29 Adaptive control of semilinear stochastic evolution equations L. Stettner Institute of Mathematics Polish Academy of Sciences Sniadeckich 8, 00-950 Warsaw, Poland Abstract In the paper a nearly self optimal strategy is constructed for the control problem of stochastic semilinear evolution equation depending on an unknown parameter with path wise average per unit time cost functional. Keywords Stochastic evolution equations, adaptive control, self-optimality 1 INTRODUCTION Let (H, I · IH) be a separable Hilbert space and (0, F, Ft, P 0 ) be a probability space. Consider the following semilinear stochastic evolution equation on H dX =(AX+ F(X))dt + B(X)dW X(O) = x E H (1) where A is a generator of a G 0 semigroup S(t) on H and W(t) is a cylindrical Wiener process on H adapted to Ft. Assume (Al) 3L > 0 such that IF(z)- F(y)IH lz- YIH and IIB(z)- B(y)IIL(H) lz- YIH, for z,y E H (A2) supzEH IIB- 1 (z)IIL(H) < oo (A3) 3(3 E (0, 1) such that for each T > 0, J{(l + < oo It follows from Da Prato and Zabczyk (1992) that under (A1)-(A3) there exists a unique mild solution X(t) to the equation (1) i.e. an H valued process X(t) such that fort;:::: 0 X(t) = S(t)x + l S(t- s)F(X(s))ds + l S(t- s)B(X(s))dW(s) (2) Moreover X(t) has a version with continuous trajectories that is a Markov process on H with transition operator P? and (see Peszat and Zabczyk (1994)) K. Malanowski et al. (eds.), Modelling and Optimization of Distributed Parameter Systems Applications to engineering © Springer Science+Business Media Dordrecht 1996