ON AN INFINITE-DIMENSIONAL DIFFERENTIAL EQUATION IN VECTOR DISTRIBUTION WITH DISCONTINUOUS REGULAR FUNCTIONS IN A RIGHT HAND SIDE MICHAEL

An infinite-dimensional differential equation in vector distribution in a Hilbert space is studied in case of an unbounded operator and discontinuous regular functions in a right-hand side. A unique solution (vibrosolution) is defined for such an equation, and the necessary and sufficient existence conditions for a vibrosolution are proved. An equivalent equation with a measure, which enables us to directly compute jumps of a vibrosolution at discontinuity points of a distribution function, is also obtained. The application of the obtained results to control theory is discussed in the conclusion.


Introduction
This paper studies an infinite-dimensional differential equation in vector distribution, whose right-hand side also contains discontinuous regular (not generalized) functions.It should be noted that a solution to a differential equation in distribution cannot be defined as a conventional solution (using the Lebesgue-Stieltjes integral) owing to multiplication of the distribution by a discontinuous regular function.Thus, the basic problems are to introduce an appropriate solution (vibrosolution), obtain the existence and uniqueness conditions for a vibrosolution, and design an equivalent equation with a measure, which enables us to directly compute jumps of a vibro- solution at discontinuity points of a distribution function.
Infinite-dimensional equations in vector distribution appear, for example, when solving the ellipsoidal guaranteed estimation problem [14] over discontinuous observations [2], or considering infinite-dimensional (solid state) impulsive Lagrangian systems [4].The definition of a unique vibrosolution to a differential equation is first introduced in the background paper [9] and is shown again in Section 3. Finite-dimensional differential equations in scalar distribution with discontinuous regular functions in right-hand sides are studied in [1].Finite-dimensional  equations in vector distribution are then considered in [3].This paper generalizes the results ob- tained in [1,3] to the case of infinite-dimensional differential equations in vector distribution.The substantiation of existence and uniqueness conditions is based [7,8] on the representation of a solution to a differential equation in a Hilbert space as a Fourier sum of solutions to finite- dimensional differential equations.
The paper is organized as follows.The problem statement is given in Section 1.In Section 2 a solution to an infinite-dimensional equation in vector distribution is introduced as a vibrosolu- tion, that is defined as a unique limit.Sections 3 and 4 present the necessary and sufficient exis- tence conditions for a vibrosolution, respectively.By definition, existence of a vibrosolution yields its uniqueness.In Section 5 an equivalent equation with a measure is designed.The application of the obtained results to control theory is discussed in the conclusion.

Problem Statement
Let us consider an infinite-dimensional differential equation in vector distribution with an unbounded operator in a right-hand side [c(t) Ax(t) + f(x, u, t) + (x, u, t)b(x, u, t)i(t), x(to) Xo, (1) where x(t) E H; A is a generator of a strongly continuous semigroup such that A) is a strongly positive operator and has a compact inverse operator A 1; I(x, u, t) e H, b(x, u, t) e L(Rm-H) are bounded continuous functions defined in the space H x Rmx R, L(A%) is a space of linear continuous operators from a space 2. to a space %; fl(x, u, t) G R is a scalar piecewise continuous in x,u,t function such that its continuity domain is locally connected; (t)= (Ul(t),...,u,(t))e R" is a vector bounded variation function which is non-decreasing in the following sense: u(t2) >_ u(tl) as t 2 >_ tl, if ui(t2) >_ ui(tl) for 1,..., m.
Let St(.):H---H be a strongly continuous semigroup generated by an operator A, and D(A) C H be a definition domain.The following conditions imposed on an initial value and a right-hand side of the equation (1): 1) St(xo) e D(A), 2) St_s(f(x,u,s)+(x,u,s)b(x,u,s) iv(s)) D(A), s < t, are assumed to hold for any absolutely continuous non-decreasing function () e/.

Definition of a Solution
Let us note that a solution to the equation (1) cannot be defined as a conventional solution owing to multiplication of distribution (t) by a discontinuous in function b(x(t), u(t), t).
If u(t) R TM is an absolutely continuous function, then an absolutely continuous solution to the equation ( 1) is dCnd in the sense of Filippov [6].Following [6], a function (x, u, t) is said to be piecewise continuous in a finite domain G C H x R m + 1 if 1) a domain G consists of a finite number of continuity domains Gi, i-1,...,m, with boundaries Fi, 2) a function (x, u, t) has finite one-side limiting values along boundaries r i,

3)
the set consisting of all boundaries r has zero measure.
A function (x,u,t) is said to be piecewise continuous in H x R 'n+ if it is piecewise continuous in each finite domain G C H x R m+ 1.
If u(t) R" is an absolutely continuous function, then a solution to the equation ( 1) is defined [6] as an absolutely continuous solution to the differential inclusion 2(t) E Ax(t)/ F(x, t), where r(,t) is minimum convex closed set containing all limiting values f(x*,u(t),t)+ /3(x*, u(t), t)b(x*, u(t), t)i(t) as x*--x, t const, while points (x*, u(t), t) are not included in a discontinuity set of the function f(x, u(t), t) + (x, u(t), t)b(x, u(t), t)it(t).
The existence and uniqueness conditions for an absolutely continuous solution to the equation (1) are given in the next lemma that is a direct corollary to theorem 1 [11].
Lemma: Let the above conditions hold, and the functions f(x, u, t), (x, u, t)b(x, u, t) satisfy the one-sided Lipschitz condition in x (x y, f(x, u, t) f(y, u, t)) <_ m (t, ?l)(x y, x y), where functions ml(t,u G R, m2(t,u G R TM are integrable in t,u; B*:H--R TM is an operator ad- joint to an operator B: Rm---H. Then there exists a unique absolutely continuous solution to the equation (1) corresponding to an absolutely continuous function u(t) G Rm.
In case of an arbitrary non-decrease function u(t)G RTM, a solution to the equation ( 1) is defined as a vibrosolution [9].A vibrosolution is expected to be a function discontinuous at discon- tinuity points of the function u(t).
Definition 1: The convergence in the Hilbert space H lim xk(t) x(t), >_ to, is said to be the ,-weak convergence if the following conditions hold 1) lira Ilxk(t0)-x(t0)l] -0, t>_to, 2) lim I I xk(t) x(t) O, >_ to, in all continuity points of the function x(t),

3)
sup kVarc[to, T]x (t) < cxz for any T > to, where a variation of a function x(t) H is defined by N Var[a,b]x(t) I I x(t)II /su p .I I x(ti)-x(ti-1)II, (2) =1 and supremum is over all possible partitions v (a to, tl,... ,t N b), [[. ]1is the norm in the space H.
Definition 2: The left-continuous function x(t) is said to be a vibrosolution to the equation (1) if the ,-weak convergence of an arbitrary sequence of absolutely continuous non-decreasing functions uk(t) R TM to a non-decreasing function u(t) and the unique limit x(t) occurs regardless of a choice of an approximating sequence {uk(t)}, k--1,2, 4. Existence of a Solution.Necessary Conditions As in case of a finite-dimensional differential equation in distribution [1, 3], existence of a vibrosolution to an equation ( 1) is closely related to the solvability of a certain associated system in differentials.
Theorem 1" Let the lemma conditions hold.
If a unique vibrosolution to the equation (1) exists, then a system of differential equations in differentials in the space H d z(, , )(, , ), () z, () is solvable inside a cone of positive directions K-{u E Rm'ui >_ w i, i-1,...,rn} with arbitrary initial values w RTM, w >_ u(to) z H, and s >_ o.
Proof: Consider a vibrosolution x(t) to an equation ( 1) with an initial value x(s)-z and a function u(t) w + (v w)x(t s), where x(t s) is a Heaviside function, w, v RTM, v >_ w, and s >_ 0. By virtue of the theorem conditions this vibrosolution exists.Let us prove that under the theorem conditions the Kurzweil equality [10] x(s-t- holds as x(s + -lim x(t), t--s +, where a limit is regarded in the norm of the space H, and y(-) By virtue of the given lemma and the theorem conditions, an absolutely continuous solution to the equation ( 5) y(') exists and is unique, if v _> w.Following the proof of theorem 1 [12], it is readily verified that under the theorem conditions, the functions y#C(r)=x#C(s+r/k), 0 _< r_< 1, k= 1,2,..., where x#c(t)are vibrosolutions to equations (1) with initial values x(s) z and absolutely continuous functions uk(t) w, if t _< s, uk(t)-wWk(v-w)(t-s), if s <_ <_ s+ 1/k, and uk(t)-v, if >_ s+ 1/k, in right-hand sides, are solutions to the equations and the following equality holds xk(s + l/k) yk(1). (6) By virtue of the theorem on continuous dependence of a solution to a differential inclusion on a right-hand side in a Banach space [11], a sequence of absolutely continuous functions yk(r) con- verges to an absolutely continuous solution to the equation ( 5) pointwise in the norm of the space H" lim I I Yk(7) y( 7)II 0, -+oo, , e [0,1].
Define the function (z,w, v,s) by 1 ((z, w, v, s) y(1) z + / (y(r), w + (v w)r, s)b(y(r), w + (v w)r, s)(v w)dr, o where y(r) is a solution to the equation ( 5).Since a solution y(r) exists and is unique under the theorem conditions, if v >_ w, the function (z, w, v, s) E H is uniquely defined inside a cone K-{uERm:ui>wi, i-1,...,m}.It only remains to prove that dv v, v, v, v, However, the proof of this correlation is quite consistent with the last part of the proof of theorem 1 [12] and can be omitted here.Thus, the function (z,w, v,s) defined by ( 7) is a unique solution to the system of equations in differentials (3) inside a cone K as s 0. Theorem 1 is proved.

Existence of a Solution. Sufficient Conditions
Let us prove that under additional conditions imposed on a function t(x, u, t)b(x, u, t) the necessary existence conditions for a vibrosolution to an equation (1) coincide with the sufficient ones.
If a system of differential equations in differentials (3) is solvable inside a cone of positive directions K {u _Rm: u >_ wi, 1,.m} with arbitrary initial values w RTM, w >_ u(to) z H, and s >_ to, then a unique vibrosolution to the equation (1) exists.
Proof: Let [uk(t)},k 1,2,..., be a sequence of absolutely continuous non-decreasing functions uk(t) RTM, which tends to a distribution function u(t) in the sense of the .-weakconvergence.Consider the equation (1) with absolutely continuous non-decreasing functions uk(t) in a right-hand side, that is ie(t) Axl(t) + f(x , uI, t) + fl(xk,uI, t)b(x I, u1, t)izl(t), xk(to) x o. (8) It should be noted that the theorem conditions (1)-(3), the lemma of Section 2, and the theorem 1 [11] yield existence and uniqueness of an absolutely continuous solution to the equation (8).As follows from [7,8], this solution can be represented as a Fourier sum in the space H on the com- plete orthonormal basis {ci}i= o generated by eigenfunctions of the operator A ()-(t)c.
{'i}=0 is a countable set [7] of eigenvalues of the operator A, and fi(x,u,t) E R, bi(x u, t) Rrn, Xio R, 0, 1, 2,..., are Fourier coefficients for a function f(x, u, t), an operator o respectively: b(x, u, t), and an initial value x 0 on the basis {ci} o, The convergence of the Fourier series ( 11) is regarded in the norms of the corresponding Hilbert spaces.
Consider an infinite (i-0, 1,2,...) number of finite-dimensional equations (10) which contain an arbitrary non-decreasing function u(t) R m in right-hand sides dxi(t Aixi(t)dt + fi(xi, u, t) + (xi, u, t)bi(xi, u, t)du(t), xi(O Xio (12) whose solutions are thus regarded as vibrosolutions.Since xi(t R are scalar functions, existence and uniqueness of solutions to the equations ( 12) are assured of the existence and uniqueness theorem for a vibrosolution [3] by virtue of the inequalities Re(,i) < 0 [7], the theorem conditions (1)-( 3), and the solvability of the system of equations in differentials (3) inside a cone K.Then, taking into account the vibrosolution definition given in Section 2, we obtain the pointwise conver- gence of absolutely continuous solutions xki(t) to the equations (10) to vibrosolutions xi(t to the equations (12) lim x(t)x(t) O, k--,c, t >_ to, O, 1,2,...,   in all continuity points of the function u(t) as -lim uk(t) u(t), tcx, where ulC(t) R m are absolutely continuous non-decreasing functions.Thus, in all continuity points of the function u(t).
Consider the Fourier sum generated by the functions xi(t on the basis {ci} c i--0 xi(t)ci. ( Let us prove that the series (13) converges in the norm of the space H. Indeed, the following in- equalities [5] hold i=N -II {x0exp((t))/ exp((t-s))[f(,u,)ds/(x,,s)b(x,u,s)d()]}cll i=N to < I I :oexp(/(()))II + I I exp(Re(,i(t-s)))fi(xi, u,s)ds I I i=N i=N o / I I exp(Re()i(t )))/(, , )(, , )d() I I < , i-N o since functions fi(x, u, t) and (x, u, t)bi(x u, t) are bounded and satisfy the one-sided Lipschitz condition, limRe(li)-as i--* [7], t-s >0, the Fourier series (11) converge, and the latter integral is with a bounded variation function u(t).Thus, the Fourier series (13) converges, i.e., there exists an H-valued function z(t) E H such that N lim I I x(t)-E xi(t)ci l] 0, N---<x. i=0 Let us finally prove that the function x(t) H obtained as a Fourier sum ( 13) is a vibrosolution to the equation (1).For any > 0 there exist a number N 1 such that the inequality holds by virtue of the convergence of a Fourier series (13), and for any > 0 there exists a number N 2 such that the inequality holds by virtue of convergence of a Fourier series (9), Moreover, for any > 0 there exist a number N-max(N1,N2) and a number K such that for any k > K the inequality holds in all continuity points of the function u(t), since i() is a vibrosolution to the equation ( 12) and {()}, -1,2,..., is a sequence of approximating solutions to the equations (10).
Thus, for any e > 0 there exist a number N and number K such that for any k K we obtain in all continuity points of the function u(t).The inequalities ( 14) yield the convergence in the norm of the space H in all continuity points of the function u(t).Moreover,   x(to) (to) o by virtue of coincidence of initial values of the equations ( 1) and ( 8), and the inequality sup Var o[to, t]xk(T) < cx for any T > o holds by virtue of the uniform boundedness of variations of absolutely continuous functions xki(t), supYar[to, T]x(t < cxz for all i-0,1,2,.., and any T >_ to, and the convergence of the Fourier series (9).Thus, the ,-weak convergence in the space H limxk(t) x(t), k--<xz, t >_ to, is proved.Since xk(t), k-1,2,..., are absolutely continuous solutions to the equations (8) that are equations (1) with absolutely continuous non-decreasing functions uk(t)E R m in right-hand sides, the vibrosolution definition implies that the function x(t) H is a vibrosolution to the equation (1).Theorem 2 is proved.
Remark: The vibrosolution definition as well as the necessary and sufficient existence conditions can also be stated for nonmonotonic functions u(t) Rm, assuming that the one-sided Lipschitz condition holds for a function (x,u,t)b(x,u,t)sign(it(t)) and approximating functions (x k, uk, t)b(xk, uk, t)itk(t) for any k 1,2, 6. Equivalent Equation with a Measure It should be noted that only vibrosolutions, which correspond to absolutely continuous functions uk(t) Rm, are absolutely continuous solutions to a differential equation in distribution (1).Therefore, it is not clear how to compute jumps of a vibrosolution to a differential equation in distribution at discontinuity points of an arbitrary non-decreasing function u(t) tm.Thus, it is helpful to design an equivalent equation with a measure whose conventional (in the sense of the definition of a solution to an ordinary differential equation with a discontinuous right-hand side that is given in Section 2) solution coincides with a vibrosolution to an equation ( 1), and which enables us to directly compute jumps of a solution at discontinuity points of an arbitrary non-decreasing function u(t) Rm.
Theorem 3: Let the theorem 2 conditions hold.
have the same unique solution regarded for in equation (1) as a vibrosolution. (15) Here G(z,v,u,s) (z,v,v + u,s) z, where (z,v,u,s) is a solution to a system of equations in differentials (3); uC(t)is a continuous component of a non-decreasing function u(t), Au(ti) u(ti+ )--u(t is a jump of a function u(t) at ti, are discontinuity points of a function u(t), x(t-ti) is a Heaviside function.
Proof: A function of jumps G(z,w, u,s) is bounded in the norm of the space H as a solution to the system (3) with a right-hand side (, u, s)b(, u, s) satisfying the one-sided Lipschitz condition.Then, by virtue of the lemma of Section 2 and the theorem 1 [11], a solution to the equation with a measure (15) exists and is unique as a bounded variation function with an absolutely continuous component in continuity intervals of the function u(t) and the jumps determined by the function G(y(t ),u(t ),Au(ti),ti) at discontinuity points of the function u(t).As follows from [7, 8], this solution can be represented as a Fourier sum in the space H on the complete orthonormal basis {ci}ic= o generated by eigenfunctions of the operator A" EYi(t)ci.
Consider an equation ( 1) with an arbitrary non-decreasing function u(t) G R TM in a right-hand side.Existence and uniqueness of a vibrosolution to such an equation have already been proved in the theorem 2. That vibrosolution can also be represented as a Fourier sum (13) on the basis {ci} i=0: where scalar functions xi(t satisfy the equations ( 12) dxi(t ixi(t)dt + fi(xi, u, t) + (xi, u, t)bi(xi, u, t)du(t), xi(O Xio. The equations ( 12) and ( 17) are finite-dimensional equations whose right-hand sides contain vector distribution and piecewise continuous regular functions satisfying the one-sided Lipschitz condition.Thus, by the virtue of theorem 2 [3], unique solutions xi(t and yi(t) to the equations ( 12) and (17) coincide as vibrosolutions for any i= 0,1,2, The vibrosolution definition given in Section 2 implies that for any 0, 1,2..., the equalities (t)v(t) 0, >_ to, (18) hold in all continuity points of the function u(t).
Let us finally prove that solutions to the equations (1) and ( 15) are indistinguishable as vibrosolutions.Let I I (t)-v(t)11 0, >_ to, for at least one continuity point of the function u(t)G RTM.Fourier series on the basis {ci}i__ 0: Expand the mentioned solutions into (t)-v(t) ((t)-v(t))c, >_ to.
i--O By virtue of the equalities (18) and the uniqueness of the expansion into a Fourier series on the given basis, the equalities I I (t)-v(t) I I <_ _, x(t)-v(t) o, >_ to, i=0 also hold in all continuity points of the function u(t).Thus, the vibrosolution definition given in Section 2 implies that the functions x(t)E H and y(t) H are indistinguishable as vibrosolutions to the equations ( 1) and (15).In other words, the equations (1) and ( 15) have the same unique vibrosolution.Theorem 3 is proved.

Conclusion
The vibrosolution definition assumes uniqueness of a vibrosolution to an equation (1).This enables us to apply the obtained sufficient existence conditions for a vibrosolution to filtering equations for an infinite-dimensional process over discontinuous observations, for example [13], in case of simultaneous impulses in all observation channels (a scalar function u(t)).However, the results of this paper also enable us to consider a case of non-simultaneous impulses in observation channels (a vector function u(t)).

i=0
Scalar functions x(t)satisfy the equations