For this section, we will again take A to be the cube [−1, 1] ^m in R^m.

DEFINITION. A control α(.) ∈ A is called bang-bang if for each time t ≥ 0 and each index i = 1, . . . ,m, we have |αⁱ(t)| = 1, where

THEOREM 1.1 (BANG-BANG PRINCIPLE). Let t > 0 and suppose x⁰ ∈C(t), for the system

Then there exists a bang-bang control α(.) which steers x⁰ to 0 at time t.

To prove the theorem we need some tools from functional analysis, among hem the Krein–Milman Theorem, expressing the geometric fact that every bounded onvex set has an extreme point.

1.1 SOME FUNCTIONAL ANALYSIS. We will study the “geometry” of certain infinite dimensional spaces of functions.

NOTATION:

DEFINITION. Let α_n ∈ L^∞ for n = 1, . . . and α ∈ L^∞. We say αn converges to α in the weak* sense, written

We will need the following useful weak* compactness theorem for L^∞:

ALAOGLU’S THEOREM. Let α_n ∈ A, n = 1, . . . . Then there exists a subsequence α_nk and α ∈ A, such that

DEFINITIONS. (i) The set K is convex if for all x, xˆ ∈ K and all real numbers  0 ≤ λ ≤ 1,

                                    λx + (1 − λ) xˆ ∈ K.

(ii) A point z ∈ K is called extreme provided there do not exist points x, xˆ ∈ K and 0 < λ < 1 such that

                        z = λx + (1 − λ) xˆ.

KREIN-MILMAN THEOREM. Let K be a convex, nonempty subset of L^∞, which is compact in the weak ∗ topology.

Then K has at least one extreme point.

1.2 APPLICATION TO BANG-BANG CONTROLS.

The foregoing abstract theory will be useful for us in the following setting. We will take K to be the set of controls which steer x⁰ to 0 at time t, prove it satisfies the hypotheses of Krein–Milman Theorem and finally show that an extreme point is a bang-bang control.

So consider again the linear dynamics

Take x⁰ ∈ C(t) and write

K = {α(.) ∈ A |α(.) steers x⁰ to 0 at time t}.

LEMMA 1.3 (GEOMETRY OF SET OF CONTROLS). The collection K of admissible controls satisfies the hypotheses of the Krein–Milman Theorem.

Proof. Since x0 ∈ C(t), we see that K = ∅.

Next we show that K is convex. For this, recall that α(.) ∈ K if and only if

Now take also αˆ ∈ K and 0 ≤ λ ≤ 1. Then

Lastly, we confirm the compactness. Let α_n ∈ K for n = 1, . . . . According to Alaoglu’s Theorem there exist n_k → ∞ and α ∈ A such that α_nk^∗⇀ α. We need

to show that α ∈ K.

Now α_nk ∈ K implies

by definition of weak-* convergence. Hence α ∈ K.

We can now apply the Krein–Milman Theorem to deduce that there exists an extreme point α^∗ ∈ K. What is interesting is that such an extreme point corresponds to a bang-bang control.

THEOREM 1.4 (EXTREMALITY AND BANG-BANG PRINCIPLE). The control α^∗(.) is bang-bang.

Proof. 1. We must show that for almost all times 0 ≤ s ≤ t and for eachi = 1, . . . ,m, we have

|α^i∗ (s)| = 1.

Suppose not. Then there exists an index i ∈ {1, . . . ,m} and a subset E ⊂ [0, t]  of positive measure such that |α^i∗(s)| < 1 for s ∈ E. In fact, there exist a number ε > 0 and a subset F ⊆ E such that

the function β in the i^th slot. Choose any real-valued function β(.) ≡ 0, such that IF (β(.)) = 0

and |β(.)| ≤ 1. Define

α₁(.) := α^∗(.) + εβ(.)

α₂(.) := α^∗(.) − εβ(.),

where we redefine β to be zero off the set F

2. We claim that

                                        α₁(.),α₂(.) ∈ K.

To see this, observe that

Note also α₁(.) ∈ A. Indeed,

But on the set F, we have |α^∗_i (s)| ≤ 1 − ε, and therefore

|α₁(s)| ≤ |α^∗ (s)| + ε|β(s)| ≤ 1 − ε + ε = 1.

Similar considerations apply for α₂. Hence α₁,α₂ ∈ K, as claimed above.

3. Finally, observe that

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