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Simplified multitime maximum principle
Constantin Udri¸ste
Abstract. Many science and engineering problems can be formulated as
optimization problems that are governed by m-flow type P DEs (multitime evolution systems) and by cost functionals expressed as multiple integrals or curvilinear integrals. Our paper discuss the m-flow type P DEconstrained optimization problems, focussing on a simplified multitime
maximum principle. This extends the simplified single-time maximum
principle of Pontryaguin in the ODEs case (curves) to include the case of
P DEs (submanifolds).
In Section 1 the idea of multitime is motivated. In Section 2 a multitime maximum principle, for the case of multiple integral functionals, is
stated and proved. A version of multitime maximum principle, for the case
of curvilinear integral functionals, is formulated in Section 3. Though a
multiple integral functional is mathematically equivalent to a curvilinear
integral functional (Section 4), their meaning is totally different in real life
problems. A multitime maximum principle approach of variational calculus is presented in Section 5.
M.S.C. 2000: 93C20, 93C35, 49K20, 49J20, 53C44.
Key words: P DE-constrained optimal control, multitime maximum principle, multiple or curvilinear integral functional, geometric evolution.
1
Multitime concept
The adjective multitime was introduced in Physics by Dirac (1932), and later was
used in Mathematics by [4], [6], [8], [12]-[16], [19], [20], [22], etc. To underline the
sense of this adjective, we collect the following remarks.
1) A space coordinate is merely an index numbering freedom degrees, and the time
coordinate is usual the physical time in which the systems evolves. Such classical
theory is satisfactory unless we turn our attention to relativistic problems (chiral
fields, sine-Gordon P DE, etc). Moreover, in some physical problems we use a twotime t = (t1 , t2 ), where t1 means the intrinsic time and t2 is the observer time. Also
there are a lot of problems where is no reason to prefer one coordinate to another. In
this sense, we refer to multitime geometric evolutions and multitime optimal control
∗
Balkan
Journal of Geometry and Its Applications, Vol.14, No.1, 2009, pp. 102-119.
c Balkan Society of Geometers, Geometry Balkan Press 2009.
°
Simplified multitime maximum principle
103
problems, where multitime means a vector parameter of evolution. Here we can include
the description of torsion of prismatic bars, the maximization of the area surface for
given width and diameter etc.
2) Multitime wave functions were first considered by Dirac in 1932 via m-time
∂ψ
evolution equations i ~ α = Hα ψ. The Dirac P DE system is consistent (completely
∂t
integrable) if and only if [Hα , Hβ ] = 0 for α 6= β. The consistency condition is easy
to achieve for non-interacting particles and tricky in the presence of interaction. But,
until now, nobody attempted to write down consistent multitime equations for many
interacting particles, although this would seem an immediate and highly relevant
problem if one desires a manifestly covariant formulation of relativistic quantum mechanics.
3) The oscillators are very important in engineering and communications. For example, voltage-controlled oscillators, phase-locked loops, lasers, etc., abound in wireless and optical systems. A new approach for analyzing frequency and amplitude
modulation in oscillators was realized recently using a novel concept, warped time,
within a multitime P DE framework. To explain this idea from our point of view, we
2π
start with a single-time wave front y(t) = sin ( 2π
T1 t) sin ( T2 t), T1 = 0.02 s; T2 = 1 s,
1
1
where the two tones are at frequencies f1 =
= 50 Hz and f2 =
= 1 Hz. Here
T1
T2
there are 50 times faster varying sinusoids of period T1 modulated by a slowly-varying
sinusoid of period T2 . Then we build a ”two-variable representation” of y(t), obtained
by the rules: for the ”fast-varying” parts of y(t), the time t is replaced by a new variable t1 ; for the ”slowly-varying” parts, by t2 . It appears a new periodic function of
2π 2
1
two variables, yˆ(t1 , t2 ) = sin ( 2π
T1 t ) sin ( T2 t ), motivated by the wide separated time
scales. Inspection of the two-time (two-variable) wavefront yˆ(t1 , t2 ) directly provides
information about the slow and fast variations of y(t) more naturally and conveniently
than y(t) itself.
4) The known evolution laws in physical theories are single-time evolution laws
(ODEs) or multitime evolution laws (PDEs). To change a single-time evolution into
a multitime evolution it is enough to change the ODEs into P DEs accepting that
the time t is a C ∞ function of certain parameters, let say t = t(s1 , ..., sm ).
The P DE constraints often present significant challenges for optimization principles [2]-[6], [8], [9]. The multivariable maximum principle was studied in the presence
of PDE constraints, starting with the papers [12]-[22]. This approach extends the
single-time Pontryaguin’s model [1], [7], [10], [11].
In this paper, we are looking for a multitime maximum principle, i.e., for necessary
conditions of optimality. Our formulation and a proof mimic those that were applied
to single-time maximum principle. A simplified version of this problem, obtained after
years of debates in my research group and in conferences, is presented in this paper.
2
m-Flow type constrained optimization problem
with multiple integral functional
Let us analyze a multitime optimal control problem based on a multiple integral cost
functional and m-flow type P DE constraints:
104
Constantin Udri¸ste
max I(u(·)) =
u(·),xt0
Z
(1)
X(t, x(t), u(t))dt
Ω0,t0
subject to
i
∂x
(t) = Xαi (t, x(t), u(t)), i = 1, ..., n; α = 1, ..., m,
∂tα
u(t) ∈ U, t ∈ Ω0,t0 ; x(0) = x0 , x(t0 ) = xt0 .
(2)
(3)
m
Ingredients: t = (tα ) = (t1 , ..., tm ) ∈ R+
is the multitime (multi-parameter of evolu1
m
m
tion); dt = dt · · · dt is the volume element in R+
; Ω0,t0 is the parallelepiped fixed
by the diagonal opposite points 0 = (0, ..., 0) and t0 = (t10 , ..., tm
0 ) which is equivalent
m
to the closed interval 0 ≤ t ≤ t0 via the product order on R+
; x : Ω0,t0 → Rn , x(t) =
(xi (t)) is a C 2 state vector; u : Ω0,t0 → U ⊂ Rk , u(t) = (ua (t)), a = 1, ..., k is a C 1
control vector; the running cost X(t, x(t), u(t)) is a C 1 nonautonomous Lagrangian;
Xα (t, x(t), u(t)) = (Xαi (t, x(t), u(t))) are C 1 vector fields satisfying the complete integrability conditions (m-flow type problem), i.e., Dβ Xα = Dα Xβ (Dα is the total
derivative operator) or
¶
µ
∂Xβ
∂Xα
∂Xα γ ∂Xβ γ ∂ua
δ −
δ
= [Xα , Xβ ] +
−
,
∂ua β
∂ua α ∂tγ
∂tα
∂tβ
where [Xα , Xβ ] means the bracket of vector fields. This hypothesis selects the set of
all admissible controls (satisfying the complete integrability conditions)
¯
©
ª
m
U = u : R+
→ U ¯ Dβ Xα = Dα Xβ
and the admissible states.
We introduce a costate variable or Lagrange multiplier matrix p = (pα
i ) and a new
Lagrangian
i
L(t, x(t), u(t), p(t)) = X(t, x(t), u(t)) + pα
i (t)[Xα (t, x(t), u(t)) −
∂xi
(t)].
∂tα
The P DE-constrained optimization problem (1)-(3) is changed into another optimization problem
Z
max
u(·),xt0
L(t, x(t), u(t), p(t))dt
Ω0,t0
subject to
u(t) ∈ U, p(t) ∈ P, t ∈ Ω0,t0 , x(0) = x0 , x(t0 ) = xt0 ,
where the set P will be defined later. The control Hamiltonian
i
H(t, x(t), u(t), p(t)) = X(t, x(t), u(t)) + pα
i (t)Xα (t, x(t), u(t)),
i.e.,
∂xi
(modified Legendrian duality),
∂tα
allows to rewrite this new problem as
H = L + pα
i
105
Simplified multitime maximum principle
max
u(·),xt0
Z
Ω0,t0
[H(t, x(t), u(t), p(t)) − pα
i (t)
∂xi
(t)]dt
∂tα
subject to
u(t) ∈ U, p(t) ∈ P, t ∈ Ω0,t0 , x(0) = x0 , x(t0 ) = xt0 .
Suppose that there exists a continuous control u
ˆ(t) defined over the parallelepiped
ˆ(t) ∈ Int U which is an optimum point in the previous problem. Now
Ω0,t0 with u
consider a variation u(t, ǫ) = u
ˆ(t) + ǫh(t), where h is an arbitrary continuous vector
function. Since u
ˆ(t) ∈ Int U and a continuous function over a compact set Ω0,t0 is
bounded, there exists ǫh > 0 such that u(t, ǫ) = u
ˆ(t) + ǫh(t) ∈ Int U, ∀|ǫ| < ǫh . This
ǫ is used in our variational arguments.
Define x(t, ǫ) as the m-sheet of the state variable corresponding to the control
variable u(t, ǫ), i.e.,
∂xi
(t, ǫ) = Xαi (t, x(t, ǫ), u(t, ǫ)), ∀t ∈ Ω0,t0
∂tα
and x(0, ǫ) = x0 . For |ǫ| < ǫh , we define the function
Z
X(t, x(t, ǫ), u(t, ǫ))dt.
I(ǫ) =
Ω0,t0
Since the function u(t, ǫ) is admissible, it follows that the function x(t, ǫ) is admissible.
On the other hand, the control u
ˆ(t) must be optimal. Therefore I(ǫ) ≤ I(0), ∀|ǫ| < ǫh .
nm
, we have
For any continuous vector function p = (pα
i ) : Ω0,t0 → R
Z
∂xi
i
pα
(t)[X
(t,
x(t,
ǫ),
u(t,
ǫ))
−
(t, ǫ)]dt = 0.
i
α
∂tα
Ω0,t0
Necessarily, we must use the Lagrange function which includes the variations
L(t, x(t, ǫ), u(t, ǫ), p(t)) = X(t, x(t, ǫ), u(t, ǫ))
i
+pα
i (t)[Xα (t, x(t, ǫ), u(t, ǫ)) −
∂xi
(t, ǫ)]
∂tα
and the associated function
I(ǫ) =
Z
L(t, x(t, ǫ), u(t, ǫ), p(t))dt.
Ω0,t0
Suppose that the costate variable p is of class C 1 . Also we introduce the control
Hamiltonian
i
H(t, x(t, ǫ), u(t, ǫ), p(t)) = X(t, x(t, ǫ), u(t, ǫ)) + pα
i (t)Xα (t, x(t, ǫ), u(t, ǫ))
corresponding to the variation. Then we rewrite
Z
∂xi
[H(t, x(t, ǫ), u(t, ǫ), p(t)) − pα
(t)
I(ǫ) =
(t, ǫ)]dt.
i
∂tα
Ω0,t0
106
Constantin Udri¸ste
To evaluate the multiple integral
Z
Ω0,t0
pα
i (t)
∂xi
(t, ǫ)dt,
∂tα
we integrate by parts, via the divergence formula
∂
∂pα
∂xi
i i
i
(pα
x + pα
,
i x )=
i
α
α
∂t
∂t
∂tα
obtaining
Z
Ω0,t0
pα
i (t)
∂xi
(t, ǫ)dt =
∂tα
Z
Ω0,t0
∂
(pα (t)xi (t, ǫ))dt −
∂tα i
Z
Ω0,t0
∂pα
i
(t)xi (t, ǫ)dt.
∂tα
Now we apply the divergence integral formula
Z
Z
∂
α
i
i
β
(p
(t)x
(t,
ǫ))dt
=
δαβ pα
i
i (t)x (t, ǫ)n (t)dσ,
α
∂Ω0,t0
Ω0,t0 ∂t
where (nβ (t)) is the unit normal vector to the boundary ∂Ω0,t0 . Substituting, we find
Z
∂pα
j
[H(t, x(t, ǫ), u(t, ǫ), p(t)) + α (t)xj (t, ǫ)]dt
I(ǫ) =
∂t
Ω0,t0
−
Z
i
β
δαβ pα
i (t)x (t, ǫ)n (t)dσ.
∂Ω0,t0
Differentiating with respect to ǫ, it follows
Z
∂pα
j
[Hxj (t, x(t, ǫ), u(t, ǫ), p(t)) + α (t)]xjǫ (t, ǫ)dt
I ′ (ǫ) =
∂t
Ω0,t0
+
Z
a
Hua (t, x(t, ǫ), u(t, ǫ), p(t))h (t)dt −
Ω0,t0
Z
i
β
δαβ pα
i (t)xǫ (t, ǫ)n (t)dσ.
∂Ω0,t0
Evaluating at ǫ = 0, we find
Z
∂pα
j
ˆ(t), p(t)) + α (t)]xjǫ (t, 0)dt
[Hxj (t, x(t), u
I ′ (0) =
∂t
Ω0,t0
+
Z
a
Hua (t, x(t), u
ˆ(t), p(t))h (t)dt −
Ω0,t0
Z
i
β
δαβ pα
i (t)xǫ (t, 0)n (t)dσ.
∂Ω0,t0
where x(t) is the m-sheet of the state variable corresponding to the optimal control
u
ˆ(t).
We need I ′ (0) = 0 for all h(t) = (ha (t)). On the other hand, the functions xiǫ (t, 0)
are the components of the solution of the Cauchy problem
∇t xiǫ (t, 0) = Xx (t, x(t, 0), u(t)) · xǫ (t, 0) + Xu (t, x(t, 0), u(t)) · h(t),
Simplified multitime maximum principle
107
t ∈ Ω0,t0 , xǫ (0, 0) = 0
and hence they depend on h(t). To overpass this difficulty, we define P as the set of
solutions of the boundary value problem
∂pα
∂H
j
(t) = − j (t, x(t), u
ˆ(t), p(t)), ∀t ∈ Ω0,t0 ,
α
∂t
∂x
β
δαβ pα
j (t)n (t)|∂Ω
0,t0
(4)
= 0, (orthogonality or tangency).
Therefore
Moreover
Hua (t, x(t), u
ˆ(t), p(t)) = 0, ∀t ∈ Ω0,t0 .
(5)
∂xj
∂H
(t) = α (t, x(t), u
ˆ(t), p(t)), ∀t ∈ Ω0,t0 , x(0) = x0 .
α
∂t
∂pj
(6)
Remarks. (i) The algebraic system (5) describes the critical points of the Hamiltonian with respect to the control variable. (ii) The P DEs (4) and (6) and the condition
(5) are Euler-Lagrange P DEs associated to the new Lagrangian.
Summarizing, we obtain a multitime maximum principle similar to the single-time
Pontryaguin maximum principle.
Theorem 1. (Simplified multitime maximum principle; necessary conditions ) Suppose that the problem of maximizing the functional (1) subject to the
P DE constraints (2) and to the conditions (3), with X, Xαi of class C 1 , has an interior solution u
ˆ(t) ∈ U which determines the m-sheet of state variable x(t). Then there
exists a C 1 costate p(t) = (pα
i (t)) defined over Ω0,t0 such that the relations (4), (5),
(6) hold.
Theorem 2. (Sufficient conditions) Consider the problem of maximizing the
functional (1) subject to the P DE constraints (2) and to the conditions (3), with
ˆ(t) ∈ U and the corresponding
X, Xαi of class C 1 . Suppose that an interior solution u
m-sheet of state variable x(t) satisfy the relations (4), (5), (6). If for the resulting
costate variable p(t) = (pα
i (t)) the control Hamiltonian H(t, x, u, p) is jointly concave
ˆ(t) and the corresponding x(t) achieve the unique
in (x, u) for all t ∈ Ω0,t0 , then u
global maximum of (1).
Proof. Let us have in mind that we must maximize the functional (1) subject to
the evolution system (2) and the conditions (3). We fix a pair (ˆ
x, u
ˆ), where u
ˆ is a
candidate optimal m-sheet of the controls and x
ˆ is a candidate optimal m-sheet of
the states. Calling Iˆ the values of the functional for (ˆ
x, u
ˆ), let us prove that
Z
ˆ − X)dt ≥ 0,
(X
Iˆ − I =
Ω0,t0
ˆ = H(ˆ
where the strict inequality holds under strict concavity. Denoting H
x, pˆ, u
ˆ) and
H = H(x, pˆ, u), we find
µ
¶
Z
i
∂x
ˆi
α ∂x
ˆ − pˆα
(H
Iˆ − I =
)
−
(H
−
p
ˆ
)
dt.
i
i
∂tα
∂tα
Ω0,t0
108
Constantin Udri¸ste
Integrating by parts, we obtain
µ
¶
Z
ˆα
ˆα
i
i
i ∂p
i ∂p
ˆ
ˆ
(H + x
ˆ α ) − (H + x α ) dt
I −I =
∂t
∂t
Ω0,t0
Z
i
β
+
(δαβ pˆα
ˆα
xi (t)nβ (t))dσ.
i (t)x (t)n (t) − δαβ p
i (t)ˆ
∂Ω0,t0
Taking into account that any admissible m-sheet has the same initial and terminal
conditions as the optimal m-sheet, we derive
µ
¶
Z
α
i
i
ˆ − H) + ∂ pˆi (ˆ
(H
x
−
x
)
dt.
Iˆ − I =
∂tα
Ω0,t0
The definition of concavity and the maximum principle imply
¶
µ
Z
∂ pˆα
i
i
i
ˆ
x − x ) dt
(H − H) + α (ˆ
∂t
Ω0,t0
Ã
!
Z
ˆ
ˆ
∂ pˆα
i
i
i ∂H
a
a ∂H
≥
(ˆ
x − x )( i + α ) + (ˆ
u − u ) a dt = 0.
∂x
∂t
∂u
Ω0,t0
This last equality is true since all ”ˆ” functions satisfy the multitime maximum principle. In this way, Iˆ − I ≥ 0.
Theorem 3. (Sufficient conditions) Consider the problem of maximizing the
functional (1) subject to the P DE constraints (2) and to the conditions (3), with
X, Xαi of class C 1 . Suppose that an interior solution u
ˆ(t) ∈ U and the corresponding
m-sheet of state variable x(t) satisfy the relations (4), (5), (6). Giving the resulting
costate variable p(t) = (pα
ˆ(t), p). If M (t, x, p) is
i (t)), we define M (t, x, p) = H(t, x, u
ˆ(t) and the corresponding x(t) achieve the unique
concave in x for all t ∈ Ω0,t0 , then u
global maximum of (1).
Remark. The Theorems 2 and 3 can be extended immediately to incave functionals.
Examples. 1) We consider the problem
Z
(x(t) + u1 (t)2 + u2 (t)2 )dt1 dt2
max I(u(·)) = −
u(·),x1
Ω0,1
subject to
∂x
(t) = uα (t), α = 1, 2, x(0, 0) = 0, x(1, 1) = x1 = free.
∂tα
This problem means to find an optimal control u = (u1 , u2 ) to bring the (P DE)
dynamical system from the origin x(0, 0) = 0 at two-time t1 = 0, t2 = 0 to a terminal
point x(1, 1) = x1 , which is unspecified, at two-time t1 = 1, t2 = 1, such as to
maximize the objective functional. Also the complete integrability condition imposes
∂u1
∂u2
= 1 . The control Hamiltonian is
∂t2
∂t
H(x(t), u(t), p(t)) = −(x(t) + u1 (t)2 + u2 (t)2 ) + p1 (t)u1 (t) + p2 (t)u2 (t).
109
Simplified multitime maximum principle
Since
∂H
∂2H
∂2H
= −2uα + pα ,
= −2 < 0,
= 0,
2
∂uα
∂uα
∂uα ∂uβ
the critical point pα = 2uα is a maximum point. Then the P DE
∂H
∂pα
=−
reduces
α
∂t
∂x
∂p1 ∂p2
+ 2 = 1. Also, since the point x(1, 1) = x1 is unspecified, the transversality
∂t1
∂t
conditions imply p1 (t)n1 (t) + p2 (t)n2 (t)|∂Ω0,1 = 0.
We continue by solving the boundary value problem
to
∂p2
∂p1
∂p2
∂p1
+ 2 = 1, 2 = 1 , p1 (t)n1 (t) + p2 (t)n2 (t)|∂Ω0,1 = 0.
1
∂t
∂t
∂t
∂t
Consequently the components of the optimal control u(t) = (u1 (t), u2 (t)) are harmonic
functions satisfying the boundary conditions u1 (0, t2 ) = u1 (1, t2 ) = 0, u2 (t1 , 0) =
u2 (t1 , 1) = 0. Also the dynamical system dx = u1 (t)dt1 + u2 (t)dt2 gives x(t) − x(0) =
Z
u1 (s)ds1 + u2 (s)ds2 .
Γ0,t
2) We consider the problem
1
1
max I(u(·)) = − x(1, 1)2 −
2
2
u(·),x1
Z
(u1 (t)2 + u2 (t)2 )dt1 dt2
Ω0,1
subject to
∂x
(t) = −uα (t), α = 1, 2, x(0, 0) = 1.
∂tα
This problem means to find an optimal control u = (u1 , u2 ) to bring the (P DE)
dynamical system from the point x(0, 0) = 1 at two-time t1 = 0, t2 = 0 to a terminal
point x(1, 1) = x1 , at two-time t1 = 1, t2 = 1, such as to maximize the objective
∂u2
∂u1
functional. Also the complete integrability condition imposes 2 = 1 . The control
∂t
∂t
Hamiltonian is
1
H(x(t), u(t), p(t)) = − (u1 (t)2 + u2 (t)2 ) − pα (t)uα (t).
2
Since
∂2H
∂2H
∂H
= −uα − pα ,
= −1 < 0,
= 0,
2
∂uα
∂uα
∂uα ∂uβ
the critical point pα = −uα is a maximum point. Then the P DE
reduces to
∂p2
∂p1
+
= 0. The transversality condition implies
∂t1
∂t2
∂pα
∂H
=−
=0
α
∂t
∂x
p1 (t)n1 (t) + p2 (t)n2 (t)|∂Ω0,1 = 0.
We continue by solving the Dirichlet problem
∂p1
∂p2
∂p1
∂p2
+ 2 = 0, 2 = 1 , p1 (t)n1 (t) + p2 (t)n2 (t)|∂Ω0,1 = 0.
1
∂t
∂t
∂t
∂t
110
Constantin Udri¸ste
Consequently the components of the optimal control u(t) = (u1 (t), u2 (t)) are harmonic
functions satisfying suitable boundary conditions. Also the dynamical system dx =
−u1 (t)dt1 − u2 (t)dt2 gives the corresponding evolution
Z
u1 (s)ds1 + u2 (s)ds2 .
x(t) − x(0) = −
Γ0,t
3) Transport P DEs for Air Traffic Flow (see also [4]). The flow of aircraft
can be analyzed and controlled using an Eulerian viewpoint of the airspace. In our
formulation we use two parameters of evolution: s=position, t=time. The variable of
the state is the density of aircraft ρ(s, t), which represents the number of aircraft per
unit length of jetway. The control variable is the speed v(s, t) which the air traffic
controller can prescribe to the aircraft located at position s and time t.
Given a speed field v(s, t), the density of aircraft ρ(s, t) satisfies the continuity
∂(ρv)
∂ρ
(s, t) +
(s, t) = 0. We would like to determine the speed field which
P DE
∂t
∂s
maximizes the number of aircraft landing at the destination airport under the constraint that the density does not exceed the safety density ρmax . Mathematically,
Z
ρ(s, t)v(s, t)dsdt
max I(v(·)) =
v(·)
Ω0,A
subject to
∂(ρv)
∂ρ
(s, t) +
(s, t) = 0, ρ ≤ ρmax , vmin ≤ v ≤ vmax ,
∂t
∂s
where A = (L, T ), L = value of s for final destination, T = final time, I(v(·))= total
number of aircraft landing, vmin , vmax = bounds of authorized aircraft speed. The
new Lagrangian
µ
¶
∂ρ
∂(ρv)
L = ρ(s, t)v(s, t) + p(s, t)
(s, t) +
(s, t) + µ1 (s, t)(ρmax − ρ(s, t))
∂t
∂s
+µ2 (s, t)(v(s, t) − vmin ) + µ3 (s, t)(vmax − v(s, t))
produces the Hamiltonian
¡
¢
H = −ρ(s, t) − µ2 (s, t) + µ3 (s, t) v(s, t) − µ1 (s, t)(ρmax − ρ(s, t))
+µ2 (s, t)vmin − µ3 (s, t)vmax ,
of degree one with respect to the control v. The sign of the switching function σ =
−ρ(s, t) − µ2 (s, t) + µ3 (s, t) decides an optimal control. The adjoint P DEs (obtained
from I ′ (0) = 0) are
∂p
∂p
∂p
+v
= v − µ1 , ρ
= ρ + µ2 − µ3 .
∂t
∂s
∂s
Open problem: Study optimal control problems subject to
P DE : Xβα (x(t), u(t))
∂f β
(x(t), u(t)) = 0.
∂tα
111
Simplified multitime maximum principle
Bibliographical note. For strong contributions to optimal control problems,
which have influenced our point of view, see [1]-[4], [6]-[11].
3
m-Flow type constrained optimization problem
with curvilinear integral cost functional
The cost functionals of mechanical work type are very important for applications
(see our papers [12]-[22]). This is the reason to analyze a multitime optimal control
problem based on a path independent curvilinear integral as cost functional and on
P DE constraints of m-flow type :
Z
max J(u(·)) =
Xα0 (t, x(t), u(t))dtα
(7)
u(·),xt0
Γ0,t0
subject to
i
∂x
(t) = Xαi (t, x(t), u(t)), i = 1, ..., n; α = 1, ..., m,
∂tα
(8)
u(t) ∈ U, t ∈ Ω0,t0 ; x(0) = x0 , x(t0 ) = xt0 .
(9)
α
m
R+
Ingredients: t = (t ) ∈
is the multitime (multi-parameter of evolution);
Γ0,t0 is an arbitrary C 1 curve joining the diagonal opposite points 0 = (0, ..., 0)
n
i
2
and t0 = (t10 , ..., tm
0 ) in Ω0,t0 ; x : Ω0,t0 → R , x(t) = (x (t)) is a C state vector;
k
a
1
u : Ω0,t0 → U ⊂ R , u(t) = (u (t)), a = 1, ..., k is a C control vector; the running cost Xα0 (t, x(t), u(t))dtα is a nonautonomous closed (completely integrable) Lagrangian 1-form , i.e., it satisfies Dβ Xα0 = Dα Xβ0 (Dα is the total derivative operator)
or
Ã
!
0
∂Xβ0
∂Xβ0
∂Xα0 γ ∂Xβ0 γ ∂ua
∂Xα0
i
i ∂Xα
δ
−
δ
=
X
−
X
+
−
;
α
α
β
∂ua β
∂ua
∂tγ
∂xi
∂xi
∂tα
∂tβ
the vector fields Xα (t, x(t), u(t)) = (Xαi (t, x(t), u(t))) are of class C 1 and satisfy the
complete integrability conditions (m-flow type problem) , i.e., Dβ Xα = Dα Xβ or
¶
µ
∂Xβ
∂Xα
∂Xα γ ∂Xβ γ ∂ua
δ
−
δ
= [Xα , Xβ ] +
−
,
α
β
a
a
γ
α
∂u
∂u
∂t
∂t
∂tβ
where [Xα , Xβ ] means the bracket of vector fields. This hypothesis selects the set of
all admissible controls (satisfying the complete integrability conditions)
¯
©
ª
m
U = u : R+
→ U ¯ Dβ Xα0 = Dα Xβ0 , Dβ Xα = Dα Xβ
and the set of admissible states.
We introduce a costate variable or Lagrange multiplier function p = (pi ) such that
the new Lagrange 1-form
Lα (t, x(t), u(t), p(t)) = Xα0 (t, x(t), u(t)) + pi (t)[Xαi (t, x(t), u(t)) −
∂xi
(t)]
∂tα
112
Constantin Udri¸ste
be closed. The P DE constrained optimization problem (7)-(9) is replaced by another
optimization problem
Z
max
Lα (t, x(t), u(t), p(t))dtα
u(·),xt0
Γ0,t0
subject to
u(t) ∈ U, p(t) ∈ P, t ∈ Ω0,t0 , x(0) = x0 , x(t0 ) = xt0 ,
where the set P will be defined later. If we use the control Hamiltonian 1-form
Hα (t, x(t), u(t), p(t)) = Xα0 (t, x(t), u(t)) + pi (t)Xαi (t, x(t), u(t)),
Hα = Lα + pi
∂xi
(nonstandard duality),
∂tα
we can rewrite
max
u(·),xt0
Z
[Hα (t, x(t), u(t), p(t)) − pi (t)
Γ0,t0
∂xi
(t)]dtα
∂tα
subject to
u(t) ∈ U, p(t) ∈ P, t ∈ Ω0,t0 , x(0) = x0 , x(t0 ) = xt0 .
ˆ(t) ∈
Suppose that there exists a continuous control u
ˆ(t) defined over Ω0,t0 with u
Int U which is optimum in the previous problem. Now consider a variation u(t, ǫ) =
u
ˆ(t)+ǫh(t), where h is an arbitrary continuous vector function. Since u
ˆ(t) ∈ Int U and
a continuous function over a compact set Ω0,t0 is bounded, there exists ǫh > 0 such
that u(t, ǫ) = u
ˆ(t) + ǫh(t) ∈ Int U, ∀|ǫ| < ǫh . This ǫ is used in the next variational
arguments.
Let us consider an arbitrary vector function h(t) and define x(t, ǫ) as the m-sheet
of the state variable corresponding to the control variable u(t, ǫ), i.e.,
∂xi
(t, ǫ) = Xαi (t, x(t, ǫ), u(t, ǫ)), ∀t ∈ Ω0,t0 , x(0, ǫ) = x0 .
∂tα
For |ǫ| < ǫh , we define the function
Z
J(ǫ) =
Xα0 (t, x(t, ǫ), u(t, ǫ))dtα .
Γ0,t0
Since the control function u(t, ǫ) is admissible, it follows that the evolution function
x(t, ǫ) is admissible. On the other hand, the control u
ˆ(t) is supposed to be optimal.
Therefore J(ǫ) ≤ J(0), ∀|ǫ| < ǫh .
For any continuous function p = (pi ) : Ω0,t0 → Rn , we have
Z
Γ0,t0
pi (t)[Xαi (t, x(t, ǫ), u(t, ǫ)) −
∂xi
(t, ǫ)]dtα = 0.
∂tα
The variations determine the closed Lagrange 1-form
113
Simplified multitime maximum principle
Lα (t, x(t, ǫ), u(t, ǫ), p(t)) = Xα0 (t, x(t, ǫ), u(t, ǫ))+pi (t)[Xαi (t, x(t, ǫ), u(t, ǫ))−
and the function
J(ǫ) =
Z
∂xi
(t, ǫ)]
∂tα
Lα (t, x(t, ǫ), u(t, ǫ), p(t))dtα .
Γ0,t0
Suppose that the costate p is of class C 1 . Also we introduce the control Hamiltonian 1-form
Hα (t, x(t, ǫ), u(t, ǫ), p(t)) = Xα0 (t, x(t, ǫ), u(t, ǫ)) + pi (t)Xαi (t, x(t, ǫ), u(t, ǫ)).
Then we rewrite
J(ǫ) =
Z
[Hα (t, x(t, ǫ), u(t, ǫ), p(t)) − pi (t)
Γ0,t0
To evaluate the curvilinear integral
Z
Γ0,t0
pi (t)
∂xi
(t, ǫ)]dtα .
∂tα
∂xi
(t, ǫ)dtα ,
∂tα
we integrate by parts, via
∂
∂pi i
∂xi
i
(p
x
)
=
x
+
p
,
i
i
∂tα
∂tα
∂tα
obtaining
Z
Γ0,t0
∂xi
pi (t) α (t, ǫ)dtα = (pi (t)xi (t, ǫ))|t00 −
∂t
Z
Γ0,t0
∂pi
(t)xi (t, ǫ)dtα .
∂tα
Substituting, we get the function
Z
∂pj
[Hα (t, x(t, ǫ), u(t, ǫ), p(t))+ α (t)xj (t, ǫ)]dtα −pi (t0 )xi (t0 , ǫ)+pi (0)xi (0, ǫ).
J(ǫ) =
∂t
Γ0,t0
It follows
J ′ (ǫ) =
+
Z
Γ0,t0
Z
[Hαxj (t, x(t, ǫ), u(t, ǫ), p(t)) +
Γ0,t0
∂pj
(t)]xjǫ (t, ǫ)dtα
∂tα
Hαua (t, x(t, ǫ), u(t, ǫ), p(t))ha (t)dtα − pi (t0 )xiǫ (t0 , ǫ) + pi (0)xiǫ (0, ǫ).
Evaluating at ǫ = 0, we find
Z
∂pj
J ′ (0) =
ˆ(t), p(t)) + α (t)]xjǫ (t, 0)dtα
[Hαxj (t, x(t), u
∂t
Γ0,t0
+
Z
Γ0,t0
Hαua (t, x(t), u
ˆ(t), p(t))ha (t)dtα − pi (t0 )xiǫ (t0 , 0),
114
Constantin Udri¸ste
where x(t) is the m-sheet of the state variable corresponding to the optimal control
u
ˆ(t). We need J ′ (0) = 0 for all h(t) = (ha (t)). This is possible if we define P as the
set of solutions of the terminal value problem
∂pj
∂Hα
(t) = − j (t, x(t), u
ˆ(t), p(t)), ∀t ∈ Ω0,t0 ; pj (t0 ) = 0.
α
∂t
∂x
(10)
ˆ(t), p(t)) = 0, ∀t ∈ Ω0,t0 .
Hαua (t, x(t), u
(11)
∂xj
∂Hα
(t) =
(t, x(t), u
ˆ(t), p(t)), ∀t ∈ Ω0,t0 ; x(0) = x0 .
∂tα
∂pj
(12)
Therefore
Moreover
Remarks. (i) The algebraic system (11) describes the common critical points of
the functions Hα with respect to the control variable u. (ii) The P DEs (10) and (12)
and the relation (11) are Euler-Lagrange P DEs associated to the new Lagrangian
1-form.
Summarizing, we obtain a new variant of multitime maximum principle.
Theorem 4. (Simplified multitime maximum principle; necessary conditions) Suppose that the problem of maximizing the functional (7) subject to the P DE
constraints (8) and to the conditions (9), with Xα0 , Xαi of class C 1 , has an interior
solution u
ˆ(t) ∈ U which determines the m-sheet of state variable x(t). Then there
exists a C 1 costate p(t) = (pi (t)) defined over Ω0,t0 such that the relations (10), (11),
(12) hold.
Theorem 5. (Sufficient conditions) Consider the problem of maximizing the
functional (7) subject to the P DE constraints (8) and to the conditions (9), with
ˆ(t) ∈ U and the corresponding
Xα0 , Xαi of class C 1 . Suppose that an interior solution u
m-sheet of state variable x(t) satisfy the relations (10), (11), (12). If, for the resulting
costate variable p(t) = (pi (t)), the control Hamiltonian 1-form Hα (t, x, u, p) is jointly
ˆ(t) and the corresponding x(t) achieve the
concave in (x, u) for all t ∈ Ω0,t0 , then u
unique global maximum of (7).
Theorem 6. (Sufficient conditions) Consider the problem of maximizing the
functional (7) subject to the P DE constraints (8) and to the conditions (9), with
ˆ(t) ∈ U and the corresponding
Xα0 , Xαi of class C 1 . Suppose that an interior solution u
m-sheet of state variable x(t) satisfy the relations (10), (11), (12). Giving the resulting
costate variable p(t) = (pi (t)), we define the 1-form Mα (t, x, p) = Hα (t, x, u
ˆ(t), p). If
the 1-form Mα (t, x, p) is concave in x for all t ∈ Ω0,t0 , then u
ˆ(t) and the corresponding
x(t) achieve the unique global maximum of (7).
Remark. The Theorems 5 and 6 can be extended immediately to incave functionals.
Example. Let t = (t1 , t2 ) ∈ Ω0,1 , where 0 = (0, 0), 1 = (1, 1) are diagonal opposite
points in Ω0,1 . Denote by Γ0,1 an arbitrary C 1 curve joining the points 0 and 1. We
consider the problem
115
Simplified multitime maximum principle
max J(u(·)) = −
u(·),x1
Z
(x(t) + uβ (t)2 )dtβ
Γ0,1
subject to
∂x
(t) = uα (t), α = 1, 2, x(0, 0) = 0, x(1, 1) = x1 = free.
∂tα
This problem means to find an optimal control u = (u1 , u2 ) to bring the (P DE)
dynamical system from the origin x(0, 0) = 0 at two-time t1 = 0, t2 = 0 to a terminal
point x(1, 1) = x1 , which is unspecified, at two-time t1 = 1, t2 = 1, such as to
maximize the objective functional. Also the complete integrability conditions impose
∂u2
∂x
∂u1 ∂u1
∂u2
∂x
+ 2u2 1 = 2 + 2u1 2 ,
= 1.
∂t1
∂t
∂t
∂t
∂t2
∂t
The control Hamiltonian 1-form is
Hβ (x(t), u(t), p(t)) = −(x(t) + uβ (t)2 ) + p(t)uβ (t).
Since
∂ 2 Hβ
∂Hβ
= −2 < 0,
= −2uβ + p,
∂uβ
∂u2β
p
∂p
∂Hα
the critical point u1 = u2 =
is a maximum point. The P DEs
= −
2
∂tα
∂x
∂p
reduces to α = 1. Also, since the point x(1, 1) = x1 is unspecified, the transversality
∂t
condition implies p(1) = 0. It follows the costate p(t) = t1 + t2 − 2, the optimal control
1
(t1 )2 + (t2 )2
u
ˆ1 (t) = u
ˆ2 (t) = (t1 + t2 − 2) and the corresponding evolution x(t) =
+
2
4
t1 t2
− (t1 + t2 ).
2
4
Equivalence between multiple and curvilinear integral functionals
A multitime evolution system can be used as a constraint in a problem of extremizing
a multitime cost functional. On the other hand, the multitime cost functionals can be
introduced at least in two ways:
- either using a path independent curvilinear integral,
Z
P (u(·)) =
Xβ0 (x(t), u(t))dtβ + g(x(t0 )),
Γ0,t0
where Γ0,t0 is an arbitrary C 1 curve joining the points 0 and t0 , the running cost
ω = Xβ0 (x(t), u(t))dtβ is an autonomous closed (completely integrable) Lagrangian
1-form, and g is the terminal cost;
- or using a multiple integral,
Z
X(x(t), u(t))dt + g(x(t0 )),
Q(u(·)) =
Ω0,t0
116
Constantin Udri¸ste
where the running cost X(x(t), u(t)) is an autonomous continuous Lagrangian, and g
is the terminal cost.
Let us show that the functional P is equivalent to the functional Q. This means
that in a multitime optimal control problem we can choose the appropriate functional
based on geometrical-physical meaning or other criteria.
Theorem 7 [23]. The multiple integral
Z
I(t0 ) =
X(x(t), u(t))dt,
Ω0,t0
with X as continuous function, is equivalent to the curvilinear integral
Z
Xβ0 (x(t), u(t))dtβ ,
J(t0 ) =
Γ0,t0
where ω = Xβ0 (x(t), u(t))dtβ is a closed (completely integrable) Lagrangian 1-form
and the functions Xβ0 have partial derivatives of the form
∂
∂
∂ m−1
,
(α
<
β),
...,
,
∂tα ∂tα ∂tβ
∂t1 ...∂tˆα ...∂tm
where the symbol ”ˆ” posed over ∂tα designates that ∂tα is omitted.
5
Multitime maximum principle approach of variational calculus
It is well known that the single-time Pontryaguin’s maximum principle is a generalization of the Lagrange problem in the single-time variational calculus and that these
problems are equivalent when the control domain is open [1], [7], [10]. Does this property survive for simplified multitime maximum principle? The aim of this Section is
to formulate an answer to this question.
In fact we show that the simplified multitime maximum principle motivates the
multitime Euler-Lagrange or Hamilton PDEs. For that, suppose that the evolution
system is reduced to a completely integrable system
∂xi
m
,
(t) = uiα (t), x(0) = x0 , t ∈ Ω0,t0 ⊂ R+
∂tα
and the functional is a path independent curvilinear integral
Z
Xβ0 (x(t), u(t))dtβ , u = (uiα ),
J(u(·)) =
(P DE)
(J)
Γ0,t0
where Γ0,t0 is an arbitrary piecewise C 1 curve joining the points 0 and t0 , the running
cost ω = Xβ0 (x(t), u(t))dtβ is a closed (completely integrable) Lagrangian 1-form.
The associated basic control problem leads necessarily to the multitime maximum
principle. Therefore, to solve it we need the control Hamiltonian 1-form
Hβ (x, p0 , p, u) = Xβ0 (x, u) + pi uiβ
117
Simplified multitime maximum principle
and the adjoint PDEs
∂Xβ0
∂pi
(t)
=
−
(x(t), u(t)).
(ADJ)
∂tβ
∂xi
Suppose the simplified multitime maximum principle is applicable (see the relation
(10))
∂Xβ0
∂Xβ0
∂Hβ
γ
γ
=
+
p
δ
=
0,
p
δ
=
−
, uiγ = xiγ .
(13)
i
i
β
β
∂uiγ
∂uiγ
∂uiγ
Suppose the functions Xβ0 are dependent on x (a strong condition!). Then (ADJ)
shows that
Z
∂Xβ0
pi (t) = pi (0) −
(x(s), u(s))dsβ ,
(14)
i
Γ0,t ∂x
where Γ0,t is an arbitrary piecewise C 1 curve joining the points 0, t ∈ Ω0,t0 .
5.1
Multitime Euler-Lagrange PDEs
From the relations (13) and (14), it follows
∂Xβ0
− i (x(t), u(t)) = δβγ pi (0) − δβγ
∂xγ
Z
Γ0,t
∂Xλ0
(x(s), u(s))dsλ .
∂xi
Suppose that Xβ0 are functions of class C 2 . Applying the divergence operator Dγ =
∂Xβ0
∂Xβ0
∂
we
find
the
multitime
Euler-Lagrange
P
DEs
−
D
= 0.
γ
∂tγ
∂xi
∂xiγ
5.2
Conversion to multitime Hamilton PDEs (canonical variables)
Let u(·) be an optimal control, x(·) the optimal evolution m-sheet, and p(·) be the
solution of (ADJ) which corresponds to u(·) and x(·). Suppose that the critical point
∂xi
condition admits a unique solution uiγ (t) = uiγ (x(t), p(t)) =
(t). Then, using a
∂tγ
path independent curvilinear integral, we can write
Z
i
i
uiγ (x(s), p(s))dsγ .
x (t) = x (0) +
Γ0,t
The control Hamiltonian 1-form Hβ = Xβ0 + pj ujβ must satisfy
relation, pi δβγ +
∂Hβ
= 0. This last
∂uiγ
∂Xβ0
= 0, defines the costate p as a moment. On the other hand
∂uiγ
∂ujβ
∂Xβ0 ∂ujγ
∂Hβ
∂xi
∂Hβ
i
=
+
u
+
p
= uiβ or β (t) =
(x(t), p(t), u(t)).
j
β
j
∂pi
∂p
∂p
∂t
∂pi
∂uγ
i
i
Now, the relation
118
Constantin Udri¸ste
∂Hβ
−
=−
∂xi
Ã
∂Xβ0
∂Xβ0 ∂ujγ
+
∂xi
∂ujγ ∂xi
!
− pj
∂ujβ
∂xi
and (ADJ) show
∂Hβ
∂pi
(t) = −
(x(t), p(t), u(t)).
∂tβ
∂xi
In this way we find the canonical variables x, p and the multitime Hamilton P DEs
∂xi
∂pi
∂Hβ
∂Hβ
(x(t), p(t)),
(t) =
(t) = −
(x(t), p(t)).
β
β
∂t
∂pi
∂t
∂xi
Remark. To make a computer aided study of P DE-constrained optimization
problems we can perform symbolic computations via Maple (see also [5]).
Acknowledgements: Partially supported by Grant CNCSIS 86/ 2008, by 15-th
Italian-Romanian Executive Programme of S&T Co-operation for 2006-2008, University Politehnica of Bucharest, and by Academy of Romanian Scientists, Splaiul
Independentei 54, 050094 Bucharest, Romania. We thank Professors Franco Giannessi, Valeriu Prepelit¸˘a, Marius R˘adulescu, Ionel T
¸ evy and Assistant Cristian Ghiu
for their suggestions improving the paper.
References
[1] D. Acemoglu, Introduction to Economic Growth, MIT Department of Economics,
2006.
[2] V. Barbu, I. Lasiecka, D. Tiba, C. Varsan, Analysis and Optimization of Differential Systems, IFIP TC7/WG7.2 International Working Conference on Analysis
and Optimization of Differential Systems, September 10-14, 2002, Constanta,
Romania, Kluwer Academic Publishers, 2003.
[3] G. Barles, A. Tourin, Commutation properties of semigroups for first-order
Hamilton-Jacobi equations and application to multi-time equations, Indiana University Mathematics Journal, 50, 4 (2001), 1523-1544.
[4] A. M. Bayen, R. L.Raffard, C. J. Tomlin, Adjoint-based constrained control of
Eulerian transportation networks: Application to air traffic control, Proceedings
of the American Control Conference, Boston, June 2004.
[5] M. T. Calapso, C. Udri¸ste, Isothermic surfaces as solutions of Calapso PDE,
Balkan Journal of Geometry and Its Applications, 13, 1(2008), 20-26.
[6] F. Cardin, C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time
equations, Duke Math. J., 144, 2 (2008), 235-284.
[7] L. C. Evans, An Introduction to Mathematical Optimal Control Theory, Lecture
Notes, University of California, Department of Mathematics, Berkeley, 2008.
[8] M. Motta, F. Rampazo, Nonsmooth multi-time Hamilton-Jacobi systems, Indiana
University Mathematics Journal, 55, 5 (2006), 1573-1614.
[9] S. Pickenhain, M. Wagner, Piecewise continuous controls in DieudonneRashevsky type problems, Journal of Optimization Theory and Applications
(JOTA), 127 (2005), 145-163.
Simplified multitime maximum principle
119
[10] L. Pontriaguine, V. Boltianski, R. Gamkrelidze, E. Michtchenko, Th´eorie
Math´ematique des Processus Optimaux, Edition MIR, Moscou, 1974.
[11] D. F. M. Torres, A. Yu. Plakhov, Optimal Control of Newton type problems of
minimal resistance, Rend. Sem. Mat. Univ. Pol. Torino, 64, 1 (2006), 79-95.
[12] C. Udri¸ste, Multi-time maximum principle, Short Communication at International Congress of Mathematicians, Madrid, August 22-30, 2006; Plenary Lecture
at 6-th WSEAS International Conference on Circuits, Systems, Electronics, Control&Signal Processing (CSECS’07) and 12-th WSEAS International Conference
on Applied Mathematics, Cairo, Egypt, December 29-31, 2007.
[13] C. Udri¸ste, I. T
¸ evy, Multi-time Euler-Lagrange-Hamilton theory, WSEAS Transactions on Mathematics, 6, 6 (2007), 701-709.
[14] C. Udri¸ste, I. T
¸ evy, Multi-time Euler-Lagrange dynamics, Proceedings of the 7th
WSEAS International Conference on Systems Theory and Scientific Computation
(ISTASC’07), Vouliagmeni Beach, Athens, Greece, August 24-26, 2007, 66-71.
[15] C. Udri¸ste, Controllability and observability of multitime linear PDE systems,
Proceedings of The Sixth Congress of Romanian Mathematicians, Bucharest,
Romania, June 28 - July 4, 2007, vol. 1, 313-319.
[16] C. Udri¸ste, Multi-time stochastic control theory, Selected Topics on Circuits, Systems, Electronics, Control&Signal Processing, Proceedings of the 6-th WSEAS
International Conference on Circuits, Systems, Electronics, Control&Signal Processing (CSECS’07), Cairo, Egypt, December 29-31, 2007, 171-176.
[17] C. Udri¸ste, Finsler optimal control and Geometric Dynamics, Mathematics and
Computers In Science and Engineering, Proceedings of the American Conference
on Applied Mathematics, Cambridge, Massachusetts, 2008, 33-38.
[18] C. Udri¸ste, Lagrangians constructed from Hamiltonian systems, Mathematics
a Computers in Business and Economics, Proceedings of the 9th WSEAS
International Conference on Mathematics a Computers in Business and
Economics(MCBE-08), Bucharest, Romania, June 24-26, 2008, 30-33.
[19] C. Udri¸ste, Multitime controllability, observability and bang-bang principle, Journal of Optimization Theory and Applications, 139, 1(2008), 141-157; DOI
10.1007/s10957-008-9430-2.
[20] C. Udri¸ste, L. Matei, Lagrange-Hamilton theories (in Romanian), Monographs
and Textbooks 8, Geometry Balkan Press, Bucharest, 2008.
[21] C. Udri¸ste, T. Oprea, H-convex Riemannian manifolds, Balkan Journal of Geometry and Its Applications, 13, 2(2008), 112-119.
[22] Ariana Pitea, C. Udri¸ste, S¸t. Mititelu, P DI&P DE-constrained optimization
problems with curvilinear functional quotients as objective vectors, Balkan Journal of Geometry and Its Applications, 14, 2 (2009), 75-88.
[23] C. Udriste, O Dogaru, I. Tevy, Null Lagrangian forms and Euler-Lagrange PDEs,
J. Adv. Math. Studies, 1, 1-2 (2008), 143 - 156.
Author’s address:
Constantin Udriste
University Politehnica of Bucharest, Faculty of Applied Sciences,
Department of Mathematics-Informatics, Splaiul Independentei 313,
Bucharest 060042, Romania.
E-mail: [email protected], [email protected]
Constantin Udri¸ste
Abstract. Many science and engineering problems can be formulated as
optimization problems that are governed by m-flow type P DEs (multitime evolution systems) and by cost functionals expressed as multiple integrals or curvilinear integrals. Our paper discuss the m-flow type P DEconstrained optimization problems, focussing on a simplified multitime
maximum principle. This extends the simplified single-time maximum
principle of Pontryaguin in the ODEs case (curves) to include the case of
P DEs (submanifolds).
In Section 1 the idea of multitime is motivated. In Section 2 a multitime maximum principle, for the case of multiple integral functionals, is
stated and proved. A version of multitime maximum principle, for the case
of curvilinear integral functionals, is formulated in Section 3. Though a
multiple integral functional is mathematically equivalent to a curvilinear
integral functional (Section 4), their meaning is totally different in real life
problems. A multitime maximum principle approach of variational calculus is presented in Section 5.
M.S.C. 2000: 93C20, 93C35, 49K20, 49J20, 53C44.
Key words: P DE-constrained optimal control, multitime maximum principle, multiple or curvilinear integral functional, geometric evolution.
1
Multitime concept
The adjective multitime was introduced in Physics by Dirac (1932), and later was
used in Mathematics by [4], [6], [8], [12]-[16], [19], [20], [22], etc. To underline the
sense of this adjective, we collect the following remarks.
1) A space coordinate is merely an index numbering freedom degrees, and the time
coordinate is usual the physical time in which the systems evolves. Such classical
theory is satisfactory unless we turn our attention to relativistic problems (chiral
fields, sine-Gordon P DE, etc). Moreover, in some physical problems we use a twotime t = (t1 , t2 ), where t1 means the intrinsic time and t2 is the observer time. Also
there are a lot of problems where is no reason to prefer one coordinate to another. In
this sense, we refer to multitime geometric evolutions and multitime optimal control
∗
Balkan
Journal of Geometry and Its Applications, Vol.14, No.1, 2009, pp. 102-119.
c Balkan Society of Geometers, Geometry Balkan Press 2009.
°
Simplified multitime maximum principle
103
problems, where multitime means a vector parameter of evolution. Here we can include
the description of torsion of prismatic bars, the maximization of the area surface for
given width and diameter etc.
2) Multitime wave functions were first considered by Dirac in 1932 via m-time
∂ψ
evolution equations i ~ α = Hα ψ. The Dirac P DE system is consistent (completely
∂t
integrable) if and only if [Hα , Hβ ] = 0 for α 6= β. The consistency condition is easy
to achieve for non-interacting particles and tricky in the presence of interaction. But,
until now, nobody attempted to write down consistent multitime equations for many
interacting particles, although this would seem an immediate and highly relevant
problem if one desires a manifestly covariant formulation of relativistic quantum mechanics.
3) The oscillators are very important in engineering and communications. For example, voltage-controlled oscillators, phase-locked loops, lasers, etc., abound in wireless and optical systems. A new approach for analyzing frequency and amplitude
modulation in oscillators was realized recently using a novel concept, warped time,
within a multitime P DE framework. To explain this idea from our point of view, we
2π
start with a single-time wave front y(t) = sin ( 2π
T1 t) sin ( T2 t), T1 = 0.02 s; T2 = 1 s,
1
1
where the two tones are at frequencies f1 =
= 50 Hz and f2 =
= 1 Hz. Here
T1
T2
there are 50 times faster varying sinusoids of period T1 modulated by a slowly-varying
sinusoid of period T2 . Then we build a ”two-variable representation” of y(t), obtained
by the rules: for the ”fast-varying” parts of y(t), the time t is replaced by a new variable t1 ; for the ”slowly-varying” parts, by t2 . It appears a new periodic function of
2π 2
1
two variables, yˆ(t1 , t2 ) = sin ( 2π
T1 t ) sin ( T2 t ), motivated by the wide separated time
scales. Inspection of the two-time (two-variable) wavefront yˆ(t1 , t2 ) directly provides
information about the slow and fast variations of y(t) more naturally and conveniently
than y(t) itself.
4) The known evolution laws in physical theories are single-time evolution laws
(ODEs) or multitime evolution laws (PDEs). To change a single-time evolution into
a multitime evolution it is enough to change the ODEs into P DEs accepting that
the time t is a C ∞ function of certain parameters, let say t = t(s1 , ..., sm ).
The P DE constraints often present significant challenges for optimization principles [2]-[6], [8], [9]. The multivariable maximum principle was studied in the presence
of PDE constraints, starting with the papers [12]-[22]. This approach extends the
single-time Pontryaguin’s model [1], [7], [10], [11].
In this paper, we are looking for a multitime maximum principle, i.e., for necessary
conditions of optimality. Our formulation and a proof mimic those that were applied
to single-time maximum principle. A simplified version of this problem, obtained after
years of debates in my research group and in conferences, is presented in this paper.
2
m-Flow type constrained optimization problem
with multiple integral functional
Let us analyze a multitime optimal control problem based on a multiple integral cost
functional and m-flow type P DE constraints:
104
Constantin Udri¸ste
max I(u(·)) =
u(·),xt0
Z
(1)
X(t, x(t), u(t))dt
Ω0,t0
subject to
i
∂x
(t) = Xαi (t, x(t), u(t)), i = 1, ..., n; α = 1, ..., m,
∂tα
u(t) ∈ U, t ∈ Ω0,t0 ; x(0) = x0 , x(t0 ) = xt0 .
(2)
(3)
m
Ingredients: t = (tα ) = (t1 , ..., tm ) ∈ R+
is the multitime (multi-parameter of evolu1
m
m
tion); dt = dt · · · dt is the volume element in R+
; Ω0,t0 is the parallelepiped fixed
by the diagonal opposite points 0 = (0, ..., 0) and t0 = (t10 , ..., tm
0 ) which is equivalent
m
to the closed interval 0 ≤ t ≤ t0 via the product order on R+
; x : Ω0,t0 → Rn , x(t) =
(xi (t)) is a C 2 state vector; u : Ω0,t0 → U ⊂ Rk , u(t) = (ua (t)), a = 1, ..., k is a C 1
control vector; the running cost X(t, x(t), u(t)) is a C 1 nonautonomous Lagrangian;
Xα (t, x(t), u(t)) = (Xαi (t, x(t), u(t))) are C 1 vector fields satisfying the complete integrability conditions (m-flow type problem), i.e., Dβ Xα = Dα Xβ (Dα is the total
derivative operator) or
¶
µ
∂Xβ
∂Xα
∂Xα γ ∂Xβ γ ∂ua
δ −
δ
= [Xα , Xβ ] +
−
,
∂ua β
∂ua α ∂tγ
∂tα
∂tβ
where [Xα , Xβ ] means the bracket of vector fields. This hypothesis selects the set of
all admissible controls (satisfying the complete integrability conditions)
¯
©
ª
m
U = u : R+
→ U ¯ Dβ Xα = Dα Xβ
and the admissible states.
We introduce a costate variable or Lagrange multiplier matrix p = (pα
i ) and a new
Lagrangian
i
L(t, x(t), u(t), p(t)) = X(t, x(t), u(t)) + pα
i (t)[Xα (t, x(t), u(t)) −
∂xi
(t)].
∂tα
The P DE-constrained optimization problem (1)-(3) is changed into another optimization problem
Z
max
u(·),xt0
L(t, x(t), u(t), p(t))dt
Ω0,t0
subject to
u(t) ∈ U, p(t) ∈ P, t ∈ Ω0,t0 , x(0) = x0 , x(t0 ) = xt0 ,
where the set P will be defined later. The control Hamiltonian
i
H(t, x(t), u(t), p(t)) = X(t, x(t), u(t)) + pα
i (t)Xα (t, x(t), u(t)),
i.e.,
∂xi
(modified Legendrian duality),
∂tα
allows to rewrite this new problem as
H = L + pα
i
105
Simplified multitime maximum principle
max
u(·),xt0
Z
Ω0,t0
[H(t, x(t), u(t), p(t)) − pα
i (t)
∂xi
(t)]dt
∂tα
subject to
u(t) ∈ U, p(t) ∈ P, t ∈ Ω0,t0 , x(0) = x0 , x(t0 ) = xt0 .
Suppose that there exists a continuous control u
ˆ(t) defined over the parallelepiped
ˆ(t) ∈ Int U which is an optimum point in the previous problem. Now
Ω0,t0 with u
consider a variation u(t, ǫ) = u
ˆ(t) + ǫh(t), where h is an arbitrary continuous vector
function. Since u
ˆ(t) ∈ Int U and a continuous function over a compact set Ω0,t0 is
bounded, there exists ǫh > 0 such that u(t, ǫ) = u
ˆ(t) + ǫh(t) ∈ Int U, ∀|ǫ| < ǫh . This
ǫ is used in our variational arguments.
Define x(t, ǫ) as the m-sheet of the state variable corresponding to the control
variable u(t, ǫ), i.e.,
∂xi
(t, ǫ) = Xαi (t, x(t, ǫ), u(t, ǫ)), ∀t ∈ Ω0,t0
∂tα
and x(0, ǫ) = x0 . For |ǫ| < ǫh , we define the function
Z
X(t, x(t, ǫ), u(t, ǫ))dt.
I(ǫ) =
Ω0,t0
Since the function u(t, ǫ) is admissible, it follows that the function x(t, ǫ) is admissible.
On the other hand, the control u
ˆ(t) must be optimal. Therefore I(ǫ) ≤ I(0), ∀|ǫ| < ǫh .
nm
, we have
For any continuous vector function p = (pα
i ) : Ω0,t0 → R
Z
∂xi
i
pα
(t)[X
(t,
x(t,
ǫ),
u(t,
ǫ))
−
(t, ǫ)]dt = 0.
i
α
∂tα
Ω0,t0
Necessarily, we must use the Lagrange function which includes the variations
L(t, x(t, ǫ), u(t, ǫ), p(t)) = X(t, x(t, ǫ), u(t, ǫ))
i
+pα
i (t)[Xα (t, x(t, ǫ), u(t, ǫ)) −
∂xi
(t, ǫ)]
∂tα
and the associated function
I(ǫ) =
Z
L(t, x(t, ǫ), u(t, ǫ), p(t))dt.
Ω0,t0
Suppose that the costate variable p is of class C 1 . Also we introduce the control
Hamiltonian
i
H(t, x(t, ǫ), u(t, ǫ), p(t)) = X(t, x(t, ǫ), u(t, ǫ)) + pα
i (t)Xα (t, x(t, ǫ), u(t, ǫ))
corresponding to the variation. Then we rewrite
Z
∂xi
[H(t, x(t, ǫ), u(t, ǫ), p(t)) − pα
(t)
I(ǫ) =
(t, ǫ)]dt.
i
∂tα
Ω0,t0
106
Constantin Udri¸ste
To evaluate the multiple integral
Z
Ω0,t0
pα
i (t)
∂xi
(t, ǫ)dt,
∂tα
we integrate by parts, via the divergence formula
∂
∂pα
∂xi
i i
i
(pα
x + pα
,
i x )=
i
α
α
∂t
∂t
∂tα
obtaining
Z
Ω0,t0
pα
i (t)
∂xi
(t, ǫ)dt =
∂tα
Z
Ω0,t0
∂
(pα (t)xi (t, ǫ))dt −
∂tα i
Z
Ω0,t0
∂pα
i
(t)xi (t, ǫ)dt.
∂tα
Now we apply the divergence integral formula
Z
Z
∂
α
i
i
β
(p
(t)x
(t,
ǫ))dt
=
δαβ pα
i
i (t)x (t, ǫ)n (t)dσ,
α
∂Ω0,t0
Ω0,t0 ∂t
where (nβ (t)) is the unit normal vector to the boundary ∂Ω0,t0 . Substituting, we find
Z
∂pα
j
[H(t, x(t, ǫ), u(t, ǫ), p(t)) + α (t)xj (t, ǫ)]dt
I(ǫ) =
∂t
Ω0,t0
−
Z
i
β
δαβ pα
i (t)x (t, ǫ)n (t)dσ.
∂Ω0,t0
Differentiating with respect to ǫ, it follows
Z
∂pα
j
[Hxj (t, x(t, ǫ), u(t, ǫ), p(t)) + α (t)]xjǫ (t, ǫ)dt
I ′ (ǫ) =
∂t
Ω0,t0
+
Z
a
Hua (t, x(t, ǫ), u(t, ǫ), p(t))h (t)dt −
Ω0,t0
Z
i
β
δαβ pα
i (t)xǫ (t, ǫ)n (t)dσ.
∂Ω0,t0
Evaluating at ǫ = 0, we find
Z
∂pα
j
ˆ(t), p(t)) + α (t)]xjǫ (t, 0)dt
[Hxj (t, x(t), u
I ′ (0) =
∂t
Ω0,t0
+
Z
a
Hua (t, x(t), u
ˆ(t), p(t))h (t)dt −
Ω0,t0
Z
i
β
δαβ pα
i (t)xǫ (t, 0)n (t)dσ.
∂Ω0,t0
where x(t) is the m-sheet of the state variable corresponding to the optimal control
u
ˆ(t).
We need I ′ (0) = 0 for all h(t) = (ha (t)). On the other hand, the functions xiǫ (t, 0)
are the components of the solution of the Cauchy problem
∇t xiǫ (t, 0) = Xx (t, x(t, 0), u(t)) · xǫ (t, 0) + Xu (t, x(t, 0), u(t)) · h(t),
Simplified multitime maximum principle
107
t ∈ Ω0,t0 , xǫ (0, 0) = 0
and hence they depend on h(t). To overpass this difficulty, we define P as the set of
solutions of the boundary value problem
∂pα
∂H
j
(t) = − j (t, x(t), u
ˆ(t), p(t)), ∀t ∈ Ω0,t0 ,
α
∂t
∂x
β
δαβ pα
j (t)n (t)|∂Ω
0,t0
(4)
= 0, (orthogonality or tangency).
Therefore
Moreover
Hua (t, x(t), u
ˆ(t), p(t)) = 0, ∀t ∈ Ω0,t0 .
(5)
∂xj
∂H
(t) = α (t, x(t), u
ˆ(t), p(t)), ∀t ∈ Ω0,t0 , x(0) = x0 .
α
∂t
∂pj
(6)
Remarks. (i) The algebraic system (5) describes the critical points of the Hamiltonian with respect to the control variable. (ii) The P DEs (4) and (6) and the condition
(5) are Euler-Lagrange P DEs associated to the new Lagrangian.
Summarizing, we obtain a multitime maximum principle similar to the single-time
Pontryaguin maximum principle.
Theorem 1. (Simplified multitime maximum principle; necessary conditions ) Suppose that the problem of maximizing the functional (1) subject to the
P DE constraints (2) and to the conditions (3), with X, Xαi of class C 1 , has an interior solution u
ˆ(t) ∈ U which determines the m-sheet of state variable x(t). Then there
exists a C 1 costate p(t) = (pα
i (t)) defined over Ω0,t0 such that the relations (4), (5),
(6) hold.
Theorem 2. (Sufficient conditions) Consider the problem of maximizing the
functional (1) subject to the P DE constraints (2) and to the conditions (3), with
ˆ(t) ∈ U and the corresponding
X, Xαi of class C 1 . Suppose that an interior solution u
m-sheet of state variable x(t) satisfy the relations (4), (5), (6). If for the resulting
costate variable p(t) = (pα
i (t)) the control Hamiltonian H(t, x, u, p) is jointly concave
ˆ(t) and the corresponding x(t) achieve the unique
in (x, u) for all t ∈ Ω0,t0 , then u
global maximum of (1).
Proof. Let us have in mind that we must maximize the functional (1) subject to
the evolution system (2) and the conditions (3). We fix a pair (ˆ
x, u
ˆ), where u
ˆ is a
candidate optimal m-sheet of the controls and x
ˆ is a candidate optimal m-sheet of
the states. Calling Iˆ the values of the functional for (ˆ
x, u
ˆ), let us prove that
Z
ˆ − X)dt ≥ 0,
(X
Iˆ − I =
Ω0,t0
ˆ = H(ˆ
where the strict inequality holds under strict concavity. Denoting H
x, pˆ, u
ˆ) and
H = H(x, pˆ, u), we find
µ
¶
Z
i
∂x
ˆi
α ∂x
ˆ − pˆα
(H
Iˆ − I =
)
−
(H
−
p
ˆ
)
dt.
i
i
∂tα
∂tα
Ω0,t0
108
Constantin Udri¸ste
Integrating by parts, we obtain
µ
¶
Z
ˆα
ˆα
i
i
i ∂p
i ∂p
ˆ
ˆ
(H + x
ˆ α ) − (H + x α ) dt
I −I =
∂t
∂t
Ω0,t0
Z
i
β
+
(δαβ pˆα
ˆα
xi (t)nβ (t))dσ.
i (t)x (t)n (t) − δαβ p
i (t)ˆ
∂Ω0,t0
Taking into account that any admissible m-sheet has the same initial and terminal
conditions as the optimal m-sheet, we derive
µ
¶
Z
α
i
i
ˆ − H) + ∂ pˆi (ˆ
(H
x
−
x
)
dt.
Iˆ − I =
∂tα
Ω0,t0
The definition of concavity and the maximum principle imply
¶
µ
Z
∂ pˆα
i
i
i
ˆ
x − x ) dt
(H − H) + α (ˆ
∂t
Ω0,t0
Ã
!
Z
ˆ
ˆ
∂ pˆα
i
i
i ∂H
a
a ∂H
≥
(ˆ
x − x )( i + α ) + (ˆ
u − u ) a dt = 0.
∂x
∂t
∂u
Ω0,t0
This last equality is true since all ”ˆ” functions satisfy the multitime maximum principle. In this way, Iˆ − I ≥ 0.
Theorem 3. (Sufficient conditions) Consider the problem of maximizing the
functional (1) subject to the P DE constraints (2) and to the conditions (3), with
X, Xαi of class C 1 . Suppose that an interior solution u
ˆ(t) ∈ U and the corresponding
m-sheet of state variable x(t) satisfy the relations (4), (5), (6). Giving the resulting
costate variable p(t) = (pα
ˆ(t), p). If M (t, x, p) is
i (t)), we define M (t, x, p) = H(t, x, u
ˆ(t) and the corresponding x(t) achieve the unique
concave in x for all t ∈ Ω0,t0 , then u
global maximum of (1).
Remark. The Theorems 2 and 3 can be extended immediately to incave functionals.
Examples. 1) We consider the problem
Z
(x(t) + u1 (t)2 + u2 (t)2 )dt1 dt2
max I(u(·)) = −
u(·),x1
Ω0,1
subject to
∂x
(t) = uα (t), α = 1, 2, x(0, 0) = 0, x(1, 1) = x1 = free.
∂tα
This problem means to find an optimal control u = (u1 , u2 ) to bring the (P DE)
dynamical system from the origin x(0, 0) = 0 at two-time t1 = 0, t2 = 0 to a terminal
point x(1, 1) = x1 , which is unspecified, at two-time t1 = 1, t2 = 1, such as to
maximize the objective functional. Also the complete integrability condition imposes
∂u1
∂u2
= 1 . The control Hamiltonian is
∂t2
∂t
H(x(t), u(t), p(t)) = −(x(t) + u1 (t)2 + u2 (t)2 ) + p1 (t)u1 (t) + p2 (t)u2 (t).
109
Simplified multitime maximum principle
Since
∂H
∂2H
∂2H
= −2uα + pα ,
= −2 < 0,
= 0,
2
∂uα
∂uα
∂uα ∂uβ
the critical point pα = 2uα is a maximum point. Then the P DE
∂H
∂pα
=−
reduces
α
∂t
∂x
∂p1 ∂p2
+ 2 = 1. Also, since the point x(1, 1) = x1 is unspecified, the transversality
∂t1
∂t
conditions imply p1 (t)n1 (t) + p2 (t)n2 (t)|∂Ω0,1 = 0.
We continue by solving the boundary value problem
to
∂p2
∂p1
∂p2
∂p1
+ 2 = 1, 2 = 1 , p1 (t)n1 (t) + p2 (t)n2 (t)|∂Ω0,1 = 0.
1
∂t
∂t
∂t
∂t
Consequently the components of the optimal control u(t) = (u1 (t), u2 (t)) are harmonic
functions satisfying the boundary conditions u1 (0, t2 ) = u1 (1, t2 ) = 0, u2 (t1 , 0) =
u2 (t1 , 1) = 0. Also the dynamical system dx = u1 (t)dt1 + u2 (t)dt2 gives x(t) − x(0) =
Z
u1 (s)ds1 + u2 (s)ds2 .
Γ0,t
2) We consider the problem
1
1
max I(u(·)) = − x(1, 1)2 −
2
2
u(·),x1
Z
(u1 (t)2 + u2 (t)2 )dt1 dt2
Ω0,1
subject to
∂x
(t) = −uα (t), α = 1, 2, x(0, 0) = 1.
∂tα
This problem means to find an optimal control u = (u1 , u2 ) to bring the (P DE)
dynamical system from the point x(0, 0) = 1 at two-time t1 = 0, t2 = 0 to a terminal
point x(1, 1) = x1 , at two-time t1 = 1, t2 = 1, such as to maximize the objective
∂u2
∂u1
functional. Also the complete integrability condition imposes 2 = 1 . The control
∂t
∂t
Hamiltonian is
1
H(x(t), u(t), p(t)) = − (u1 (t)2 + u2 (t)2 ) − pα (t)uα (t).
2
Since
∂2H
∂2H
∂H
= −uα − pα ,
= −1 < 0,
= 0,
2
∂uα
∂uα
∂uα ∂uβ
the critical point pα = −uα is a maximum point. Then the P DE
reduces to
∂p2
∂p1
+
= 0. The transversality condition implies
∂t1
∂t2
∂pα
∂H
=−
=0
α
∂t
∂x
p1 (t)n1 (t) + p2 (t)n2 (t)|∂Ω0,1 = 0.
We continue by solving the Dirichlet problem
∂p1
∂p2
∂p1
∂p2
+ 2 = 0, 2 = 1 , p1 (t)n1 (t) + p2 (t)n2 (t)|∂Ω0,1 = 0.
1
∂t
∂t
∂t
∂t
110
Constantin Udri¸ste
Consequently the components of the optimal control u(t) = (u1 (t), u2 (t)) are harmonic
functions satisfying suitable boundary conditions. Also the dynamical system dx =
−u1 (t)dt1 − u2 (t)dt2 gives the corresponding evolution
Z
u1 (s)ds1 + u2 (s)ds2 .
x(t) − x(0) = −
Γ0,t
3) Transport P DEs for Air Traffic Flow (see also [4]). The flow of aircraft
can be analyzed and controlled using an Eulerian viewpoint of the airspace. In our
formulation we use two parameters of evolution: s=position, t=time. The variable of
the state is the density of aircraft ρ(s, t), which represents the number of aircraft per
unit length of jetway. The control variable is the speed v(s, t) which the air traffic
controller can prescribe to the aircraft located at position s and time t.
Given a speed field v(s, t), the density of aircraft ρ(s, t) satisfies the continuity
∂(ρv)
∂ρ
(s, t) +
(s, t) = 0. We would like to determine the speed field which
P DE
∂t
∂s
maximizes the number of aircraft landing at the destination airport under the constraint that the density does not exceed the safety density ρmax . Mathematically,
Z
ρ(s, t)v(s, t)dsdt
max I(v(·)) =
v(·)
Ω0,A
subject to
∂(ρv)
∂ρ
(s, t) +
(s, t) = 0, ρ ≤ ρmax , vmin ≤ v ≤ vmax ,
∂t
∂s
where A = (L, T ), L = value of s for final destination, T = final time, I(v(·))= total
number of aircraft landing, vmin , vmax = bounds of authorized aircraft speed. The
new Lagrangian
µ
¶
∂ρ
∂(ρv)
L = ρ(s, t)v(s, t) + p(s, t)
(s, t) +
(s, t) + µ1 (s, t)(ρmax − ρ(s, t))
∂t
∂s
+µ2 (s, t)(v(s, t) − vmin ) + µ3 (s, t)(vmax − v(s, t))
produces the Hamiltonian
¡
¢
H = −ρ(s, t) − µ2 (s, t) + µ3 (s, t) v(s, t) − µ1 (s, t)(ρmax − ρ(s, t))
+µ2 (s, t)vmin − µ3 (s, t)vmax ,
of degree one with respect to the control v. The sign of the switching function σ =
−ρ(s, t) − µ2 (s, t) + µ3 (s, t) decides an optimal control. The adjoint P DEs (obtained
from I ′ (0) = 0) are
∂p
∂p
∂p
+v
= v − µ1 , ρ
= ρ + µ2 − µ3 .
∂t
∂s
∂s
Open problem: Study optimal control problems subject to
P DE : Xβα (x(t), u(t))
∂f β
(x(t), u(t)) = 0.
∂tα
111
Simplified multitime maximum principle
Bibliographical note. For strong contributions to optimal control problems,
which have influenced our point of view, see [1]-[4], [6]-[11].
3
m-Flow type constrained optimization problem
with curvilinear integral cost functional
The cost functionals of mechanical work type are very important for applications
(see our papers [12]-[22]). This is the reason to analyze a multitime optimal control
problem based on a path independent curvilinear integral as cost functional and on
P DE constraints of m-flow type :
Z
max J(u(·)) =
Xα0 (t, x(t), u(t))dtα
(7)
u(·),xt0
Γ0,t0
subject to
i
∂x
(t) = Xαi (t, x(t), u(t)), i = 1, ..., n; α = 1, ..., m,
∂tα
(8)
u(t) ∈ U, t ∈ Ω0,t0 ; x(0) = x0 , x(t0 ) = xt0 .
(9)
α
m
R+
Ingredients: t = (t ) ∈
is the multitime (multi-parameter of evolution);
Γ0,t0 is an arbitrary C 1 curve joining the diagonal opposite points 0 = (0, ..., 0)
n
i
2
and t0 = (t10 , ..., tm
0 ) in Ω0,t0 ; x : Ω0,t0 → R , x(t) = (x (t)) is a C state vector;
k
a
1
u : Ω0,t0 → U ⊂ R , u(t) = (u (t)), a = 1, ..., k is a C control vector; the running cost Xα0 (t, x(t), u(t))dtα is a nonautonomous closed (completely integrable) Lagrangian 1-form , i.e., it satisfies Dβ Xα0 = Dα Xβ0 (Dα is the total derivative operator)
or
Ã
!
0
∂Xβ0
∂Xβ0
∂Xα0 γ ∂Xβ0 γ ∂ua
∂Xα0
i
i ∂Xα
δ
−
δ
=
X
−
X
+
−
;
α
α
β
∂ua β
∂ua
∂tγ
∂xi
∂xi
∂tα
∂tβ
the vector fields Xα (t, x(t), u(t)) = (Xαi (t, x(t), u(t))) are of class C 1 and satisfy the
complete integrability conditions (m-flow type problem) , i.e., Dβ Xα = Dα Xβ or
¶
µ
∂Xβ
∂Xα
∂Xα γ ∂Xβ γ ∂ua
δ
−
δ
= [Xα , Xβ ] +
−
,
α
β
a
a
γ
α
∂u
∂u
∂t
∂t
∂tβ
where [Xα , Xβ ] means the bracket of vector fields. This hypothesis selects the set of
all admissible controls (satisfying the complete integrability conditions)
¯
©
ª
m
U = u : R+
→ U ¯ Dβ Xα0 = Dα Xβ0 , Dβ Xα = Dα Xβ
and the set of admissible states.
We introduce a costate variable or Lagrange multiplier function p = (pi ) such that
the new Lagrange 1-form
Lα (t, x(t), u(t), p(t)) = Xα0 (t, x(t), u(t)) + pi (t)[Xαi (t, x(t), u(t)) −
∂xi
(t)]
∂tα
112
Constantin Udri¸ste
be closed. The P DE constrained optimization problem (7)-(9) is replaced by another
optimization problem
Z
max
Lα (t, x(t), u(t), p(t))dtα
u(·),xt0
Γ0,t0
subject to
u(t) ∈ U, p(t) ∈ P, t ∈ Ω0,t0 , x(0) = x0 , x(t0 ) = xt0 ,
where the set P will be defined later. If we use the control Hamiltonian 1-form
Hα (t, x(t), u(t), p(t)) = Xα0 (t, x(t), u(t)) + pi (t)Xαi (t, x(t), u(t)),
Hα = Lα + pi
∂xi
(nonstandard duality),
∂tα
we can rewrite
max
u(·),xt0
Z
[Hα (t, x(t), u(t), p(t)) − pi (t)
Γ0,t0
∂xi
(t)]dtα
∂tα
subject to
u(t) ∈ U, p(t) ∈ P, t ∈ Ω0,t0 , x(0) = x0 , x(t0 ) = xt0 .
ˆ(t) ∈
Suppose that there exists a continuous control u
ˆ(t) defined over Ω0,t0 with u
Int U which is optimum in the previous problem. Now consider a variation u(t, ǫ) =
u
ˆ(t)+ǫh(t), where h is an arbitrary continuous vector function. Since u
ˆ(t) ∈ Int U and
a continuous function over a compact set Ω0,t0 is bounded, there exists ǫh > 0 such
that u(t, ǫ) = u
ˆ(t) + ǫh(t) ∈ Int U, ∀|ǫ| < ǫh . This ǫ is used in the next variational
arguments.
Let us consider an arbitrary vector function h(t) and define x(t, ǫ) as the m-sheet
of the state variable corresponding to the control variable u(t, ǫ), i.e.,
∂xi
(t, ǫ) = Xαi (t, x(t, ǫ), u(t, ǫ)), ∀t ∈ Ω0,t0 , x(0, ǫ) = x0 .
∂tα
For |ǫ| < ǫh , we define the function
Z
J(ǫ) =
Xα0 (t, x(t, ǫ), u(t, ǫ))dtα .
Γ0,t0
Since the control function u(t, ǫ) is admissible, it follows that the evolution function
x(t, ǫ) is admissible. On the other hand, the control u
ˆ(t) is supposed to be optimal.
Therefore J(ǫ) ≤ J(0), ∀|ǫ| < ǫh .
For any continuous function p = (pi ) : Ω0,t0 → Rn , we have
Z
Γ0,t0
pi (t)[Xαi (t, x(t, ǫ), u(t, ǫ)) −
∂xi
(t, ǫ)]dtα = 0.
∂tα
The variations determine the closed Lagrange 1-form
113
Simplified multitime maximum principle
Lα (t, x(t, ǫ), u(t, ǫ), p(t)) = Xα0 (t, x(t, ǫ), u(t, ǫ))+pi (t)[Xαi (t, x(t, ǫ), u(t, ǫ))−
and the function
J(ǫ) =
Z
∂xi
(t, ǫ)]
∂tα
Lα (t, x(t, ǫ), u(t, ǫ), p(t))dtα .
Γ0,t0
Suppose that the costate p is of class C 1 . Also we introduce the control Hamiltonian 1-form
Hα (t, x(t, ǫ), u(t, ǫ), p(t)) = Xα0 (t, x(t, ǫ), u(t, ǫ)) + pi (t)Xαi (t, x(t, ǫ), u(t, ǫ)).
Then we rewrite
J(ǫ) =
Z
[Hα (t, x(t, ǫ), u(t, ǫ), p(t)) − pi (t)
Γ0,t0
To evaluate the curvilinear integral
Z
Γ0,t0
pi (t)
∂xi
(t, ǫ)]dtα .
∂tα
∂xi
(t, ǫ)dtα ,
∂tα
we integrate by parts, via
∂
∂pi i
∂xi
i
(p
x
)
=
x
+
p
,
i
i
∂tα
∂tα
∂tα
obtaining
Z
Γ0,t0
∂xi
pi (t) α (t, ǫ)dtα = (pi (t)xi (t, ǫ))|t00 −
∂t
Z
Γ0,t0
∂pi
(t)xi (t, ǫ)dtα .
∂tα
Substituting, we get the function
Z
∂pj
[Hα (t, x(t, ǫ), u(t, ǫ), p(t))+ α (t)xj (t, ǫ)]dtα −pi (t0 )xi (t0 , ǫ)+pi (0)xi (0, ǫ).
J(ǫ) =
∂t
Γ0,t0
It follows
J ′ (ǫ) =
+
Z
Γ0,t0
Z
[Hαxj (t, x(t, ǫ), u(t, ǫ), p(t)) +
Γ0,t0
∂pj
(t)]xjǫ (t, ǫ)dtα
∂tα
Hαua (t, x(t, ǫ), u(t, ǫ), p(t))ha (t)dtα − pi (t0 )xiǫ (t0 , ǫ) + pi (0)xiǫ (0, ǫ).
Evaluating at ǫ = 0, we find
Z
∂pj
J ′ (0) =
ˆ(t), p(t)) + α (t)]xjǫ (t, 0)dtα
[Hαxj (t, x(t), u
∂t
Γ0,t0
+
Z
Γ0,t0
Hαua (t, x(t), u
ˆ(t), p(t))ha (t)dtα − pi (t0 )xiǫ (t0 , 0),
114
Constantin Udri¸ste
where x(t) is the m-sheet of the state variable corresponding to the optimal control
u
ˆ(t). We need J ′ (0) = 0 for all h(t) = (ha (t)). This is possible if we define P as the
set of solutions of the terminal value problem
∂pj
∂Hα
(t) = − j (t, x(t), u
ˆ(t), p(t)), ∀t ∈ Ω0,t0 ; pj (t0 ) = 0.
α
∂t
∂x
(10)
ˆ(t), p(t)) = 0, ∀t ∈ Ω0,t0 .
Hαua (t, x(t), u
(11)
∂xj
∂Hα
(t) =
(t, x(t), u
ˆ(t), p(t)), ∀t ∈ Ω0,t0 ; x(0) = x0 .
∂tα
∂pj
(12)
Therefore
Moreover
Remarks. (i) The algebraic system (11) describes the common critical points of
the functions Hα with respect to the control variable u. (ii) The P DEs (10) and (12)
and the relation (11) are Euler-Lagrange P DEs associated to the new Lagrangian
1-form.
Summarizing, we obtain a new variant of multitime maximum principle.
Theorem 4. (Simplified multitime maximum principle; necessary conditions) Suppose that the problem of maximizing the functional (7) subject to the P DE
constraints (8) and to the conditions (9), with Xα0 , Xαi of class C 1 , has an interior
solution u
ˆ(t) ∈ U which determines the m-sheet of state variable x(t). Then there
exists a C 1 costate p(t) = (pi (t)) defined over Ω0,t0 such that the relations (10), (11),
(12) hold.
Theorem 5. (Sufficient conditions) Consider the problem of maximizing the
functional (7) subject to the P DE constraints (8) and to the conditions (9), with
ˆ(t) ∈ U and the corresponding
Xα0 , Xαi of class C 1 . Suppose that an interior solution u
m-sheet of state variable x(t) satisfy the relations (10), (11), (12). If, for the resulting
costate variable p(t) = (pi (t)), the control Hamiltonian 1-form Hα (t, x, u, p) is jointly
ˆ(t) and the corresponding x(t) achieve the
concave in (x, u) for all t ∈ Ω0,t0 , then u
unique global maximum of (7).
Theorem 6. (Sufficient conditions) Consider the problem of maximizing the
functional (7) subject to the P DE constraints (8) and to the conditions (9), with
ˆ(t) ∈ U and the corresponding
Xα0 , Xαi of class C 1 . Suppose that an interior solution u
m-sheet of state variable x(t) satisfy the relations (10), (11), (12). Giving the resulting
costate variable p(t) = (pi (t)), we define the 1-form Mα (t, x, p) = Hα (t, x, u
ˆ(t), p). If
the 1-form Mα (t, x, p) is concave in x for all t ∈ Ω0,t0 , then u
ˆ(t) and the corresponding
x(t) achieve the unique global maximum of (7).
Remark. The Theorems 5 and 6 can be extended immediately to incave functionals.
Example. Let t = (t1 , t2 ) ∈ Ω0,1 , where 0 = (0, 0), 1 = (1, 1) are diagonal opposite
points in Ω0,1 . Denote by Γ0,1 an arbitrary C 1 curve joining the points 0 and 1. We
consider the problem
115
Simplified multitime maximum principle
max J(u(·)) = −
u(·),x1
Z
(x(t) + uβ (t)2 )dtβ
Γ0,1
subject to
∂x
(t) = uα (t), α = 1, 2, x(0, 0) = 0, x(1, 1) = x1 = free.
∂tα
This problem means to find an optimal control u = (u1 , u2 ) to bring the (P DE)
dynamical system from the origin x(0, 0) = 0 at two-time t1 = 0, t2 = 0 to a terminal
point x(1, 1) = x1 , which is unspecified, at two-time t1 = 1, t2 = 1, such as to
maximize the objective functional. Also the complete integrability conditions impose
∂u2
∂x
∂u1 ∂u1
∂u2
∂x
+ 2u2 1 = 2 + 2u1 2 ,
= 1.
∂t1
∂t
∂t
∂t
∂t2
∂t
The control Hamiltonian 1-form is
Hβ (x(t), u(t), p(t)) = −(x(t) + uβ (t)2 ) + p(t)uβ (t).
Since
∂ 2 Hβ
∂Hβ
= −2 < 0,
= −2uβ + p,
∂uβ
∂u2β
p
∂p
∂Hα
the critical point u1 = u2 =
is a maximum point. The P DEs
= −
2
∂tα
∂x
∂p
reduces to α = 1. Also, since the point x(1, 1) = x1 is unspecified, the transversality
∂t
condition implies p(1) = 0. It follows the costate p(t) = t1 + t2 − 2, the optimal control
1
(t1 )2 + (t2 )2
u
ˆ1 (t) = u
ˆ2 (t) = (t1 + t2 − 2) and the corresponding evolution x(t) =
+
2
4
t1 t2
− (t1 + t2 ).
2
4
Equivalence between multiple and curvilinear integral functionals
A multitime evolution system can be used as a constraint in a problem of extremizing
a multitime cost functional. On the other hand, the multitime cost functionals can be
introduced at least in two ways:
- either using a path independent curvilinear integral,
Z
P (u(·)) =
Xβ0 (x(t), u(t))dtβ + g(x(t0 )),
Γ0,t0
where Γ0,t0 is an arbitrary C 1 curve joining the points 0 and t0 , the running cost
ω = Xβ0 (x(t), u(t))dtβ is an autonomous closed (completely integrable) Lagrangian
1-form, and g is the terminal cost;
- or using a multiple integral,
Z
X(x(t), u(t))dt + g(x(t0 )),
Q(u(·)) =
Ω0,t0
116
Constantin Udri¸ste
where the running cost X(x(t), u(t)) is an autonomous continuous Lagrangian, and g
is the terminal cost.
Let us show that the functional P is equivalent to the functional Q. This means
that in a multitime optimal control problem we can choose the appropriate functional
based on geometrical-physical meaning or other criteria.
Theorem 7 [23]. The multiple integral
Z
I(t0 ) =
X(x(t), u(t))dt,
Ω0,t0
with X as continuous function, is equivalent to the curvilinear integral
Z
Xβ0 (x(t), u(t))dtβ ,
J(t0 ) =
Γ0,t0
where ω = Xβ0 (x(t), u(t))dtβ is a closed (completely integrable) Lagrangian 1-form
and the functions Xβ0 have partial derivatives of the form
∂
∂
∂ m−1
,
(α
<
β),
...,
,
∂tα ∂tα ∂tβ
∂t1 ...∂tˆα ...∂tm
where the symbol ”ˆ” posed over ∂tα designates that ∂tα is omitted.
5
Multitime maximum principle approach of variational calculus
It is well known that the single-time Pontryaguin’s maximum principle is a generalization of the Lagrange problem in the single-time variational calculus and that these
problems are equivalent when the control domain is open [1], [7], [10]. Does this property survive for simplified multitime maximum principle? The aim of this Section is
to formulate an answer to this question.
In fact we show that the simplified multitime maximum principle motivates the
multitime Euler-Lagrange or Hamilton PDEs. For that, suppose that the evolution
system is reduced to a completely integrable system
∂xi
m
,
(t) = uiα (t), x(0) = x0 , t ∈ Ω0,t0 ⊂ R+
∂tα
and the functional is a path independent curvilinear integral
Z
Xβ0 (x(t), u(t))dtβ , u = (uiα ),
J(u(·)) =
(P DE)
(J)
Γ0,t0
where Γ0,t0 is an arbitrary piecewise C 1 curve joining the points 0 and t0 , the running
cost ω = Xβ0 (x(t), u(t))dtβ is a closed (completely integrable) Lagrangian 1-form.
The associated basic control problem leads necessarily to the multitime maximum
principle. Therefore, to solve it we need the control Hamiltonian 1-form
Hβ (x, p0 , p, u) = Xβ0 (x, u) + pi uiβ
117
Simplified multitime maximum principle
and the adjoint PDEs
∂Xβ0
∂pi
(t)
=
−
(x(t), u(t)).
(ADJ)
∂tβ
∂xi
Suppose the simplified multitime maximum principle is applicable (see the relation
(10))
∂Xβ0
∂Xβ0
∂Hβ
γ
γ
=
+
p
δ
=
0,
p
δ
=
−
, uiγ = xiγ .
(13)
i
i
β
β
∂uiγ
∂uiγ
∂uiγ
Suppose the functions Xβ0 are dependent on x (a strong condition!). Then (ADJ)
shows that
Z
∂Xβ0
pi (t) = pi (0) −
(x(s), u(s))dsβ ,
(14)
i
Γ0,t ∂x
where Γ0,t is an arbitrary piecewise C 1 curve joining the points 0, t ∈ Ω0,t0 .
5.1
Multitime Euler-Lagrange PDEs
From the relations (13) and (14), it follows
∂Xβ0
− i (x(t), u(t)) = δβγ pi (0) − δβγ
∂xγ
Z
Γ0,t
∂Xλ0
(x(s), u(s))dsλ .
∂xi
Suppose that Xβ0 are functions of class C 2 . Applying the divergence operator Dγ =
∂Xβ0
∂Xβ0
∂
we
find
the
multitime
Euler-Lagrange
P
DEs
−
D
= 0.
γ
∂tγ
∂xi
∂xiγ
5.2
Conversion to multitime Hamilton PDEs (canonical variables)
Let u(·) be an optimal control, x(·) the optimal evolution m-sheet, and p(·) be the
solution of (ADJ) which corresponds to u(·) and x(·). Suppose that the critical point
∂xi
condition admits a unique solution uiγ (t) = uiγ (x(t), p(t)) =
(t). Then, using a
∂tγ
path independent curvilinear integral, we can write
Z
i
i
uiγ (x(s), p(s))dsγ .
x (t) = x (0) +
Γ0,t
The control Hamiltonian 1-form Hβ = Xβ0 + pj ujβ must satisfy
relation, pi δβγ +
∂Hβ
= 0. This last
∂uiγ
∂Xβ0
= 0, defines the costate p as a moment. On the other hand
∂uiγ
∂ujβ
∂Xβ0 ∂ujγ
∂Hβ
∂xi
∂Hβ
i
=
+
u
+
p
= uiβ or β (t) =
(x(t), p(t), u(t)).
j
β
j
∂pi
∂p
∂p
∂t
∂pi
∂uγ
i
i
Now, the relation
118
Constantin Udri¸ste
∂Hβ
−
=−
∂xi
Ã
∂Xβ0
∂Xβ0 ∂ujγ
+
∂xi
∂ujγ ∂xi
!
− pj
∂ujβ
∂xi
and (ADJ) show
∂Hβ
∂pi
(t) = −
(x(t), p(t), u(t)).
∂tβ
∂xi
In this way we find the canonical variables x, p and the multitime Hamilton P DEs
∂xi
∂pi
∂Hβ
∂Hβ
(x(t), p(t)),
(t) =
(t) = −
(x(t), p(t)).
β
β
∂t
∂pi
∂t
∂xi
Remark. To make a computer aided study of P DE-constrained optimization
problems we can perform symbolic computations via Maple (see also [5]).
Acknowledgements: Partially supported by Grant CNCSIS 86/ 2008, by 15-th
Italian-Romanian Executive Programme of S&T Co-operation for 2006-2008, University Politehnica of Bucharest, and by Academy of Romanian Scientists, Splaiul
Independentei 54, 050094 Bucharest, Romania. We thank Professors Franco Giannessi, Valeriu Prepelit¸˘a, Marius R˘adulescu, Ionel T
¸ evy and Assistant Cristian Ghiu
for their suggestions improving the paper.
References
[1] D. Acemoglu, Introduction to Economic Growth, MIT Department of Economics,
2006.
[2] V. Barbu, I. Lasiecka, D. Tiba, C. Varsan, Analysis and Optimization of Differential Systems, IFIP TC7/WG7.2 International Working Conference on Analysis
and Optimization of Differential Systems, September 10-14, 2002, Constanta,
Romania, Kluwer Academic Publishers, 2003.
[3] G. Barles, A. Tourin, Commutation properties of semigroups for first-order
Hamilton-Jacobi equations and application to multi-time equations, Indiana University Mathematics Journal, 50, 4 (2001), 1523-1544.
[4] A. M. Bayen, R. L.Raffard, C. J. Tomlin, Adjoint-based constrained control of
Eulerian transportation networks: Application to air traffic control, Proceedings
of the American Control Conference, Boston, June 2004.
[5] M. T. Calapso, C. Udri¸ste, Isothermic surfaces as solutions of Calapso PDE,
Balkan Journal of Geometry and Its Applications, 13, 1(2008), 20-26.
[6] F. Cardin, C. Viterbo, Commuting Hamiltonians and Hamilton-Jacobi multi-time
equations, Duke Math. J., 144, 2 (2008), 235-284.
[7] L. C. Evans, An Introduction to Mathematical Optimal Control Theory, Lecture
Notes, University of California, Department of Mathematics, Berkeley, 2008.
[8] M. Motta, F. Rampazo, Nonsmooth multi-time Hamilton-Jacobi systems, Indiana
University Mathematics Journal, 55, 5 (2006), 1573-1614.
[9] S. Pickenhain, M. Wagner, Piecewise continuous controls in DieudonneRashevsky type problems, Journal of Optimization Theory and Applications
(JOTA), 127 (2005), 145-163.
Simplified multitime maximum principle
119
[10] L. Pontriaguine, V. Boltianski, R. Gamkrelidze, E. Michtchenko, Th´eorie
Math´ematique des Processus Optimaux, Edition MIR, Moscou, 1974.
[11] D. F. M. Torres, A. Yu. Plakhov, Optimal Control of Newton type problems of
minimal resistance, Rend. Sem. Mat. Univ. Pol. Torino, 64, 1 (2006), 79-95.
[12] C. Udri¸ste, Multi-time maximum principle, Short Communication at International Congress of Mathematicians, Madrid, August 22-30, 2006; Plenary Lecture
at 6-th WSEAS International Conference on Circuits, Systems, Electronics, Control&Signal Processing (CSECS’07) and 12-th WSEAS International Conference
on Applied Mathematics, Cairo, Egypt, December 29-31, 2007.
[13] C. Udri¸ste, I. T
¸ evy, Multi-time Euler-Lagrange-Hamilton theory, WSEAS Transactions on Mathematics, 6, 6 (2007), 701-709.
[14] C. Udri¸ste, I. T
¸ evy, Multi-time Euler-Lagrange dynamics, Proceedings of the 7th
WSEAS International Conference on Systems Theory and Scientific Computation
(ISTASC’07), Vouliagmeni Beach, Athens, Greece, August 24-26, 2007, 66-71.
[15] C. Udri¸ste, Controllability and observability of multitime linear PDE systems,
Proceedings of The Sixth Congress of Romanian Mathematicians, Bucharest,
Romania, June 28 - July 4, 2007, vol. 1, 313-319.
[16] C. Udri¸ste, Multi-time stochastic control theory, Selected Topics on Circuits, Systems, Electronics, Control&Signal Processing, Proceedings of the 6-th WSEAS
International Conference on Circuits, Systems, Electronics, Control&Signal Processing (CSECS’07), Cairo, Egypt, December 29-31, 2007, 171-176.
[17] C. Udri¸ste, Finsler optimal control and Geometric Dynamics, Mathematics and
Computers In Science and Engineering, Proceedings of the American Conference
on Applied Mathematics, Cambridge, Massachusetts, 2008, 33-38.
[18] C. Udri¸ste, Lagrangians constructed from Hamiltonian systems, Mathematics
a Computers in Business and Economics, Proceedings of the 9th WSEAS
International Conference on Mathematics a Computers in Business and
Economics(MCBE-08), Bucharest, Romania, June 24-26, 2008, 30-33.
[19] C. Udri¸ste, Multitime controllability, observability and bang-bang principle, Journal of Optimization Theory and Applications, 139, 1(2008), 141-157; DOI
10.1007/s10957-008-9430-2.
[20] C. Udri¸ste, L. Matei, Lagrange-Hamilton theories (in Romanian), Monographs
and Textbooks 8, Geometry Balkan Press, Bucharest, 2008.
[21] C. Udri¸ste, T. Oprea, H-convex Riemannian manifolds, Balkan Journal of Geometry and Its Applications, 13, 2(2008), 112-119.
[22] Ariana Pitea, C. Udri¸ste, S¸t. Mititelu, P DI&P DE-constrained optimization
problems with curvilinear functional quotients as objective vectors, Balkan Journal of Geometry and Its Applications, 14, 2 (2009), 75-88.
[23] C. Udriste, O Dogaru, I. Tevy, Null Lagrangian forms and Euler-Lagrange PDEs,
J. Adv. Math. Studies, 1, 1-2 (2008), 143 - 156.
Author’s address:
Constantin Udriste
University Politehnica of Bucharest, Faculty of Applied Sciences,
Department of Mathematics-Informatics, Splaiul Independentei 313,
Bucharest 060042, Romania.
E-mail: [email protected], [email protected]