LINEAR TIME-OPTIMAL CONTROL-THE MAXIMUM PRINCIPLE FOR LINEAR TIME-OPTIMAL CONTROL |
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The really interesting practical issue now is understanding how to compute an optimal control α∗(.).
DEFINITION. We define K(t, x0) to be the reachable set for time t. That is, K(t, x0) = {x1| there exists α(.) ∈ A which steers from x0 to x1at time t}.
Since x(.) solves (ODE), we have x1 ∈ K(t, x0) if and only if
for some control α(.) ∈ A.
THEOREM 1.1 (GEOMETRY OF THE SET K). The set K(t, x0) is convex and closed.
Proof. 1. (Convexity) Let x1, x2 ∈ K(t, x0). Then there exists α1,α2 ∈ A such that
and hence λx1 + (1 − λ)x2 ∈ K(t, x0).
2. (Closedness) Assume xk ∈ K(t, x0) for (k = 1, 2, . . . ) and xk → y. We must show y ∈ K(t, x0). As xk ∈ K(t, x0), there exists αk(.) ∈ A such that
According to Alaoglu’s Theorem, there exist a subsequence kj → ∞ and α ∈ Asuch that αk∗⇀α. Let k = kj → ∞ in the expression above, to find
Thus y ∈ K(t, x0), and hence K(t, x0) is closed.
NOTATION. If S is a set, we write ∂S to denote the boundary of S.
Recall that τ∗ denotes the minimum time it takes to steer to 0, using the optimal control α∗. Note that then 0 ∈ ∂K(τ∗, x0).
THEOREM 1.2 (PONTRYAGIN MAXIMUM PRINCIPLE FOR LINEAR TIME-OPTIMAL CONTROL). There exists a nonzero vector h such that
for each time 0 ≤ t ≤ τ∗.
INTERPRETATION. The significance of this assertion is that if we know h then the maximization principle (M) provides us with a formula for computing α∗(.), or at least extracting useful information.
We will see in the next chapter that assertion (M) is a special case of the general Pontryagin Maximum Principle.
Proof. 1. We know 0 ∈ ∂K(τ ∗, x0). Since K( ∗, x0) is convex, there exists a supporting plane to K(τ∗, x0) at 0; this means that for some g = 0, we have
g. x1 ≤ 0 for all x1 ∈ K(τ∗, x0 ).
2. Now x1 ∈ K(τ∗, x0) if and only if there exists α(.) ∈ A such that
Since g . x1 ≤ 0, we deduce that
for all controls α(.) ∈ A.
3. We claim now that the foregoing implies
for almost every time s.
For suppose not; then there would exist a subset E ⊂ [0, τ∗] of positive measure, such that
for s ∈ E. Design a new control αˆ (.) as follows:
where α(s) is selected so that
This contradicts Step 2 above.
For later reference, we pause here to rewrite the foregoing into different notation; this will turn out to be a special case of the general theory developed later in Chapter
4. First of all, define the Hamiltonian
H(x, p, a) := (Mx + Na) .p (x, p ∈ Rn, a ∈ A).
THEOREM 1.3 (ANOTHER WAY TO WRITE PONTRYAGIN MAXIMUM PRINCIPLE FOR TIME-OPTIMAL CONTROL). Let α∗(.) be a time
optimal control and x∗(.) the corresponding response.
Then there exists a function p∗(.) : [0, τ∗] → Rn, such that
We call (ADJ) the adjoint equations and (M) the maximization principle. The function p∗(.) is the costate.
Proof. 1. Select the vector h as in Theorem 1.2, and consider the system
The solution is
and hence
2. We know from condition (M) in Theorem 1.2 that
3. Finally, we observe that according to the definition of the Hamiltonian H,
the dynamical equations for x∗(.), p∗(.) take the form (ODE) and (ADJ), as stated in the Theorem.
References
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