6), and (2.7) hold and (2.12) as well. Let u0 , w0 be a pair of initial data such that (u0 ,w0 ) ∈ Ᏺ. According to the a priori estimates (2.8) for solutions, we may assume, without loss of generality, that the functions ai , bi , i = 1,2, g and h̃ are bounded on R2 . Consider approximate problems (Pλ ), which consist of (5.1), (5.2), and (5.3), with parameter λ ∈ (0,λ0 ] for a positive number λ0 small enough, to find a pair of functions {u,w} satisfying d J(u,w) = g(u,w) a.e. on [0,T], dt w + ∂Iuλ (w) = h̃(u,w) a.e. on [0,T], a1 (u,w)u + a2 (u,w) (5.1) (5.2) subject to the initial conditions u(0) = u0 , w(0) = w0 , (5.3) where J(u, ·) is the projection mapping from R onto [ f∗ (u), f ∗ (u)], namely J(u,w) = w − w − f ∗ (u) + + f∗ (u) − w + ∀w ∈ R (5.4) and ∂Iuλ is the Yosida approximation of ∂Iu , namely ∂Iuλ (w) = w − f ∗ (u) λ + − f∗ (u) − w λ + ∀w ∈ R. (5.5) Emil Minchev et al. 573 Clearly, J is Lipschitz continuous on R2 and nondecreasing in both variables, and ∂Iuλ is Lipschitz continuous on R2 and nondecreasing in w. Also, ∂Iuλ (·) is the subdiﬀerential of the convex function w − f ∗ (u) 2λ Iuλ (w) := + 2 + f∗ (u) − w 2λ + 2 ∀w ∈ R. (5.6) Lemma 5.1. For each parameter λ ∈ (0,λ0 ], problem (Pλ ) has a solution {uλ ,wλ } such that uλ and wλ are Lipschitz continuous on [0,T]. Moreover, the following uniform estimates hold: there is a positive constant R1 independent of λ ∈ (0,λ0 ] such that uλ (t) + wλ (t) ≤ R1 uλ (t) ≤ R1 ∀t ∈ [0,T], wλ for a.e. t ∈ [0,T], L2 (0,T) ≤ R1 . (5.7) Proof. Let Jε be the regularization J by means of the usual mollifier ρε , 0 < ε ≤ ε0 (ε0 is a positive number close enough to 0), namely Jε (u,w) := R2 J(ζ,ξ)ρε (u − ζ,w − ξ)dζ dξ ∀(u,w) ∈ R2 . (5.8) Consider the further regularized approximate equation of (5.1) a1 (u,w)u + a2 (u,w) d Jε (u,w) = g(u,w) on [0,T]. dt (5.9) This is written in the form a1 (u,w) + a2 (u,w) ∂ ∂ Jε (u,w) u + a2 (u,w) Jε (u,w)w = g(u,w). ∂u ∂w Note here that ∗ f (u) (resp., 0) ∂ ∂ J(u,w) resp., J(u,w) = 0 (resp., 1) ∂u ∂w f (u) (resp., 0) ∗ (5.10) if w > f ∗ (u), if f∗ (u) < w < f ∗ (u), if w < f∗ (u), (5.11) which shows that 0≤ ∂ Jε (u,w) ≤ f ∗ (u) resp., f∗ (u) + τε ∂u (5.12) in the εneighborhood of {(u,w); w ≥ f ∗ (u) (resp., w ≤ f∗ (u))}, where τε := sup J u1 ,w1 − J u2 ,w2 ; u1 − u2 2 + w1 − w2 2 1/2 ≤ ε ; (5.13) clearly τε → 0 as ε → 0. Therefore, on account of (2.4), (2.7), and (2.11), we see that the coeﬃcient a1 (u,w) + a2 (u,w)(∂/∂u)Jε (u,w) of u in (5.10) is bounded 574 Systems with hysteresis eﬀect from below by a positive constant δ0 on R2 , where δ0 is independent of λ ∈ (0,λ0 ] and ε ∈ (0,ε0 ]. Therefore, system (5.9) and (5.2) is written in the form g(u,w) − a2 (u,w)(∂/∂w)Jε (u,w) h̃(u,w) − ∂Iuλ (w) , u = a1 (u,w) + a2 (u,w)(∂/∂u)Jε (u,w) w = h̃(u,w) − ∂Iuλ (w) (5.14) and the righthand sides are Lipschitz continuous in (u,w). By the general existenceuniqueness theorem for ODEs, problem (5.9) and (5.2) with initial condition (5.3) admits a unique solution, denoted by {uλε ,wλε }. Next, we give uniform estimates for {uλε ,wλε } in ε and λ. To do so, multiply (5.9) by (uλε − N0 )+ , with N0 := k0 + 1 + u0 . Then, just as (3.1) of Lemma 3.1, we see that uλε (t) ≤ K2 , Jε uλε (t),wλε (t) ≤ K2 , 0 ≤ ∀t ≤ T, (5.15) for a positive constant K2 independent of λ and ε. From these uniform estimates and (5.14) with (5.11) we derive easily that for each λ > 0 small enough, there is a positive constant K3 (λ) independent of ε such that uλε ≤ K3 (λ), d Jε uλε ,wλε ≤ K3 (λ), dt ≤ K3 (λ), wλε a.e. on [0,T]. (5.16) Therefore, it is possible to extract a null sequence {εn } such that uλεn and wλεn converge to some Lipschitz continuous functions uλ and wλ uniformly on [0,T], respectively, as n → +∞. Moreover, it is easy to see that the pair {uλ ,wλ } is a solution of (Pλ ); note from (5.15) that uλ (t) ≤ K2 , 0 ≤ ∀t ≤ T. (5.17) Taking account of (5.17) and the expression h̃ uλ ,wλ d J uλ ,wλ = f ∗ uλ uλ dt f uλ u ∗ λ a.e. on t; f∗ uλ ≤ wλ ≤ f ∗ uλ , a.e. on t;w > f ∗ uλ , a.e. on t;w < f ∗ uλ , (5.18) just as (3.10) in the proof of Lemma 3.3, we obtain by condition (2.7) that uλ ≤ K4 a.e. on [0,T] (5.19) for a positive constant K4 independent of λ > 0. Finally, multiplying (5.2) by wλ , we have wλ 2 + ∂Iuλλ wλ · wλ = h̃wλ . (5.20) Emil Minchev et al. 575 Here, use the following inequality which is derived directly from the expressions (5.4), (5.5), and (5.6): d λ I wλ − ∂Iuλλ wλ · wλ ≤ L0 uλ dt uλ ∂Iuλλ wλ a.e. on [0,T]. (5.21) Then, noting this with (5.19), we immediately obtain wλ 2 + d λ a.e. on [0,T]. Iuλ wλ ≤ L0 K4 wλ + h̃ dt (5.22) From this inequality it is easy to get a uniform estimate for wλ L2 (0,T) with respect to λ, since h̃ := h̃(uλ ,wλ ) is uniformly bounded and Iuλ0 (w0 ) = 0. Thus uniform estimates of the form (5.7) hold for a positive constant R1 independent of λ. By virtue of Lemma 5.1 there is a null sequence {λn } such that the solution un := uλn , wn := wλn of (Pλn ) constructed above converge to some functions u, w uniformly on [0,T] as n → +∞. Simultaneously, since ∂Iuλnn (wn ) (= −wn + h̃(un ,wn )) is bounded in L2 (0,T), we see from (5.5) that + + − f∗ un − wn = λn ∂Iuλnn wn −→ 0 in L2 (0,T) (as n −→ +∞), wn − f ∗ un (5.23) so that f∗ (u) ≤ w ≤ f ∗ (u) on [0,T] and J un ,wn −→ w un −→ u uniformly on [0,T], weakly in L2 (0,T) (5.24) as well as d J un ,wn −→ w dt weakly in L2 (0,T). (5.25) Therefore, the limit {u,w} gives a solution of the original problem. Thus the existence proof is now complete. 6. Some numerical simulations Another objective of this paper is to verify by some numerical simulations that our system (1.1), (1.2) is useful enough as the mathematical description of various inputoutput relations u → w of hysteresis type appearing in many nonlinear phenomena, and what is the influence of the choice of the coeﬃcients ai , bi , i = 1,2, and of functions g, h on the behaviour of the orbit (u(t),w(t)). In order to catch the main trends of the inputoutput relation u → w we simply take constants as the coeﬃcients ai (u,w), bi (u,w), i = 1,2, and mostly linear 576 Systems with hysteresis eﬀect functions as g(u,w), h(u,w). The constants are denoted by the same notations ai , bi , and they are chosen so as to satisfy that a1 > 0, b2 > 0, a1 b2 − a2 b1 > 0. (6.1) Also, in our simulation the functions f∗ and f ∗ are fixed as follows: −1 2 5u + 16u + 11.8 f ∗ (u) := 2u + 2 −5u2 − 4u + 0.2 1 if u < −1.6, if − 1.6 ≤ u < −1.4, if − 1.4 ≤ u < −0.6, if − 0.6 ≤ u < −0.4, if − 0.4 ≤ u, −1 2 5u − 4u − 0.2 f∗ (u) := 2u − 2 −5u2 + 16u − 11.8 1 (6.2) if u < 0.4, if 0.4 ≤ u < 0.6, if 0.6 ≤ u < 1.4, if 1.4 ≤ u < 1.6, if 1.6 ≤ u. As was mentioned above, the following system is considered in this section: a1 u + a2 w = g(u,w), 0 < t < T, a1 g(u,w) − b1 h(u,w) , a1 b2 − a2 b1 w(0) = w0 , u(0) = u0 , w + ∂Iu (w) h̃(u,w) := (6.3) 0 < t < T, (6.4) (6.5) where ∂Iu (·) is the subdiﬀerential of Iu (·) associated with functions f ∗ and f∗ given by (6.2). Note that (6.4) is equivalent to the original form b1 u + b2 w + ∂Iu (w) h(u,w). Now let λ and ∆t be positive numbers small enough, and put t k = k∆t (k = 1,2,...). Then the diﬀerence scheme for our numerical simulation is of the form a1 uk+1 − uk a2 wk+1 − wk = g uk ,w k , + ∆t ∆t wk+1 − wk + ∂Iuλk wk+1 = h̃ uk ,wk , k = 0,1,2,..., ∆t w0 = w0 , u 0 = u0 , (6.6) where ∂Iuλk wk+1 = wk+1 − f ∗ uk λ + f∗ uk − wk+1 − λ + . The graphs of Iuλ and ∂Iuλ are illustrated in Figures 6.1 and 6.2, respectively. (6.7) Emil Minchev et al. 577 Iuλ f ∗ (u) f∗ (u) w Figure 6.1 ∂Iuλ (w) f∗ (u) f ∗ (u) w Figure 6.2 In the numerical computations which are performed below, we take ∆t = 1 , 1000 λ= 1 , 10000 (6.8) and we examine the following items. (i) By choice of coeﬃcients ai , bi and functions g, h, system (6.3), (6.4) creates various behaviours of the solution pair (u(t),w(t)). In particular, it is possible to control by them the trend of clockwise or anticlockwise behaviour of (u(t),w(t)). (ii) Fixing the coeﬃcients ai , bi and the initial data, we investigate the influence of the functions g, h on the behaviour of (u(t),w(t)). (iii) We investigate the asymptotic behaviour of (u(t),w(t)) as t → +∞ for various initial data when any other data are fixed. 578 Systems with hysteresis eﬀect w w u u Figure 6.3 Figure 6.4 w w u u Figure 6.5 Figure 6.6 Simulation 1. These experiments show that the clockwise and anticlockwise behaviour of the orbit (u(t),w(t)) is created by choosing suitable coeﬃcients for given initial data. It seems that the orbit is periodic in time after a certain time. Data table a1 a2 b1 b2 h(u,w) g(u,w) u0 , w0 Figure 6.3 1 1 −1 1 u+w u−w u0 = −0.7, w0 = −0.8 Figure 6.4 1 1 −1 1 −u − w −u + w u0 = −0.7, w0 = −0.8 Figure 6.5 1 1 −1 1 u + 3w u − 2w u0 = −1.8, w0 = −1.0 Figure 6.6 1 1 −3 1 −u − w −u + w u0 = −1.8, w0 = −1.0 Emil Minchev et al. 579 w w u u Figure 6.7 Figure 6.8 w w u u Figure 6.10 Figure 6.9 Simulation 2. These experiments show that b1 = −1 gives an anticlockwise (resp., a clockwise) periodic behaviour of the orbit (u(t),w(t)) in time. When the value of b1 becomes smaller (resp., larger), the orbit asymptotically converges anticlockwise (resp., clockwise) to a stable point along a spiral curve. Data table a1 a2 b1 b2 h(u,w) g(u,w) u0 , w0 Figure 6.7 1 1 −1 1 u+w u−w u0 = −0.7, w0 = −0.8 Figure 6.8 1 1 −2 1 u+w u−w u0 = −0.7, w0 = −0.8 Figure 6.9 1 1 −1 1 −u − w −u + w u0 = −0.7, w0 = −0.8 Figure 6.10 1 1 −0.5 1 −u − w −u + w u0 = −0.7, w0 = −0.8 580 Systems with hysteresis eﬀect w w u u Figure 6.11 Figure 6.12 w w u u Figure 6.13 Figure 6.14 Simulation 3. These experiments show that for fixed coeﬃcients ai , bi , i = 1,2 and initial data as in the table, we can create various (asymptotically) anticlockwise periodic orbits only by the choice of linear functions g, h. Also we observe the similar behaviours of orbits in the clockwise case. Data table a1 a2 b1 b2 h(u,w) g(u,w) u0 , w0 Figure 6.11 1 1 −1 1 u+w u−w u0 = −0.7, w0 = −0.8 Figure 6.12 1 1 −1 1 u + 2w u − 2w u0 = −0.7, w0 = −0.8 Figure 6.13 1 1 −1 1 2u + 3w −w u0 = −0.7, w0 = −0.8 Figure 6.14 1 1 −1 1 u + 2w u−w u0 = −0.7, w0 = −0.8 Emil Minchev et al. 581 w w u u Figure 6.16 Figure 6.15 w w u u Figure 6.18 Figure 6.17 Simulation 4. These experiments suggest that the largetime behaviour of orbits (u(t),w(t)) is one of the following two cases: (1) the orbit converges to a periodic one as t → +∞; (2) the orbit diverges to the point (−∞, −1) or (1,+∞) as t → +∞. Data table a1 a2 b1 b2 h(u,w) g(u,w) u0 , w0 Figure 6.15 1 1 −1 1 u + 2w u−w u0 = −0.7, w0 = −0.8 Figure 6.16 1 1 −1 1 u + 2w u−w u0 = 0.4, w0 = 0.3 Figure 6.17 1 1 −1 1 u + 2w u−w u0 = 1.2, w0 = 0.9 Figure 6.18 1 1 −1 1 u + 2w u−w u0 = −1.3, w0 = −0.8 582 Systems with hysteresis eﬀect w w u u Figure 6.20 Figure 6.19 w w u u Figure 6.21 Figure 6.22 Simulation 5 (nonlinear case of h and g). These experiments show that for fixed coeﬃcients ai , bi , i = 1,2 as in the table, we can create various behaviours diﬀerent from the linear case of h and g. Figures 6.19 and 6.20 are periodic orbits of (u(t),w(t)) in time. Figure 6.21 shows that the orbit diverges to the point (1,+∞) as t → ∞. Figure 6.22 shows that the orbit converges to the point (−1.2, −0.4) as t → ∞. Data table a1 a2 b1 b2 h(u,w) g(u,w) Figure 6.19 1 1 −1 1 − sin(u+w)+cos(u+w) −u+1.5w Figure 6.20 1 1 −1 1 sin(uw) − cos(uw) u − 1.5w Figure 6.21 1 1 −1 1 uw uw − u + w Figure 6.22 1 1 −1 1 2uw u − 3w u0 , w0 u0 = −0.7, w0 = −0.8 u0 = −0.7, w0 = −0.8 u0 = 0.8, w0 = −0.1 u0 = 0.2, w0 = −0.8 Emil Minchev et al. 583 References [1] [2] [3] [4] [5] [6] [7] [8] P. Bénilan, Equations d’évolution dans un espace de Banach quelconque et applications, Ph.D. thesis, Univ. ParisSud, 1972. M. Brokate and J. Sprekels, Hysteresis and Phase Transitions, Applied Mathematical Sciences, vol. 121, SpringerVerlag, New York, 1996. P. Colli, N. Kenmochi, and M. Kubo, A phasefield model with temperature dependent constraint, J. Math. Anal. Appl. 256 (2001), no. 2, 668–685. M. Hilpert, On uniqueness for evolution problems with hysteresis, Mathematical Models for Phase Change Problems (Óbidos, 1988) (J. F. Rodrigues, ed.), Internat. Ser. Numer. Math., vol. 88, Birkhäuser, Basel, 1989, pp. 377–388. M. A. Krasnosel’skiı̆ and A. V. Pokrovskiı̆, Systems with Hysteresis, SpringerVerlag, Berlin, 1989, translated from the Russian by Marek Niezgódka. P. Krejčı́, Hysteresis operators—a new approach to evolution diﬀerential inequalities, Comment. Math. Univ. Carolin. 30 (1989), no. 3, 525–536. A. Visintin, Diﬀerential Models of Hysteresis, Applied Mathematical Sciences, vol. 111, SpringerVerlag, Berlin, 1994. , Models of Phase Transitions, Progress in Nonlinear Diﬀerential Equations and Their Applications, vol. 28, Birkhäuser Boston, Massachusetts, 1996. Emil Minchev and Takanobu Okazaki: Department of Mathematics, Graduate School of Science and Technology, Chiba University, 133 Yayoichō, Inageku, Chiba, 2638522, Japan Nobuyuki Kenmochi: Department of Mathematics, Faculty of Education, Chiba University, 133 Yayoichō, Inageku, Chiba, 2638522, Japan Advances in Operations Research Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Advances in Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Applied Mathematics Algebra Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Probability and Statistics Volume 2014 The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Differential Equations Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Submit your manuscripts at http://www.hindawi.com International Journal of Advances in Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences Mathematical Problems in Engineering Journal of Mathematics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Discrete Mathematics Journal of Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Discrete Dynamics in Nature and Society Journal of Function Spaces Hindawi Publishing Corporation http://www.hindawi.com Abstract and Applied Analysis Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Journal of Stochastic Analysis Optimization Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014