The risk-sensitive homing problem

Abstract

The ‘homing' optimal control problem, described in Whittle and Gait (1970), is given a risk-sensitive formulation. It is shown that the reduction of an optimally controlled homing problem to the treatment of an uncontrolled process, demonstrated by Whittle and Gait, can be achieved in the risk-sensitive case. Two scalar problems are analyzed in detail.

Solving an LQG homing problem via a different uncontrolled process

“…(5) Finally, we mention the fact that Kuhn (1985) has generalized Whittle's theorem to the risk-sensitive case. However, when the special relation between the noise and the control matrices is not satis® ed (as in this note), we are almost forced to limit ourselves to the risk-neutral case.…”

Probabilistic Solutions to Optimal Control Problems

“…Then the process (x 1 (t), x 2 (t)) is a controlled two-dimensional standard Brownian motion. Furthermore, we take b 1 [x i (t)] b 2 [x i (t)] 1 and we choose q i x i t x 2 i t, for i 1, 2, in the cost function de®ned in (3). Finally, we suppose that @D in the de®nition of the ®rst passage time T(x 1 , x 2 ) is given by @D fz 1 ; z 2 P R 2 : z 2 1 z 2 2 k 2 g; 7…”

“…The Wiener or Brownian motion process has also been used, in particular, by Whittle (1982, p. 290) and Kuhn (1985) as a rudimentary model for the trajectory of an aircraft. They found the control that minimizes the expected value of a cost criterion for which the objective was to obtain an optimal landing of the aircraft.…”

“…Next, Kuhn (1985) has given the LQG homing problem a risksensitive formulation and Lefebvre (1997a) has shown that a result related to that of Whittle can be obtained in a special one-dimensional case, even if the relation that corresponds to (5) is not satis®ed. Moreover, Lefebvre (1997b) has used the extension of Whittle's theorem to obtain the optimal control of a Brownian motion process by using a geometric Brownian motion, and conversely.…”

Exact solutions to two-dimensional homing problems

“…Then, the objective is to minimize the expected value of the time spent by the controlled process inside the continuation region … d 1; d 2 † , while taking the quadratic control costs into account. Next, the problem set up by Whittle was given a risksensitive formulation by Kuhn (1985) and was extended in various ways by the author (see, in particular, Lefebvre 1989Lefebvre , 1997.…”