Controllability and Optimization: Lectures Held at the - download pdf or read online

By Angelo Marzollo

ISBN-10: 3211811230

ISBN-13: 9783211811238

ISBN-10: 3709129591

ISBN-13: 9783709129593

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Extra resources for Controllability and Optimization: Lectures Held at the Department for Automation and Information September – October 1969

Sample text

1 , to be meaningful, the region as it was in Chapter I; indeed, if some ~· ! at time t 0 into n , the same 1 ! 1 , ! 0 , 11 "transfers" X _o t 1 in the presence of a certain noise at ~· cannot effect the same trasfer for an appr£ priately different noise all € !! 1 applying the same reasoning to ; k , U. , the system would not be controllable in the present situation no matter how we choose ! 0 , ;1, \<. , u For semplicity, we shall here choose for the sphere Ae ( ~ 1) of radius A - AF.. 1) = {!

G 1 .. 1\ ! _ General solution of the previous problem 53 tion II(34) becomes or where At Y ( t) are the non-null (and real, since Y (t) . :·-~d. , and from II(33) we see that the optimal control law is II (41) where ~1 ~t •••• are arbitrary constants. The expression II(40) is of course valid also if Y (f) equivalent to II(37), with A 1 . . 2i. (t)~(t)= ~~ At are the (positive) eigenvalues of 3- n the components of ~ (t) , where Y (1) along the eigenvectors y (i) . 1 Let us now notice from II(40) that if 0 eigenvalue of and Y (t) with multiplicity !

A1 The diagonal matrix 5 A= - and = I ~ -~ II -- s-" = II ~ , Wl.. o• ! 1 • From the previous position ! (T) = ~1- V(T)~ 0 we have and therefore £(T)= o~(T)~*(T) ... _ ~2. z (T) = 58 where Solution of the exercise c1(T) and c 2 (T) are functions obtainable from E( 4) , but which are not relevant for our purposes. From E( 6) we see that C (T) has the same eigenvalues as the matrix C (T) has the same character as 'R ( T) as far and therefore as positive or negative semifinitness. By inspecting R(T)we easily see that a necessary condition for it to be positive semidefinite is that 16-8 e2T be zero, and this is verified only for -t T = -ln 2 = t 2 * .

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Controllability and Optimization: Lectures Held at the Department for Automation and Information September – October 1969 by Angelo Marzollo


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