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Hamiltonian format of Pontryagin's maximum principle Revaz V. Gamkrelidze Steklov Mathematical Institute, RAS, Moscow, Russia e-mail: gam@mi.ras.ru

1. Hamiltonian format is native for the maximum principle regardless of any special restrictions imposed on the control equations of the optimal problem under consideration. It assigns canonically to any optimal problem a Hamiltonian system with parameters, complemented with the maximum condition, which "dynamically" eliminates the parameters in the process of solving the initial value problem for the Hamiltonian system as we proceed along the tra jectory. Thus the extremals of the problem are generated as simultaneous solutions of a regular Hamiltonian system with parameters and the maximum condition (to which all singularities of the problem are "relegated"), and not as a Hamiltonian flow, i.e. as a family of solutions of an initial value problem of a Hamiltonian system (without parameters), to which the Euler-Lagrange equation could be reduced in the regular case of the classical calculus of variations. 2. I shall give here an invariant formulation of the Hamiltonian format of the maximum principle for the time-optimal problem. Let the control system be given by the equation dx = X (x, u) = X, u U, dt where X is a vector field on the configuration space M , x M , u is the control parameter, U is the set of admissible values of u. With the vector field X we canonically associate a scalar-valued fiberwise linear function HX on the cotangent bundle T M , HX ( , u) = , X ( , u) , T M , u U, where : T M - M is the canonical pro jection. Hence a family of - Hamiltonian vector fields H X defined by the family of Hamiltonians HX canonically corresponds to the time-optimal problem. Every initial value T M , M , defines an extremal of the time optimal / problem as a tra jectory (t), (0) = , of the Hamiltonian vector
def


- field H X , from which the parameter u U is "dynamically" eliminated by the maximum condition HX ( (t), u(t)) = max HX ( (t), v ).
v U

Every solution of the optimal problem could be obtained in the described way. 3. Since the time-optimal problem is completely defined by the vector field X (x, u), it is natural to expect that every first order "infinitesimal ob ject" invariantly connected with our problem should be canonically (tensorially) expressed through the differential of the flow etX . To give this expression in our case, we first remark that the Hamilto- nian vector field H X coincides with the vector field adX on the cotan gent bundle T M , which is uniquely defined by the equations adX a = X a a C (M ), adX Y = [X, Y ] Y V ect M .
tL
X

Let LX be the Lie derivative over the field X , e is the differential of the flow etX on M .

=e

tX L

, where e and e (M ),

tX

According to the existing duality between the flows et expressed by the identity e
tX

X

tadX

, X = et

L

X

, e

tad

X

Y
X

Y V ect M ,


(1)

we have

etadX = e

- tL

=e

tL

X

-1

.

- Hence the flow generated by the Hamiltonian vector field H X = adX is inverse to the conjugate of the differential etX = etLX , in particular, it is a bundle isomorphism of the cotangent bundle T M - over the flow etX for t, and the vector field H X is a "Hamiltonian lift" of the vector field X . Differentiating the above identity with respect to t we establish the "infinitesimal" duality between LX and adX (the generalized Leibnitz rule), X , Y = LX , Y + , adX Y Y V ect M , (1) (M ). The indicated relations completely identify the Hamiltonian vector - field H X , hence the Hamiltonian format of the maximum principle. 4. Whereas the Lie derivative LX and the flow it generates on the tangent bundle T M , the differential etX = etLX , are, in one or another 2


way, the ob jects of everyday mathematical practice, the dual vector - - field to LX , the Hamiltonian lift HX = adX on the cotangent bundle T M , and the corresponding flow etadX were first introduced for computational purposes only in 1956 by L. S. Pontryagin under the name of "conjugate system" , and through the discovery of the maximum principle became since then a standard computational tool in engineering practice. Today, they are absolutely inevitable in optimization problems related to tra jectory variations. I think, it would be historically justified to baptize the vector field adX , (considered precisely as a vector field on T M , and not as a derivation on the C (M )-module of vector fields on M ), as the Pontryagin derivative.

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