Introduction

In the present paper we consider an optimal control problem associated with equation ( 1): more precisely we wish to minimize the cost functional

proximation of problem ( l ) - (2), in order to obtain the optimal pair and the Riccati operator relative to problem (l)-(2) as a limit of the optimal pairs and

' yu ( t ,x ) - A y ( t ,x ) y(0 ,x ) = yo(x), j / t (M ) = 2/1 (*) y ( t ,x ) = 0

( t ,x ) e ]0,T [ x i2 x e f i ( t ,x ) e ] o , T [ x r 0 ( t ,x ) e ]o ,T [xT i

(i)

Jo Jn Jo Jri

overall controls u E L2(0 , T; Z/2(r i) ) , with y subject to (1). Actually we are mainly interested in studying parabolic regularization and ap-

the solutions to Riccati equations associated to a sequence of suitable problems of parabolic type. Our motivation comes from well known regularity properties of optimal solutions and Riccati operators in the case of parabolic-like dynamics.