solution  to the corresponding adjoint equation (5.4) is given by
(_ t) = h(t);[f + ug;h]z(t)i; (5.22)
where
[f;h](z) = Dh(z)f(z)Df(z)h(z) (5.23)
denotes the Lie bracket of the vector �elds f and h.
5.4 Singular optimal control
Now that we have the necessary tools to di�erentiate our switching function (5.13), we will
compute the derivatives of the switching function until we reach an explicit form for the
singular control. We will also compute the corresponding singular curve, and the Legendre-
Clebsch condition will be used to con�rm the optimality of the singular control.
Let I be an interval of positive measure where the switching function is identically zero
(t) = k Gq2 = 0; t 2 I: (5.24)
Expressing (t) in vectors forms, we get the following expression
2 3T 2 3
1 0(t) =6 7 6 7+ k (5.25)
4 5 4 5
2 Gq
and
(t) = h;g(z(t))i+ k = 0: (5.26)
Hence h;g(z)i is constant. Therefore,
(_ t) = h(t);[f;g]z(t)i = 0; (5.27)
29
Page 1  |  Page 2  |  Page 3  |  Page 4  |  Page 5  |  Page 6  |  Page 7  |  Page 8  |  Page 9  |  Page 10  |  Page 11  |  Page 12  |  Page 13  |  Page 14  |  Page 15  |  Page 16  |  Page 17  |  Page 18  |  Page 19  |  Page 20  |  Page 21  |  Page 22  |  Page 23  |  Page 24  |  Page 25  |  Page 26  |  Page 27  |  Page 28  |  Page 29  |  Page 30  |  Page 31  |  Page 32  |  Page 33  |  Page 34  |  Page 35  |  Page 36  |  Page 37  |  Page 38  |  Page 39  |  Page 40  |  Page 41  |  Page 42  |  Page 43  |  Page 44  |  Page 45  |  Page 46  |  Page 47  |  Page 48  |  Page 49  |  Page 50  |  Page 51  |  Page 52  |  Page 53  |  Page 54  |  Page 55  |  Page 56