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If there is a control satisfying this requirement then its size has to be large and this fact can cause problems in realization. This means that there are often apriori limitations on the size of admissible controls. We can express such restrictions by requirement that controls take their values in a bounded set M ⊂ RM . The additional assumption that M is also a symmetric and convex neighborhood of 0 does not seem to be too restrictive. The case x1 = 0 is more transparent. We denote MT =L∞ (0, T ; M ) T CT ={x0 ∈ Rn ; ∃u ∈ MT such that x0 + e−sA Bu(s)ds = 0} 0 C= CT .

To avoid difficulties which can be seen from the further computation, we compute the directional derivative for a special class of directions. Moreover, we take u ∈ K which is piecewise continuous and left-continuous on (t0 , t1 ]. Define admissible directions in the following way (Pontryagin’s 40 idea of a needle variation). Let u ∈ K and x be a solution to (2,3,1). Choose τ ∈ (t0 , t1 ] and v ∈ M. For sufficiently small ε > 0 a solution to y˙ =f (t, x, v) y(τ − ε) =x(τ − ε) does exist on [τ − ε, τ ].

Let u be piecewise continuous and left-continuous on the interval [t0 , t1 ] in which φ(x(t1 )) takes its minimum over u ∈ L∞ (t0 , t1 , M ) (where x is a solution to (2,3,1)). Let w be a solution to (2,3,3). Then for all v ∈ M and τ ∈ [0, t1 ] the following inequality holds < w(τ ), f (τ, x(τ ), u(τ ) > ≥ < w(τ ), f (τ, x(τ ), v >, (2,3,4) where x is a solution to (2,3,1) corresponding to u. The function H(t, x, u) :=< w(t), f (t, x, u) > is often called the Hamiltonian and the necessary condition reads as max H(t, x(t), v) = H(t, x(t), u(t)).

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