Marco teórico
1.5 Desarrollo territorial rural
Given a control system we define an optimal control problem adding a cost function whose integral must be minimized over solutions of the control system. First, we consider an optimal control problem with a fully actuated nonholonomic mechanical control system. The equiv- alence of this system with a kinematic system, that is, a control–linear system, is known by Theorem 7.1.2. It should be useful to find a cost function for the kinematic system such that some connection between the optimal solutions to both problems may be established.
Let us point out the importance of relating those two optimal control problems. In [Liu and Sussmann 1994b; 1995] the strict abnormal minimizers have been described for the problem of shortest–paths in subRiemannian geometry. The control system in subRiemannian geometry is control–linear, so it can be understood as a kinematic system that comes from a nonholonomic mechanical control system as the example in §7.2.4 shows. Thus it might be expected to char- acterize abnormal extremals for mechanical control systems using the well–known abnormal minimizers in subRiemannian geometry.
Let us consider a cost function F : T Q × U → R for the mechanical control system. The optimal control problem for (7.1.2) is stated as follows.
Statement 7.1.3. (Nonholonomic optimal control problem) Given xa, xb∈ Q, find (γ, u) : I →
Q × U such that
1. γ satisfies the endpoint conditions on Q, i.e. γ(a) = xa,γ(b) = xb;
2. ˙γ is an integral curve of Y in (7.1.2), i.e. ¨γ(t) = Y ( ˙γ(t), u(t)); 3. ( ˙γ, u) minimizesR
IF ( ˙γ(t), u(t))dt among all the curves satisfying 1 and 2.
This optimal control problem is denoted by Σm = (ΣD, F ). In optimal control theory, it
150 7.1. Optimal control problem: nonholonomic versus kinematic
§4.1.2 for more details. In this way, all the elements in the optimal control problem are included in a control system, usually called the extended system [Athans and Falb 1966, Lee and Markus 1967, Pontryagin et al. 1962]. Nevertheless, the minimization of the functional must be added to the extended system, what turns out to be the minimization of the new coordinate; see Statement 4.1.2.
In the case of mechanical control systems, two new coordinates are added in order to main- tain the second–order condition of the vector field (7.1.2). Let bQ = R × Q, then the cost function is considered as a vector field along the projectionbπm: T bQ × U → T bQ with local expression F ∂/∂x0. Then (7.1.1) becomes
b ∇˙ b γbγ = F ◦˙ bγ + n−k X r=1 µrZr◦bγ + k X s=1 usYs◦γ + F ◦ ( ˙b bγ, u) ∂ ∂x0 ˙ b γ (7.1.4)
whereγ : I → bb Q is a differentiable curve and b∇ denotes the Levi–Civita connection extended to bQ considering the natural product connection on bQ, taking ∇ and the trivial connection on R. Moreover, we have π2◦ ˙bγ = ˙γ ∈ D with the projection π2: T bQ = T R × T Q → T Q.
The second–order differential equation (7.1.4) admits a first–order differential equation given by the vector field
b Y = c Zg z }| { v0 ∂ ∂x0 + Zg +F V + F ∂ ∂v0 + n−k X r=1 µrZrV + k X s=1 usYsV (7.1.5)
along the projectionπbm: T bQ×U → T bQ, where cZgis the geodesic spray of the above–mentioned
extended connection b∇. The differential equations added to (7.1.2) are ˙
x0 = v0,
˙v0 = F , (7.1.6)
taking into account the extension of the Levi–Civita connection to bQ. The value that must be minimized in the optimal control problem is v0=R
IF dt.
Now consider the kinematic system (7.1.3) with a cost function G : Q × V → R such that the problem to be solved is the following one.
Statement 7.1.4. (Kinematic optimal control problem) Given xa, xb ∈ Q, find (γ, w) : I →
Q × V such that
1. γ satisfies the endpoint conditions on Q, i.e., γ(a) = xa,γ(b) = xb;
2. γ is an integral curve of X =Pk
s=1wsYs, i.e., ˙γ(t) = X(γ(t), w(t));
3. (γ, w) minimizesR
IG(γ(t), w(t))dt among all the curves satisfying 1 and 2.
control system and the kinematic control system considered here are equivalent, see Theorem 7.1.2.
Remark 7.1.5. The problems in Statements 7.1.3 and 7.1.4 are called fixed optimal control problems because the domain of definition of the curves is given. However, the free optimal control problems may also be defined, analogously to Statement 4.3.1.
As before, let us extend the control system to the manifold bQ = R × Q such that we look for integral curves of the vector field
b X = G ∂ ∂x0 + k X s=1 wsYs (7.1.7)
defined alongπbk: bQ × V → bQ. The differential equation added to (7.1.3) is ˙
x0= G (7.1.8)
and the value to be minimized is x0 =RIGdt.
By Theorem 7.1.2 we know that (7.1.1) and (7.1.3) are equivalent. We are interested in establishing a connection between (7.1.6) and (7.1.8) in such a way that it is obtained a rela- tionship between the problems Σmand Σkin Statements 7.1.3 and 7.1.4, respectively.
In some sense, G = v0=R F, but this equality must be properly understood. Observe that G is a function on Q × V , meanwhile F is a function on T Q × U . Hence, some simplifications must be considered. Before proceeding with the exact interpretation of G = R F, note we also have to check what happens with the minimization conditions when G = R F; that is, if the curves minimizingR G determine the curves minimizing R F and/or in the other way round. Proposition 7.1.6. Let G : I × Q → R. If ( ˙γ, u) : I → T Q × U is an optimal curve of a nonholonomic mechanical control system with cost function F = ∂G/∂t + vi∂G/∂xi =
c
dG : I × T Q → R, then there exists w : I → V such that (γ, w) is an optimal curve of the associated kinematic system with cost functionG.
(Proof ) If ( ˙γ, u) : I → T Q × U is an integral curve of (7.1.2), then by Theorem 7.1.2 there exist w : I → V such that (γ, w) is an integral curve of (7.1.3). Thus, it only remains to prove that the optimality condition for F implies the optimality condition for G.
As ( ˙γ, u) minimizesR F, then, for any other integral curve ( ˙eγ,u) of the vector field (7.1.2)e satisfying the endpoint conditions, we have
G(t, γ(t)) − G(a, γ(a)) =Rt adG(s, ˙γ(s)) =c Rt aF (s, ˙γ(s))ds < <Rt aF (s, ˙eγ(s))ds = Rt adG(s, ˙c eγ(s)) = G(t,eγ(t)) − G(a,eγ(a)).
As γ and eγ satisfy the endpoint conditions and none of the cost functions depends on the controls, we have
152 7.1. Optimal control problem: nonholonomic versus kinematic
for any t ∈ I, thenRIG(t, γ(t))dt <R
IG(t,γ(t))dt by the monotony property of the integral.e The result just proved holds provided that the cost function for the nonholonomic optimal control problem is the total derivative of the cost function for the associated kinematic optimal control problem. Observe that both cost functions are independent of the controls.
Remark 7.1.7. Necessary conditions for a curve to be an optimal solution for a nonholonomic optimal control problem is to be an optimal solution to the optimal control problem for the associated kinematic system.
Remark 7.1.8. The inverse implication of Proposition 7.1.6 is not necessarily true. If (γ, w) is an optimal curve for the kinematic system, then for any other integral curve (eγ,w) of thee kinematic system Z I dt Z t a F (s, ˙γ(s))ds = Z I G(t, γ(t))dt < Z I G(t,eγ(t))dt = Z I dt Z t a F (s, ˙eγ(s))ds .
The monotone property of the integral is satisfied only in one direction. We should think of conditions such that
“ Z I f < Z I
g ⇒ f < g , almost everywhere (a.e.)”.
In general, we cannot expect better results than a.e., hence we will have optimal curves in a weak sense. For instance, if f and g are both positive or both negative, the implication is satisfied. Moreover, if f and g are continuous functions, then the inequality is satisfied everywhere.
Definition 7.1.9. A nonholonomic optimal control problem Σm is equivalent to an optimal
control problemΣkfor the associated kinematic control system if there exists a curveγ on Q
and controlsu : I → U and w : I → V such that ( ˙γ, u) is a solution to Σm and(γ, w) is a
solution toΣk.
Once the equivalence between the optimal control problems Σmand Σkhas been defined,
we can give the following result.
Proposition 7.1.10. The nonholonomic time–optimal control problem Σm = (ΣD, 1) is equiv-
alent to the associated kinematic optimal control problemΣk= (Σm, G = t).
(Proof ) A solution of the Σm gives a solution of Σk because of Proposition 7.1.6. Let us
prove now that the optimal curves for kinematic systems with G = t are optimal curves for the nonholonomic time–optimal problem.
If (γ, w) : I → Q × V is a minimizer ofRItdt = t2/2 satisfying the kinematic system, then by Theorem 7.1.2 there exist u : I → U such that ( ˙γ, u) is an integral curve of the non- holonomic mechanical control system. For any other integral curve of the kinematic system with the same endpoint conditions as γ,
t2/2 <et
As t, et are positive numbers, t < et. That is ( ˙γ, u) is a minimizer of the nonholonomic time–optimal control problem because of the equivalence of integral curves of (7.1.2) and (7.1.3) given by Theorem 7.1.2 and because of the nature of the cost function. The cost function G = t is positive, so we are in one of the cases where the reverse implication of the monotony property of the integral in Remark 7.1.8 is satisfied.
The optimal control problems considered in Proposition 7.1.10 are free in the sense given in Statement 4.3.1.
Remark 7.1.11. Indeed, it is feasible to consider the time–optimal problem for both control systems, and then Σm = (ΣD, 1) and Σk = (Σm, 1) are equivalent because the time is pos-
itive. Thus, to minimize the time or to minimize the square of the time is exactly the same. Moreover, the curve on the configuration manifolds are related to the same curve on Q since the equations defined by (7.1.2) and (7.1.3) also appear in the extended systems (7.1.5) and (7.1.7), respectively.
The following corollary links with the fact that some optimal control problems can be un- derstood as time–optimal control problems, as for instance happens in the problem of shortest paths in subRiemannian geometry [Liu and Sussmann 1995]. There, to minimize the functional given by the square of the length of a path for a fixed optimal control problem generates the same solutions as to solve the time–optimal control problem with the same control system but with some restriction on the control set and assuming the control vector fields to be orthonormal by the metric.
Definition 7.1.12. Two optimal control problems for a control system on Q are equivalent if the sets inQ given by the image of the solutions to both problems are the same.
Corollary 7.1.13. If Σm = (ΣD, F ) is an optimal control problem equivalent to the time–optimal
control problemΣ0m= (ΣD, 1), then Σmis equivalent to the associated kinematic optimal con-
trol problemΣk= (Σm, 1).
(Proof ) Observe that the first “equivalent” in the statement is the notion in Definition 7.1.12, but the second “equivalent” is related to the notion in Definition 7.1.9. Keeping this in mind, the proof of this corollary is obtained from Proposition 7.1.10 and Remark 7.1.11.
7.2
Hamiltonian problems for nonholonomic mechanical systems
versus kinematic systems
In order to make use of the optimal control problems defined in §7.1.2, let us associate them with a Hamiltonian problem in the sense of Pontryagin’s Maximum Principle given in §4.1.4.
As explained in §4.2, the proof of this Principle consists of choosing the initial condition for the fibers of the cotangent bundle in a suitable way. In fact, at the final time b we ask that
hbλ(b),bv(b)i ≤ 0,