Nonlinear Optimal Control Theory

The Maximum Principle and Some of Its Applications 199

Hence

W

t

= −

1

2

hx, Xxi +

1

2

hW

x

, BR

−1

B

t

W

x

i − hW

x

, Axi. (6.9.18)

The form of Eq. (6.9.18) leads to the conjecture tha t there exists a solution

of the Hamilton-Jacobi equation (6.2.11) of the form

W (t, x) =

1

2

hx, P (t)xi, (6.9.19)

where for each t, P (t) is a symmetric ma trix. Fo r then

W

x

= P x W

t

=

1

2

hx, P

′

(t)xi, (6.9.20)

and for proper choice of P (t) we would have a quadratic form in the left equal

to a quadratic form on the right.

If we assume a solution of the form (6.9.19), substitute (6.9.20) into

(6.9.18), and r ecall that P

t

= P , we get

1

2

hx, P

′

xi = −

1

2

hx, Xxi +

1

2

hx, P BR

−1

B

t

P xi − hx, P Axi. (6.9.21) ...