The Final Value Problem for Sobolev Equations

Let A and B be m-accretive linear operators in a complex Hubert space H with D{A) C D(B). The method of quasi-reversibility is used to obtain a solution to the Sobolev equation (d/dt)[(I + B)u(t)] + Au(t) = 0, 0 < l < 1, which approximates a specified final value u(\) = f. In general, when D(A) C D(B), it is not possible to find a solution which achieves exactly the final value w(l) = /. 1. Let A and Bhe a linear m-accretive operators in a complex Hilbert space H with D(A) G D(B). The purpose of the present note is to show how the method of quasi-reversibility [4] can be used to treat the final value problem (1.1) Lu = (d/dt)[(I + B)u(t)] + Au(t) = 0, 0 < t < 1, (1.2) «(1)=/. Since this problem is not well posed, in general, when D(A) G D(B), one may consider instead the problem of approximation of the final value, that is, given p > 0, find, if possible, a solution up of (1.1) such that ||wp(l) — /|| < p. Quasireversibility is a constructive method of determining such a solution. In this method, one approximates the operator L by a nearby operator Lp such that the final value problem for Lp is well posed (although the initial value problem may be ill posed; hence the term quasi-reversibility). The value v(0) of the solution of Lpv = 0, v(1) = /, is then used as an initial value in solving (1.1). Of course, various approximating operators Lp may be used. Here we approximate (1.1) by (1.3) Lpv = (d/dt)[(I + B + eA)v(t)] + Av(t) = 0, e = e(p). For this choice of L both the initial and final value problems are well posed. Furthermore, this type of approximation is stable in a sense to be made precise. Our choice of (1.3) is suggested by the results of [6] where such an approximation procedure is used to treat the special case B = 0. In fact, we shall show how the results of [6] can be used to obtain estimates in the general case as well. An additional condition imposed on the operators A and B is a sector condition: Received by the editors June 16, 1975. AMS (MOS) subject classifications (1970). Primary 35R20, 35R25; Secondary 47A50. © 1976, American Mathematical Society 247 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use