On adaptive control problems of continuous-time stochastic systems

Let x = (xt)t≥0 be a scalar observed process with control u = (ut)t≥0 described by the stochastic differential equation dxt = θxtdt + utdt + dwt, t ≥ 0, driven by the standard Wiener process (wt)t≥0. Assume the parameter θ to be unknown. The problem solved in this paper is to approximate the process x to the stable Ornstein-Uhlenbeck process x = (xt )t≥0 with the given dynamic parameter a < 0, satisfying the equation dxt = ax 0 t dt + dwt, t ≥ 0, by choosing the control process u. More precisely, based on continuous observation of x, for given ε > 0 an adaptive control law u = (ut )t≥0 is constructed, such that the corresponding observed process x = (xt )t≥0, x ε t = xt(u ) satisfies the following relations sup t≥0 E(xt ) 2 ≤ L and sup t≥tε E(xt − xt ) ≤ ε, where L is some constant (independent of ε) and tε is an unboundedly increasing as ε → 0 non-random function. The first relation ensures the stability of the object x. The rate ε−1 ln ε−1 of increase of tε is obtained. A similar problem is solved for a stochastic delay differential equation with an unknown parameter.