Nonlinear Optimal Control Theory

Existence Theorems; Compact Constraints 99

will be convex. To se e this, let by = (y

0

1

, y

1

) and by

2

= (y

0

2

, y

2

) be two points in

Q

+

(t, x). Then there exist points z

1

and z

2

in Ω(t), such that

y

0

1

≥ f

0

(t, x, z

1

) y

1

= A(t)x + B(t)z

1

+ h(t) (4.5.3)

y

0

2

≥ f

0

(t, x, z

2

) y

2

= A(t)x + B(t)z

2

+ h(t). (4.5.4)

Let α and β be two real numbers such that α ≥ 0, β ≥ 0, and α + β = 1.

If we multiply the relations in (4.5.3) by α, the relations in (4.5.4) by β, and

add, we get

αy

0

1

+ βy

0

2

≥ αf

0

(t, x, z

1

) + βf

0

(t, x, z

2

)

αy

1

+ βy

2

= A(t)x + B(t)(αz

1

+ βz

2

) + h(t).

Since Ω(t) is convex, there exists a point z

3

in Ω (t) such that z

3

= αz

1

+ βz

2

.

From the convexity of f

0

in z, we g et

αf

0

(t, x, z

1

) + βf

0

(t, x, z

2

) ≥ f

0

(t, x, αz

1

+ βz

2

) = f

0

(t, x, z

3

).

Hence

αy

0

1

+ βy

0

2

≥ f

0

(t, x, z

3

)

αy

1

+ βy

2

=