By V. N. Fomin (auth.)

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The control problem will be studied in the first two intervals of time (t = 0,1). ) is a non-negative objective function given for a pair of states of the control object. Because of the imposed conditions, the set of state variables X will have five values, viz. X = { -2, -1,0, I, 2}. ,x,). (a) Program Control. The class of program control strategy elements va ={("a,u = {(I, I), I )} (1,-1), (-1, I), (-l,-l)}. va consists of four 20 Chapter 2 Figure 2: Graphical Representation of Transition Dynamics of the State for the Example under Consideration It may be recalled that the programmed strategy realises the control law under the conditions that when the output ofthe control object (here itis y, =x,)is not being observed and hence the signal u, = ±1 is performed independent of what states the control object attains at t = 1.

The control signal is given by the relations u,=Kx" X'+1 =Ax,+Bu,-L(y,-Cx,), where K and L are matrices that determine the control strategy. 40) v-e y-,-t- where v~ is the set of possible noises v-, V, = (v,', v,"), and r is a given constant. Liapunov's method makes it possible to pick out the non-null set of matrices {KL). control objective. Application of Liapunov function to the given problem will now be explained. 39) can be interpreted as the algorithm for prediction of state X'+1 from observation of y', u'.

T. t\Y(,,-i _ ,U . 1,-1-1 ,W (li. 6) U.. 1). 2 Bellman's equation/or stochastic control systems Dynamic programming as given above is closely related to Bellman's functional U·- equation. -1-1. -,-1) = . 5). 15). -21_ )- . 9) Finite Time Period Control 23 that follows eqns. 7). The structure of eqn. 8) is considerably more complex than that of eqn. ; U'·-'-I) at any instant t is not only the function of the variables z"-', U'·-'-1 but also of the preceding control strategy U', -, -1 0 . 8) there is averaging by ensemble operation.