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Download PDF by Vladimir Igorevich Arnold (auth.): Catastrophe Theory

By Vladimir Igorevich Arnold (auth.)

ISBN-10: 354012859X

ISBN-13: 9783540128595

ISBN-10: 364296799X

ISBN-13: 9783642967993

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Singularities of Accessibility Boundaries A control system in phase space is defined as follows: at every point of the space we have not one velocity vector (as in the usual evolutionary system) but a whole set of vectors called the indicatrix of permissible velocities (Fig. 49). Fig. 49 The control problem is to choose at each moment of time a velocity vector from the indicatrix and by doing this to reach a target (for example to reach some subset of phase space in the shortest possible time).

52 illustrates one of the variants of this situation. Here it is clear that the T-marked accessibility boundary is 53 formed by the projections of separaticies of saddles (the outer points) under the mapping of a two-fold covering surface onto the plane. Over point 4 there is a node, into this node comes two saddle separaticies from different directions. At the node these two curves have a common tangent and (generically) can be represented in a suitable co-ordinate system by the equations of parabolas of degree a> 1 : y=A Ixl a for x:O:;;;O, y=Blxl a for x>O.

The network of limit curves is indicated in Fig. 49 together with the indicatrices of permissible velocities (being ellipses) and the angles formed by the possible directions of motion. The boundary of the of the accessability domain consists of segments of limit curves (and possibly segments of the target when the target does not lie within the region of complete controllability, see Fig. 50). These segments meet at points which are singularities of the accessibility domain boundary. Fig. 50 In Fig.

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Catastrophe Theory by Vladimir Igorevich Arnold (auth.)


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