Exponential dichotomy, topological equivalence and structural stability for differential equations with piecewise constant argument

In this paper we study the exponential dichotomy and the structiiral stability for a class of differential equations with piecewise constant argument. We prove that an equation belonging to the above class has an exponential dichotomy if and only if it is topologically equivalent to the equation i p ' ( t ) = diag(ci,c2, • • • , C k ) i p ( t ) where Cj = 1 or Cj = -1, i = 1, .. .,k. Finally we show that if an equation has an exponential dichotomy then it is structurally stable. AMS Classification: 34K05. INTRODUCTION. Differential eqiiations with piecewise constant argument have applications in certain biomedical models [1], The strong interest in these equations is motivated that they describe hybrid dynamical systems ( a combination of continuous and discrete) and therefore combine properties of both differentiaJ and difference equations. Results concerning differential equations with piecewise constant argument are included in the papers [2], [7], [8], [9] and the references cited therein. In what follows we denote by | • | the Euclidean norm and by [•] the greatest-integer function. Consider the differential equation (1) x'{t) = A(t)®(t), te]R= ( 0 0 , oo) where A(t) isakxk continuous matrix ahd let X ( t ) be the fundamental matrix Solution of (1) such that J f (0) = 4 , is the fc x fc identity matrix. We define the space C of differential equations with piecewise constant argument of the form (2) 3/'W = A(t)y(t) + S(t)y([t]), teR 240 Papaschinopoulos where A{t), B{t) axe ky.k continuous and bounded matrices for t € Ml and the matrix (3) G(t)=xit)(x-\n) +j'X-\u)B{u)du^ , teM, n = [t], n<t<n + l is invertible for all t e iR with bounded inverse on R . A function j/: iR —» JR* is a Solution of (2) if the foUowing conditions are satisfied: (i) y is continuous on M, (ü) the derivative 3/' of y exists on M except possibly at the points t = n, n E Z = { . . . , -1,0,1,... } where one-sided derivatives exist, (iii) y satisfies (2). Consider the difference equation (4) y(n + 1) = r(n)j,(n), neZ where T{n) ia a. k x k invertible matrix on Z. We say that (4) has an exponential dichotomy with a projection of rank equal to r if there exist a projection P (P^ = P) having rank equal to r and constants Ä" > 1, o > 0 such that n> m ^^^ ll'WC/* < m>n where Y{n) is the fundamented matrix Solution of (4) such that y ( 0 ) = J*. An equation (2) which belongs to C is said to have an exponential dichotomy with a projection of rank equal to T if the difference equation (4) where T{n) = G{n + 1), G{t) is defined in (3) has AM exponential dichotomy with a projection of rank equal to T. Suppose that the differential equation (6) z'{t) = C{t)z{t) + D{t)z{[t]), teM belongs to C. We say that (2) and (6) are topologically equivalent if there exists a continuous function h: Mx iR* —» JJ* with the foUowing properties: (i) h(t,0) = 0, t G M and if |a ;| 0 (resp. |a;|-> 00) then |/i (t ,x)|0 (resp. |/i(t,a;)| —• 00) uniformly with respect to t, (ii) the map ht{-) = h{t, •) from i l * to Ji* is a homeomorphism for every t G jR, (iii) the function g : Mx M'' —» iR* defined by = /i^'(-) is continuous on Mx Jl* and satisfies (i), (iv) if y(t) is a Solution of (2) then h{t, y{t)) is a Solution of (6). Equation (2) is caUed structurally stable if there exists a constant ß > 0 such that if \A{t)-C{t)\<6, \B(t)~D{t)\<6 where C(t), D(t) are k x k continuous and bounded matrices, then equation (6) belongs to C and (2) is topologicaUy equivalent to (6). Piecewise constant eirgument 241 In Proposition 1 of this paper we prove that rf (2) belongs to C then for every s e R and ^ e H* there exists a unique Solution y{t) of (2) such that y{s) = In Proposition 2 we give a criterion for the exponential dichotomy of (2). We also prove that equation (2) belongiog to C has an exponential dichotomy with a projection of rank equal to T if and only if it is topologically equivalent to the equation (7) V'W = diag(ci, C2, • • •, CkMt), t e R where CJ = — 1, i = l ,2, . . . , r , CJ = l , i = T + 1, ... ,k (see Proposition 3 below). Finally in Proposition 4 we show that if an equation (2) which belongs to C has an exponential dichotomy then it is structurally stable. MAIN RESULTS. We prove now our main results. In the first proposition we study the existence and the uniqueness of the solutions of (2). PROPOSITION 1. Suppose that (2) belongs to C. Then for every s e JR and ^ e JR* there exists a unique Solution y{t) of (2) such that y{s) = ^ which satisfies the relation where n = [t], m = [s], t e R, n<t<n+l, T{n) = G{n + 1), n G Z, G(t) is defined in (3) and Tfu\ / n<t<n + l t = n PROOF. By the Variation of constants formula we have that y{t) satisfies the relation (9) y{t) = R{t)y{n), t e R, n = [t], n < t < n + 1. By tzdcing as t —» n + 1 in (9) sind since y{t) is a continuous function on jR it is obvious that y(n) satisfies the difference equation (4) where T{n) = G(n + 1), n G Z and so if [s] = m it holds , . , . _ r r ( n 1) • • • T{m)y{m), n>m Using (9) we get y{m) = R~^{s)y{s). Therefore relations (9) and (10) imply that (8) is satisfied. This completes the proof of the proposition. Using the same argument as in the proof of the Proposition 3 [7] we obtain the following proposition which presents a criterion for the existence of the exponential dichotomy of (2). We note that this proposition is not used in the sequel of this paper. 2 4 2 Papaschinopoulos PROPOSITION 2. Consider equation (2) where A(t), B(t) aiekxk continuous andbounded matrices for t e iR satisfying \A{t)\<M, \B{t)\<S where M, 6 are positive constants. Suppose that (1) has an exponeutial dichotomy with a projection P of rank equal to r , that is, there exist constants Ä" > 1, o > 0 such that |A-(t)PX-i(s)| < Ke-'^^-'l t > s \X{t){h < s > t