Appl. Math. Optim. (1983) 10:275-286 Applied Mathematics and Optimization ©1983 Springer-Verlag New York Inc. Technical Note Regularity of Hyperbolic Equations Under L2(0 , T; L2( F))-Diriehlet Boundary Terms* I. Lasiecka and R. Triggiani Department of Mathematics, University of Florida, Gainesville, FL 32611 Communicated by A. V. Balakrishnan Abstract. With f~ an open bounded domain in R n with boundary F, let f(t; fo, fl; u) be the solution to a second order linear hyperbolic equation defined on ~, under the action of the forcing term u(t) applied in the Dirichlet B.C., and with initial data fo ~L2(~) and fl c H-l(~) - In a previous paper [6], we proved (among other things) that the map u -~ f®ft, from the Dirichlet input into the solution is continuous from L2(0, T; L2(F)) into L2(0, T; L2(~))®L2(0, T; H-I(~)). Here, we make crucial use of this result to present the following marked improvement: the map u ~ f®ft is continuous from L2(0 , T; L2(F)) into C([0, T]; L2(~))®C([0, T]; H-I(~)). Our approach uses the cosine operator model introduced in [6], along with crucial relevant fact from cosine operator theory. A new trace theory result, on which we base our proof here, plays also a decisive role in the correspond- ing quadratic optimal control problem [7]. When u, instead, acts in the Neumann B. C. and ~2 is either a sphere or a paralMepiped, the same approach leads to the same improvement over results obtained in [6] to the regularity in t of the solution (i.e., from L2(0 , T) to C[0, T]). 1. Introduction, Preliminaries, and Statement of Main Result The present note is a successor to (a subset of) [6]: it completes and markedly refines the authors' results in [6] regarding the regularity of solutions to (say) *This research was supported in part by the National Science Foundation under grant MCS 81-02837.