Ważewski, T.

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Arzela-like theorem with applications to differential equations and control theory

Institute of Mathematics AS CR, v. v. i.

EQUADIFF 2Tadeusz WażewskiArzela-like theorem with applications to differential equations and control theoryIn: Valter Šeda (ed.): Differential Equations and Their Applications, Proceedings of the Conferenceheld in Bratislava in September 1966. Slovenské pedagogické nakladateľstvo, Bratislava, 1967. ActaFacultatis Rerum Naturalium Universitatis Comenianae. Mathematica, XVII. pp. 155--159.Persistent URL: http://dml.cz/dmlcz/700223Terms of use:© Comenius University in Bratislava, 1967Institute of Mathematics of the Academy of Sciences of the Czech Republic provides access todigitized documents strictly for personal use. Each copy of any part of this document must containthese Terms of use.This paper has been digitized, optimized for electronic delivery and stampedwith digital signature within the project DML-CZ: The Czech Digital MathematicsLibrary http://project.dml.czACTA FACULTATIS KERU3I XATURALIUM UNIVEKSITATIS COMEXIANAE MATHEMATICA XVII - 1907 ARZELA - LIKE THEOREM WITH APPLICATIONS TO DIFFERENTIAL EQUATIONS AND CONTROL THEORY T. WAZEWSKI, Krak6u-§ 1. In numereous problems concerning given differential equations or control systems the sequences of functions (i.i) y = gm, ( » = - i , 2 , . . . ) approximating solutions of respective equations for instance one equation (i.2) y'=f(*,y) play an important role. Usually one starts with equation (1.2) and then determine suitable sequences (i.i). Our paper deals with the following opposite: Problem Z. Given the sequence (1.1), such an equation (1.2) called "asymptotically inducted by sequence (1.1)" is to be found, that the limits of all convergent subsequences y = kt(x) of (1.1) satisfy (1.2). I t is evident that the required function f(x, y) can be determined exclusively on the accumulation set L for sequence of curves (1.1). I t can happen that L reduces to a single arc. Problem Z is considered globally. We shall give conditions for subsequences k% to converge towards an "extensive solution" of (1.1) i.e. solution tending to the boundary of open set W (containing L) at both ends. Such conditions are given in Theorem A which proved to be very convenient for didactic purposes because of many applications (for instance global existence theorems, continuous dependence on and differentiability in respect to initial values, constructing of approximate solutions). The curves (1.1) occuring in Theorem A are arcs. In some applications of the classical Arzela's theorem on equicontinuous sequences (1.1) is of no use because one is obliged to use functions (1.1) with 155 graphs consisting of finite number of points. In this case we replace the notion of equicontinuity by notion of asymptotic smoothness and extensivity (Theorem B). Theorem C gives a construction of a control system "asymptotically in-duced" by the sequence (1.1). § 2. Theorem A. Let W e B2 be an open non-empty set. Consider a sequence of real functions gi(x) defined, continuous and having the right-hand (finite) derivative D^gi on open intervals Ju (i == 1,2, . . . ) . Denote by gi the graph of gx(x) and put (2.1) L - {(x, y) : (x, y) e W, lim inf r((x, y), gt) = 0} i—>oo where r((x, y), B) denotes the distance of point (x, y) from set B. Suppose the following implication: if for any subsequence g^i) (2.2) (xi,gk(i)(xi))->(x,y)eW, then two following conditions (2.3), (2.4) hold: (2.3) there exist a neighbourhood of x contained for large i in Jjc(i), (2.4) lim D+ gic(i)(xi) exists and is finite. Under above assuynptions: the limit (2.4) depends on (x, y) only and is independent of the particular choice of the subsequence gm). This limit denoted by F(x, y) is defined and continuous on L [see (2.1)]. (2.5) For any point (xQ, yQ) e L there exists such a subsequence gm(i) that lim gm{i)(x) = h(x), where h(x) is an "extensive" solution of the equation (2.6) y' = F(x, y), i.e. h(x) is an open arc "tending" at both ends to the boundary of W. Moreover if through each point of L passes a unique solution of (2.6) and lim r((xQ, yQ), gt) = 0 then the original sequence gt is convergent to the extensive solution passing through (*o> 2/o)-§ 3 . R e m a r k 1. The sequence gi satisfying the implication (2.2) => (2.3) will be called expansive on W. The differential equation (2.6) will be called to be "induced by sequence g". This notions will be generalized in the following. 156 R e m a r k 2. The proof of this theorem can be based on the classical Arzela's Lemma and on the theorem on differentiability of the limit. Theorem A can serve as the starting point for generalizations. For this purposa we introduce some definitions. § 4. Limits restreint, complete and exact in IlausdorfMike sense. Let A = {Ai} be a sequence of sets At C P = R2. We define (4.1) Z(A) = lim restr At = {z: z e P, Km r(z, At) = 0}, (4.2) Y](A) = lim compl At = {z:z eP, lim inf r(z, Ai) ==0}. If C(-4) = ri(A) we say that A is H-convergent and we define (4.3) X(A) = lim exact At = Z(A) = r](A). (4.4) P r o p o s i t i o n . For Ai =£ 0 there exists //-convergent subsequence of {At}. § 5. Univalent sets and smooth functions. For B c P we define by B1 = projection of B on #-axis, /J11 = projection of B on y-axis. We say that set B is univalent if p e B, qeB, p1 = q1 => p = q. The univalent sets will be considered as functions of variable x. The whole of such functions (or sets) will be denoted by Unival. Let / e Unival and put g = f n W. f is called extensive if g is continuous, g1 is open and (x, g(x)) tends to the boundary of W9 as x tends to the boundary of g1. f is called smooth if g is continuous and closed in W. § 6. Suppose tha t 1)9= {9i}> 9ie Unival, 2) (x0, y0) is an arbitrary point of W, 3) k = {ki} is an arbitrary H-convergent subsequence of g for which («bi Vv) e X{k). (6.1) g is called asymptotically smooth in W if [Xi e k\, Xi -> x0] => [ki(xt) -> y0]. (6.2) g is called asymptotically extensive in W if x0 is an interior point of h(*)p. 157 (6.3) g is called asymptotically lipschitzian in W if [Xi e k}, si e k\, xt -> x0, st -> x0, h(xi) -> y0] => => [lim sup \(ki(si) — ki(xi))l(si — a*)| < +cx>]. (6.4) Obviously if g is asymptotically lipschitzian it is asymptotically smooth. Theorem B. If gt e Unival, g = {gi} is asymptotically smooth and asymptoti-cally extensive (in W), (x0, y0) e rj(g) then there exists such an H-convergent subsequence k of g that (x0, y0) e X(k) and X(k) is univalent, extensive and smooth in TV. Remark 3. If moreover g is asymptotically lipschitzian then X(k) is locally lipschitzian. Remark 4. The theorem B generalizes the classical lemma of Arzela on equicontinuous sequences of functions. It is used in the following. § 7. For G c P we define Convex G = closed convex hull of G. For a e P, b e P, a1 ^ b1 we put slope (a, b) = (611 - «n)/(&i - at), life Unival, Q is open, Q ^ P, we define slope (/, Q) = u slope (a, b), for aef r\Q, bef C\Q. For g = {gi}, where gi e Unival, g asymptotically lipschitzian in W, 0 ^Q open, we put B(i, Q) = yj slope (gh Q). in For (x, y) e £(gr) n TV we define (7.1) C(x, y) = Convex [ lim exact B(i, Q)]. The relation (7.1) means that for any such sequence of open non-empty sets Qi that (x, y) e Qu diameter Qi -> 0, we have: G(x, y) = Convex [lim exact B(i, Qi)]. i-»oo (7.2) Definition. The contingent condition (7.3) D*y(x)eC(x,y(x)), where D* denotes the contingent derivative is called contingent equation asymptotic-ally induced by sequence g. Theorem C. Suppose that g is asymptotically lipschitzian and asymptotically extensive in W. 158 Then C(x, y) is ^lpper semicantinuous (in respect to inclusion) in 11(g) n W. If (xo> yo) e V(9) n W *hen there exists such a H-convergent subsequence h of g, that K(k) n TV is an extensive solution of contingent equation (7.3) passing through (x0i y0). (7.4) Remark 5. In our case the classical assumption of Zaremba— Marchaud theory on contingent and paratingent equations are satisfied. (7.5) Remark 6. The condition (7.3) can be considered as a control system with eliminated control variables. § 8 . Remark 7. Theorem B of § 6 can be easily reformulated for the case P == Rm x Rn, where m, n are arbitrary positive integers. I t can be even generalized for the case P = H x V, H and V being suitable topological spaces. 169 

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Arzela-like theorem with applications to differential equations and control theory

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Institute of Mathematics AS CR, v. v. i.