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Additional resources for Topics in Control Theory: A Seminar given at the Forschungsinstitut für Mathematik, ETH, in 1964

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PROPOSITION (Roxin [ 8 where aA(x ) u E ,O,t) J). If 1-l- steering Im a X c A(x ) n u 0 a (t ) e A(x ) , u interior *) of 0 U t e A(x ) n aA(x ) , 0 A(x ) A(x ) . e [0 1 t Let 0 u sufficiently small 1 then 0 U 0 lex ) 0 t E 0 [0 1 t u ) denotes the be a neighborhood of A(x ) . Since ], 0 0 aA(x ) . where 0 a (t ) contained in u 0 for all 1 to 0 PROOF. 8 that for ~ •t u~ *) The interior of a set 0 1 t u (U) is a neighborhood of M will be written 0 M or Int M. - 33 - is in [t , t ] U 0 Ui, to the interval u x .

1). 1. 2. DEFINITION. A subclass U,. 1. If if it satisfies the following conditions: 1 u : I an interval of by = v { t) 1L. 2. If u u is in~~ w e Q ~ Q then the mapping Rl . 3. 4. U If I 1 I E U I u Q ~ is J defined n J u its restriction to 1 belongs to U. u is in 2i and defined by u u : I = [a 1 b] u ~ Q s e R, the w (t) = u (t-s) s is a mapping defined on a compact and if there exists a such that the restrictions of are in . v and U . 3. REMARK. L. • is obviously admissible. e. the mappings u : I u n -+ for which there exists a finite strictly increasing sequence = Iu [a ,a ] o n such that u of points a.

U is in 2i and defined by u u : I = [a 1 b] u ~ Q s e R, the w (t) = u (t-s) s is a mapping defined on a compact and if there exists a such that the restrictions of are in . v and U . 3. REMARK. L. • is obviously admissible. e. the mappings u : I u n -+ for which there exists a finite strictly increasing sequence = Iu [a ,a ] o n such that u of points a. I E l. u , is constant on each open (ai, ai+ 1 ), 0 ~ i ' n-1. e. 4. DEFINITION. A control vector field on {with control space s : X x Q -+ T (X) and 7l vector field is a controllable vector field on are called the controls of su with u n.

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