17.1.1The problem

We consider the following optimization problem:

minimize

subject to

The final time T can be fixed or variable, M is a n-dimensional connected paracompact smooth manifold M and the state endpoint constraints N0 and Nf are smooth connected embedded submanifolds of M. We assume that c0, cf , f 0, f are defined on open sets and that they are C∞.

We take smooth time-independent data because we are interested in the irregularities arising from the maximization of the Hamiltonian.

The couple (ξ, υ) is called admissible if it is a solution of (1.2a), which satisfies the constraints (1.2b)–(1.2c)–(1.2d); in this case we refer to ξ as an admissible trajectory and to υ as the associated control.

We consider as a candidate optimal solution a given reference admissible trajectory ̂ξ, which is identified by the triple (̂T, , ̂υ) and we study its strong local optimality according to the following Definition 1.1.

Definition 1.1 (Strong local minimizer).

The reference admissible trajectory is a strong local minimizer of the above considered problem (1.1), if there are neighborhoods U ⊆ ℝ× M of the graph of ̂ξ, denoted by Γ, and ? ⊆ ℝ×M of (̂T, (̂T)) such that ̂ξ is a minimizer among all the admissible trajectories ξ satisfying

independently of the associated control.

When the final time is fixed, this definition reduces to the usual definition of strong local minimum which uses the C0 topology to describe a neighborhood of the admissible trajectories; whereas in the case of variable end time our definition is local with respect to the graph of ̂ξ and also with respect to both the final time and the final point. We need the neighborhood ? ⊆ ℝ × M of (̂T, (̂T)), since (T, ξ(T)) may belong to U being T ≪ ̂T.

This notion has been called time-state-local optimality in [23, 24], where a stronger version of optimality, called state-local optimality, is also used.

17.2.1Symplectic notations

For a general introduction to symplectic geometry and its application to variational problems we refer to [7] while for more specific applications to optimal control we refer to [1].

Denote by π : T∗M → M the cotangent bundle, it is well known that T∗M possesses a canonically defined symplectic structure, given by the symplectic form σℓ = ds(ℓ), where ℓ denotes an element of T∗M and s is the Liouville canonical one-form s(ℓ) = ℓ ∘ π∗.

If qi are local coordinates on the base manifold M and pi the fiber coordinates, then in local coordinates we can write

where d denotes the exterior derivative and ∧ denotes the exterior product.

We recall that thanks to the symplectic structurewe can associate, with each timedependent locally defined Hamiltonian Ht : T∗M → ℝ, a unique vector field on T∗M defined by the action

This vector field defines a corresponding Hamiltonian system

and we will denote its flow by (t, ℓ) → ℋ(t, ℓ), that is, ℋ(t, ℓ) is the solution at time t of system (2.1). Note that for any time-dependent object we will use the notation

whenever one of the variables is to be considered as fixed.

17.2.2The Pontryagin maximum principle

We assume that the candidate optimal solution is a state extremal, ξ̂that is, it satisfiesthe PMP, see Assumption 2.2.

For simpler notations we set

To state the PMP we introduce three Hamiltonians; if U × U is the common domain ofboth f and f 0 in M × U, let Ω := π−1(U)⊆ T∗M.

We define

(1) the (control-dependent) pre-Hamiltonian F : Ω × U →ℝas

F : (ℓ, u) → ⟨ℓ, f(π(ℓ), u)⟩ − p0 f 0(π(ℓ), u) ,

(2) the time-dependent reference Hamiltonian ̂F : [0, ̂T] × Ω → ℝ as

̂F : (t, ℓ) → F(ℓ, ̂υ(t)) ,

(3) the maximized Hamiltonian as

Let us remark that the maximized Hamiltonian could take the value +∞.

All the above Hamiltonians depend on the parameter p0, which can take the values {0, 1} characterizing, respectively, the abnormal and normal case. We have assumed p0 ≥ 0 to state the PMP in its standard form, some authors assume p0 ≤ 0 in which case you have to substitute p0 by −p0 in all the statements to obtain a Pontryagin minimum principle.

Assumption 2.2 (Pontryagin Maximum Principle). There is a nontrivial couple (p0,), where p0 ∈ {0, 1} and: [0, ̂T] → T∗Mis a lifting of ̂ξ to T∗M, (i.e., satisfying the Hamiltonian system

the transversality conditions

and the maximization property

Moreover ̂Ft((t)) is constant and it is zero when the end time T is variable.

is the Pontryagin extremal associated to the state extremal ξ̂For simpler notations we set

It is not difficult to see that the two functions p0 c0 and p0 cf act on N 0 and N f , respectively, but they can be extended to an open set in M in such a way that the transversality conditions (2.3) and (2.4) hold on the whole tangent space. Namely we denote by α, β : M → ℝ two locally defined functions such that

The transversality conditions can also be expressed as

where “ ⊥” means orthogonal with respect to the dual coupling.

In the normal case α, β are cost functions equivalent to the original ones while in the abnormal case they are extensions of the zero function on the constraints.

When p0 = 0 all the costs disappear from our conditions and indeed we will studya problem with a zero cost. Proving that is a strict minimizer will imply that it isisolated among the admissible trajectories.

17.3.1The Cartan form

Let α : M → ℝbe a smooth function and let Λ be the graph of dα on a contractible open set. Λ is a Lagrangian submanifold of T∗M and it is known that

Let ? be an open set in T∗M and J ⊆ ℝbe an open interval, with J ⊃ [0, ̂T], and ? ⊃ Λ, we consider a time-dependent Hamiltonian

Associated to this Hamiltonian we consider the Cartan form on J × ?

The following assumption describes a set of minimal regularity conditions which the Hamiltonian H has to verify to develop our approach. In specific examples, the properties of H will be different but they will imply this kind of regularity.

Assumption 3.1. Assume that

1.The flow

(t, ℓ) ∈ J × Λ → ℋ(t, ℓ) ∈ T∗M

is well defined and Lipschitz continuous.

2.The function

is Lipschitz–Carathéodory (Definition 2.1).

Thanks to Assumption 3.1 the Cartan form defines a Lipschitz function θ : J × Λ → ℝ, which will play a crucial role. Let

and hence

Let us now prove that, also in this case, the classical property of the Cartan form holds, that is, the form ℋ∗ω is exact on J × Λ.

We will consider differential one-forms with possibly L∞ coefficients; these forms are called (Whitney) flat forms and their properties are presented in [15]. The general theory is fully presented in Whitney’s monograph [28], see also [13]. We do not use the results in their full extent, what we really need is the chain rule for Lipschitz functions as it can be found in Lemma 4.6.3 in [17] and a suitable version of the Stokes theorem; an explicit proof is in [4] but the essential elements needed for the proof are contained in the cited books, [13] for the general case or [15] for the Lipschitz one.

The following lemma is the first step to prove that the form ℋ∗ω is exact on J × Λ.

Lemma 3.2. For every given t ∈ J the form s is exact on Λ and ∗

Proof. We can equivalently prove that for every Lipschitz curve γ : [0, 1] → Λ

Let us consider the set

and the map

By the Stokes theorem, we have

moreover

by equating the two right-hand sides we obtain

We are now able to prove the main result of this section.

Lemma 3.3. The differential form ℋ∗ω is exact on J × Λ and

Proof. We can equivalently prove that for every Lipschitz curve

we have

By definition, we have that

and by (3.2) and (3.3)

When the state projection of the Hamiltonian flow is invertible we obtain some important properties of the function θ.We recall that a homeomorphism between metric spaces is said to be bi-Lipschitz if it is Lipschitz with Lipschitz inverse.

Lemma 3.4. Suppose that, for a given t ∈ J, the function πℋt is bi-Lipschitz then on πℋt(Λ)

Proof.

Since θt ∘ (πℋt)−1 is Lipschitz then by the chain rule we can prove (i) by proving thattheir integrals coincide over any Lipschitz curve γ : [0, 1] → πℋt(Λ).By (3.3) we have

which proves statement (i). (ii) follows immediately from (i). To prove (iii) it is sufficient to notice that

17.3.2The super-Hamiltonian and its properties

Throughout this section we assume that α satisfies (2.5) and that Λ is the graph of dα on a contractible neighborhood of ̂x0. Moreover we assume that H satisfies Assumption 3.1 and the following Assumption 3.5, this last motivating the name super-Hamiltonian. We underline that these assumptions concern jointly the super-Hamiltonian and the horizontal Lagrangian manifold Λ.

Assumption 3.5. The super-Hamiltonian H satisfies the following:

As a consequence the Pontryagin extremalis also a solution of the system

Remark 3.6. If the maximized Hamiltonian Fmax satisfies Assumption 3.1, then it satisfies also Assumption 3.5; for example, in the classical case when F max is C2 we can take H := Fmax for any Λ.

From Assumption 3.1 it follows that there exists an interval I := [0, T] with T > ̂T such that ℋt(̂ℓ0) is defined for t ∈ I and we have ℋt(ℓ̂0) =(t) on [0, ̂T]. Moreover from the compactness of the time interval I, it follows that there is an open neighborhood ?ℓ0of ̂ℓ0 such that, without loss of generality, we can redefine Λ := Λ ∩̂?ℓ0, to obtain that

ℋ is defined on I × Λ and is defined on [0, ̂T] × Λ, where ℱis ̂̂the flow of

To prove the main theorem we need a final crucial assumption concerning the invertibility of the flow projection of the Hamiltonian onto the state space.

Assumption 3.7. The function

is invertible and bi-Lipschitz between I × Λ and an open set U of I × M containing thegraph of.

To verify this assumption one can use one of the available inverse function theorems for locally Lipschitz functions based on the invertibility of π∗ℋt∗ in [0, ̂T] × {̂ℓ0}. We refer, for example, to the one proposed by F. Clarke in [12].

Remark 3.8. For OCPs where the initial point is free the problem is always normal, moreover the initial Lagrangian manifold Λ0 is itself horizontal and we take it as Λ. In general, however, this is not the case and one has to define an appropriate α; in our approach we expect that the suitable α can be obtained from the second-order conditions as it is the case in the applications we describe in the final Section 17.4.

We can now state the main theorem, which compares the reference cost with the cost of an admissible neighboring trajectory.

Theorem 3.9. Under Assumption 3.1–3.5–3.7, let ξ : [0, T] → M be an admissible trajectory whose graph is contained in U and let ρ(t) := (πℋt)−1 ∘ ξ(t) then

Proof. Let μ := ℋt ∘ ρ be the lift of ξ to T∗M, by Lemma 3.3 we obtain

If moreover we isolate, on the left-hand side, the terms that describe the costs of the two trajectories we obtain

where we have used the definitions of F and ̂F. Now, by the (3.4a) property of the super-Hamiltonian, by adding to both sides β(ξ(T)) − β(̂xf ), and recalling that α = p0 c0 on the initial manifold and β = p0 cf on the final manifold we obtain

17.3.3Abstract sufficient optimality conditions

We are now able to state an abstract theorem, which reduces the strong local optimality of an admissible reference trajectory ̂ξ to the local optimality of a suitable function of the right endpoint (̂T, ̂xf ) of the graph of.Let

where αt = ℋt∘(πℋt)−1 was defined in part (ii) of Lemma 3.4. By means of this function we can rewrite the inequality (3.6) as

This relation will allow us to characterize the minima of the function p0J by studying the function Φ at the reference point (̂T, ̂xf ). Indeed we can now state a sufficient optimality condition.

Theorem 3.10. In the normal case, p0 = 1, under Assumptions 3.1–3.5–3.7, if there exists a neighborhood ? of (̂T, ̂xf ) in Nf such that (̂T, ̂xf ) is a minimizer of Φ restricted to ? then (̂T, ̂ξ , ̂υ) is a strong local minimizer.

Proof. It follows immediately from (3.7).

Remark 3.11. When the initial point is free we have that α = c0, if, moreover, the final point and time are fixed then the right-hand side of (3.7) is zero and we can prove that ̂ξ is a strong local minimizer by only verifying Assumptions 3.1–3.5–3.7 for the given Λ. In the other cases we have to find a suitable Λ and to prove the minimality property of Φ at (̂T, ̂xf ).

Remark 3.12. Concerning the abnormal case, p0 = 0, if one can prove that the minimum is strict then, as a by-product, it follows that ̂ξ is isolated among the admissible trajectories. To prove that ̂ξ is a strict strong local minimizer the first step is to require that (̂T, ̂xf ) is a strict local minimizer for Φ, in this case, if we have another minimizer (T, ξ, υ), we can conclude that T = ̂T and ξ(T) = ̂xf . From the proof of Theorem 3.9 wehave

By Assumption (3.4a) we obtain that

On the other hand, since μ = ℋt ∘ (πℋt)−1 ∘ ξ we have that

If

then one can conclude that μ = ̂λ since they both satisfy the same Hamiltonian equation with the same boundary conditions. Unfortunately Assumption (3.4a) is too mild to prove (3.8). We can strengthen it by assuming that

where ? is a neighborhood of the range of ; we note that this is true if Ht = Fmax. From this new assumption we deduce that

hence by (3.10)

where ∂υ is the vertical derivative along the fiber. This is not the only way to obtain local uniqueness of the strong minimizer; for the case of singular control see for example [10, 24].

To state necessary and/ or sufficient condition for (̂T, ̂xf ) to be a local minimizer for Φ we compute its first and second derivatives. We note that, by Lemma 3.4, the function Φ is a C1 function of x at t fixed and Lemma 3.13 states that the point (̂T, ̂xf ) is a critical point for the function Φ without further assumptions on the data. On the other hand to obtain the existence of second derivatives we will require stronger regularity assumptions on the data which here we do not specify. We underline that since the first derivatives are zero, then the second ones are well defined as a quadratic function on ℝ× T̂xfM.

Lemma 3.13. Under Assumptions 3.1–3.5–3.7, we have that

Moreover

Proof. (i) follows immediately from Lemma (3.4). To prove (ii) from (3.2) and Lemma (3.4) we have

Computing these derivatives at (̂T, ̂xf ), by the transversality condition (2.4), we obtain

By Assumption (3.4b) and by the PMP with variable final time we have

When the final time is fixed we are not interested in this last derivative.

Lemma 3.14. Let Assumptions 3.1–3.5–3.7 hold true and assume moreover that Φ is C2 in a neighborhood ? of (̂T, ̂xf ), then

Notice that since α̂T = ℋ̂T ∘ (πĤT)−1 then (dα̂T)∗ = ℋ̂T ∗ (πĤT)−∗ 1 .

Proof.

1. It follows directly from Lemma 3.13, part (i).

2./3. From Lemma 3.13, part (ii), taking into account that and some known symplectic equalities, that can be easily obtained in coordinates, we get

and

17.3.4The minimum time problem

In this section, we apply Theorem 3.9 to the minimum time problem which can be obtained from the general one by setting

For the minimum time problem with fixed final endpoint, directly from Theorem3.9,we obtain a sufficient optimality condition which requires only mild regularity of the super-Hamiltonian “near the final time.”

Lemma 3.15. Let Assumptions 3.1–3.5–3.7 hold true and let the function

be Lipschitz in a neighborhood of ̂T in ℝ. If p0 = 1 (normal case) then (̂T, ) is a strong local minimum for the minimum time problem with fixed final endpoint.

This lemma is a consequence of the following theorem for the minimum time problem with variable endpoint where we require a similar regularity assumption as in Lemma 3.15 and a second geometric assumption which is verified when the final point is fixed. This theorem is motivated by the results in [24] and [10].

Theorem 3.16. Let Assumptions 3.1–3.5–3.7 hold true and assume moreover that

if p0 = 1 (normal case) then (̂T, ̂ξ) is a strong local minimum for the minimum timeproblem.

Proof. Let k be the dimension of the submanifod Nf , we can take local coordinates centered at ̂xf such that Nf is diffeomorphic to the plane given by the last n − k coordinates equal to zero so that we can consider the problem on ℝ×ℝn with ̂xf = 0 and ξ(T) ∈ ℝk.

For the minimum time problem, in the normal case, equation (3.7) reads

By contradiction assume there exist admissible trajectories ξn : [0, tn] → M, tn < ̂T, whose graph is contained in U such that

Consider the curve

which is such that γ(tn) = ξn(tn) and γ(̂T) = 0. By Lemma 3.3 and Assumption (ii) weget from equation (3.12)

Let L > 0 be the Lipschitz constant of assumption (i). Since (πĤT)−1 (̂xf) = ̂ℓ0 by (3.4b) and Assumption (i) we can write

Dividing by ̂T − tn > 0 we obtain

which yield a contradiction by (3.13).