Speaker: B. Maslowski

Title: Ergodic Control of Semilinear Stochastic Evolution Equations

Abstract: These results concern the ergodic control probles for stochastic evolution equations of semilinear type and the associated Hamilton-Jacobi-Bellman equation. The basic example is the controlled stochastic reaction-diffusion equation. A standard method of solving the above control problem in finite dimensional state spaces is based on the Dynamic Programming Principle by means of which the corresponding HJB equation is derived. In finite dimensions, the HJB equation associated with the ergodic problem reads

\begin{equation}
{1\over 2}\hbox{\rm tr}\left(D^2v(x)\right)+\left\langle Ax+F( x),Dv(x)\right\rangle +f(x)-H(Dv(x))-\rho =0,
\end{equation}

respectively, where $H$ denotes the Hamiltonian of the problem and $D$ and $D^2$ stands for the Fr\'echet derivatives. The unknown in the first of the above equations is the pair $(v, \rho )$, where $v$ is a sufficiently smooth function and $\rho $ is a real constant. In finite dimensional case it is known that the optimal cost and control may be expressed in terms of the solution to the abovec HJB equation. In infinite dimensions, however, the above equation is just formal because of the trace term in the equation (typically, infinite-dimensional Laplacian) and the unboundedness of the operator $A$.

Our aim is to give a rigorous sense to the above HJB equation by means of the generator of the corresponding OU semigroup. Then the existence of the generalized solution as well as uniqueness up to an additive constant in the first component is proved by a suitable limit passage in the "discounted" HJB equations as the discount tends to zero. Finally, the optimal control and optimal cost are found analogously to the finite-dimensional case.

(These results were obtained jointly with Ben Goldys).