Robust Optimal Stopping-Time Control for Nonlinear Systems

Robust Optimal Switching Control for Nonlinear Systems

“…The derivation of this characterization of V γ presented in this paper is a direct argument which parallels the argument given in [1] for the analogous result for optimal stopping-time problems. An alternative derivation of this characterization relying on a general comparison principle for viscosity super-and subsolutions of SQVI is given in [2].…”

“…While in principle it should be possible to solve the problem by using the construction in [1] to perform each iterative step in the procedure outlined in Section 5.1, it turns out that, for the example which we discuss here, one can solve explicitly by a direct, geometric, noniterative procedure which we now describe. We consider the special case of the general problem where there are only two controls A = {a 1 , a 2 }, with respective system dynamics given by…”

“…All other assumptions are satisfied with the exception that B = IR is not compact; to alleviate this difficulty, one can restrict B to a large finite interval [−M, M]; to live with this restriction, one must adjust the definition of the hamiltonian functions H 1 (x, p) and H 2 (x, p) in the discussion to follow. We will construct a solution to the SQVI (5.1) for this example via a variation of the algorithm presented in [1]; rather than proving that the solution so constructed is the minimal nonnegative supersolution of SQVI (5.1), we verify directly that it is the lower value function V γ = (V 1 γ , V 2 γ ) of the switching-control differential game (2.1). Because we will take µ > 1, for large |y| the control a 2 will drive the state toward 0 more strongly than a 1 .…”

“…Thus one can view stopping-time problems as having pedagogical value as stepping stones to the more complicated impulse-control and switching-control problems. In a companion paper [1], we treat a deterministic robust-control version of such optimal stopping-time problems.…”