Nonlinear Optimal Control Theory

194 Nonlinear Optimal Control Theory

Assumption 6.9.1. (i) The matrices A, B, X, and R in (6.9.1) and (6.9.2)

are C

(1)

on an interval [a, b], as is the function d in (6.9.1).

(ii) For each t in [a, b] the matrix X(t) is symmetric, positive semi-definite

and the matrix R(t) is symmetric, p ositive definite.

(iii) For each t in [a, b], Ω(t) = O, where O is a fixed open set in R

m

.

(iv) The set B is the n-dimensional manifold cons isting of all points

(t

0

, x

0

, t

1

, x

1

) with (t

0

, x

0

) fixed and (t

1

, x

1

) in a specified n-dimens ional

manifold J

1

.

(v) The function g : (t

1

, x

1

) → g(t

1

, x

1

) is C

(1)

on R

n+1

.

(vi) The controls u are in L

2

[a, b].

The problem to be studied ...