0

Consider the ergodic, infinite time optimal control problem: dx = [F(x) + G1 u]dt + G2 dW J = lim T->infinity E{ 1/T\int_0^T [Q(x) + u'Ru]dt}, F(0) = 0, Q(0) =0, Q(x) >= 0 Now suppose that Q(x) is replaced by Q1(x); Q1(0) = 0, Q1(x) >= Q(x). Then one would expect that V, and V1, the solutions of the HJB equation for the two problems would satisfy V1(x) >= V(x). Although in its LQG version the proof of this fact is trivial, I have been unable to prove it for the present more general case. Any help would be very much appreciated

george
  • 1
  • 1
    Please clarify your specific problem or provide additional details to highlight exactly what you need. As it's currently written, it's hard to tell exactly what you're asking. – Community Mar 10 '22 at 11:40
  • I did not specify it, but W is a Wiener process. A candidate solution of the optimal ergodic control is obtained by solving the Hamilton-Jacobi-Bellman (HJB) partial differential equation. The unknown in the HJB equation is V. In contrast to the finite time case, V cannot be identified as the cost-to-go. I am trying to show that in a second problem with Q1(x) replacing Q(x), with Q1(x) >= Q(x), then the solution of the corresponding second HJB equation, V1(x) satisfies V1(x) >= V(x). – george Mar 10 '22 at 20:38

0 Answers0