Extrema with Constraints on Points and/or Velocities

C. Udrişte, O. Dogaru, M. Ferrara and I. Ţevy

Dedicated to the Memory of Grigorios TSAGAS (1935-2003), President of Balkan Society of Geometers (1997-2003)

Abstract

The main results of this paper refer to three ideas:

- to replace the constraints in an optimum problems by a selector of curves;

- to reformulate and study extremum problems with point constraints and/or velocity constraints;

- to extend the saddle point theory and the Kuhn-Tucker theory to extrema with nonholonomic constraints.

Mathematics Subject Classification 2000: 90C30, 49K24 Key words: nonlinear programming, point constraints, velocity constraints, selector of curves

1 Extremum constrained by a selector of curves

Let D be an open set of R^{n}. For each point x ∈ D, we denote by Γ_{x} the set of all C^{1} parametrized curves α : I → D which passs through the point x at a given moment t ∈ I. 1.1. Definition. Any function

Γ̂ : D → ⋃_{x∈D} Γ_{x}, Γ̂(x) ⊂ Γ_{x},

is called selector of curves on D. The elements of Γ̂(x) are called admissible curves at the point x. 1.2. Definition. Let f : D → R be a function and Γ̂ be a selector of curves on D. If

f(α(t)) ≥ f(x_{0}), ∀t ∈ [t_{0}, t_{0} + ε), ∀α ∈ Γ̂(x_{0}), x_{0} = α(t_{0}),

then x_{0} ∈ D is called a minimum point of f constrained by the selector Γ̂. Examples. Suppose Γ_{x} is either the set of regular curves at x or the set of C^{2} curves having x either as a regular point or as singular point of order 2. Define the selector Γ̂ by Γ̂(x) = Γ_{x}.

∗ Balkan Journal of Geometry and Its Applications, Vol.8, No.1, 2003, pp. 115-123.

\begin{matrix}c \\ ◯ \end{matrix} Balkan Society of Geometers, Geometry Balkan Press 2003.

1) In ([11]) was proved the following 1.3. Theorem. 1) x_{0} ∈ D is a minimum point of f iff x_{0} ∈ D is a minimum point of f constrained by the selector Γ̂.

2) Let g^{a} : D → R, a = \overline{1, p}, p < n be C^{1} functions. These functions can be used to create equality constraints (equations) or inequality constraints (inequations) on points.

a) The equations g^{a}(x) = 0 introduce the partial selectors

Γ̂^{a}(x) = \{ \begin{matrix}\{α ∈ Γ_{x}|g^{a}(Im α) = 0\} & for x ∈ D with g^{a}(x) = 0 \\ ∅ & for x ∈ D with g^{a}(x) ≠ 0. \end{matrix}

These produce the general selector

Γ̂(x) = ⋂_{a=1}^{p} Γ̂^{a}(x).

Now we can formulate 1.4. Theorem. ([16]) Suppose g^{a} are C^{1} functions on D satisfying rank [\frac{∂g^{a}}{∂x^{j}} (x)] =

p, at any point x of D, and the subset defined by g^{a}(x) = 0, a = \overline{1, p} is nonvoid. Then x_{0} ∈ D is a minimum point of a function f : D → R constrained by the selector Γ̂ iff g^{a}(x_{0}) = 0, a = \overline{1, p} and x_{0} is a minimum point for f constrained by g^{a}(x) = 0, a = \overline{1, p}.

b) The inequations g^{a}(x) ≥ 0 give the partial selectors

Γ̂^{a}(x) = \{ \begin{matrix}\{α ∈ Γ_{x}|g^{a}(α(t)) ≥ 0, t ≥ t_{0})\} & for x ∈ D with g^{a}(x) = 0 \\ Γ_{x} & for x ∈ D with g^{a}(x) > 0 \\ ∅ & for x ∈ D with g^{a}(x) < 0. \end{matrix}

The general selector is

Γ̂(x) = ⋂_{a=1}^{p} Γ̂^{a}(x).

1.5. Theorem. ([16]) Suppose g^{a} are C^{1} functions on D satisfying

rank [\frac{∂g^{a}}{∂x^{j}} (x)] = p,

at any point x of D, and the subset defined by g^{a}(x) ≥ 0 is nonvoid. Then x_{0} is a minimum point of a continuous function f : D → R constrained by the selector Γ iff x_{0} is a minimum point for f constrained by g^{a}(x) ≥ 0, a = \overline{1, p}.

3) Let ω^{a}(x) = ∑_{j=1}^{n} ω_{j}^{a}(x)dx^{j}, a = \overline{1, p}, p < n be C^{1} Pfaff forms. These Pfaff forms

can be used to create equality constraints (Pfaff equations) or inequality constraints (for example, Pfaff inequalities) on velocities.

a) The Pfaff equations generate the partial selectors

Γ̂^{a}(x) = {α ∈ Γ_{x}|α is an integral curve of the Pfaff equation ω^{a}(x) = 0}, which produce the general selector (associated to the Pfaff system)

Γ̂(x) = ⋂_{a=1}^{p} Γ^{a}(x).

The previous selector is connected to extrema with nonholonomic constraints (see [2], [10], [11]), by the following 1.6. Theorem. x_{0} ∈ D is a minimum point of the function f : D → R constrained by the selector Γ̂ iff x_{0} ∈ D is a minimum point for f constrained by the Pfaff system ω^{a}(x) = 0, a = \overline{1, p}.

b) The primitive of each Pfaff form ω^{a}(x) defines the partial selectors

Γ̂^{a}(x_{0}) = {α ∈ Γ_{x_{0}}| ∫_{t_{0}}^{t}(ω^{a}(α(u)), α′(u))du ≥ 0, ∀t ∈ [t_{0}, t_{0} + ε)} ,

where α(t_{0}) = x_{0}. From this point of view, the selector associated to all Pfaff forms is

Γ̂(x_{0}) = ⋂_{a=1}^{p} Γ̂^{a}(x_{0}).

1.7. Definition. x_{0} ∈ D is a minimum point of the function f : D → R constrained by ω^{a} ≥ 0, a = \overline{1, p} if x_{0} is a minimum point of f constrained by the selector Γ̂.

This type of extremum was studied in [11]. Remark. Generally, there are two types of constraints. One given by constraints on points and other as constraints on velocities. A equation constraint on points induces an equation constraint on velocities; this last constrained does not contribute for finding critical points, but contribute in establishing the nature of critical points. The converse is not true. In the case of extrema of type 2), the point constraints select certain semicurves. In the case of extrema of type 3), the point constraints dissapear, any point of D being susceptible to be an extremum point. In the sequel we introduce a type of extremum where the point constraints and velocity constraints are not correlate.

2 Extremum with constraints on points and / or velocities

Let ω(x) = ∑_{j=1}^{n} ω_{j}(x)dx^{j} be a C^{1} Pfaff form with rank [ω_{j}(x)] = 1. Let M = S ∪ bS

be a subset of D, where S and bS are disjoint.

The pair (ω, M) determines the following selector of curves:

Γ̂(x_{0}) = \{ \begin{matrix}Γ_{x_{0}} if x_{0} ∈ S \\ \{α ∈ Γ_{x_{0}}| ∫_{t_{0}}^{t}(ω(α(u)), α′(u))du ≥ 0, ∀t ∈ [t_{0}, t_{0} + ε)\} if x_{0} ∈ bS \\ ∅ if x_{0} ∈ D \ M. \end{matrix}

Similarly, for each a = \overline{1, p}, we consider the pair (ω^{a}, M_{a}), where ω^{a}(x) =

∑_{j=1}^{n} ω_{j}^{a}(x)dx^{j} are C^{1} Pfaff forms, and M_{a} = S_{a} ∪ bS_{a}. The pair (ω^{a}, M_{a}) determines

the selector Γ̂^{a}. If we denote ω = \{ω^{a}|a = \overline{1, p}\}, S = ⋂_{i=1}^{p} S_{a}, bS = ⋃_{a=1}^{p} bS_{a}, M = S∪bS,

then the pair (ω, M) induces the selector

Γ̂(x_{0}) = ⋂_{a=1}^{p} Γ̂^{a}(x_{0}), ∀x_{0} ∈ D.

Let f : D → R be a function. If x_{0} ∈ M is a minimum point of f constrained by the selector Γ̂, then we say that x_{0} is a minimum of f constrained by the pair (ω, M). Specifying M_{a} = S_{a} ∪ bS_{a}, we obtain all the examples in §1.

Example 1: S_{a} = D, bS_{a} = ∅.

Example 2a): S_{a} = ∅, bS_{a} = \{x ∈ D|g^{a}(x) = 0, a = \overline{1, p}\}, ω^{a} = dg^{a}.

Example 2b): S_{a} = {x ∈ D|g^{a}(x) > 0}, bS_{a} = {x ∈ D|g^{a}(x) = 0}, ω^{a} = dg^{a}.

Example 3a): S_{a} = ∅, bS_{a} = D.

Γ̂^{a}(x_{0}) = {α ∈ Γ_{x_{0}}| ∫_{t_{0}}^{t}(ω^{a}(α(u)), α′(u))du = 0, ∀t ∈ [t_{0}, t_{0} + ε)} .

Example 3b): S_{a} = ∅, bS_{a} = D. Remark. In the Theorems 2.1 and 2.2, the set Γ_{x} is the family of all C^{1} parametrized curves passing by x and regular at x. In Theorem 2.2, the set Γ_{x} can be also the family of all C^{2} parametrized curves passing by x and with singularities of order 2 at x. 2.1. Theorem. Let f : D → R be a C^{1} function, and (ω, M) be the pair described above. If x_{0} ∈ M is a minimum point of f constrained by the pair (ω, M), then there

exist λ_{a} ≥ 0, a = \overline{1, p} such that df(x_{0}) = ∑_{a=1}^{p} λ_{a}ω^{a}(x_{0}). Moreover, if λ_{a} ≠ 0, then

x_{0} ∈ bS_{a}. Proof. Let B(x_{0}) = {a|x_{0} ∈ bS_{a}}. Let v ≠ 0 be a vector of R^{n} such that

(ω^{a}(x_{0}), v) ≥ 0, ∀a ∈ B(x_{0}).

We denote

J(x_{0}) = {a ∈ B(x_{0})|(ω^{a}(x_{0}), v) = 0}.

Let α be an integral curve of the Pfaff system ω^{a}(x) = 0, a ∈ J(x_{0}), satisfying α(t_{0}) = x_{0}, α′(t_{0}) = v. The existence of this curve is ensured by the hypothesis on the rank of the Pfaff form (In case J(x_{0}) = ∅, the curve α is arbitrary, with α(t_{0}) = x_{0},

α′(t_{0}) = v). If a ∈ J(x_{0}), then it follows ∫_{t_{0}}^{t}(ω^{a}(α(u)), α′(u))du = 0, ∀t ∈ [t_{0}, t_{0} + ε);

if a ∈ B(x_{0}) \ J(x_{0}), then ∫_{t_{0}}^{t}(ω^{i}(α(u)), α′(u))du ≥ 0, ∀t ∈ [t_{0}, t_{0} + ε). Since x_{0} is a

minimum point of f constrained by (ω, M), i.e., f(α(t)) ≥ f(α(t_{0})), ∀t ∈ [t_{0}, t_{0} + ε), we have (grad f(x_{0}), v) ≥ 0. By the Farkas Lemma we find df(x_{0}) = ∑_{a∈B(x_{0})} λ_{a}ω^{a}(x_{0})

with λ_{a} ≥ 0. For a ∉ B(x_{0}), we take λ_{a} = 0.

The hypothesis regarding the rank of the Pfaff forms can be replaced with a regularity condition of Kuhn-Tucker type. 2.2. Definition. We say that (ω, M) satisfies the Kuhn-Tucker regularity condition at x_{0} ∈ bS if for each vector v ≠ 0 with ⟨ω^{a}(x_{0}), v⟩ ≥ 0, ∀a ∈ B(x_{0}) = {a|x_{0} ∈ bS_{a}}, there exists a parametrized curve α ∈ Γ̂_{x_{0}}(α(t_{0}) = x_{0}) such that α′(t_{0}) = v. 2.3. Theorem. Let f : D → R be a C^{1} function and (ω, M) be the pair described above. If (ω, M) satisfies the Kuhn-Tucker regularity condition at x_{0} ∈ M and x_{0} is a minimum point of f, then there exist λ_{a} ≥ 0 such that

df(x_{0}) = ∑_{a=1}^{p} λ_{a}ω^{a}(x_{0}).

Moreover, if λ_{a} ≠ 0, then x_{0} ∈ bS_{a}. Proof. Let v ≠ 0 be a vector of R^{n} with (ω^{a}(x_{0}), v) ≥ 0, ∀a ∈ B(x_{0}). By the Kuhn-Tucker regularity condition there exists a parametrized curve (α ∈ Γ̂_{x_{0}} (α(t_{0}) = x_{0})

with α′(t_{0}) = v. Consequently, ∫_{t_{0}}^{t}(ω^{i}(α(u)), α′(u))du ≥ 0, ∀t ∈ [t_{0}, t_{0} + ε). In rest,

the same arguments as in the proof of the Theorem 2.1. 2.4. Theorem . Let Γ_{x} be the set of all C^{2} parametrized curves passing by the point x and being regular at x. Suppose the Pfaff forms ω and the function f be of class C^{1} on D. Let (ω, M) be the pair described above and x_{0} ∈ M.

Suppose: i) there exist λ_{a} ≥ 0, a = \overline{1, p}, such that

df(x_{0}) = ∑_{a=1}^{p} λ_{a}ω^{a}(x_{0}),

and, if λ_{a} ≠ 0, then x_{0} ∈ bS_{a}; ii) the restriction of the quadratic form

d^{2}f(x_{0}) − \frac{1}{2} ∑_{a=1}^{p} λ_{a} ∑_{j,k=1}^{n} (\frac{∂ω_{j}^{a}}{∂x^{k}} + \frac{∂ω_{k}^{a}}{∂x^{j}} ) (x_{0})dx^{j}dx^{k}

to the subspace

∑_{j=1}^{n} ω_{j}^{a}(x_{0})dx^{j} = 0, a ∈ J^{1} = {a ∈ B(x_{0})|λ_{a} > 0}

is positive definite.

Then x_{0} is a minimum point of f constrained by the pair (ω, M). Proof. Let α : I → D be a C^{2} curve with α(t_{0}) = x_{0}, regular at the point x_{0} and α ∈ Γ̂(x_{0}). It follows

∫_{t_{0}}^{t}(ω^{a}(α(u)), α′(u))du ≥ 0, ∀a ∈ [t_{0}, t_{0} + ε), ∀j ∈ B(x_{0}) = {a = |x_{0} ∈ bS_{a}}.

Case 1. If there exists a_{0} ∈ J′ with (ω^{a}(x_{0}), α′(t_{0})) > 0, then df(x_{0})(α′(t_{0})) =

∑_{a=1}^{p} λ_{a}(ω^{a}(x_{0}), α′(t_{0})) > 0. Using Taylor expansion

f(x) − f(x_{0}) = df(x_{0})(x − x_{0}) + O(∥x − x_{0}∥)

and

α(t) − α(t_{0}) = α′(t_{0})(t − t_{0}) + β(t) · (t − t_{0}),

with lim_{t→t_{0}} β(t) = 0, we find

f(α(t)) − f(α(t_{0})) = (t − t_{0})df(x_{0})(α′(t_{0})) + (t − t_{0})df(x_{0})(β(t))+

+O(∥α(t) − α(t_{0})∥) = (t − t_{0})df(x_{0})(α′(t_{0})) + O(t − t_{0}) ≥ 0, ∀t ∈ [t_{0}, t_{0} + ε). Case 2. Suppose

(ω^{a}(x_{0}), α′(t_{0})) = 0, ∀a ∈ J′.

The function

φ(t) = f(α(t)) − ∑_{a=1}^{p} λ_{a} ∫_{t_{0}}^{t}(ω^{a}(α(τ)), α′(τ))dτ.

has the derivative

φ′(t) = ∑_{i=1}^{n} \frac{∂f}{∂x^{i}} (α(t))\frac{dx^{i}}{dt} − ∑_{a=1}^{p} λ_{a} ∑_{i=1}^{n} ω_{i}^{a}(α(t))\frac{dx^{i}}{dt} ,

and hence

φ′(t_{0}) = ∑_{i=1}^{n} (\frac{∂f}{∂x^{i}} (x_{0}) − ∑_{a=1}^{p} λ_{a}ω_{i}^{a}(x_{0})) \frac{dx^{i}}{dt} =

= (df(x_{0}) − ∑_{a=1}^{p} λ_{a}ω^{a}(x_{0}))(α′(t_{0})) = 0.

Also

φ′′(t) = ∑_{i,j=1}^{n} \frac{∂^{2}f}{∂x^{i}∂x^{j}} (α(t))\frac{dx^{i}}{dt} \frac{dx^{j}}{dt} + ∑_{i=1}^{n} \frac{∂f}{dx^{i}} (α(t))\frac{d^{2}x^{i}}{dt^{2}} −

−\frac{1}{2} ∑_{a=1}^{p} λ_{a} ∑_{i,j=1}^{n} (\frac{∂ω_{i}^{a}}{∂x^{j}} + \frac{∂ω_{j}^{a}}{∂x^{i}} ) (α(t))\frac{dx^{i}}{dt} \frac{dx^{j}}{dt} + ∑_{a=1}^{p} λ_{a} ∑_{i=1}^{n} ω_{i}^{a}(α(t))\frac{d^{2}x^{i}}{dt^{2}} .

Then

φ′′(t_{0}) = d^{2}f(x_{0}) − \frac{1}{2} ∑_{a=1}^{p} λ_{a} ∑_{i,j=1}^{n} (\frac{∂ω_{i}^{a}}{∂x^{j}} + \frac{∂ω_{j}^{a}}{∂x^{i}} ) (x_{0})\frac{dx^{i}}{dt} (t_{0})\frac{dx^{j}}{dt} (t_{0})+

+ ∑_{i=1}^{n} (\frac{∂f}{∂x^{i}} (x_{0}) − ∑_{a=1}^{p} λ_{a}ω_{i}^{a}(x_{0})) \frac{d^{2}x^{i}}{dt^{2}} (t_{0}) =

= d^{2}f(x_{0}) − \frac{1}{2} ∑_{a=1}^{p} λ_{a} ∑_{i,j=1}^{n} (\frac{∂ω_{i}^{a}}{∂x^{j}} + \frac{∂ω_{j}^{a}}{∂x^{i}} ) (x_{0})\frac{dx^{i}}{dt} (t_{0})\frac{dx^{j}}{dt} (t_{0}).

Finally,

φ(t) − φ(t_{0}) = \frac{1}{2}φ′′(t_{0})(t − t_{0})^{2} + O((t − t_{0})^{2}),

whence φ(t) ≥ φ(t_{0}), ∀t ∈ (t_{0} − ε, t_{0} + ε). But φ(t_{0}) = f(x_{0}) and, for t ≥ t_{0}, f(α(t)) ≥ φ(t) so that there follows f(α(t)) ≥ f(x_{0}) for t ∈ [t_{0}, t_{0} + ε).

3 Extremum constrained by inequalities associated to a family of primitives of a Pfaff form

Let D be an open set in R^{n} and ω(x) = ∑_{j=1}^{n} ω_{j}(x)dx^{j} be a C^{0} Pfaff form. Let Γ be

the family of all C^{1} parametrized curves α : I → D. Each curve α generates a family {g_{α}} of functions,

g_{α} : I → R, g′_{α}(t) = (ω(α(t)), α′(t)),

called the primitives of ω along α. On the other hand, each curve α defines an equivalence class {β = α ∘ φ|φ : J → I} is C^{1} diffeomorphism. 3.1. Definition. Let g: be a function which associates to each parametrized curve α a function g_{α} from the family {g_{α}}. If g_{β} = g_{α} ∘ φ, for any equivalent curves α and β, then the function g is called system of ω-primitives.

For a Pfaff form ω and its associated system of primitives we can associate the set M = S ∪ bS, where

\begin{matrix}bS = \{x_{0} ∈ D|∃α ∈ Γ_{x_{0}}, α(t_{0}) = x_{0}, g_{α}(t_{0}) = 0\} \\ S = \{x_{0} ∈ D \ bS|∃α ∈ Γ_{x_{0}}, α(t_{0}) = x_{0}, g_{α}(t_{0}) > 0\}. \end{matrix}

The pair (ω, M) induces a selector Γ̂ of curves.

Similarly, for each a = \overline{1, p}, we consider the pair (ω^{a}, M_{a}), where ω^{a}(x) =

∑_{j=1}^{n} ω_{j}^{a}(x)dx^{j} are C^{0} Pfaff forms, and M_{a} = S_{a} ∪ bS_{a} are defined using the system

of ω^{a}-primitives g^{a}. The pair (ω, M) induces a selector of curves via the system of primitives g = (g^{a}).

Let f : D → R be a C^{0} function. Using the previous ingredients we define the Lagrange function

L_{α}(t, λ) = f(α(t)) − ∑_{a=1}^{p} λ_{a}g_{α}^{a}(t), ∀t ∈ I, ∀λ_{a} ≥ 0.

This function is defined along each curve α : I → D, using the restriction of function f to α and the primitives of the Pfaff forms ω^{a} along α. In this way obtain a family of Lagrange functions, which will satisfy conditions of saddle point type. 3.2. Definition. Let x_{0} ∈ D, and λ^{0} = (λ_{a}^{0}) with λ_{a}^{0} ≥ 0, a = \overline{1, p}. The point (x_{0}, λ^{0}) is called saddle point for the family of all Lagrange functions L_{α} if

a) L_{α}(t_{0}, λ^{0}) ≤ L_{α}(t, λ^{0}), ∀α ∈ Γ_{x_{0}}, α(t_{0}) = x_{0}, ∀t ∈ [t_{0}, t_{0} + ε);

b) there exists α ∈ Γ_{x_{0}} with α(t_{0}) = x_{0} such that

L_{α}(t_{0}, λ^{0}) ≥ L(t_{0}, λ), ∀λ = (λ_{a}) ≥ 0.

3.3. Lemma. The condition b) in the Definition 3.2 is equivalent to:

b′) there exists α ∈ Γ_{x_{0}}, α(t_{0}) = x_{0}, such that g_{α}^{a}(t_{0}) ≥ 0, a = \overline{1, p} and λ_{a}^{0}g_{α}^{a}(t_{0}) = 0 (no summ).

This condition afirms: if x_{0} ∈ M and if λ_{a}^{0} > 0, then x_{0} ∈ bS_{a}.

Proof. Suppose b) is true: ∃α ∈ Γ_{x_{0}} with α(t_{0}) = x_{0}, and L_{α}(t_{0}, λ^{0}) ≥ L(t_{0}, λ),

∀λ = (λ_{a}) ≥ 0. It follows (∗) ∑_{a=1}^{p}(λ_{a}−λ_{a}^{0})g_{α}^{a}(t_{0}) ≥ 0, ∀λ_{a} ≥ 0. Suppose, ∃a_{0} ∈ \overline{1, p} with

g_{α}^{a_{0}}(t_{0}) < 0; taking λ_{a_{0}} > λ_{a_{0}}^{0} and λ_{a} = λ_{a}^{0} for a ≠ a_{0} we obtain a contradiction with

(∗). Taking λ_{a} = 0, ∀a = \overline{1, p}, in (∗), we obtain ∑_{a=1}^{p} λ_{a}^{0}g_{α}^{a}(t_{0}) ≤ 0, i.e., λ_{a}^{0}g_{α}^{a}(t_{0}) = 0

for each a = \overline{1, p}.

The converse is automatically true. 3.4. Theorem. Let x_{0} ∈ D. If there exists λ^{0} = (λ_{a}^{0}) ≥ 0, a = \overline{1, p} such that (x_{0}, λ^{0}) is a saddle point for the family for all Lagrange functions L_{α}, then x_{0} is a minimum point of f constrained by (ω, M). Proof. The condition b′) from Lemma 4.3 shows that x_{0} ∈ M. Suppose λ_{a}^{0} = 0, ∀a = \overline{1, p}. From the condition a) it follows: for each α ∈ Γ_{x_{0}}, with α(t_{0}) = x_{0}, f(α(t)) ≥ f(x_{0}), ∀t ∈ [t_{0}, t_{0} + ε). Hence, x_{0} is a minimum point of f constrained by (ω, M); the point x_{0} is a free minimum point if Γ_{x_{0}} is the set of all C^{1} parametrized curves, regular at x_{0} or the set of all C^{2} parametrized curves having x_{0} as a regular point or as a singular point of order 2 (Theorem 1.3.)

Suppose J′ = {a|λ_{a}^{0} > 0} is nonvoid. According the condition b′), it follows

x_{0} ∈ bS_{a}, ∀a ∈ J′. Let α ∈ Γ̂_{x_{0}}, with α(t_{0}) = x_{0}. Since ∫_{t_{0}}^{t}(ω^{a}(α(u)), α′(u))du ≥ 0,

∀t ∈ [t_{0}, t_{0} + ε), ∀a ∈ J′, we find g_{α}^{a}(t) − g_{α}^{a}(t_{0}) ≥ 0, ∀t ∈ [t_{0}, t_{0} + η), ∀a ∈ J′. From the condition a) we get

f(α(t)) − f(x_{0}) ≥ ∑_{a∈J′} λ_{a}^{0}(g_{α}^{a}(t) − g_{α}^{a}(t_{0})).

Consequently x_{0} is also a minimum point of f constrained by the pair (ω, M).

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[12] O. Dogaru and V. Dogaru, Extrema Constrained by C^{k} Curves, B.J.G.A., vol. 4, no.1 (1999), 45-52.

[13] C. Udrişte, I. Ţevy and M. Ferrara, Nonholonomic Economic Systems, Manuscript Notices, unpublished.

[14] C. Udrişte, I. Ţevy, Geometry of Test Functions and Pfaff Equations, see [15], 151-165.

[15] C. Udrişte, O. Dogaru and I. Ţevy, Extrema with Nonholonomic Constraints, Monographs and Textbooks 4, Geometry Balkan Press, 2002.

[16] C. Udrişte, O. Dogaru, M. Ferrara and I. Ţevy, Pfaff inequalities and semi-curves in optimum problems (manuscript).

University Politehnica of Bucharest Department of Mathematics I Splaiul Independentei 313 77206 Bucharest, Romania email:udriste@mathem.pub.ro