By Angelo Marzollo
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Extra resources for Controllability and Optimization: Lectures Held at the Department for Automation and Information September – October 1969
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 ! , is an there are oot OQ 54 Continuatio n and conclusion timal control laws, all of them transferring trols which have the same norm ~0 affect the norm e" ~i with con- e. A ~1 We may check that the value of not in simply by computing •••• e ; A ~'II does from II(35) and II(40) we have from which II(42) t=O The formula II(42) with also when is of course valid Y ( t) is non-singula r.
The results of the previous chapter, namely theorem 1,4, will be heavily exploited. We shall exclude from our considerations the trivial case in which, if time interval, there exists therefore the function ! 1 Let us first prove a theorem concerning time optimal controls. o and~~ , time interval K = [to, t1] and the class U! of admissible controls, then there exists a least interval k = [t 0 , t ] , ( t 0 < t ~ words I if a system is t 1 ) in which it is controllable. In other (~0 I! 1 I u; I K) controllable I then there exists a time optimal control.
Let us distinguish two cases: in the first one the system I(l) is controllable at time points of R'TI. e2 -i! (i) is positive semide- is still controllable with respect to therefore the matrix C(t) = V(t)- finite. We shall see that it is still possible to find an explicit expression of the optimal control law. To do that, we need the following Theorem 11,5 Let Then, i f R [v (t)] be the range of the matrix C (t) is positive semidefinite, the vector i! V (t). (t) belongs to R[Y(t)] Proof Let N [Y (t)] the set of vectors ~ such that the null space N [ Y ( t) as an eigenvalue of be the null space of Y (t) ~ J is equal Y ( t) =0 .
Controllability and Optimization: Lectures Held at the Department for Automation and Information September – October 1969 by Angelo Marzollo