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Quadratic cost optimal controls can be solved by penalizing the governing linear differential equation [2], [9]. In this paper, we study the numerical analysis of this approach using finite elements. We formulate the geometric condition (H) which requires that pairs of certain related finite-dimensional approximation spaces form "angles" which are bounded away from the " 180? angle". Under condition (H), we prove that the penalty parameter E and the discretization parameter h are independent in the error bounds, thereby giving sharp asymptotic error estimates. This condition (H) is shown to be also a necessary condition for such independence. Examples and numerical evidence are also provided. 0. Introduction. Consider the optimal control problem: Given the quadratic cost functional J, J(x, u) _JT[(x, N,x )Rn ? Kx, N2x)Rn + (u, Mu) Rm] dt, solve (0.1) Min J(x, u) (x, u) E Hon X Lr subject to (0.2) {x(t) = A(t)x(t) + B(t)u(t) + f(t), t E [0, T], x )= 0, where x(t) E Rn is the state at time t, u(t) E R' is the control at t, A(t) and B(t) are, respectively, n X n and n X m time-varying matrices, and f is the inhomogeneous forcing term. In the cost functional J, we assume NI, N, ~M are constant n X n, n X n, and m X m symmetric (0.3) positive semi-definite matrices, {0K3) x, N1x )Rn > 1PIIXI12 n, (U, MU)Rm >. VIIUII|R for all x E Rn, u E R' where v > 0, is independent of x and u. The standard Sobolev norms and spaces used are as follows. Received November 17, 1981; revised February 26, 1982 and May 3, 1982. 1980 Mathematics Subject Classification. Primary 34H05, 49D30, 65N30; Secondary 41A65. * Supported in part by NSF Grant MCS 81-01892. Work completed while the third author was visiting Purdue University. * * Current address: Research and Development Department, Standard Oil of Ohio, Cleveland, Ohio 44128. (D1 983 American Mathematical Society 0025-571 8/82/0000-0724/$07.50

Sharp error estimates for a finite element-penalty approach to a class of regulator problems