Nonlinear Analysis, Theory , Methods & Applications, Vol. 5, No. 11, pp. 1213- 1224, 1981. Printed in Great Britain. 0362-546X/81/1 11213-12 $02 00/O 0 1981 Pergamon Press Ltd zyxwvutsr EXISTENCE AND APPROXIMATION FOR STATIONARY HAMILTON JACOBI EQUATIONS V. BARBU Faculty of Mathematics, University of Iasi, Iasi 6600, Romania and G. DA PRATO Scuola Normale Superiore, 56100 Pisa, Italy (Receiwd 18 February 1981) Key words andphrases: Optimal control, nonlinear operators 1. INTRODUCTION WE ARE HERE concerned with the operator equation: @P(x)l” + (Ax, Vcp(x)) = g(x), x E %4), (1.1) in a real Hilbert space H. Here -A is the infinitesimal generator of a strongly continuous semi- group of linear bounded operators on H and g is a continuous convex positive function on H. By Vq we have denoted the gradient of the unknown function cp. Under supplementary conditions on A and g, Theorem 1 below gives existence and unique- ness of solutions in the class of convex, Gateaux differentiable functions cpsatisfying the condition: ~(0) = Oandcp 3 OonH. Theorem 2 shows that this solution can be obtained as a limit for t + cc of the solution cp(t, x) to the Hamilton-Jacobi equation: cp,(r, 4 + :IVXcp(4 x)12 + (Ax, V,cp(t, 4) = g(x); x E %4), t 3 0, cp(O> 4 = cp,(x). i (1.2) Theorem 3 is concerned with the convergence of the sequence {cp,} defined by the implicit schema cp,,i(X) + @P,+i(412 + (&Vcp,+,(x)) = g(x) + cp,(x); n = 192,. (1.3) to the solution cp of (1.1). If A is positive definite then existence and uniqueness for (1.1) follows directly by using different approximating process already used in [ 11. It must be emphasized that (1.1) is relevant in control theory on infinite interval (see [2, 31) and in several other problems of physical significance. For comparison with other literature on stationary Hamilton-Jacobi equations most closely to present paper the works [4-71 which are concerned with Riccati stationary equations must be cited. Now we shall briefly recall some definitions and notations that will be used in the sequel. H is a real Hilbert space with norm 1. ) and inner product (. , . 1. Given a lower semicontinuous 1213