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By J. Topuszanski

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7) p As a consequence oz p the saddle point nature oz Ix, p) the eigenvalues 39 of ~ cannot be purely complex and periodic sinusoidal motions(1) are ruled out in the vicinity of (x, p), since closed contours of constant H cannot exist in the neighborhood of (x, p). This result will be of far reaching consequences. An optimal trajectory when the horizon is infinite has to satisfy the EulerLagrange equation, if the problem can be formulated as a problem of Lagrange with unbounded time interval. The preceding result suggests that the extremal trajectory will either converge toward x or diverge depending on the correct choice of the initial value p(O) for the costate variable.

21> 0 where k is the one-dimensional state variable, c is the onedimensional control variable and Wt is the accrued (discounted) z utility up to time t z • Let k* : [O,~) ~ R be a trajectory emanating zrom k o and generated by an admissible control c* : [O,~) ~ R which is optimalover an Thus, according to the maximum principle, the inzinite time horizon. 36». 23) constitute a complete system which would characterize k*, c*, A iz a complete set (k o ' Ao ) oz initial conditions were specizied. The dizziculty is that only k o is known.

O - + E. 8 Jzo(x*(t),u*(t),tldt o E. 14). (iil tory. zind Clearly ve have a eontradietion. Mov ve extend the prooz to the ease oz an agreeable trajee- Suppose this is not true. E. ,H lim W ~ H (2. 16) ~ ~: ~* W (x It:f) ~ ~ > 0 (2. 4. 1 illustrates this theorem. Generally speaking, the theorem shows that i:f a trajeetory is optimal any seetion o:f this trajeetory, emanating :from XO is optimal, in the elassieal sense, in the elass o:f all trajeetories which have the same :fixed end-point. 3 A maximum principle :for an in:finite horizon control problem Now the main theorem can be proved.

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An Intro to Symmetry and Supersym. in Quantum Field Theory by J. Topuszanski


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