EXTREMALS OF A QUADRATIC COST OPTIMAL PROBLEM ON THE REAL PROJECTIVE LINE

Let Σ be a bilinear control system on R whose matrices generate the Lie algebra sl(2) of the Lie group Sl(2) : the group of order two real matrices with determinant 1. In this work we focus on the extremals of a quadratic cost optimal problem for the angle system PΣ defined by the projection of Σ onto the real projective line P. It has been proved in [2] that through the Cartan-Killing form the cotangent bundle of P can be identified with a cone C in sl(2). Via the Pontryagin Maximum Principle, we explicitly show the extremals by using the mentioned identification and the special form of the trajectories associated with the lifting of vector fields on PΣ. We analyze both: the controllable case and when the system bfPΣ give rise to control sets. Some examples are shown. Subject classification : 35B50, 49J15, 93C15.


Introduction
Optimal control problems with quadratic cost have already been analyzed with great success for linear control systems, see [6].However, a great number of applications are modelled by bilinear dynamics.For instance, different classes of biological and chemical processes has been analyzed in [5], [8].On the other hand, the state of the art of this theory is far from a sufficient understanding and require new approaches to meet the challenges in this field, see [4] for a nice exposition of control systems on fiber bundles.This paper analyze the quadratic cost optimal problem for a class of systems induced by a bilinear one.
Let Σ be a bilinear control system on R 2 whose matrices generate the Lie algebra sl(2) of the Lie group Sl(2) : the group of order two real matrices with determinant 1. Precisely, Σ is determined by the family of differential equations: Here, A, B ∈ sl(2), and u is an element of the admissible restricted controls class U={u : R → [−1, 1] , u locally integrable} .In this work we focus on the extremals of the quadratic cost optimal problem for the angle system PΣ defined by the projection of Σ onto the real projective line P 1 , as follows PΣ : ṡ(t) = h(A, s(t)) + u(t)h(B, s(t)), s ∈ P 1 (1.3) The projected dynamics on the sphere is induced by the formula h(X, s) = (X − (s > X s)Id)s with Id the identity matrix, u an element of the restricted admissible control set and s > X s corresponds to the radial component of the vector field Xs and () > denotes transpose.
In other words, given two points p and q in P 1 our aim is to find a trajectory of PΣ starting on p and ending on q minimizing the functional J over all such a curves.
It has been proved in [2] that through the Cartan-Killing form, the cotangent bundle of P 1 can be identified with a cone C in sl (2).Via the Pontryagin Maximum Principle, we explicitly show the extremals by using the mentioned identification and the special form of the trajectories associated with the lifting of vector fields on PΣ to the cone.We analyze both: the controllable case and when the system PΣ give rise to control sets.
Since the projective real line is a compact manifold, it follows that given any two arbitrary points in P 1 there exists an optimal path connecting them.
Through the paper we assume that Σ satisfy the Lie algebra rank condition, (LARC), which means that the Lie algebra generated by A and B coincides with sl(2).Furthermore, we follow the references [1] and [3] which give an algebraic and geometric condition to the controllability of PΣ, respectively.
The paper is organized as follows.Section 2 contains the Hamiltonian formalism on the cone in sl(2) and the trajectories associated with the lifting of vector fields on PΣ.In Section 3 we describe the associated Hamiltonian and the explicitly form of the optimal control.In Section 4 we analyze the synthesis of the problem.We also give examples.

Hamiltonian Formalism and Trajectories on C
In order to have an appropriated frame for the understanding of our problem, we first recall some results established by the authors in [2].
Let G be a Lie group and H ⊂ G a closed Lie subgroup.We consider the homogeneous space G/H = {zH : z ∈ G}.Denotes by g and h the Lie algebras of G and H respectively.Each X ∈ g induces a vector field e X on G/H and its flow is given by e X t (x) = exp (tX) x, x ∈ G/H.
If X is a vector field on a differentiable manifold N , there exists a lifting X * of X on the cotangent bundle T * N.Moreover, X * is a Hamiltonian vector field and the corresponding Hamiltonian function Here, π : is the fiber bundle projection.
As has been showed in [2], the elements of the cotangent bundle T * (G/H) can be represented by elements of T * G.We specialize this identification to semisimple Lie groups.Through the Cartan-Killing form K the Lie algebra g of G is identified with its dual g * (see [10]).Let X ∈ g and e X its induced vector field on the homogeneous space G/H.We have Proposition 2.1.Let g be a semisimple Lie algebra and h ⊂ g the Lie algebra of H.Then, the lifted vector field e X * on T * (G/H) induced by X ∈ g is given by e X * (Y ) = ad (X) (Y ) with flow e X * t (Y ) = Ad (exp tX) Y.
On the other hand, it is well known that in the semisimple Lie algebra sl (2) , the form K is a multiple of the trace form.Precisely, where k 6 = 0 is a constant.In [1] is showed that the zeros set ) is a cone with respect to the basis of sl (2).In fact, in the (z 1 , z 2 , z 3 )-coordinates the equation of C with respect to (2.1) is given by z 2 1 + z 2 2 = z 2 3 .The set C − {0} has two connected components, we distinguish them by putting C + and C − for the one which contains the matrix respectively.Just observe that the rotation group turns around C + while the group of diagonal matrices is transitive along the ray of upper triangular matrices in C + .Thus, Sl (2) acts transitively on C + .Furthermore, the action of Sl (2) into the set [C + ] of rays of C + is equivalent to the action of Sl (2) in the projective line P 1 .We denote by C + int , C + ext the regions of R 3 inside and outside the cone, respectively.In [2] an algebraic picture of the cone is given as follows Proposition 2.2.The entire cone C is identified with the cotangent bundle T * (P 1 ).
For P ∈ C + the tangent plane T P C + of C + at P is Notice that any Z ∈ sl (2) defines, through the linear application ad Z in sl (2) , a linear differential equation in sl (2) , explicitly given by .P = [Z, P ], P ∈ sl (2), whose trajectories are exp (t ad Z) P = Ad(exp tZ)(P ), with P ∈ sl (2) and t ∈ R.
In particular, Z ∈ sl (2) induces also a differential equation on C + .Just observe that < P = [Z, P ] > = 0.In the sequel, we will describe these trajectories in C + as the intersections of C + with the planes orthogonal to Z with respect to the trace form.
Remark 2.1.In practice, with respect to the trace form, the plane orthogonal to any matrix of trace zero can be seen with the aid of the inner product (• , •) in sl (2) which is defined by ´.
Actually, if we denote by Z (⊥) the orthogonal plane to Z with respect to ´>.On the other hand, transposition is obtained by a reflection through the plane s of symmetric matrices.Therefore, Z ⊥ is the reflection through s of the plane orthogonal to Z with respect to (• , •).The authors in [1] give the following description of the trajectories on C. Proposition 2.3.According to the location of Z respect to the cone, the trajectories of

The Hamiltonian and the Optimal Control
In this section, we establish the Pontryagin Maximum Principle (PMP) for the angle system PΣ on P 1 determined by a bilinear control system Σ in the plane.From the principle, we characterize the optimal control for the quadratic cost.
The dynamic here described is determined by the bilinear control system (1.1),where A and B are two matrices of trace zero, with running cost (1.2) and bounded control | u |≤ 1.
For our purposes, the suitable version of the PMP provided in [7], gives a first order necessary conditions for the optimality of quadratic control problems.In our particular case the principle reads: Theorem 3.1.If u * is an optimal control and x * is the associated trajectory of PΣ on P 1 , there exists a constant λ 0 ≥ 0 and an absolutely continuous function λ : [0, T ] −→ T * P 1 , such that for almost every t ∈ Dom(x * ), (λ, λ 0 ) never vanishing and the adjoint equation, is satisfied.Furthermore, the optimal control u * (t) minimizes the Hamiltonian The pair (u * , x * ) is called extremal pair if u * is an admissible control and the corresponding trajectory x * satisfy the PMP conditions.We normalize λ 0 to 1.
By Proposition 2.2, the form of the Hamiltonian H and by using the invariance of the Cartan-Killing form K, it turns out that the equation (3.1) can be rewritten as An admissible constant control u induces the vector field X u = A+uB ∈ g.On the other hand, since that the cotangent bundle T * P 1 is identified with the cone C, each curve [0, T ] −→ T * zH P 1 given by the PMP is identified with a curve in C + .We obtain: the Hamiltonian for the quadratic optimal control problem where e X u is the lifted vector field on C + induced by The Hamiltonian H is strictly convex in the control u and its minimum over [−1, 1] is attained with In particular, the optimal control u * minimizing H over We call the function ϕ(t) the switching function of the control.Thus, the application of the PMP to any control system lead controls belonging to the interior or to the boundary of the control range [−1, 1].

The Quadratic Optimal Control Synthesis
In this section we analyze, on the real projective line, the extremals for the quadratic cost.These trajectories are obtained by the radial projection of the extremals in C + defined by the adjoint system (3.2).Precisely, given p and q two arbitrary points on P 1 our goal is to find the extremals of the quadratic cost connecting these two points.Since in our case, the number of extremals are finite, it turns out that it is possible to find the optimal path in a closed analytic form.In future research, we hope to apply numerical algorithms to approach the optimal solutions.We identify P 1 with the ordered circle n . According to the shape of the optimal controls determined by the PMP we need to consider the planes hP, Bi = 0, hP, Bi = ±1 and the intersection of these planes with C + .We have the following relative positions.Namely, The synthesis of the optimal control will be established by considering the following cases Case 1 hP, Bi = 0 doesn´t meet the cone.In this case, hP, Bi = 1 or hP, Bi = −1 intersects C + int but no both.
For Z ∈ sl (2) , the linear differential equation Ṗ = [Z, P ] on sl (2) induces a vector field on C + .Proposition 2.3 gives the form of the trajectories on C + and their projection on P 1 are as follows: Ellipses onto the circle, points onto singularities, parabolas onto segments, lines onto singularities and semi-hyperbolas onto segments.
Via the adjoint action of Sl(2, R) on the cone, any rotation on C + is conjugated to the matrix In particular, they move in the counterclockwise direction.
In the next sections the synthesis of the optimal problem is established according to the relative position of the planes hP, Bi = 0, ±1 and hP, A ± Bi = 0.

The Controllable Case
Through this section we assume that the projected control system (1.3) is controllable.Since the projective real line is a compact manifold, it follows that given any two arbitrary points in P 1 there exists an optimal path connecting them and this path is coming from the Pontryagin Maximum Principle.
Theorem 5.2 in [1] gives an algebraic characterization of the controllability property of (1.1).From a geometric point of views, this result says that the bilinear control system is controllable in R 2 \ {0} if and only if the segment Of course, if (1.1) is controllable then the projected control system (1.3) is also controllable, see for instance [9].
Proof: Since the system Σ is controllable, from Theorem 5.2 in [1] it follows that the segment A + uB : We obtain two regions, the lower and the upper region.See Figure 4.2 We claim: given two arbitrary points p, q ∈ P 1 show an optimal path connecting them.In fact, We have the possibilities: The trajectories of the adjoint system (3.2) on C + induced by the bang control u * (t) = 1 are ellipses which we denote by ε + .Denote by β ϕ the trajectories of the adjoint system for u * (t) = ϕ(t) ∈ (−1, 1).
1.2.Assume ζ 0 ∈ U belongs to ε + which intersects the plane hP, Bi = 1.Let us assume that the fiber of π −1 (q) intersects ε + at the point ξ a) If ξ ∈ clU, and the control u = 1 connect ζ 0 with ξ.Then, as in the previous case the projection of the dynamic determined by u * = 1 gives the optimal path.b) If ξ ∈ L, then ζ 0 moves under the influence of u * = 1 up to the intersection with the plane hP, Bi = 1.From this intersection point, it follows the trajectory corresponding to the differential equation induced by the control ϕ(t) = − hP (t), Bi moving on L until intersects the fiber of q.The projection of β ϕ • ε + gives rise the optimal path connecting them, starting on p and ending on q.
c) If ξ ∈ clU, and the control u = 1 connect ξ with ζ 0 , we follows the path β ϕ • ε + up to the second intersection with the plane hP, Bi = 1.After that we use again u = 1 to reach ζ 0 .So, the projection of the absolutely continuous curve . Eventually, we could reach q from p as follows: through the projection of the differential equation determined by the optimal control ϕ(t) = − hP (t), Bi ∈ (−1, 1) with initial condition ζ 0 and ending point ξ.Or, starting with the control ϕ(t) = − hP (t), Bi up to the curve meets the plane hP, Bi = 1 at the point η, to continue with a particular class of ellipses ε + .Depending on the class, η could be move towards L or U .In fact, in the first case, the curve will reach ξ inside of L,i.e., the positive direction of the ellipses (derivative at the point η point out towards L).In the second case, the curve will continue with the ellipse up to meet the fiber of q.The projection of the last two mentioned curves will give other extremals.
b) If ξ ∈ U, we first need to meet the plane hP, Bi = 1 at η with an appropriate β ϕ .Then, to continue with the class of ellipses which carry on η up to ξ.Just observe that the other class of ellipses will not be useful for our purposes.
We notice that in the previous analysis the plane hP, Bi = 1 intersect the cone C + and A − B ∈ C + ext determine two singularities on the projective line.Fortunately, we do not need to care about.In fact, the optimal control given by the PMP consider u * (t) = 1 or u * (t) = −hP (t), Bi or combinations of these controls.Suppose the plane hP, Bi = −1 intersect the cone C + .
The optimal paths coming from the PMP just consider u * = −1 or u * = ϕ(t).We define L and As before, the analyze follows by considering the cases 1.
In this situation, the synthesis of the extremals should be realized taking in account the singularities of A − B which determine the regions R 1 = π −1 (q − −1 , q + −1 ] and R 2 = π −1 (P 1 \R 1 ).1.1 If p and q lies in the same region, the analysis is analogously to the case hP, Bi = 1, (1.) .
1.2 If p and q lies in different regions, for instance, p ∈ R 1 and q ∈ R 2 .Recall that q − −1 is an attractor of A − B, in particular it is not possible to cross R 2 from R 1 with u = −1.
a) Let ζ 0 ∈ cl(L) and ξ ∈ cl(L).We just reach q from p through the projection of the differential equation determined by the optimal control ϕ(t) = − hP (t), Bi ∈ (−1, 1) with initial condition ζ 0 and ending point ξ.
b) If ζ 0 ∈ cl(L) and ξ ∈ U we start with u = − hP (t), Bi moving up to the point ξ 0 : the intersection between plane hP, Bi = −1 in R 2 and the semi-hyperbola h− ξ through ξ.So, the projection of h− ξ • β ϕ send p to q. c) The case ζ 0 ∈ U and ξ ∈ R 2 is not possible.Actually, we can not leave R 1 . 2.
This case is analogously to the case (1.) taking care of the singularities of A + B.  We denote by We begin the analysis when the fibers of both p and q intersect the same region.For instance, if π −1 (p), π −1 (q) intersect L. We have two possibilities.The first one consider the projection of the solution with initial condition ζ 0 ∈ π −1 (p) and ending state ξ ∈ π −1 (q), through the adjoint differential equation determined by u = ϕ inside of L. If we need to scape from L, then starting on ζ 0 we move first under the influence of u = ϕ up to ξ 0 : the intersection between the ϕ-integral curve and the plane hP, Bi = 1.From the new initial condition ξ 0 , we move on the region U through the ellipse ε + which intersect the plane hP, Bi = 1 at the point ξ 1 .From ξ 1 we enter inside of L by a β ϕ trajectory up to reach the line π −1 (q).Therefore, by an appropriated projection of the curve β ϕ • ε + • β ϕ we obtain the desired extremal on the projective line.
The case π −1 (p), π −1 (q) ∈ cl(U ) is analogous.Finally, if the fibers of both p and q intersect different regions.For instance, if π −1 (p) intersect L, and π −1 (q) intersect U, the analysis is similar to the previous cases.In fact, the extremals are given by the projection of Assume now the plane hP, Bi = −1 intersect the cone C + This situation is not possible.As before According to the PMP, on the region L we always need to use the control u = −1.However, the line π −1 (q − −1 ) is an attractor of the adjoint equation.So, the trajectory will never leave L and its projection will remains in (q + −1 , q − −1 ). 2. For −B ∈ C + and A ∈ sl (2) the synthesis follows in the same way as (1.) when hP, Bi = −1 .But, hP, Bi = 1 it is not possible.a) The fiber of p and q intersect the same region.Suppose, this region is R. We have two different kind of dynamics If both fibers meet L or the same connected component of M the synthesis is analogous.Assume, π −1 (p) and π −1 (q) belong two the different connected components of M, the possible dynamics are Therefore, the projections of these classes of curves give us the extremals on P 1 .
b) If the fibers intersects two any different regions, the case (a) give the way to compute the extremals. 2.
In principle, we have two situations 2.1.The plane hP, A − Bi = 0 intersects hP, Bi = −1 and hP, Bi = 1 on C + .Assume π −1 (q − −1 ) meets R and π −1 (q + −1 ) meets L. According to the PMP, this situation is not possible.In fact, starting from a state in R we can not reach any element of L.
2.2.The planes hP, A − Bi = 0 , hP, Bi = −1 and hP, Bi = 1 are parallels.This case is also not possible.Actually, under these hypothesis it follows that hP, Ai = 0.Then, the matrices A, B and A − B are linear dependent which is a contradiction.
Summarizing, in the controllable case we have Theorem 4.2.Assume that the projected control system (1.3) is controllable on P 1 .Given any two arbitrary points p and q on the real projective line, there exists an optimal control u p, q steering p to q, minimizing the functional J (1.2).The synthesis of u p, q is given through the Pontryagin Maximum Principle, as follows J(u p, q ) = min {J(u) : u ∈ U determines an extremal} .
Next, we give some examples in which the projected system (1.3) is controllable.
int then, the system is controllable.This situation correspond to Case 3, (1).

The Non Controllable Case
In this Section do not assume the controllability property of the projected system on the projective real line.The synthesis is established by considering the existence of two control sets of PΣ on P 1 .
ii) there exists a control u ∈ U with χ(t, p, u) ∈ D for all t ≥ 0, and iii) with respect to the set inclusion, D is maximal with the properties (i) and (ii).
As usual, clM denotes the closure of the set M , the positive orbit S(p) is the action of the semigroup of the angle system on the state p ∈ P 1 and χ(t, p, u) is the solution of the projected system with initial condition p and control u.A main control set is a control set with nonvoid interior.
Since we assume that (1.1) satisfy the Lie algebra rank condition, the semigroup S of the bilinear control system is a proper subsemigroup of Sl (2) with int S 6 = ∅, see [11], [12].Then, Proposition 3.1 in [1], shows that there are exactly two control sets on P 1 , denoted by I ± .They satisfy the following properties: iii) If g ∈ S is diagonalizable then its attractor belongs to I − and its repeller is an element of the closure of I + As before, we consider matrices A, B with det [A, B] 6 = 0.If PΣ is not controllable on P 1 then A + B and A − B must be outside of C + int .In the sequel, we just analyze the case, B ∈ C + ext From a dynamic point of view, the planes hP, A + Bi = 0 and hP, A − Bi = 0 determine the control sets on P 1 , as follows ´.
where the boundary point of the intervals are the singularities of A + B and A − B as explained above.
From the optimality point of view, the planes hP, Bi = ±1 give rise to the regions L, M and R in C + , and four intersection points on P 1 .All together determines the following eight regions on the projective line, which we describe in as in the Figure (4.4) : ´.In terms of those regions and according to the PMP, we must apply the control u = −1 on L, u = ϕ(t) = − hP (t), Bi on M and u = 1 inside of R. For p < q, we get in the table 1 for the possible trajectories on the cone C + Here p ext −→ q denotes an extremal path on P 1 starting on p and ending on q with p < q.
As a consequence, there exists an extremal path starting on any point of I + and ending on an arbitrary point in the invariant control set I − .The optimal path is constructed using bang controls as well as controls with values in the interior of [−1, 1].However, this is not the case in the other direction.
Next, we illustrate an example in the noncontrollable case

Case 3
hP, Bi = 0 divides the Cone C + in two regions.So, the planes hP, Bi = ±1 meet C + int , see Figure 4.1.

Figure 4 . 1 :
Figure 4.1: Intersection planes hP, Bi = 0, ±1 with the cone C + According to the proposition 2.3, if Z ∈ C + ext the trajectories of the adjoint equation (3.2) are the rays in Z ⊥ ∩ C + \ {0} or the hyperbolas {hZ, P i = c}∩C + .Recall that the rays are determined by the corresponding eigenvectors of the matrices.We denote by l + +1 and l − +1 the rays of Z = A + B and by l − −1 and l + −1

Remark 4 . 1 .
If Σ is controllable and −B ∈ C + int according to Proposition 4.1 A − B ∈ C + int and the synthesis is analogous.4.1.1.Case 1. hP, Bi = 0 does not meet C + Let us assume B ∈ C + int .We follows the synthesis by the relative position of the planes hP, Bi = ± 1 with the cone.Suppose the plane hP, Bi = 1 intersect the cone C + .

Figure 4 . 2 :
Figure 4.2: Intersection of the plane hP, Bi = 1 with the cone C +

2 .
A − B ∈ C + int and A + B ∈ C + ext According to Remark 4.1, this situation is analogously to the case 1. 3. A + B, A − B ∈ C + int a) If ζ 0 , ξ ∈ cl(U ) the projection of any ellipse inside of U starting on ζ 0 and ending in ξ will do the job b) The case ζ 0 ∈ cl(U ) and ξ ∈ L is analogous to 1.2 (b) c) The case ζ 0 ∈ L and ξ ∈ cl(L) ∪ U is analogous to 1.3 (a) and (b).

3 .
A + B, A − B ∈ C + int Analogous to the case (1.) of hP, Bi = 1.4.1.2.Case 2. hP, Bi = 0 is tangent to C + In this subsection we consider two cases 1. B ∈ C + and A ∈ sl (2).Without lost of generality we assume A + B ∈ C + int .In particular, we get A − B ∈ C + ext .Assume the plane hP, Bi = 1 intersect the cone C + .

Example 4 . 3 .
Let Σ be determined by A = H 2 + 2R and B = H + 2R, see (2.1).Since, the segment A + uB with −1 ≤ u ≤ 1, intercepts C + int the system is controllable.IN fact, B, A + B ∈ C + int belong to C + int and A − B ∈ C + ext .Since the plane hP, Bi = 1 meet the cone.This example corresponds to the Case 1, (1.) .

Example 4 . 4 .
Let Σ be determined by B = R − H ∈ C + and A = R, see (2.1).Since, A ∈ C + int the system is controllable, (4.1).On the other hand, A + B ∈ C + int and A − B ∈ C + ext and the plane hP, Bi = 1 meets C + .So, this is an example of Case 2, (1).Example 4.5.Consider Σ defined by B = H ∈ C + ext and A = 2R, see (2.1).Thus, A + B ∈ C + int and A −

Example 4 . 7 .I⎦
Consider Σ with the basis vectors A = S and B = H ∈ C + ext , see (2.1).The system Σ satisfies LARC but is not controllable.The plane B ⊥ splits C + in two regions cone∩ {z 1 > 0} , and cone ∩ {z 1 < 0} and both Planes hP, A ± Bi = 0 intersect to hP, Bi = ±1 on C + .Since, A + B as well as A − B belong to C + ext .Then the control sets are given by where k • k denotes the usual norm for vectors in R 2 .(see Figure 4.4).