Functional determinants for the second variation

We study the determinant of the second variation of an optimal control problem for general boundary conditions. Generically, this operators are not trace class and the determinant is defined as a principal value limit. We provide a formula to compute this determinant in terms of the linearisation of the extremal flow. We illustrate the procedure in some special cases, proving some Hill-type formulas.


Introduction
The main focus of this paper is to study the spectrum of a particular class of Fredholm operators that arise in the context of Optimal Control.Our main result is a formula which relates the determinant of these operators to the fundamental solutions of an ODE system in a finite dimensional space, much in the spirit of Gelfand-Yaglom Theorem.
For operators of the form 1 + K, where K is a compact operator, various ways of defining a determinant function can be found in the literature.Going all the way back to Poincaré, Fredholm and Hilbert.If the operator K one considers is in the so called trace class, i.e. the sequence of its eigenvalues (with multiplicity) gives an absolutely convergent series, a definition of determinant which involves the (infinite) product of its eigenvalues is possible.
In our case, however, the classical approach is not immediately applicable since, typically, the operators one encounters are not trace class.Under some technical assumptions, the operators arising as second variation have a symmetric spectrum, as shown in [3] and [9].Generically, there is a non negative real number ξ(K), which we call capacity, for which the following asymptotic for the ordered sequence of the eigenvalues holds: This symmetry allows us to talk about trace and determinant of the operators K and 1 + K in the sense of principal value limits.A similar approach has been independently adopted in the works [19,18] to study the spectrum of Hamiltonian differential operators.
There are, of course, many other ways to define a determinant function for classes of Fredholm operators.For instance one could apply the theory of regularized determinants, see [26], or rely on the so called ζ−regularization, see [14] for details.The literature concerning these topics is vast.To mention a few works, one could refer to [24,23] for some results about Sturm-Liouville problems, to [16] for graphs and to [15] results proved in the general framework of elliptic operators on section of vector bundles.A relation between regularized determinants and ζ-regularization is given in [17].
Our approach, even if less general, provides an actual extension of the definition of determinant given for trace class operators.It involves some principal value limit of the product of the eigenvalues.Thus, whenever the compact part of the second variation is trace class, it gives exactly the usual Fredholm determinant defined for trace class operators.
It is worth pointing out another feature of the construction.Our formula relates the determinant of 1 + K to the fundamental solution of a finite dimensional system of (linear) ODEs.This provides a way to actually compute the determinant and allows to recover some classical results such as Hill's formula for periodic trajectories.This kind of formulas have important applications since allow to relate variational properties of an extremal (i.e. the eigenvalues of the second variation), to dynamical properties such as stability.These properties are usually expressed through the eigenvalues of the linearisation of the Hamiltonian system of which the extremal we are considering is solution.Take the case of a periodic, non degenerate trajectory.On one hand, knowing the sign of the determinant amounts to know the parity of the Morse index of the extremal.On the other hand this sign is completely determined by the number of positive eigenvalues of the linearisation grater than one.Applications of these kind of ideas go back to Poincaré's result about the instability of closed geodesics and can be found for example in [12] or [20].For several interesting examples of the interplay between parity of Morse index and stability see for instance [10,11] or [8].Other related works in this direction are [25,27] and [21] .
We stress that our results are formulated in a quite general framework which encompasses Riemannian, sub-Riemannian, Finsler geometry and mechanical systems on manifolds to name a few.Moreover the techniques can be applied without virtually any modifications to treat constrained variational problems on compact graphs as already done in [7] to compute the Morse index.
The structure of the paper is the following: in Section 1.1 we recall the notation and the setting we will use through out the paper and give the full statement of our results.Section 2 contains some information about the second variation and the structure of the space of variation we will employ.
In Section 3 we deal with a couple of applications, such as Hill's formula and the eigenvalue problem for Schrödinger operators.The results deserve some interest on their own, however, the main focus of the section is to provide a worked out example of how to apply the formulas to concrete situations.
The last part of the paper is Sections 4.1 and 4.2.They are devoted to the proof of Theorem 1.We first prove the result for boundary conditions of the type N 0 × N 1 (Section 4.1) and than extend it to the general case (Section 4.2).Finally we give formulas to compute the trace of the compact part of the second variation K (Lemmas 3 and 4) and some normalization constant appearing in Theorem 1.

Problem statement and main results
We begin this section recalling briefly the setting and the notations that will be used throughout the paper.The reader is referred to [6,4] for more information on optimal control and sub-Riemannian problems.By an optimal control problem we mean the following data: a configuration space, i.e. some smooth manifold M and a family of smooth (and complete if M is non compact) vector fields f u .They depend on a parameter u living in some open set U ⊆ R k .We will always assume that the f u are smooth jointly in both variables, i.e. they belong to C ∞ (U × M, T M ).We can think of the parameter u as our way of interacting with the system and moving a particle from one state to another.
To any function u(•) ∈ L ∞ ([0, 1], U ) we can associate a trajectory in the configuration space considering the solution of: We will usually call the function u(•) control.
We say that a control u is admissible if the corresponding trajectory, denoted by γ u , is defined on the whole interval [0, 1].We can impose further restrictions on the trajectory γ u specifying proper boundary conditions.The most general situation that we are going to treat in the paper, is the case in which the boundary conditions are given by a submanifold N ⊆ M × M .We say that (a Lipschitz continuous) γ is admissible if γ = γ u for some admissible control u and if (γ(0), γ(1)) ∈ N .
Given a smooth function ϕ : U × M → R, we are interested in the following minimization problem on the space of admissible curves: It is customary to parametrized the space of admissible curves using the velocity, i.e. the control function u(•), and a finite dimensional parameter space which takes into account initial data and boundary conditions.We are going to follow this approach.However, this is just a technical point which is independent of the main statements.Hence we postpone the discussion of the structure of our space of variations to Section 2. Let us just mention that, under some natural assumptions, the space of variations can be endowed, locally, with a smooth Banach manifold structure.Thus, it makes sense to consider the tangent space to the space of variations.It is a finite codimension subspace (N ) .Suppose that u is critical point of the functional J restricted to our space of variations and consider the Hessian of J .It is a quadratic form on V. We denote this quadratic form by ).Instead of working with L ∞ topology, we will work with the weaker L 2 one since everything extends by continuity.For an appropriate choice of scalar product on is compact, but in general not trace-class.For a detailed account on the spectrum of the second variation the reader is referred to [3,9].Given an eigenvalue of Q, λ, denote by m(λ) its multiplicity.We define the determinant of the second variation as the following limit: λ) , where λ ∈ Spec(Q).
As already stated in the introduction, the computation of this determinant for general boundary conditions is the main contribution of this work.We provide a formula for this determinant involving essentially two ingredients: • the fundamental solution of a linear (non autonomous) system of ODE's which we call Jacobi equation; • the annihilator to the boundary condition manifold, a Lagrangian submanifold of T * M .
To state the main Theorem and define precisely the objects above, we need to introduce a little bit of notation.We will just sketch here what is needed to this purpose, further details are collected in A or given along the proofs.From now on, assume that a strictly normal extremal λ t with optimal control ũ is fixed (see A).
The first tool we introduce is the following family of Hamiltonians.It is strictly related to Legendre transform and quite useful when dealing with problems in the cotangent bundle.Given our optimal control ũ(•) define: Denote its flow at time t by Φt and its differential by ( Φt ) * .Pontryagin Maximum Principle (see A) tells us that normal extremals λ t satisfy λ t = Φt (λ 0 ).We will use the map Φt to connect the tangent spaces to each point of λ t to the starting one, λ 0 .This flow, in some sense, plays the role of the choice of a connection (or parallel transport as in [12]).
The second object we are going to introduce, is a kind of quadratic approximation of our starting system.It is given by a quadratic Hamiltonian on T λ0 T * M (see for detail [1] or [6][Chapters 20 and 21]).To define it we need to introduce two matrix valued functions Z t ∈ Mat k×2 dim(M) (R) and H t ∈ Mat k×k (R).The precise way to compute them is given in (32).However, for the moment, a precise understanding of how this matrices are obtained is not strictly needed.Heuristically, the matrix Z t represents a linear approximation of the Endpoint map of the original system whereas H t is a quadratic approximation of the Lagrangian ϕ along the extremal.
Let π : T * M → M be the natural projection and set Π := ker π * , the fibres.Define δ s as the dilation by s ∈ R of Π.It is determined by the relations π * (δ s w) = π * (w) for all w ∈ T λ0 T * M and δ s v = sv for all v ∈ ker(π * ).Let J be some coordinates representation of the standard symplectic form on T λ0 T * M .Let us define the following quadratic form: We will call Jacobi (or Jacobi type) equation the following ODEs system on and denote its fundamental solution at time t as Φ s t .Here, and for the rest of the paper, we will call fundamental solution any family Θ t , t ∈ R of linear maps which satisfies a linear ODE and have initial condition Θ 0 = I.Remark 1. Whenever PMP's maximum condition determines a C 2 function h on T * M , normal extremals satisfy a Hamiltonian ODE on the cotangent bundle of the form λ = h(λ).Jacobi equation for s = 1 is closely related to the linearisation of h along the extremal we are fixing.Suppose local coordinates are fixed and let d 2 λt h be the Hessian of h along the extremal.Let Ψ t be the fundamental solution of: It can be shown (see for example [6][Proposition 21.3]) that: The last maps we will need are a family of symplectomorphism of T λ0 T * M and T λ1 T * M .Their definition depends on the choice of a scalar product on each tangent space.Let g 0 and g 1 be two scalar products on these spaces.Assume that at each λ i , Π i := ker(π * ) ⊆ T λi T * M has a Lagrangian orthogonal complement with respect to g i which we denote by Π ⊥ i .For a subspace V , denote by pr V the orthogonal projection onto V .We set: The datum of the boundary condition is encoded in a Lagrangian submanifold of T * (M × M ), (−σ) ⊕ σ , the annihilator of N .It can be thought of as the symplectic version of the normal bundle in Riemannian geometry and is define as follows.Take a sub-manifold N ⊆ M × M and consider: In light of PMP (see A), critical points of J with boundary conditions given by N , lift to the cotangent bundle to curves λ t such that (λ 0 , λ 1 ) ∈ Ann(N ).
Fix now a complement to T (λ0,λ1) Ann(N ), say V N , and denote by π N the projection on V N having T (λ0,λ1) Ann(N ) as kernel.We are ready now to define a function that plays the role of the characteristic polynomial of the Hessian of J .For a map f denote by Γ(f ) its graph, set: ).With this notation, our main result reads as follows: Theorem 1. Suppose that λ t is a strictly normal extremal for problem (3) and ũ is its optimal control.Moreover, suppose that λ t satisfies Legendre strong condition, i.e. that ∃α > 0 such that, ∀v ∈ R k and that at least one of the following holds: • the maps t → Z t and t → H t are piecewise analytic in t; • the dimension of the space of controls is k ≤ 2; • the operator I − Q is trace class; Let λ ∈ Spec(Q) and m(λ) be its multiplicity, the limits: are well defined and finite.Moreover, for almost any choice of metrics g 0 and g 1 we have that p Q (0) = 0 and that: Remark 2. The hypothesis about the regularity of Z t and H t are needed to obtain the asymptotic for the spectrum of Q − I that guarantees the existence of the trace and of the determinant as limits.They can be weakened somehow by requiring that the skew-symmetric k × k matrix Z * t JZ t is continuously diagonalizable (see [3]).
Remark 3. The constants p Q (0), p ′ Q (0) and tr(Q − I) are completely explicit and are given in terms of iterated integrals in Lemmas 3 and 4.
In particular we have the following corollary: Corollary 1.Under the assumption above, the determinant of the second variation Q satisfies: Where Ψ t = Φ * Φ 1 and coincides with the fundamental solution of the linearisation of the extremal flow, whenever the latter is defined.

The Second Variation
The aim of this section is threefold: to define precisely what we mean by d 2 J | V , to define precisely its domain and to provide the integral representation of this quadratic form we will use throughout the proof section of the paper.Before going on, a little remark about topology is in order.Up until now we have considered Lipschitz continuous curves and L ∞ controls.Hence, it would be natural to work on the Banach space L ∞ ([0, 1], R k ) ⊕ R dim (N ) .However it turns out that, even if d 2 J | V is defined on the latter space, it extends to a continuous quadratic form on For this reason (and Fredholm alternative), we will work with L 2 controls for the rest of the paper.
Let n 0 , n 1 ∈ N and consider the Hilbert space (its scalar product will be defined in the next section).Let (Σ, σ) be a symplectic space and consider a linear map Z : H → Σ defined as: Suppose that Π ⊂ Σ is a Lagrangian subspace transverse to the image of the map Z and define V = Z −1 (Π).For an appropriate choice of Z and Π which depends on (2) and (3), the second variation (at a strictly normal critical point) is the quadratic form given in the following definition.
Definition 1 (Second Variation).The second variation at ũ is the quadratic form defined on V ⊆ H: The definition of this quadratic form may be a bit strange at first glance.Despite the appearances, the way one gets to such an expression is quite natural.The construction is explained in detail in [7].We will sketch here just the main features, essentially to introduce the notation needed.
The idea is to reduce the problem with boundary conditions N to a fixed points (or Dirichlet) problem for an appropriate auxiliary system.We will consider just the case of separated boundary conditions N 0 × N 1 .The general case reduces to this one using the procedure explained in Section 4.2.
The first step of the construction is to build the auxiliary system.We always work with a fixed strictly normal extremal λ t .Fix local foliations in neighbourhoods of its initial and final points having a portion of N 0 and N 1 as leaves.This determines two integrable distributions in a neighbourhood of those points.Suppose that said distributions are generated by some fields {X j i } dim(Nj ) i=1 and j = 0, 1.Consider the extended system: . Denote the initial and final points of our original extremal curve by (q 0 , q 1 ).We will use controls that are locally constant on [0, 1] c , this will be enough to reach any neighbouring point of (q 0 , q 1 ) in N 0 × N 1 .Minimizing our original functional is equivalent to minimize, with Dirichlet boundary conditions, the following one: The second step is to differentiate the Endpoint map (see A) of the auxiliary system.We employ the machinery of Chronological Calculus (see also [6][Section 20.3]), which is standard for fixed endpoints.One of the main steps of this differentiation, is to use a suitable family of symplectomorphism to trivialize the cotangent bundle along the curve we are fixing.This allows to write all the equations in the tangent space to the initial point, T λ0 (T * M ).Let us consider the following functions depending on the parameter u: . When an optimal control ũ(t) is given, we consider h t ũ(t) (λ) and the Hamiltonian system: Φt = h t ũ(t) ( Φt ).We then define the following functions: The asymptotic expansions of Chronological Calculus tell us that the second variation at ũ is the following quadratic form: It is defined on the tangent space V to the variations fixing the endpoints of our curve.This space can be described explicitly as V = {v : The third step is to specialize this representation to our auxiliary system.Notice that an extremal of the original problem lifts naturally to an extremal of the auxiliary one.If ũ is the original optimal control, extending it by zero on [0, 1] c gives the optimal control for the auxiliary problem.Applying the construction just sketched to the extended system, we find that Z t and H t are locally constant on [0, 1] c .We denote Z 0 to be its value on [−1, 0) and Z 1 the value on (1,2].H t is zero outside [0, 1].Thus, after substitution, we recover precisely the operator given in (5).
Remark 4. We always assume that our extremal is strictly normal and satisfies Legendre strong condition.In terms of the matrices Z t and H t this means that for t ∈ [0, 1]: As a last remark, notice that, by the first order optimality conditions, the map Z 0 takes values in the space T λ0 Ann(N 0 ) and the map Z 1 in the space ( Φ−1 1 ) * (T λ1 Ann(N 1 )) ( see PMP in A and [7]).

The scalar product on the space of variations
As already mentioned, we will assume through out this paper Legendre strong condition.The matrix −H t is positive definite on [0, 1], with uniformly bounded inverse.This allows to use −H t to define an Hilbert structure on L 2 ([0, 1], R k ) equivalent to the standard one.We have still to define the scalar product on a subspace transversal to V 0 = {u 0 = u 1 = 0}.A natural choice would be to introduce two metrics on T λ0 T * M and T λ1 T * M and pull them back to the space of controls using the maps Z 0 and Φ * Z 1 : R n → T λi T * M .Let us call any such metrics g 0 and g 1 .
Definition 2. For any u, v ∈ H define: Since the symplectic form σ is a skew-symmetric bilinear form, there exists a g i −skew-symmetric linear operator J i such that: In terms of the symplectic form the scalar product can be written as: Now, we use the Hilbert structure just introduced to write the operator K associated to the quadratic form Q − I, which is compact.To simplify notation, we can perform the change of coordinates in In this way the Hilbert structure on the interval becomes the standard one.
We introduce a further piece of notation, call pr 0 (respectively pr 1 ) the orthogonal projection on Im(Z 0 ) (respectively Im( Φ * Z 1 )) with respect to scalar product g 0 (respectively g 1 ).Let L be a partial inverse to Φ * Z 1 i.e. a map L : Lemma 1.The second variation, as a bilinear form, can be expressed as: where u, v ∈ V and K is the operator defined by: where Λ(u) is given above, in eq.(6).
Proof.A quick manipulation of the expression involving the symplectic form in Definition 1, yields the following: Recall that Z t is constant on [0, 1] c .Moreover the images of the maps Z 0 and Z 1 are isotropic subspaces.We used this fact to simplify the expression in the first line.Now, it is clear that in the last term: only the projection onto the image of Φ * Z 1 plays a role.It is straightforward to check that: Recall that we have normalized H t to −1, thus the first summand can be rewritten as follows: Adding and subtracting g(Z 0 u 0 , Z 0 v 0 ) and g(Z 1 u 1 , Z 1 v 1 ) to single out the identity, we obtain the formula in the statement.

Hill-type formulas
Before going to the proof of Theorem 1 we present here some applications of the main result.We deduce Hill's formula for periodic trajectory and specify it to the eigenvalue problem for Schrödinger operators.In the second sub-section we present a variation of the classical Hill formula for systems with drift.We will mainly deal with periodic and quasi-periodic boundary conditions.Namely, we consider the case N = Γ(f ) for a diffeomorphism of the state space f : M → M .
The proofs of this section rely quite heavily on the machinery introduce in Section 4.2, in particular in Lemmas 3 and 4. The statements, on the other hand, do not and could shed some light on Theorem 1.Despite the appearance, proofs are rather simple.They reduce to a (long) computation of the normalizing factors appearing in the statement of Theorem 1 and can be skipped at first reading.

Driftless systems and classical Hill's formula
In this section we consider driftless systems with periodic boundary conditions on R n and specify the formulas of Theorem 4 for this class of problems.
First of all let us explain what we mean by driftless systems.Let t → R t a continuous family of symmetric matrices of size n × n and let us denote by u a function in Consider the following family of vector fields f u (q), their associated trajectories q u (t) and the action functional A(u): We impose periodic boundary conditions, i.e. we take N = ∆ = {(q, q) ∈ R 2n : q ∈ R n }.The Hamiltonian coming from the Maximum Principle takes the form: Let us denote the flow generated by H by Ψ t .Fix a normal extremal λ t for periodic boundary conditions and its optimal control ũ(t).
Theorem 2 (Hill's formula).Let I + K be the second variation at ũ and Ψ the fundamental solution of Ψ = HΨ.The following equality holds: where G is a scalar product on the tangent space to the initial point.We can apply the previous result to study boundary value problems for Sturm-Liouville operators.Let us illustrate the case of Schrödinger equation with periodic boundary conditions.Fix, without loss of generality, the normal extremal (p(t), q(t)) = (0, 0) and with relative optimal control ũ = 0. Consider the cost Rt = R t + λ, for λ ∈ R. Consider the second variation of the functional at the point ũ = 0.It is given by the operator 1 + K λ where: We have the following corollary: The determinant of the operators 1 + K λ can be expressed as: where G as in the previous statement.
Proof of Theorem 2. We are going now to describe explicitly all the objects involved in Theorem 1.Let us start with the flow Φ we use to re-parametrize the space and its differential.It is given by the Hamiltonian: The matrix Z t is the following: To simplify notation, let us call R = − 1 0 R τ dτ .The annihilator of the diagonal is simply the graph of the identity.Hence the following map, defined on (T λ0 T * M ) 2 , has the latter as kernel: We will now define Q s as in eq.(28) (actually up to a scalar, but this is irrelevant).For η ∈ T λ0 T * M set: It is clear that the kernel of Q s is precisely the intersection of the graph with the diagonal subspace.Since we are working on R 2n we can define the determinant of this map as: As already mentioned in Section 1.1 this function is a multiple of the characteristic polynomial of K.It satisfies (see sections 4.1 and 4.2): Let us compute the normalization factors.To do so, we have to evaluate det(Q s ) and its derivative at s = 0.This will give us the relations: We have to work a bit to write down precisely all the quantities appearing in the formulas above.It is straightforward to compute the matrix representations of the maps A s 0 and A s 1 .In this setting the projections onto Π 0 and Π ⊥ 0 are given by: Recall that the definition of A s 0 and A s 1 given in (4) depends on the choice of two scalar products g 0 and g 1 .Denote by G 0 and G 1 their restriction to Π ⊥ 0 and Π ⊥ 1 respectively.We have that: The value of Φ s t and its derivative at s = 0 is given in Lemma 4. Here, for the submatrices of Φ 0 t and ∂ s Φ s t | s=0 , we use the notation defined in the Lemma 4. In this case, since Y t = t 0 R τ dτ and X t = 1, we obtain: Let us compute the value of det(Q s ) in zero.Putting all together we have: After a little bit of computation we find that Q s | s=0 satisfies: We can compute the derivative det(Q s ) at s = 0, we find that: We use now Jacobi formula for the derivative of the determinant of a family of invertible matrices.It reads: Without going into the detail of the actual computation, which at this point is just matrix multiplication, we have that: The last quantity we have to compute is tr(K).To do so we use Lemma 3. Mind that in the statement of the Lemma one works with twice the variables, taking as state space R n × R n and using the symplectic form The quantities with ˜on top always refer to the system in R 4n , where we have a trivial dynamic on the first factor and the boundary condition we impose are in this case ∆ × ∆ (see the beginning of Section 4.2 for more details).The formula given in the lemma reads: Let us explain all the objects appearing in the formula.πi * denotes the differential of the natural projection on the i−th factor.The matrix Φ * is given here by: Moreover the matrices Z0 , Zt and Z1 are: The map pr 1 denotes the orthogonal projection onto the image of Z1 .We are using the scalar product g 0 ⊕ g 1 on T λ0 T * M × T λ1 T * M to define it.One can check that the following map is the coordinate representation of pr 1 : Now everything reduces to some tedious matrix multiplications.The second term in the right hand side of (10) simplifies as follows: For the third term notice that (π 2 * − π 1 * )( Φ * ) −1 pr 1 is identically zero since Φ * does not change the projection on the horizontal part and we are working with periodic boundary conditions.It follows we are left with tr(Γ −1 Ω).Summing up we have that: It is natural to think of g 0 and g 1 as restrictions of globally defined Riemannian metrics.Doing so, since we are working with periodic boundary conditions, amounts to choose G 0 = G 1 .Hence the result.

System with drift and Hill-type formulas
In this section we give a version of Hill's formula for linear systems with drift.They are again linear system with quadratic cost of the following form: Here A t is n × n matrix and B t a n × k one, both with possibly non-constant (but continuous) coefficients.The maximized Hamiltonian takes the form: Denote by Φt the fundamental solution of q = A t q at time t.We can lift this map to a symplectomorphism of the cotangent bundle which we denote by Φ t .As boundary conditions we take the following affine subspace of R 4n : Notice that, since only the tangent space matters in our formulas, the translation is irrelevant and it would be the same as if we considered Γ( Φ).An obvious choice of extremal is the point λ t = (0, 0), with control ũ = 0.
Theorem 3 (Hill's formula with drift).Suppose that a critical point of the functional given in (11) is fixed and let ũ be its optimal control.Let G 0 and G 1 be our choices of scalar product, Γ := Let I + K be the second variation at ũ.The following equality holds: Proof.The proof is completely analogous to the one of Theorem 2. First of all the Hamiltonian we use to re-parametrize is given as follows: Hence the flow and its differential are given by: We have to define the map Q s .Similarly as the previous case, we can define a map having as kernel the annihilator to the boundary conditions as follows: Φτ dτ , the upper right minor of Φ * .A quick computation shows that: Hence, up to renaming G 1 and Γ in the proof of Theorem 2, we have: Now we have to apply Lemma 3 to compute the trace of the compact part of the second variation.Here pr 1 and Z1 are different since we have changed boundary conditions.However we have the same kind of simplification as in the previous case.Let us write explicitly the new objects: In the end, the trace reads: Contrary to the previous section the two term do not simplify.We are left with the following equation for b: Hence, the statement follows evaluating det(Q s ) at s = 1:

Proof of the main Theorem
In this section we provide a proof of Theorem 1.First we work with separated boundary conditions and then reduce the general case to the former.The proof is a bit long so we try to give here a concise outline.The idea is to construct an analytic function f which vanishes precisely on the set {−1/λ : λ ∈ Spec(K)} ⊆ R.
Particular care is needed to show that the multiplicity of the zeros of this function equals the multiplicity of the eigenvalues of K. We do this in Proposition 1 and Proposition 2 respectively.We show that this function decays exponentially and use a classical factorization Theorem by Hadamard to represent it as f (s) = as k e bs λ∈Spec(K) To prove the general case, we double the variables and consider general boundary conditions as separated ones.In this framework we compute the value of the parameters a, b and k appearing in the factorization.

Separated boundary conditions
We briefly recall the notation.We are working with an extremal λ t with initial and final point (λ 0 , λ 1 ) ∈ Ann(N ), where N = N 0 × N 1 are product boundary conditions.We are assuming that λ t is strictly normal and satisfies Legendre strong conditions.We work in a fixed tangent space, namely T λ0 T * M , to do so we backtrack our curve to its starting point λ 0 using the flow generated by the time dependent Hamiltonian: We denote the differential of said flow by Φ * .We have a scalar product g i on T λi T * M , for each i = 0, 1.We assume that the orthogonal complement to the fibre at λ i , Π i = T λi T * π(λi) M is a Lagrangian subspace and that the range of Z 0 (and Φ * Z 1 respectively) is contained in Π ⊥ 0 (resp.Π ⊥ 1 ).Remark 6.If we fix Darboux (i.e.canonical ) coordinates coming from the splitting Π i ⊕ Π ⊥ i it is straightforward to check that g i takes a block diagonal form with symmetric n × n matrix G j i on the main diagonal.Similarly we can write down the coordinate representation of the matrix J i and find: For s ∈ R (or C) we introduce the following symplectic maps: Notice that the transformation A s i are indeed symplectomorphisms.In (canonical) coordinates given by Π i and Π ⊥ i they have the following matrix representation: The last map we are going introduce is two families of dilation in T λ0 (T * M ), one of the vertical subspace and one of its orthogonal complement.Let s ∈ R (or C) and let us define the following maps: Proposition 1.Let A s i be the maps given in eq. ( 12) and let Φ s 1 be the fundamental solution of the system: The operator I + sK restricted to V has non trivial kernel if and only if there exists a non zero (η 0 , η 1 ) ∈ T (λ0,λ1) (Ann(N )) such that In particular, the geometric multiplicity of the kernel of I +sK equals the number of linearly independent solutions of the above equation.
Proof.I + sK has a non trivial kernel if and only if Equivalently if and only if ∃u such that u + sKu ∈ V ⊥ (see Lemma 2 below for a description of V ⊥ ).This is in turn equivalent to the following system Let us substitute Z t with Z s t = δ s Z t .It is straightforward to check, using the definition in (13), that: All the computations we will do from here on are aimed at rewriting ( 14) as a boundary value problem in T λ0 T * M × T λ0 T * M .Let us start with the second equality in (14).Define: Last equality being due to the fact that the image of Φ * Z 1 is isotropic and thus J 1 Im( Φ * Z 1 ) ⊂ Im( Φ * Z 1 ) ⊥ .Moreover Φ * Z 1 u 1 coincides with the projection of − Φ * η(1) on Π ⊥ 1 .Thus we are left with: It is straightforward to check that pr 1 J 1 Φ * pr Π0 Z 1 u 1 depends only on the projection of and using the relation , the second equality in ( 16) can be rewritten as: the equations reduce to pr 0 (J 0 η(0)) = 0 and pr 1 J 1 Φ * η(1) = 0. Consider the first case, the relation is equivalent to: Thus we are looking for solution starting from T λ0 Ann(N 0 ).Similarly setting s = 1 in the second equality we find that η(1) must lie inside T λ1 Ann(N 1 ).Now, for s = 1, we want to interpret the boundary conditions as an analytic family of Lagrangian subspaces depending on s.To do so we employ the following linear map defined in (12): Ann(N 0 ) we have that pr Π ⊥ 0 η = pr 0 η and pr 0 J 0 η = 0 and thus: So we have shown that A s 0 (T λ0 Ann(N 0 )) is precisely the space satisfying the first set of equations.A similar argument works for the final point.Let us recall the definition of A s 1 given in (12): . Now, we check that the boundary condition for the final point are satisfied if and only if A s 1 • Φ * η(1) ∈ T λ1 Ann(N 1 ).In fact, take any η in T λ1 Ann(N 1 ), it holds: Hence, the subspace {u : Hence v t = Z * t Jν.Now, take u ∈ V and compute: Since u ∈ V, we have: It follows that: Since we are assuming that 1 0 X t X * t dt > 0, the image of the map u t → π * 1 0 Z t u t dt is the whole T q0 M .In particular, for any u 0 , is possible to find variations of the form (u 0 , u t , 0) ∈ V.An analogous statements holds for variations of the form (0, u t , u 1 ).
Moreover, this implies that v 0 and v 1 are completely determined by the value of ν.Finally notice that: Remark 7. If we complexify all the subspaces involved in the proof of Proposition 1, i.e. tensor with C we can take also s ∈ C.
We can reformulate the intersection problem in the statement of Proposition 1 as follows.Let π N1 the orthogonal projection, with respect to g 1 , onto the subspace T λ1 Ann(N 1 ) ⊥ and define a map Q s as: Let us fix now two bases, one of T λ0 Ann(N 0 ) and one of T λ1 Ann(N 1 ).Construct two 2n × n matrices using the elements of the chosen basis.Let us call the resulting objects T 0 and T 1 respectively.It follows that J 1 T 1 is a basis of T λ1 Ann(N 1 ) ⊥ .Define the function det(Q s ) as the determinant of the n × n Clearly, different choices of basis give simply a scalar multiple of det(Q s ) and thus is well defined: Moreover det(Q s )| s=s0 = 0 if and only if there exists at least a solution to our boundary problem.Notice that map s → det(Q s ) is analytic in s since the fundamental matrix is an entire map in s (see [3][Proposition 4]).The following Proposition shows that the multiplicity of any root s 0 = 0 is equal to the number of independent solutions to the boundary value problem.
Proposition 2. The multiplicity of any root s 0 = 0 of det Q s equals the dimension of the kernel of Q s .
Proof.The proof is done in two steps.First we show that the equation det(Q s ) = 0 is equivalent to det(R s ) = 0 where R s is a symmetric matrix, analytic in s.
Once one knows this, it suffices to compute ∂ s R s and show that it is non degenerate to prove that the multiplicity of the equation is the same as the dimension of the kernel.
Step 1: Replace Q s with a symmetric matrix Let us restrict to the case s ∈ R, since all the roots are real.As shown in (18) and remarked above, the determinant of the matrix Q s is zero whenever the graph of A s 1 Φ * Φ s 1 A s 0 intersect the subspace L 0 = T (λ0,λ1) (Ann(N 0 × N 1 )).Suppose that s 0 is a time of intersection and choose as coordinates in the Lagrange Grassmannian L 0 and another subspace L 1 transversal to both L 0 and Λ s := Γ(A s 1 Φ * Φ s 1 A s 0 ).This means that, if (T λ0 T * M ) 2 ≈ {(p, q)|p, q ∈ R 2n }, we identify L 0 ≈ {q = 0} and L 1 ≈ {p = 0}.
In this coordinates Λ s is given by the graph of a symmetric matrix, i.e. is the following subspace Λ s = {(p, R s p)} where again R s is analytic in s.
The quadratic form associated to the derivative ∂ s R(s) can be interpreted as the velocity of the curve s → Λ s inside the Grassmannian, it is possible to compute it choosing an arbitrary base of Λ s and an arbitrary set of coordinates.Invariants such as signature and nullity do not change (see for example [5, ?] or [2]).Take a curve λ s = (p s , R s p s ) inside Λ s then one has: Recall that we will be using the symplectic form given by (−σ λ0 ) ⊕ σ λ1 , in order to have that graph of a symplectic map is a Lagrangian subspace.
Step 2: Replace Λ s with a positive curve We slightly modify our curve to exploit an hidden positivity of the Jacobi equation.We substitute the fundamental solution Φ s 1 with the following map: It is straightforward to check that Ψ s is again a symplectomorphism and that it is the fundamental solution of the following ODE system at time t = 1: On one hand we are introducing a singularity at s = 0 but on the other hand we are going to show that the graph of Ψ s becomes a monotone curve and its velocity is fairly easy to compute.
First of all, hoping that the slight abuse of notation will not create any confusion, let us introduce a family of dilations similar to the δ s ,δ s also in T λ1 T * M .The definition is analogous to the one in (13) but with Π 1 and Π ⊥ 1 instead of Π 0 and Π ⊥ 1 .We will denote them with the same symbol.Let us consider the following symplectomorphisms: Notice that the dilations δ s preserve the subspaces T λi Ann(N i ) and thus the intersection points between the graph of the above map and the subspace T (λ0,λ1) Ann(N 0 × N 1 ) are unchanged.Let us rewrite the maps δ s A s 0 δ 1 s and δ s A s 1 Φ * δ 1 s .For the former: For the latter, a computation in local coordinates and the fact that the dilations δ s and Φ * do not commute yield: Thus we take, for s = 0, as curve Λ s := Γ(B s 1 Φ * Ψ s B s 0 ), the graph of the symplectomorphism just introduced.Notice that Ψ s is actually analytic, the singularity at s = 0 comes only form the maps B s i .
Step 3: Computation of the velocity Now we compute the velocity of the graph of B s 1 Φ * Ψ s B s 0 .Take a curve λ s = (η, B s 1 Φ * Ψ s B s 0 η) inside the Λ s and let us compute the quadratic form associated to the velocity: Let us consider the terms of the type σ(B s i x, ∂ s B s i x).It immediate to compute the derivative in this case, recall that thus the first and last term read as: Notice we used the fact that J i (and thus J −1 i ) is g i −skew symmetric.Now we rewrite the middle term.We present it as the integral of its derivative using the equation for Ψ s t .Let us use the shorthand notation x = B s 0 η.We obtain: The first and second term have opposite sign and thus cancel out.What remains is: Integrating over [0, 1] and using the fact that ∂ s Ψ s t | (0,0) = 0 we get that: Using the notation ||•|| to denote the norm with respect to the corresponding metric and summing everything up we find the following expression for the velocity of our curve: Since each term of the sum is non negative S s (λ s ) is zero if and only if each term is zero.From the first one we obtain that η must be contained in the fibre.Notice that B s 0 acts as the identity on Π 0 and thus in this case η is a solution of the Jacobi equation ( 15) starting and reaching the fibre (recall that Φ * (Π 0 ) = Π 1 ).Let us consider now the third piece, since the integrand is positive it must hold that for almost any t, Z * t JΨ s t η = 0.If we multiply this equation by Z t we find that: It follow that we are dealing with a constant solution starting and reaching the fibre.However this contradicts the assumption that the matrix 1 0 X t X * t dt is non degenerate.In fact, if we substitute a non zero constant solutions starting from the fibre in (15), we find that pr The following proposition is proved in [3].
Proposition 3.There exists c 1 , c 2 > 0 such that: Moreover Φ s t is analytic and the function s → det(Q s ) is entire and satisfy the same type of estimate.
This fact tells us that det Q s is an entire function of order ρ ≤ 1.We know its zeros, which are determined by the eigenvalues of K, and thus we can apply Hadamard factorization theorem (see [13]) to present it as an infinite product.It follows that we have the following identity: where m(λ) is the geometric multiplicity of the eigenvalue λ.To determine the remaining parameters it sufficient to know the value of det(Q s ) and a certain number of its derivatives at s = 0 (depending on the value of k).Assume for now that k = 0, a straightforward computation shows that: We will compute these quantities in Lemmas 3 and 4 for general boundary conditions.The proofs can be adapted to the case of separated conditions easily.

General boundary condition
In this section we prove a determinant formula for general boundary conditions N ⊆ M × M .First, we reduce this case to the case of separate boundary conditions.We have to slightly modify the proof of Proposition 1 since, after this reduction, the Endpoint map will not be a submersion any more.Then, we compute the normalization factors given in (22).
Let us consider M × M as state space, with the following dynamical system: and boundary conditions ∆ × N .With this definition, any extremal between two points q 0 and q 1 lifts naturally to an extremal between (q 0 , q 0 ) and (q 0 , q 1 ).However, the Endpoint map of the new system is no longer surjective.In fact, any trajectory is confined to a submanifold of the form {q}×M .Thus, even if we started with a strictly normal extremal, we do not get a strictly normal extremal of the new system.However, there is no real singularity of the Endpoint map here: we have just introduced a certain number of conserved quantities.All the proofs presented above work also in this case.We are going to discuss briefly how to adapt them.Let us start with Pontryagin maximum principle.It implies that the lift of the extremal curve q(t) = (q 0 , q(t)) is the curve λ(t) = (−λ 0 , λ(t)).This is because the initial and final covector of the lift must annihilate the tangent space of the boundary conditions manifold.In this case N 0 = ∆ and the annihilator of the diagonal subspace is {(λ, −λ) : λ ∈ T λ0 T * q0 M }.Moreover, by the orthogonality condition in PMP (see A), we know that (−λ(0), λ(1)) annihilates the tangent space of N .
Thus, if we want to work in one fixed tangent space, we have to multiply the first covector by −1.This changes the sign of the symplectic form and we are thus brought to work on T λ0 T * M × T λ0 T * M with symplectic form (−σ) ⊕ σ.
With this change of sign, the tangent space to the annihilator of the diagonal gets mapped to the diagonal subspace of T λ0 T * M × T λ0 T * M and the tangent space to the annihilator of the boundary conditions N is mapped to the tangent space of: Let us make now some notational remarks.We will still denote by Φ * the map 1 × Φ * , which correspond to the new flow we are using to backtrack our trajectory to the starting point.
As explained in Section 2, to define the scalar product on our space of variations, it is necessary to introduce two metrics on the tangent spaces to the endpoints of our curve.We will choose them of the form g0 = g 0 ⊕ g 0 and g1 = g 0 ⊕ g 1 where g 0 and g 1 are two metrics on T λ0 T * M and T λ1 T * M respectively.Now we compute the second variation of the new system (23).As a general rule, we will denote all the quantities relative to (23) on M × M , putting a˜on top.We have: Notice that Φ * Z1 maps R dim(N ) to the tangent space to A(N ) and we can assume that its image is contained in Π ⊥ 0 × Π ⊥ 1 .We will denote by pr 1 the orthogonal projection onto the image of Z1 .
The domain of the second variation is the subspace Clearly, this equation is equivalent to: It follows that the control u 0 is completely determined by u 1 .Moreover we can assume that Z 0 u 0 = −Z 0 1 u 1 since we are free to choose any system of coordinates and any trivialization of the tangent bundle of the manifolds ∆ and N .
Let A s i be the maps given in (12).The following proposition is the counterpart of Proposition 1 for general boundary conditions.Proposition 4. Let Φ s 1 be the fundamental solution of the Jacobi system: The operator 1 + sK restricted to V has non trivial kernel if and only if there exists a non zero (η 0 , η 1 ) ∈ T (λ0,λ1) A(N ) such that The geometric multiplicity of the kernel equals the number of linearly independent solutions of the above equation.
Proof.The proof is completely analogous to the one of Proposition 1.However, some slight modifications are in order since the Endpoint map is not surjective in this case.
Step 1: Characterize V ⊥ .The orthogonal complement to V is given by: The proof is the same as the one of Lemma 2 and yields v t = Z * t J ν. Here, however, we can not separate v 0 from v 1 .Take u ∈ V and v ∈ V ⊥ : Hence: Step 2: Derivation of Jacobi equation.
Now we can write down Jacobi equation in a fashion similar to the one of Proposition 1.The system reads: Now we define an analogous map to the one in eq.(19).Let π N be the orthogonal projection on the space T (λ0,λ1) A(N ) ⊥ and consider the map: Let T = (T 0 , T 1 ) be any linear invertible map from R 2n to the tangent space T (λ0,λ1) A(N ).We denote by J the map (−J 0 ) ⊕ J 1 representing the symplectic form (−σ λ0 ) ⊕ σ λ1 .As in the previous section we define the following function: One could also define (28) as a bilinear form, using just the symplectic pairing.In fact, for (η, (ξ 0 , ξ 1 )) in T λ0 T * M × T (λ0,λ1) A(N ), define: This form is degenerate exactly when Γ(A s 1 Φ * A s 0 ) ∩ A(N ) = (0).Proposition 5.The multiplicity of any roots s 0 = 0 of the equation det(Q s ) is equal to the geometric multiplicity of the boundary value problem.
Proof.The same proof of Proposition 2 works verbatim.Indeed, we are working with the same curve, Γ(A s 1 Φ * A s 0 ).In the remaining part of this section we carry out the computation of the normalizing factors of the function det(Q s ).As already mentioned at the end of the previous section a classical factorization theorem by Hadamard (see [13]) tells us that: where m(λ) is the geometric multiplicity of the eigenvalue.We are now going to compute the values of a, b ∈ C and k.Theorem 4. For almost any choice of metrics g 0 , g 1 on T λi T * M , det(Q s | s=0 ) = 0. Whenever this condition holds, the determinant of the second variation is given by: Proof.We prove the first assertion: for almost any choice of scalar product, k = 0 and thus a = det(Q s 1 | s=0 ) = 0.This is equivalent to a transversality condition between the graph of the symplectomorphism A s 1 Φ * Φ s A s 0 and the annihilator of the boundary conditions N .
We can argue as follows: consider the following family of maps acting on the Lagrange Grassmannian of T λ0 T * M × T λ1 T * M depending on the choice of scalar products G 0 and G 1 : It is straightforward to see that they define a family of algebraic maps of the Grassmannian to itself.For any chosen subspace L . Using the formula in Lemma 4 one has that Γ( Φ * Φ s ) is transversal to Π 0 × Π 1 and thus to F −1 G (L 0 ) for any fixed L 0 and G i sufficiently large.Now, since everything is algebraic in G and there is a Zariski open set in which the transversality condition holds, the possible choices of G i for which k > 0 are in codimension 1.
Let us assume that k = 0 and compute b.Differentiating the expression for det(Q s ) in eq. ( 21) at s = 0 we find that: An integral formula for the trace of K is given in Lemma 3. The derivative of det(Q s ) can be computed using Jacobi formula: An explicit expression of the derivatives of the map Q s can be computed using Lemma 4. It follows that b = tr(∂ s Q s (Q s ) −1 ) − tr(K) and we obtain precisely the formula in the statement.
Before giving the explicit formula for tr(K) and the derivatives of the fundamental solution to Jacobi equation at s = 0 we need to make some notational remark and write down a formula for the second variation in the same spirit of Section 2 and eq.( 5).We are working on the state space M × M with twice the number of variables of the original system and trivial dynamic on the first factor and separated boundary conditions.The left boundary condition manifold is the diagonal of M × M and the right one is our starting N .We apply the formula in eq. ( 5) to this particular system, we denote by Zt and Zi the matrices for the auxiliary problem, in general everything pertaining to it will be marked by a tilde.Identifying T (λ0,λ0) T * (M × M ) with T λ0 T * M × T λ0 T * M we have that: We still work on the subspace V = {(u 0 , u t , u 1 ) : Z0 u 0 + 1 0 Zt u t + Z1 u 1 ∈ Π}.However it is clear that the this equation implies that: It follows that control u 0 is completely determined by u 1 .Moreover we can assume that Z 0 u 0 = −Z 0 1 u 1 since we are free to choose any system of coordinates and any trivialization of the tangent bundle of the manifolds ∆ and N .Technically we are working with different scalar products on each of the copies of T λ0 T * M .However it is easy to see that on the space V only the sum of this metrics plays a role.We will denote it g 0 .Now we are ready to state the following: A similar strategy applied to V ′ ∩ {ν = 0} tells us that the last contribution for the trace is given by the following map: It is worth pointing out that indeed the trace does not depend on X 0 and that the vector 1 0 Zt Z t Jνdt is the following: In particular if the boundary conditions are separated (i.e.N = N 0 × N 1 ) the part of the trace coming from V ′ ∩ {u 0 = 0} depends only on the projection onto T λ1 N 1 .
Lemma 4. The flow Φ s t | s=0 and its derivative ∂ s Φ s t | s=0 are given by: Proof.It is straightforward to check that Φ s 1 | s=0 and ∂ s Φ s 1 | s=0 solve the following Cauchy problems: Solving the ODE one obtains the formula in the statement.

A
In this appendix we collect some information concerning Pontryagin Maximum Principle (PMP) and the differentiation of the endpoint map used and mentioned throughout the text.Everything is fairly standard material in geometric control theory, the reader is referred to [6,22,4] for further details.

A.1 Pontryagin Maximum Principle
Let us introduce a useful family of Hamiltonian functions on T * M .They generate a family of Hamiltonian flows which we use to backtrack admissible trajectories γ to their initial point.Moreover, they appear in the formulation of PMP and extend the flow of the fields f u(t) to the cotangent bundle.Set: h t u : T * M → R, h t u (λ) = λ, f u + νϕ(u, π(λ)), ν ≤ 0.
In particular, if γ is and admissible curve, we can build a lift, i.e. a curve λ in T * M such that π( λ) = γ, solving λ = h u (λ).The following wellknown theorem, Pontryagin Maximum Principle, gives a characterization of critical points of J (as defined in (3)), for any set of boundary conditions N .
We call q(t) an extremal curve (or trajectory) and λ(t) an extremal.
There are essentially two possibility for the parameter ν, it can be either 0 or, after appropriate normalization of λ t , −1.The extremals belonging to the first family are called abnormal whereas the ones belonging to second normal.

A.2 The Endpoint map and its differentiation
In this subsection we write down the integral expression for the first and second derivative of the endpoint map.Further details can be found in [6][ Section 20.3].Denote by U q0 ⊂ L ∞ ([0, 1], U ) be the space of admissible controls at point q 0 and define the following map: It takes the control u and gives the position at time t of the solution starting from q 0 of: q = f u(τ ) (q).
We call this map Endpoint map.It turns out that E t is smooth, provided that the fields f u (q) are smooth too.

Remark 5 .
If we are working on the interval [0, T ] instead of [0, 1] everything remains essentially unchanged.The only difference is that extra factor T −n appears in the right hand side.In the notation of the proof below this corresponds to det(Γ) −1 .