A6struct-The problem of residence time controllability in dynamical systems with stochastic perturbations is formulated. The solution is given for linear systems with small, additive, white noise perturbation. It is shown that the existence of the desired residence time controller depends on the relationship between the column spaces of the control and noise matrices. If the former includes the latter, any residence time is possible. If this incusion does not occur, the achievable residence time is bounded, and we give lower and upper estimates of this bound. For each of these cases, controller design techniques are suggested and illustrative examples are considered. The development is based on an asymptotic version of the large deviations theory.
I. THE PROBLEMIVEN a controlled dynamical system with states x(t) E W", G control u(t) E W", and disturbances [ ( t ) E W r , assume its desired behavior is specified by a pair {Q, 7) where Q C W" is the domain to which the states x(t) should be confined and 7 is the
W+.For example, in the problem of telescope pointing [l], the domain Q is defined by the size of the film grain, and 7 is defined by the time of the exposure. In the laser beam pointing problem [2], Q is defined by the cross section of the beam and the size of the target, whereas 7 is defined by the duration of the pulse. In the gun pointing problem [3], Q is defined by the size of the target and the power of the explosives, whereas 7 is defined by the incidence time, i.e., time during which the shell travels in the barrel. In the robot arm pointing problem [4], Q is defined by the relative sizes of the gripper and the object to be manipulated, and 7 is defined by the duration of the task. In the aircraft landing problem [ 5 ] , 62 is defined by the parameters of the aircraft and the touchdown area, whereas 7 is defined by the landing period. In the missile terminal guidance problem [6], Q is defined by the domain to which the line of sight rate should be confined, and 7 is the period of the intercept. Given a pair { Q , 7}, the problem of residence time control is formulated as the problem of choosing a feedback control law, so as to force the states x to remain, at least on the average, in Q during period 7, in spite of the disturbances [ ( t ) that are acting on the system.Modem control theory does not offer tools for a direct solution of this problem. Indirect approaches, such as pole placement, LQG design, covariance control, H,, and H , minimization techniques do not seem to give explicit relationships with the residence time. Therefore, given a relative importance of pointing problems in modem technology, the development of a control theory for processes specified by a pair { Q , 7 } seems desirable.In this paper such a theory is developed for linear systems with small, additive, stochastic perturbations under the assumption that all states are available for control and the control law is of the Manuscript
