1 Multitime maximum principle

with area integral costs on boundary

Constantin Udri¸ste, Andreea Bejenaru

Abstract. This paper joins some concepts that appear in Mechanics, Field Theory, Differential Geometry and Control Theory in order to solve multitime optimal control problems with area integral costs on boundary.

Section 1 recalls the multitime maximum principle in the sense of the first author. The main results in Section 2 include the needle-shaped control variations, the adjoint PDEs, the behavior of infinitesimal deformations and other ingredients needed for the multitime maximum principle in case of no running cost and in case of running cost. Section 3 solves the previ- ous multitime control problems based on techniques of variational calculus.

Section 4 shows that concavity is a sufficient condition in multitime opti- mal control theory. Section 5 contains an example illustrating the utility of such a multitime optimal control theory.

M.S.C. 2010: 49J20, 49J40, 70H06, 37J35.

Key words: multitime maximum principle, multitime needle variations, boundary optimal problems.

1 Multitime maximum principle

Let N = R^m with global coordinates (t¹, ..., t^m), M = Rⁿ with global coordinates (x¹, ..., xⁿ) and R^k having global coordinates (u¹, ..., u^k). We consider the hyper- parallelepipedT = Ω0t0 ⊂R^m defined by the opposite diagonal points 0 = (0, ...,0) and t0 = (t¹₀, ..., t^m₀) and a subset U ⊂ R^k. For the multi-times s = (s¹, ..., s^m) and t = (t¹, ..., t^m) we denote s ≤ (<)t if and only if s^α ≤ (<)t^α, α = 1, ..., m (product order). We also consider the L - type set [s] = {t ∈R^m₊|t ≤sand∃α = 1, msuch thatt^α=s^α}. We shall use the followingL - type intervals:

([s],[t]] = Ω0,t\Ω0,s; [[s],[t]] = ([s],[t]]∪[s]; ([s],[t]) = ([s],[t]]\[t].

Let Xα = (X_αⁱ) : T ×M ×U → Rⁿ be C¹ vector fields. For a given control functionu:T →R^k, suppose the evolution PDEs system (controlled m-flow) (P DE) ∂xⁱ

∂t^α(t) =X_αⁱ(t, x(t), u(t)), x(0) =x0, t∈Ω0t0⊂R^m₊.

Balkan Journal of Geometry and Its Applications, Vol.16, No.2, 2011, pp. 138-154.∗

°c Balkan Society of Geometers, Geometry Balkan Press 2011.

has solution. As it is wellknown, this PDEs system has solutions if and only if the complete integrability conditions

(CIC) ∂X_αⁱ

∂t^β +∂X_αⁱ

∂x^j X_β^j +∂X_αⁱ

∂u^a

∂u^a

∂t^β = ∂X_βⁱ

∂t^α +∂X_βⁱ

∂x^j X_α^j +∂X_βⁱ

∂u^a

∂u^a

∂t^α are satisfied. The relations CIC define the set ofadmissible controls

U ={u(·) :R^m₊ →U|u(·) is constrained by (CIC)}.

The multitime evolution system (PDE) is used as a constraint when we want to optimize amultitime cost functional

(J) J[u(·)] =

Z

Ω0t0

X(t, x(t), u(t))dt+ Z

∂Ω0t0

g(t, x(t))dσ,

where the running cost X : N ×M ×U → R is a C² function (nonautonomous Lagrangian), andg:∂N×M →Ris aC¹ boundary cost.

Multitime optimal control problem. Find maxu(·) J[u(·)] =

Z

Ω0t0

X(t, x(t), u(t))dt+ Z

∂Ω0t0

g(t, x(t))dσ

subject to ∂xⁱ

∂t^α(t) =X_αⁱ(t, x(t), u(t)), i= 1, ..., n, α= 1, ..., m, u(t)∈ U, t∈Ω0t0, x(0) =x0.

The multitime maximum principle (necessary condition) asserts that the existence of an optimal controlu^∗(·) implies the existence of costate vector functions (p^∗₀, p^∗)(·) = (p^∗₀(·), p^∗α_i (·)), which together with the optimalm-sheetx^∗(·) satisfy a suitable PDEs system. Similar to single-time theory, this multitime maximum principle involves an appropriatecontrol Hamiltonian

H(t, x, p0, p, u) =p0X(t, x, u) +p^α_iX_αⁱ(t, x, u).

Theorem 1.1. (multitime maximum principle) Suppose u^∗(·) is optimal for (P DE),(J)and thatx^∗(·)is the corresponding optimalm-sheet. Then there exists a map(p^∗₀, p^∗) = (p^∗₀, p^∗_i) : Ω0t0 →R^mn+1 such that

(P DE) ∂x^∗i

∂t^α (t) = ∂H

∂p^α_i (t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t)),

(ADJ) ∂p^∗α_i

∂t^α (t) =−∂H

∂xⁱ(t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t)) and

(M) ∂H

∂u^a(t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t)) = 0, ∀t∈Ω0t0.

Finally, the boundary conditions

(t0) nαp^∗α_i |∂Ω0t0 = ∂g

∂xⁱ|∂Ω0t0

are satisfied, where, for each multi-time s, n denotes the covector corresponding to the unit normal vector on∂Ω0s, that is n= (nα), where

nα(t) =





1, if t^α=s^α;

−1, if t^α= 0;

0, otherwise.

We callx^∗(·) thestateof the optimally controlled system and (p^∗₀, p^∗(·)) thecostate map. Even more, we can considerp^∗₀= 1.

Remark 1.2. 1) Explicitly, (P DE)means the identities

∂x^∗i

∂t^β (t) =X_βⁱ(t, x^∗(t), u^∗(t)), β= 1, . . . , m; i= 1, . . . , n, and(ADJ)means the identities

∂p^∗α_i

∂t^α (t) =− µ

p^∗0(t)∂X

∂xⁱ +p^∗α_j (t)∂X_α^j

∂xⁱ

¶

(t, x^∗(t), u^∗(t)).

2) The identities (P DE) reveal the controlled evolution PDEs, the identities(ADJ) suggest the adjoint PDEs, the relation(M) represents the multitime maximum prin- ciple and the relation(t0)means the transversability (boundary) condition.

3) The multitime maximum principle states necessary conditions that must hold on an optimalm-sheet of evolution.

2 Needle-shaped control variations and adjoint PDEs

The general proofs of multitime maximum principle rely on a special type of variations, calledneedle-shaped control variations.

Supposeu^∗(·) is a candidate optimal control and that x^∗(·) is the corresponding m-sheet. Fixing a multitimes ∈ ([0],[t0]) and u(·) ∈ U, an m-needle variation is a family of controls u² obtained replacing u^∗ with u on ([s−²],[s]]. In other words, given the multitimes∈([0],[t₀]) and an admissible control u(t), we set ²∈[[0],[s]]

and define the modified control u²(t) =

½ u(t) ift∈([s−²],[s]]

u^∗(t) otherwise.

We also denotex²(·) the corresponding response of our system, i.e.

∂xⁱ_²

∂t^α(t) =X_αⁱ(t, x²(t), u²(t)), x²(0) =x0, t∈Ω0t0⊂R^m₊. Let theny_αⁱ(t) = ∂xⁱ_²(t)

∂²^α |²=0be the infinitesimal gradient deformation of them-sheet x^∗(t) induced by the previous control variation.

Lemma 2.1. Letϕ: Ω0,s×(−δ, δ)^m→R, ϕ=ϕ(t, ²)be a differentiable parametrized function. Then

∂

∂²^α Z

[[s−²],[s]]

ϕ(t, ²)dt|²=0= Z

[s]

ϕ(t,0)nα(t)dσ.

Proof. Successively, we can write

∂

∂²^α Z

[[s−²],[s]]

ϕ(t, ²)dt|²=0= ∂

∂²^α

"Z

Ω0s

ϕ(t, ²)dt− Z

Ω0s−²

ϕ(t, ²)dt

#

|²=0

= ∂

∂²^α

"Z

Ω0s

ϕ(t, ²)dt−

Z _s^m_−²^m

0

...

Z _s¹_−²¹

0

ϕ(t, ²)dt¹...dt^m

#

|²=0

= Z

Ω0s

∂ϕ

∂²^α(t,0)dt− Z

Ω0s−²

∂ϕ

∂²^α(t, ²)dt|²=0

+

Z _s^m_−²^m

0

...

Z _s^α+1_−²^α+1

0

Z _s^α−1_−²^α−1

0

...

Z _s¹_−²¹

0

ϕ(t¹, ..., s^α−²^α, ...t^m, ²)dtα|²=0

= Z

[s]

ϕ(t,0)nα(t)dσ.

¤ Lemma 2.2. The infinitesimal deformation y induced by the needle-shaped control variation satisfies the following relations:

y_αⁱ(t) = 0, if t∈[[0],[s]), Z

[s]

yⁱ_β(t)nα(t)dσ= Z

[s]

[X_αⁱ(t, x∗(t), u(t))−X_αⁱ(t, x^∗(t), u^∗(t))]nβ(t)dσ,

∂yⁱ_β

∂t^α(t) =∂X_αⁱ

∂x^j (t, x^∗(t), u^∗(t))y_β^j(t), if t∈([s],[t0]],

∀α, β= 1, ..., m, ∀i= 1, ..., n.

Proof. We recall yⁱ_α(², t) = ∂xⁱ_²

∂²^α(t). Since x²(t) =x^∗(t), ∀t ∈ [[0],[s−²]], we have yα(², t) = 0, ∀t∈[[0],[s−²]]. Let us considert∈([s−²],[s]) a fixed multi-time. The PDE ∂xⁱ_²

∂t^α(t) =X_αⁱ(t, x²(t), u(t)) generates the variational PDE

∂y_βⁱ

∂t^α(², t) = ∂X_αⁱ

∂x^j (t, x²(t), u(t))y^j_β(², t).

Since we are interested on what it happens starting with the multi-time s, we can choseyα(², t) = 0, ∀t∈([s−²],[s]). Thereforeyα(², t) = 0, ∀t∈[[0],[s]) and, when making²= 0, we obtainyα(t) = 0, ∀t∈[[0],[s]).

In the following, we taket=s. Successively, we have Z

[[s−²],[s]]

[X_αⁱ(t, x²(t), u(t))−X_αⁱ(t, x^∗(t), u^∗(t))]dt

= Z

[[s−²],[s]]

µ∂xⁱ_²

∂t^α −∂x^∗i

∂t^α

¶ dt=

Z

∂[[s−²],[s]]

(xⁱ_²−x^∗i)nα(t)dσ

= Z

[s]

(xⁱ_²−x^∗i)nα(t)dσ

and, by applying Lemma 2.1, it follows Z

[s]

yⁱ_β(t)nα(t)dσ= Z

[s]

[X_αⁱ(t, x^∗(t), u(t))−X_αⁱ(t, x^∗(t), u^∗(t))]nβ(t)dσ.

On ([s],[t0]], we have again

∂yⁱ_β

∂t^α(t) = ∂X_αⁱ

∂x^j (t, x^∗(t), u^∗(t))y^j_β(t).

On the other hand, them-dimensional flow of the infinitesimal deformation,

∂yⁱ_α

∂t^β(t) =y_β^j(t)∂X_αⁱ

∂x^j (x(t)) or ∂yⁱ_α

∂t^β(t) =y_α^j(t)∂X_βⁱ

∂x^j (x(t)), on the jet bundle of order oneJ¹(T, M), determines adualm-flow

(ADJ) ∂p^α_i

∂t^α(t) =−p^α_j(t)∂X_α^j

∂xⁱ (x(t))

on the dual spaceJ^1∗(T, M). These PDEs systems areadjointin the sense ofzero total divergenceof the tensor field Q^α_β = p^α_iy_βⁱ produced by their solutions. The adjoint system (ADJ) has solutions since it containsnPDEs withnmunknown functionsp^α_i.

¤

2.1 Free boundary problem, no running cost

The basic problem here is to find

maxu(·) J[u(·)] = Z

∂Ω0t0

g(t, x(t))dσ

subject to ∂xⁱ

∂t^α(t) =X_αⁱ(t, x(t), u(t)), i= 1, ..., n, α= 1, ..., m, u(t)∈ U, t∈Ω0t0, x(0) =x0.

We denote byu^∗(·) respectivelyx^∗(·) the optimal control and the optimalm-sheet of this problem and we consider the control Hamiltonian

(H) H(t, x, p, u) =p^α_iX_αⁱ(t, x, u), p= (p^α_i).

Lemma 2.3. (fundamental inequality)Let ϕ: Ω0t0 →Rbe a continuous (mea- surable) function. If Z

[s]

ϕ(t)nα(t)dσ≥0, ∀s∈Ω0t0,

then Z

Ω0s

ϕ(t)dt≥0, ∀s∈Ω_0t0. Proof. Since we can write

Z

[s]

ϕ(t)nα(t)dσ= ∂

∂s^α ÃZ

[[0],[s]]

ϕ(t)dt

!

= ∂

∂s^α µZ

Ω_0s

ϕ(t)dt

¶ ,

the hypotheses ensure us thats→ Z

Ω0s

ϕ(t)dtis a partial increasing function. There-

fore Z

Ω0s

ϕ(t)dt≥0, ∀s∈Ω0t0.

¤ Theorem 2.4. (multitime maximum principle, no running cost)Supposeu^∗(·) is optimal for(P DE),(J)and thatx^∗(·)is the corresponding optimalm-sheet. Then there exists the dual functionsp^∗α_i : Ω0t0 →Rsuch that

(P DE) ∂x^∗i

∂t^α (t) = ∂H

∂p^α_i (t, x^∗(t), p^∗(t), u^∗(t)),

(ADJ) ∂p^∗α_i

∂t^α (t) =−∂H

∂xⁱ(t, x^∗(t), p^∗(t), u^∗(t)),

(M) ∂H

∂u^a(t, x^∗(t), p^∗(t), u^∗(t)) = 0, ∀t∈Ω0t0

and satisfy the boundary conditions

(t0) nαp^∗α_i |∂Ω0t0 = ∂g

∂xⁱ|∂Ω0t0.

Proof. For each mapp, the control HamiltonianH(t, x, p, u) =p^α_iX_αⁱ satisfies (1) H(t, x^∗(t), p(t), u^∗(t)) +∂p^α_i

∂t^α(t)x^∗i(t) = ∂(p^α_ix^∗i)

∂t^α , ∀t∈[[0],[t0]];

(2) H(t, x²(t), p(t), u(t)) +∂p^α_i

∂t^α(t)xⁱ_²(t) =∂(p^α_ixⁱ_²)

∂t^α , ∀t∈([s−²],[s]];

(3) H(t, x²(t), p(t), u^∗(t)) +∂p^α_i

∂t^α(t)xⁱ_²(t) =∂(p^α_ixⁱ_²)

∂t^α , ∀t∈([s],[t0]].

Therefore, by taking the difference (2)−(1) on ([s−²],[s]] and integrating after- wards, we obtain

Z

[s]

(xⁱ_²−x^∗i)p^α_inαdσ = Z

[[s−²],[s]]

[H(t, x²(t), p(t), u(t))

− H(t, x^∗(t), p(t), u^∗(t)) +∂p^α_i

∂t^α(t)(xⁱ_²−x^∗i)]dσ.

Computing the partial derivative with respect to²^β (see Lemma 2.1), we obtain Z

[s]

y_βⁱp^α_inαdσ= Z

[s]

[H(t, x^∗(t), p(t), u(t))−H(t, x^∗(t), p(t), u^∗(t))]nβdσ.

If the costate vectorp^∗ is the solution for the adjoint system (ADJ) with boundary conditions (t0), then, on ([s],[t0]], we have ∂(p^∗α_i yⁱ_β)

∂t^α = 0.Denoting by nthe normal vector field on∂Ω0t0, respectively on ∂Ω0s, we obtain

0 = Z

([s],[t0])

∂(p^∗α_i y_βⁱ)

∂t^α dt= Z

∂Ω0t0

yⁱ_βp^∗α_i nαdσ− Z

[s]

yⁱ_βp^∗α_i nαdσ

= Z

∂Ω0t0

∂g

∂xⁱyⁱ_βdσ− Z

[s]

y_βⁱp^∗α_i nαdσ.

Since u^∗ is an optimal control, it follows that ² = 0 is a maximum point for the function²→

Z

∂Ω0t0

(g◦xⁱ_²)dσand, therefore, Z

∂Ω0t0

∂g

∂xⁱyⁱ_βdσ≤0. We find Z

[s]

[H(t, x^∗(t), p^∗(t), u(t))−H(t, x^∗(t), p^∗(t), u^∗(t))]nβdσ≤0.

Applying Lemma 2.3, we obtain the maximum principle inequality in functional in- tegral form. Consequently, using the Euler-Lagrange relation, it appears the critical

point condition. ¤

2.2 Free boundary problem, with running cost

We suppose that the functional includes a running cost, i.e.,

(J) J[u(·)] =

Z

Ω0t0

X(t, x(t), u(t))dt+ Z

∂Ω0t0

(g(t, x(t))dσ.

In this case, the control Hamiltonian has the following expression:

(H) H(t, x, p0, p, u) =p0X(t, x, u) +p^α_iX_αⁱ(t, x, u).

Theorem 2.5. (Multitime maximum principle with running cost) Suppose u^∗(·)is optimal for(P DE),(J)and thatx^∗(·)is the corresponding optimal m-sheet.

Then there exist some functionsp^∗₀, p^∗α_i : Ω0t0→R such that

(P DE) ∂x^∗i

∂t^α (t) = ∂H

∂p^α_i (t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t)),

(ADJ) ∂p^∗α_i

∂t^α (t) =−∂H

∂xⁱ(t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t)) and

(M) ∂H

∂u^a(t, x^∗(t), p^∗(t), u^∗(t)) = 0, ∀t∈Ω0t0

Finally, the boundary conditions

(t0) nαp^∗α_i |∂Ω0t0 = ∂g

∂xⁱ|∂Ω0t0

are satisfied.

Proof. We begin by adding new variables in order to transform the running cost into a terminal cost. We introduce some supplementary state-variablesx^(α) : Ω0t0 →R, solutions for the PDE system

(4) ∂x^(α)

∂t^β (t) = 1

mδ^α_βX(t, x(t), u(t)), x^(α)(0) = 0, t∈Ω0t0. We also consider

x= (x^(α), xⁱ); x0= (0, x0); x(·) = (x^(α)(·), xⁱ(·)) and

Xα(t, x, u) = 1

mδ_α^βX(t, x, u) ∂

∂x^(β)+X_αⁱ(t, x, u) ∂

∂xⁱ; g(t, x) =nβ(t)x^(β)+g(t, x).

Then (P DE) and relation (4) give the dynamics

P DE ∂x

∂t^α =Xα(t, x(t), u(t)), x(0) =x0, t∈Ω0t0.

Consequently, the initial control problem transforms into a new control problem with boundary cost functional

(J) J[u(·)] =

Z

∂Ω0t0

g(t, x(t))dσ.

The control Hamiltonian associated to this new problem is H(t, x, p, u) = 1

mp^α_βδ_α^βX(t, x, u) +p^α_iX_αⁱ(t, x, u) =H(t, x, p0, p, u), wherep0= _m¹T r(p^α_β).

We apply the multitime maximum principle with no running cost and we obtain a costate vectorp^∗= (p^∗α_β , p^∗α_i ) such thatH satisfyes (P DE), (ADJ), (M) and (t0).

Letp^∗₀=_m¹T r(p^∗α_β ). Relation (P DE) can be rewritten as

∂x^∗i

∂t^α (t) = ∂H

∂p^α_i (t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t)) and

∂x^∗(β)

∂t^α (t) = ∂H

∂p^α_β(t, x^∗(t), p^∗(t), u^∗(t)) = 1 mδ_β^α∂H

∂p0

(t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t)).

The immediate consequence of the previous relation is

∂x^∗(α)

∂t^α (t) = ∂H

∂p0(t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t)).

We analyze next the adjoint equation (ADJ). We obtain

(ADJ) ∂p^∗α_i

∂t^α (t) =−∂H

∂xⁱ(t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t))

and ∂p^∗α_β

∂t^α (t) = 0.

The maximization principle can be rewritten

(M) ∂H

∂u^a(t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t)) = 0, ∀t∈Ω0t0

and the boundary conditions are (t0). nαp^∗α_i |∂Ωt0 = ∂g

∂xⁱ|∂Ωt0; nαp^∗α_β |∂Ωt0 =nβ

Gathering together the restrictions related top^∗α_β , p^∗₀= 1

mtr(p^∗α_β ); ∂p^∗α_β

∂t^α (t) = 0; nαp^∗α_β |∂Ω_t0 =nβ,

the immediate consequence is that we can choosep^∗α_β =δ^α_β andp^∗₀= 1. ¤

3 Optimal control theory based on techniques of variational calculus

We consider a smooth vector field v= (v^a) : Ω0t0 →R^k satisfyingv^a(0) = 0, ∀a= 1, ..., k.Letu^∗(·)∈ U denote the optimal control. Moreover, we suppose thatu^∗(t)∈ IntU . Then, we consider the control variation

uδ(t) =u^∗(t) +δv(t).

Sinceu^∗(t)∈IntU, there isδ0>0 such thatuδ(t)∈IntU, ∀ |δ|< δ0. Letxδ(t) be the state variable corresponding to the control variableuδ(t), that isxδ(t) is solution for the following PDEs system

(P DE) ∂xⁱ_δ

∂t^α(t) =X_αⁱ(t, xδ(t), uδ(t)), xδ(0) =x0, t∈Ω0t0 ⊂R^m₊. Ify=∂x

∂δ|δ=0 is the infinitesimal deformation induced by the previous control varia- tion, then

∂yⁱ

∂t^β(t) =∂X_βⁱ

∂x^j(t)y^j(t) +∂X_βⁱ

∂u^a (t)v^a(t), yⁱ(0) = 0, t∈Ω0t₀.

3.1 Free boundary problem, no running cost

In this subsection, we consider again the boundary cost functional

(J) J[u(·)] =

Z

∂Ω0t0

g(t, x(t))dσ,

restricted by (P DE). We use again the control Hamiltonian (H) H(t, x, p, u) =p^α_iX_αⁱ(t, x, u).

We also introduce thecontrol tensor field

T_β^α(t, x, p, u) =p^α_iX_βⁱ(t, x, u) and we prove next a simplified maximum principle.

Theorem 3.1. (simplified multitime maximum principle, no running cost) Suppose u^∗(·) is an interior optimal control for (P DE), (J) and that x^∗(·) is the corresponding optimalm-sheet. Then there exist the dual functions p^∗α_i : Ω0t0 → R such that

(P DE) ∂x^∗i

∂t^α (t) = ∂H

∂p^α_i (t, x^∗(t), p^∗(t), u^∗(t)),

(ADJ) ∂p^∗α_i

∂t^α (t) =−∂H

∂xⁱ(t, x^∗(t), p^∗(t), u^∗(t)),

(M) ∂H

∂u^a(t, x^∗(t), p^∗(t), u^∗(t)) = 0 and satisfy the boundary conditions

(t0) nαp^∗α_i |∂Ω0t0 = ∂g

∂xⁱ|∂Ω0t0. Moreover

Dα[T_β^α(t, x^∗(t), p^∗(t), u^∗(t))] = ∂H

∂t^β(t, x^∗(t), p^∗(t), u^∗(t)), whereDα denotes the total derivative with respect to t^α.

Proof. From (H) and (P DE) we have H(t, xδ(t), p(t), uδ(t)) +∂p^α_i

∂t^α(t)xⁱ_δ(t) =∂(p^α_ixⁱ_δ)

∂t^α (t), therefore

∂(p^α_iyⁱ)

∂t^α (t) = [∂H

∂xⁱ(t, x^∗(t), p(t), u^∗(t)) +∂p^α_i

∂t^α(t)]yⁱ(t)

+ ∂H

∂u^a(t, x^∗(t), p(t), u^∗(t))v^a(t).

We choosep^∗ solution for (ADJ), (t0). When integrating on Ω0t0, we obtain Z

Ω0t0

∂H

∂u^a(t, x^∗(t), p^∗(t), u^∗(t))v^a(t)dt= Z

Ω0t0

∂(p^∗α_i yⁱ)

∂t^α (t)

= Z

∂Ω0t0

yⁱ(t)p^∗α_i (t)nα(t)dσ= Z

∂Ω0t0

∂g

∂xⁱ(t, x^∗(t))yⁱ(t)dσ.

Since u^∗(·) is an optimal control, it follows that δ = 0 is a critical point for the function

Z

Ω0t0

g(xδ(t))dσ and, therefore Z

∂Ω_0t0

∂g

∂xⁱ(t, x^∗(t))yⁱ(t)dσ= 0.

It follows that Z

Ω0t0

∂H

∂u^a(t, x^∗(t), p^∗(t), u^∗(t))v^a(t)dt= 0 and, sincev is an arbitrary vector field, we conclude that

∂H

∂u^a(t, x^∗(t), p^∗(t), u^∗(t)) = 0, ∀t∈Ω0t0. Let us compute next the divergence of the control tensor field,

DαT_β^α = ∂p^α_i

∂t^αX_βⁱ +p^α_iDαX_βⁱ =∂p^α_i

∂t^αX_βⁱ +p^α_iDβX_αⁱ

= −∂H

∂xⁱX_βⁱ +∂H

∂xⁱX_βⁱ + ∂H

∂u^a

∂u^a

∂t^β +∂H

∂t^β =∂H

∂t^β.

¤ Remark 3.2. If the control Hamiltonian is autonomous, that is H doesn’t depend explicitly ont, then we obtain a conservation law asserting that the control tensor has null divergence.

3.2 Free boundary problem, with running cost

We suppose the functional includes a running cost:

(J) J[u(·)] =

Z

Ω0t0

X(t, x(t), u(t))dt+ Z

∂Ω0t0

g(t, x(t))dσ.

In this case, the control Hamiltonian has the following expression:

(H) H(t, x, p0, p, u) =p0X(t, x, u) +p^α_iX_αⁱ(t, x, u).

Theorem 3.3. (simplified multitime maximum principle with running cost) Suppose u^∗(·) is an interior optimal control for (P DE), (J) and that x^∗(·) is the corresponding optimal m-sheet. Then there exist some costate functions p^∗₀, p^∗α_i : Ω0t0 →Rsuch that

(P DE) ∂x^∗i

∂t^α (t) = ∂H

∂p^α_i (t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t)),

(ADJ) ∂p^∗α_i

∂t^α (t) =−∂H

∂xⁱ(t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t)) and

(M) ∂H

∂u^a(t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t)) = 0.

Finally, the boundary conditions

(t₀) n_αp^∗α_i |_∂Ω0t

0 = ∂g

∂xⁱ|_∂Ω0t

0

are satisfied. Moreover, the control tensor field

T_β^α(t, x, p0, p, u) =p0δ_β^αX(t, x, u) +p^α_iX_βⁱ(t, x, u) satisfies the relation

Dα[T_β^α(t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t))] = ∂H

∂t^β(t, x^∗(t), p^∗₀(t), p^∗(t), u^∗(t)).

Proof. Same arguments as in the proof of Theorem 2.5. ¤

4 Sufficient conditions in multitime optimal control theory

If we add some concavity restrictions to the components of the control tensor, to the boundary cost and the constrained set, then we can prove the sufficiency of the conditions of multitime maximum principle.

Definition 4.1. A function f : Rⁿ → R is called concave if its Hessian matrix is negative definite at each point.

A concave function satisfies the inequality

f(y)−f(x)≤dfx(y−x).

We consider the more general optimal control problem with running cost and we shall use the control Hamiltonian

H(t, x, p0, p, u) =p0X(t, x, u) +p^α_iX_αⁱ(t, x, u).

Moreover, we can suppose thatp0= 1.

Theorem 4.1. (sufficient condition in multitime optimal control)If(x^∗, p^∗, u^∗) satisfies the conditions of simplified multitime maximum principle and the control Hamiltonian evaluated at p = p^∗ is (strictly) concave in the pair (x^∗, u^∗) and the boundary costg is (strictly) concave atx^∗, then (x^∗, p^∗, u^∗)is the (unique) solution of the control problem.

Proof. We must maximize the functional J(u(·)) =

Z

Ω0t0

X(t, x(t), u(t))dt+ Z

∂Ω0t0

g(t, x(t))dσ.

subject to the evolution PDEs system. We fix a pair (x^∗, u^∗), whereu^∗is a candidate optimalm-sheet of the controls andx^∗ is a candidate optimalm-sheet of the states.

CallingJ^∗ the value of the functional for (x^∗, u^∗), let us prove that J^∗−J =

Z

Ω0t0

(X^∗−X)dt+ Z

∂Ω0t0

(g^∗−g)dσ≥0,

where the strict inequality holds under strict concavity. DenotingH^∗=H(t, x^∗, p^∗, u^∗) andH=H(t, x, p^∗, u), we find

J^∗−J = Z

Ω0t0

(X^∗−X)dt+ Z

∂Ω0t0

(g^∗−g)dσ

= Z

Ω0t0

µ

(H^∗−p^∗α_i ∂x^∗i

∂t^α)−(H−p^∗α_i ∂xⁱ

∂t^α)

¶ dt+

Z

∂Ω0t0

(g^∗−g)dσ.

Integrating by parts, we obtain J^∗−J =

Z

Ω0t0

µ

(H^∗+x^∗i∂p^∗α_i

∂t^α )−(H+xⁱ∂p^∗α_i

∂t^α )

¶ dt

+ Z

∂Ω0t0

¡(g^∗−g)−(x^∗i−xⁱ)p^∗α_i nα

¢dσ.

Taking into account thatp^∗ satisfyes the boundary condition (t0), we infer J^∗−J =

Z

Ω0t0

µ

(H^∗−H) +∂p^∗α_i

∂t^α (x^∗i−xⁱ)

¶ dt+

Z

∂Ω0t0

µ

(g^∗−g)−∂g^∗

∂xⁱ(x^∗i−xⁱ)

¶ dσ.

The definition of concavity implies Z

∂Ω0t0

µ

(g^∗−g)−∂g^∗

∂xⁱ(x^∗i−xⁱ)

¶ dσ≥0

and Z

Ω0t0

µ

(H^∗−H) +∂p^∗α_i

∂t^α (x^∗i−xⁱ)

¶ dt

≥ Z

Ω0t0

µ

(x^∗i−xⁱ)(∂H^∗

∂xⁱ +∂p^∗α_i

∂t^α ) + (u^∗a−u^a)∂H^∗

∂u^adt

¶

= 0.

This last equality follows from the fact that all ”∗” variables satisfy the conditions of the multi-time maximum principle. In this way,J^∗−J ≥0. ¤

5 Example of optimal control problem with area integral cost on boundary

In the previous sections, we considered the parallelepiped Ω0t0 to be the domain of multitimes. Next, we give an idea of how to extend our theory for arbitrary compact domains fromR^m. Let Ω⊂R^mbe a connected and compact subset, with a piecewise smooth (m−1)-dimensional boundaryU =∂Ω. The new optimal control problem with area integral boundary costs asks for finding

maxu(·) J[u(·)] = Z

Ω

X(t, x(t), u(t))dt+ Z

U

g(t, x(t))dσ

subject to ∂xⁱ

∂t^α(t) =X_αⁱ(t, x(t), u(t)), i= 1, ..., n, α= 1, ..., m, u(t)∈ U, t∈Ω, x(0) =x0.

Solving this problem using needle-shaped control variations requires the introduc- tion of a temporal orientation on Ω. In order to do so, we consider a fixed point t0 ∈Ω. Moreover, for simplicity, we assume t0 = 0. For each point t ∈ Ω, we de- note by αt : [0,1] → Ω the line segment starting from 0, passing through t, such that αt(1) ∈U. We also consider the function τ : Ω →[0,1] satisfying the relation αt(τ(t)) =t. For two multitimessandt in Ω, we denotes <(≤)t ifτ(s)<(≤)τ(t).

Using the functionτ, we can also define the∂-type set [s] ={t∈Ω|τ(t) =τ(s)}

and the∂-type intervals

[[0],[s]] ={t∈Ω|0≤τ(t)≤τ(s)}; ([s],[t]] = [[0],[t]]−[[0],[s]].

By considering needle-shaped control variations relative to the above intervals, we regain the multitime maximum principle.

There is some time now since we look for a variational proof of the fact that, amongst all the bodies of constant surface, the sphere maximizes the volume. Recently,

we have solved this problem, using multitime calculus of variations and taking the normal vector field as state variable. In our opinion, this is an important example since it emphasis’s the utility of considering and studying a multitime variational theory. We reconsider this problem now, using multitime optimal control theory.

If D is a compact set of R^m = {(t¹, ..., t^m)} with a piecewise smooth (m−1)- dimensional boundary ∂D, we can write the volume

Z

D

dt¹...dt^m of the domain D using the position vectort= (t^α) and the exterior unit normal vector fieldN = (N^β) on∂D, via Gauss-Ostragradski formula, as

m Z

D

dt¹...dt^m= Z

∂D

δαβt^αN^βdσ.

Moreover, the area of ∂D is Z

∂D

dσ. Introducing a parametrization on D, whose domain is Ω⊂R^m and denotingU =∂Ω, we havedσ=||N ||dη, whereN =||N ||N andη is a differential (m−1)-form.

Let us show next, that of all solids having a given surface area, the sphere is the one having the greatest volume. To prove this statement, we formulate the multitime optimal control problem with isoperimetric constraint

maxN

Z

U

δαβt^αN^β(t)dη subject to Z

U

q

δαβN^α(t)N^β(t)dη=const.

In order to solve this problem, we also add the evolution system

∂N^α

∂t^β (t) =u^α_β(t), ∀t∈Ω,

which does not interfere with the quality of the solutions on the boundary. Using the Hamiltonian

H=p^β_αu^α_β and the boundary costg(t,N) =δαβt^αN^β−pp

δαβN^αN^β, p=const., the critical point condition, in the multitime maximum principle, gives

∂H

∂u^α_β =p^β_α(t) = 0, ∀t∈Ω and the boundary condition writes

0 =p^α_βN^β= ∂g

∂N^α =t^α−pN^α, ∀t∈U.

Since the boundary cost g is a concave function of N, the critical point is a maximum point. This confirms thatDis the sphere ||t||²≤p²in R^m.

Acknowledgments. Partially supported by University Politehnica of Bucharest, and by Academy of Romanian Scientists, Bucharest, Romania. We express kind regards to professor I. T¸ evy for the valuable suggestions and comments which led to the improvement of this paper.