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For technical reasons one makes the restriction on the controllable thrust. g. e. v(t 1 ) = 0, m(t 1 ) = m1 . Among all possible control functions find the one that maximizes h(t 1 )· 2) We consider a forced oscillation, which can be given by a linear differential equation of second order: ~. + a(t)i + b(t)x u (t) . This process is controlled by the external force U; therefore u is called the control function. Let the initial condition of the system be determined by (** ) x(O) O. g. continuous.

Again we begin with the primal problem in normal form: (p) Let us define the set b, x ~ a}. (b-Ax,cTx) E mm x ]R : x ~ O}. A obviously has the following properties: 1. e. A is convex. For if x 1 ,x 2 T (b-AX 1 ,C x 1 ), have T with x 2. ~ 0, then (b-AX 2 ,c x 2 ) E A and for any A E [0,1] we = (1-A)X 1 + AX 2 ~ O. e. A is a "cone with vertex at (b ,0)") . For if x ~ 0, then (b-Ax,cTx) E A and for A ~ 0 we have (1-A) (b,O) + A(b-Ax,cTx) Thus A looks rough~yas = (b-A(AX),CT(AX)) EA. follows (b, 0) Hence we can formulate the primal problem in the following terms: (P) Minimize B subject to (O,B) E A.

P) is therefore geometrically the lowest intersection point of A and the m-axis . For given fixed (y,a) E mm x m 35 is a hyperplane in :mm x :m, which is not parallel to JR • is the closed nonnegative halfspace generated by this hyperplane. To (P) we now associate the following problem: (0) Maximize a on N := {(y,a) € :mm x :m: A C H+(y,a)}. Formulated in words: Among all hyperplanes not parallel to :m and containing A in the nonnegative closed halfspace they generate find the one whose intersection point with the :m-axis is as large as pos- sible.

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